No title

Letters to the Editor

The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor-in-chief, Chandler Davis.

But, then, other properties of life are

A Computer Scientist's View of Evolution

equally strange. Computer scientists

Granville Sewell's article (vol. 22 (2000),

find it hard to believe that a moderate

no. 4, 5-7) attempts to show difficulties

number of very slow components, neu­

in evolutionary theory that are missed

rons, can be combined into a comput­

by biologists. I have several reactions. 1. Philosophers

and

mathemati­

cians (e.g., Brouwer and the intuition­ ists)

have

long

discussed

the

ing device that is able to perform pat­ tern

recognition,

understanding

of

logical natural

reasoning, language,

gap

etc.-and moreover can tolerate dam­

between finite and infinite. Now com­

age to a significant fraction of its neu­

puting practice and computer science

rons.

have showed that another gap is of

3. Sewell ends by arguing that the

philosophical importance, namely, the

second law of thermodynamics is vio­

gap between feasible and infeasible

lated by the development of life. Surely

"The U niverse is not o n ly stranger than we i mag i n e - it is stranger than we are capable of imag i n i ng." -J.B.S. H aldane polynomial

more careful wording is needed. Strange

A biologist in former times could

will reorganize the basic particles of Na­

finite integers:

between

and exponential growth.

it may be that "basic forces of Nature

consider the number of molecules and

ture into libraries full of encyclopedias."

the number of years involved in evolu­

We don't know any dynamical system,

tion as almost infinite. This is now

random or deterministic, that exhibits

clearly seen as wrong. These numbers

similar behavior. But there is no mathe­

are negligible compared to 2n, where n

matical theorem (or clear theory) called

is the number of bits that might be al­

"the second law of thermodynamics"

tered in a substantial mutation.

that prohibits it.

2. The analogy between the genetic code and code of a program sounds

A. Shen

convincing. However, from a computer

Institute for Problems of Information

scientist's point of view, the genetic code has very strange properties. As Sewell says,

if you mix different parts

Transmission Ermolovoi 1 9 K-51 Moscow GSP-4, 1 0 1 447

of a PDE-solver code, in superficial

Russia

analogy to mixing parents' genes, you

e-mail: [email protected]

will get something non-functional. We apparently must suppose that a small

How Anti-Evolutionists

mutation sometimes changes the per­

Abuse Mathematics

formance of the genetic program sig­

The Reverend William A Williams was

nificantly yet in self-consistent ways. If

not one of Darwin's bigger fans. In [21]

this seems miraculous, so do other

he wrote

properties of the genetic mechanism. Sewell is at pains to show that evolu­

The evolution theory, especially as

tion is unbelievable; to me, its opera­

applied to man, likewise is dis­

tion-and for that matter its formation

proved by mathematics. The proof

from simple chemical processes-is

is overwhelming and decisive. Thus

even more unbelievable than he says.

God makes the noble science of

© 2001 SPRINGER-VERLAG NEW YORK. VOLUME 23, NUMBER 4, 2001

3

mathematics bear testimony in favor of the true theo­ ries and against the false theories. Needless to say, this will come as news to most biologists. The Reverend, writing in 1925, relied heavily on the au­ thority of the Bible in making his arguments. That same year saw biology teacher John Scopes hauled into a Ten­ nessee courtroom, charged with teaching scientific theo­ 1 ries that were in conflict with scripture. Modem critics of Darwinism take a more subtle approach, preferring to cloak their dubious religious arguments in the raiment of science. They call themselves Intelligent-Design Theorists (IDTs), the term "creationist" being now somewhat disreputable. Granville Sewell of the University of Texas at El Paso is one representative of this movement. In [19] he opined, bas­ ing himself on Michael Behe [ 1], "I believe there are two central arguments against Darwinism, and both seem to be most readily appreciated by those in more mathematical sci­ ences." The two arguments were that natural selection is not capable of building complex organisms, and that Darwinism is in conflict with the second law of thermodynamics. In making these arguments he simply ignored the vast litera­ ture addressing both subjects, so as to give the impression that logical fallacies obvious to you or me have somehow eluded our benighted colleagues in the life sciences.It is an arrogance typical of the ID movement; armchair philoso­ phers believing they can refute in a day what thousands of scientists have built over the course of a century. ID theorists offer a wide array of arguments in defense of their position, some of them explicitly mathematical.I will consider some of these arguments here. The hemoglobin in our blood is comprised of 574 amino acids arranged in a precise sequence.Any major deviation from this sequence leads to a nonfunctional molecule. We also note that there are twenty sorts of amino acids used by living organisms. Is it plausible that a mechanism based on chance, as Dar­ winism plainly is, could have produced hemoglobin? Mathematician David Foster doesn't think so.In [7] he offers the following: The basic argument from improbability:

The specificity of hemoglobin is described by the im­ probability of the specific amino acid sequence occur­ ring by random chance. Such specificity is capable of exact calculation in the permutation formula:

P=

N! -----

nr!n2! ...n2o!

...In the case of hemoglobin, and substituting in the above formula the specific numerical value of the solu­ tion, P 10654• =

Of course, N denotes the total number of amino acids in the sequence; while ni denotes the number of occurrences of the i-th amino acid. Hemoglobin is a dummy variable in this argument, any other complex organic molecule or system would have worked just as well. The logic is always the same: the n parts of the complex system are identified as the points of a probability space.This space is then equipped with the uniform distribution. The origin of the system is modeled as the event of choosing the appropriate nt- uple out of this space. If the system in question is at all complex, the prob­ ability of this event will invariably prove to be too small to be worth bothering with. This argument is a mainstay of creationist literature; it has been applied to DNA, the hu­ man eye, and the origin of life in [7], [12], and [14], re­ spectively, among many others.I will refer to it as the Ba­ sic Argument from Improbability (BAI). David Foster [7] is confused on many points (one of them being the difference between a permutation and a combination), but the most important error is the portrayal of Darwinism as fundamentally a theory of chance. Dar­ winism, as described in [9], has three components: 1. Organisms produce more offspring than can possibly survive. 2. Organisms vary, and these variations are at least partly heritable by their offspring. 3. On average, offspring that vary most strongly in direc­ tions favored by the environment will survive and prop­ agate. Favorable variation will therefore accumulate in populations. Part one is a simple empirical fact.Part two is the realm of chance; the genetic variations exhibited by an organism are random with respect to the needs of that organism.But part three is the antithesis of chance. Natural selection is a lawlike process.It is this aspect of Darwinism that gets left out of the BAl. Foster's argument assumes that evolution proceeds by "single-step selection." But if the preliminary stages of a complex system are preserved by selection, then com­ plexity can be explained as the end result of a step-by-step 2 process. Improving the BAI: Perhaps we could develop a more sophisticated probabilistic model of evolution. For example, Darwinism can be viewed as a Markov chain.The states of the chain are the genotypes3 of the organisms that have existed throughout history; the transition probabilities . are the chances of an organism with genotype E1 leaving offspring with genotype E2. Denote by C§ the set of all genotypes. Defme a function J.t: C§ X C§ __,. [0, 1] which denotes the degree of difference between two genotypes, say t:1 and t:2.

1 The question of whether Darwinism is genuinely in conflict with the Bible was not addressed at the trial. 2Popular-level treatments of the power of cumulative selection versus single-step selection, and Darwinian explanations of complexity can be found in the books by Dawkins [4] and [5]. 3The genotype of an organism is the sum total of its genes.

4

THE MATHEMATICAL INTELLIGENCER

If /.L = 0 then t:1 = E2. If /.L = 1 then E 1 and E2 share no genes. Let the random variable �(t) represent the state of the sys­ tem in time t. A central tenet of Darwinism asserts that the relevant genetic variations between parent and child are small relative to the size of the genome, so Prob{W + 1)

=

E J I �(t)

=

Ek} � 0 as p,(E J, Ek) � 1.

Let us take �(0) = Eo as representing the genotype of some ancient organism, one that is simple relative to the complexity we see today. The evolutionary path followed by the descendants of this organism trace out a path through our Markov chain, �(0)

�

�(1) � �(2) �

.

.

.

.

Given our present understanding of genetics, we can say that the future states of the random variable � are inde­ pendent of its past states, the hallmark of a Markov process. We need one more ingredient to transform our Markov chain into a model for Darwinism.Let f: C§ � IR associate to each genotype its fitness.4 Now let E be a genotype con­ taining a system composed of the parts p1, P 2 , ... , Pk; we will write E = {pi, . . . , Pkl· The state E is a descendant of the state Eo, which we assume did not contain the Pi· For selection to preserve the parts of the system as they ap­ peared, we must have the following: f(E)

> f({pl, P2,

·

·

·

,

Pk- I}) > · · · > f({pi}) > f(Eo)

The addition of each part must increase the fitness of the genotype. Further, we can assert thatfsatisfies some sort of additivity law, since each part of the system can be viewed as increasing the fitness of the system.Say: We can say that a particular state E is accessible to a Dar­ winian mechanism if there is a path through our chain on whichfsatisfies the above conditions. This line of argument is pursued by David Berlinski in [3]. So far it is simply a mathematical framework within which to model Darwinian explanations of complexity. The alleged refutation of Darwinism arises from the following definition: 1. A system {p1, p2, ..., Pnl is irreducibly com­ hereafter denoted IC, iff(E) = 0 for aU E E C§ such that Pi E E and PJ ti E for some 1 :=::; i, j :=::; n. If E is a state containing such an irreducibly complex system, then we wiU say that E is irreducibly complex. DEF1NITION

plex,

THEOREM 1. If t: is irreducibly complex then it is not ac­ cessible to a Darwinian mechanism.

Do IC systems exist in nature? Well, Berlinski's definition of IC is a mathematization of a definition given by biochemist Michael Behe in [1]. Behe defined a system as IC if it involves several parts working together to perform some function,

such that the removal of any part from the system results in the nonfunctionality of the machine. Examples of such sys­ tems are the human blood clotting cascade5, or the flagellae used for locomotion by some bacteria Thus, by taking p1, . . . , Pn to be the various parts of, say, the blood-clotting cascade, we have our example of a system satisfying Berlinski's defmition of IC.It follows that the natural world is replete with systems inaccessible to Darwinian pathways. It's an impressive argument, but wrong for at least three reasons. In [3] Berlinski claims that his definition of IC entails Behe's, but this is not correct. A system is IC in Behe's sense if the removal of one part of the system results in the non­ functionality of the system. It is IC in Berlinski's sense if the organism can derive no benefit from possessing only one part of the multipart system.These are plainly not the same. There are at least two sorts of explanation for how the in­ dividual pieces of an IC system can benefit an organism, even without the other parts of the system in place: 1. They might perform the same function in isolation as they do in the fmished system, but not as well.This mode of explanation is used by Miller in [17], in the case of the clotting cascade, and by Dawkins in [5] in the case of the vertebrate eye. 2. They might initially have performed a different function but have been later coopted for their present purpose. In [11] paleontologists Stephen Jay Gould and Elisabeth Vrba coined the term "exaptation" to describe this phe­ nomenon. Two examples are the evolution of the three bones in our inner ear from homologous bones in the reptilian jaw6 as described in [9], and the origin of the Krebs cycle7 as described in [16]. In 1996 Behe [1] made the audacious claim that the tech­ nical literature on evolution is silent with regard to the for­ mation of irreducibly complex systems. This charge was shamelessly repeated by Sewell in 2000 [19]-though Ken­ neth Miller [17] had meanwhile cited numerous examples from the technical literature to show this to be false. The point is that Berlinski's definition of IC is far more re­ strictive than Behe's. Thus, systems that are IC in Behe's sense are known to exist but are not inaccessible to Dar­ winian mechanisms. Systems that are IC in Berlinski's sense are inaccessible to Darwinian mechanisms, but are not known to exist. This is the most serious flaw in Berlinski's model, but there are two others worth mentioning. The first is that notions of irreducible complexity treat the parts of a complex system as if they are discrete entities that either exist in their complete, perfected glory, or do not exist at all. This is not realistic.The parts of a complex system become gradually differentiated

4The fitness of a genotype depends partly on the environment in which that genotype finds itself, but that is ignored for the moment. 5The details of the clotting cascade and a detailed discussion of its evolution can be found in [17]. This fine book contains a chapter refuting Behe's arguments. 6There is an extraordinary series of fossils documenting this change. 7This refers to the series of chemical reactions that releases energy from food.

VOLUME 23, NUMBER 4, 2001

5

over the course of many generations. Therefore, asking what happens to a system when one of its parts is summarily re­ moved is a question of little evolutionary importance. Finally, Berlinski's argument given here is one of a class of arguments based on the proposition that "genotype space" is too vast to be searched effectively by natural se­ lection acting on chance variations. Complex organisms represent islands of functionality in a sea of nonfunctional genotypes, you see. This brings us to the second difficulty with Berlinski's framework. His insistence that the fitness functionjbe properly increasing on any sequence of adja­ cent states in a Darwinian pathway ignores the possibility that mutations can be neutral. In other words, we might havej(Ej) j(EJ+t) for somej. The overwhelming majority of mutations are neutral in this sense. This vastly increases the number of genotypes that are accessible to Darwinian pathways. Two examples of the importance of neutral mu­ tations in molecular evolution are given by (6] and [15].8 =

Sewell also argued that Darwinism runs afoul of the laws of thermodynamics. Evolution requires a decrease in entropy over time, whereas a cherished principle of physics says that is impossible. Since Sewell recognizes that the second law applies only to closed systems (which the Earth is not), it is difficult to understand the difficulty. His claim that "natural forces do not cause extremely improbable things to happen" is pure gibberish. Does Sewell invoke supernatural forces to explain the winning numbers in last night's lottery? The fact is that natural forces routinely lead to local de­ creases in entropy. Water freezes into ice and fertilized eggs turn into babies. Plants use sunlight to convert carbon diox­ ide and water into sugar and oxygen, but Sewell does not invoke divine intervention to explain the process. Certainly the question of how the input of energy into the environ­ ment of the early Earth led to the creation of all that we see around us is a fascinating and important one. That ex­ plains the large number of scholarly articles published on the subject every year. But thermodynamics offers nothing to dampen our confidence in Darwinism. Thermodynamics:

introduction to population genetics: The ability of natural selection to craft complex adaptations out of chance variations is contingent upon two assumptions:

An

1. Beneficial mutations occur with sufficient frequency.

2.

A beneficial mutation, once it occurs in an individual, will spread through the population.

Biologists have developed mathematical models to aid in ad­ dressing these points. The subdiscipline of biology devoted to analyzing such models is called population genetics. I begin with a very simple model. Our genes are found in long strings, called chromosomes, in the nuclei of our cells. Typically we imagine a chromosome divided into individual

regions called loci. The bit of DNA found at a particular lo­ cus is referred to as an allele. Let us consider a single locus which, in each individual in the population, contains one of two alleles. Denote these alleles by A1 and A2•9 Assume that the species in question reproduces sexu­ ally and that the offspring inherit two copies of each gene, one from each parent. Then members of the population will either possess two copies of the A 1 allele, two copies of the A2 allele, or one copy of each. I will refer to these three cases as genotypes A1A�o A�2, and A1A2, respectively. Let us further assume that the A1 allele appears with frequency p in the population, and A2 appears with frequency q = 1 p. We can think of p and q as representing the probability that a randomly chosen allele is A1 or A2, respectively. 2. (Hardy-Weinberg) Let A1, A 2, p, and q be as above, and assume that the population mates randomly with respect to this allele. Then in the next generation the genotypes A1A1o A1A 2 , and A�z will appear with fre­ quencies p2, 2pq, and q2, respectively. THEOREM

Of course, this theorem is elementary. Given the sim­ plicity of the model, it is surprising that the Hardy-Wein­ berg law has proven invaluable in explaining observed data in wild populations. Next we try to quantify the effect of selection on the fre­ quencies of the alleles A1 and A 2. Imagine that the three possible genotypes initially appear with the frequencies de­ termined by the Hardy-Weinberg law. Then the extent to which a particular allele is represented in the next genera­ tion is proportional to its representation in the current gen­ eration and the probability that an individual possessing that allele survives long enough to reproduce. Let us de­ note the constant of proportionality by w. This constant is often referred to as the mean fitness of the population. Denote by Wij, with i, j E (1, 2}, the probability that an individual of genotype AiAJ survives to reproduce. If we now let f(A0J ) denote the frequency of genotype AiAJ in the next generation, we find

f(A 1A 1) =

p2wu {J)

'

f(A 1A2) =

2pqw 1 2 :¥>2) ' f(A -.1 {J)

=

q2W22 {J)

0

Since the sum of the three frequencies should be 1, set

w

=

p2wu + 2pqw 1 2

+

q2ltJ2z.

Let us denote by p' the frequency of the new generation. Then we can say

A1

allele in the

(Note that each A1A2 individual posseses only one copy of the A1 allele). So what can we say about the change in frequency of the A1 allele as time passes? One further calculation yields

8Berlinski presses his argument further by introducing ideas from the theories of finite-state automata and linguistics, but these arguments are no better than the ones considered here. "The following mathematical arguments are drawn from the excellent text by Gillespie [8]

6

THE MATHEMATICAL INTELLIGENCER

!:lp = p'- p P2 wu + p qw 1 2 - p w w

4.Let ProbfixCP) denote the probability that the allele Ah appearing with an initial frequency of p, be­ comes fixed in a population of size N. Then THEOREM

=

- pq [p(wl l - WJz) + q(w 1 22 - W22)] p2 w1 1 + Zpqw12 + q W22

ProbfixCP)

_

The quantity pq is referred to as the genetic variation of the population. It is maximized when p Suppose now that the

A1

=

q = t.

allele confers a selective ad­

vantage on the individuals that possess it. Specifically, as­

wu > w12 > W2z. In that case we see that !:lp > 0, A1 will tend to increase in succeeding generations. By contrast, if A1 is at a selective disadvantage, so that w 1 1 < w1 2 < W2z , then we have !:lp < 0 and the frequency of A2 will tend to decrease. This ob­ sume that

indicating that the frequency of

servation can be expressed more succinctly in the equation !:l

P

=

pq 2w

( dw)' dp

and in words in the following theorem: THEOREM 3. (Fundamental Theorem of Natural Selection) Natural selection always increases the mean fitness of the population, and does so at a rate proportional to the ge­ netic variation.

A victory for evolution, right? Beneficial mutations will

=

- e-2Nsp , 1 - e-2Ns

1

where s denotes the selective advantage conferred by the allele A1. If we assume that A1 initially appears in a single individ­ �. Since s is assumed to be small, ual, then we will have p we can say e-s 1- s. If we then assume that N 0, we conclude that is large enough so that e-2Ns s. So most beneficial mutations are lost without Probfix(p) =

=

=

=

ever having a chance to become fixed in the population. Hoyle concludes from this that it is effectively impossible to string together a large number of beneficial mutations. Fred Hoyle is no kind of creationist.He doubts neither the truth of evolution nor the existence of a fully naturalistic ex­ planation for it. Indeed, he offers a rather imaginative alter­ native to Neo-Darwinism based on the premise that the Earth is periodically bombarded with storms of genetic material from outer space. For a brief discussion of why biologists are generally skeptical of this possibility, see

[18]

and [20].

Hoyle's argument is wrong for many reasons, the most fun­ damental being the absurdity of extrapolating to geologic time a mathematical model that is reliable only for short-term data. The dynamics of gene frequencies in wild populations

tend to become fixed in the population, and over long pe­

are governed by so many variables that a mathematical model

riods will accumulate to produce complex adaptations.

for describing them in the long term is impossible. For ex­

Not so fast. Randomness also has a role to play in the

ample, the selective value of a particular allele changes with

change of gene frequencies over time. For example, sup­

the environment.The population size, and therefore the fre­

pose a single individual in a population has a beneficial mu­

quency of a particular allele within it, changes as subpopu­

tation. The probability is only one-half that any particular

lations of animals migrate away from the ancestral stock An­

child born to that individual will inherit the mutation.So it

imals interact with other animals, which are themselves

is entirely possible that the mutation will be flushed out of

evolving. Consider also that we have been focusing on one

the population before it has a chance to spread.

locus, when in reality the selective value of the allele at that

This is one example of a more general phenomenon

locus is certainly affected by the alleles at other loci.

called genetic drift. Selection is tending to cause beneficial

There are other problems. Early in his book Hoyle states,

mutations to spread through a population, while drift is

"...a considerable fraction of individuals born in every gen­

tending to remove them. Perhaps a more sophisticated

eration exhibit some new mutation, the great majority being

model of population dynamics would have shown that drift

harmful in some degree." This premise is entirely false. As in­

is powerful enough to overcome selection, thus effectively

dicated earlier, most mutations are neutral.And what of the

falsifying the Darwinian premise of complexity arising from

small probability that a beneficial mutation will become fixed

the gradual accretion of small, chance variations.

in a population? That only applies to very large populations.

This line of argument is pursued by physicist Fred Hoyle

Most evolutionists believe that periods of speciation, during

[13].

His starting point is the assumption that mutations

which directional evolutionary change accumulates very

are far more likely to be harmful than beneficial. How do

quickly, occur when small "founder" populations become

the handful of beneficial mutations avoid being swamped

geographically isolated from the ancestral stock

in

by the more numerous harmful ones? The answer, known

This leads us to the most insidious aspect of Hoyle's

for decades by population geneticists but presented as rev­

work His book offers no index, no bibliography, and only

elation by Hoyle, is that the mechanics of sexual repro­

the briefest mention of any other work in population ge­

duction allow beneficial mutations to become "decoupled"

netics. Most of his book is spent rederiving old results, with­

drift, which tends to deplete variation.

A lay reader will inevitably get the impression that the for­

from the harmful ones. 10 But sexual reproduction leads to Hoyle then points to results like the following:

out giving any indication that they are not original to him. midable mathematical machinery employed by Hoyle, cou-

10A mathematical derivat1on of this fact can be found in any text on population genetics, [8] being a particularly good one.

VOLUME 23, NUMBER 4, 2001

7

pled with his dismissals of work that came before him, constitutes a devas­ tating attack on Neo-Darwinism. It doesn't.

[1 1 ] Gould, S.J. , and Vrba, E . , "Exaptation: A

on this topic. I consider that the main point in my article was the second one. Paleobiology 8 (1 982), 4-1 5. Mathematicians are trained to value [1 2] Hanegraaf, Hank, The F.A. C.E. that simplicity. When we have a simple, Demonstrates the F.A.R.C.E. of Evolution, clear proof of a theorem, and a long, Word Publishing, 1 998. Pseudomathematics: As an academic complicated counter-argument, full of dispute, all this is minor. But it plays in [13] Hoyle, Fred, Mathematics of Evolution, hotly debated and unverifiable points, Acorn Enterprises LLC, 1 999. public. ID theorists, much like the we accept the simple proof, even be­ 4] Huse. Scott M . , The Collapse of Evolution, [1 creationists before them, know they fore we fmd the errors in the compli­ 3rd ed., Baker Books, 1 997. will not convince scientifically cated argument. That is why I prefer [1 5] Huynen, Martijn A., "Exploring Phenotype knowledgeable people. Instead, they not to extend here the long-standing Space Through Neutral Evolution," Jour­ market their ideas to a public untrained debate over the first point, but to dwell nal of Molecular Evolution 43 (1 996) in both the methods and findings of further on the much simpler and 165-1 69. science. And all too often theirs is the clearer second point of my article, [1 6] Melendez Hevia, Waddell, Cascante, "The only viewpoint that is readily available. which is that the increase in order ob­ Puzzle of the Krebs Citric Acid Cycle: As­ When scientists are presented with served on Earth (and here alone, as far sembling the Pieces of Chemically Feasi­ subjects that invoke the terminology of as we know) violates the laws of prob­ ble Reactions, and Opportunism in the science to defend nonsense, like as­ ability and the second law of thermo­ Design of Metabolic Pathways During Evo­ trology or creationism, they use the dynamics in a spectacular fashion. lution, " Journal of Molecular Evolution 43 term pseudoscience. I suggest we need Evolutionists have always dis­ (1 996), 293-303. a similar term, pseudomathematics missed this argument by saying that the perhaps, to describe mathematical for­ [1 7] Miller, Kenneth R. , Finding Darwin's God, second law of thermodynamics only H arper Collins, 1999. malism used to promote bad argu­ dictates that order cannot increase in ments. As professional mathemati­ [ 1 8] Pigliucci, Massimo, "Impossible Evolution? an isolated (closed) system, and the Another Physicist Challenges Darwin , " cians, we all have an interest in Earth is not a closed system-in par­ Skeptic 8(4)(2001 ), 54-57. protecting the integrity of our subject. ticular, it receives energy from the Sun. We have an obligation to be aware of [1 9] Sewell, Granville, "A Mathematician's View The second law allows order to in­ of Evolution," The Mathematical lntelli­ how mathematics is being used in the crease locally, provided the local in­ gencer 22 (2000), 5-7. public square. When we see pseudo­ crease is offset by an equal or greater mathematics, we should not be afraid [20] Walsh, J. Bruce, "No Light from the Black decrease in the rest of the universe. Cloud," Evolution 54 (2000), 1 461 -1 4 62. to identify it. This always seems to be the end of the [21] Williams, William A, The Evolution of Man argument: order can increase (entropy Scientifically Disproved, in Fifty Argu­ REFERENCES can decrease) in an open system, there­ ments, Privately published, 1 925. [1 ] Behe, Michael, Darwin's Black Box, The fore, anything can happen in an open Free Press, 1 996. system, even the rearrangement of Jason Rosenhouse [2] Berlinski, David, "The Deniable Darwin," atoms into computers, without violat­ Department of Mathematics Commentary, June 1 996. ing the second law. Kansas State University [3] Berlinski, David, Gode/'s Question, in Mere It requires only a modicum of com­ Manhattan, KS 66506-2602, USA Creation: Science, Faith, and Intelligent mon sense to see that it is extremely Design, Wm. Dembski ed. , Inter Varsity [email protected] improbable that atoms should re­ Press, 1998. arrange themselves into mammalian [4] Dawkins, Richard, The Blind Watchmaker brains, computers, cars, and airplanes, 2nd ed. , Norton, 1 996. even if the Earth does receive energy [5] Dawkins, Richard, Climbing Mount Im­ Can Anything Happen in an from the Sun. We will see that the idea probable, Norton, 1 996. Open System? that anything can happen in an open [6] Dean, A.M . , "The Molecular Anatomy of an Critics of my Opinion piece "A Mathe­ system is based on a misunderstanding Ancient Adaptive Event," American Scien­ matician's View of Evolution" [1] have of the second law; that order can in­ tist 86, Jan-Feb 1998. focused primarily on my first point, crease in an open system, not because [7] Foster, David, "Proving God Exists, " The which deals with whether or not major the laws of probability are suspended Saturday Evening Post, December 1 999. evolutionary improvements can be when the door is open, but simply be­ [8] Gillespie, John H . , Population Genetics: A built up through many minor improve­ cause order may walk in through the Concise Guide, The Johns Hopkins Univ. ments. It is clear to me that they can­ door. Let us look first at a form of "or­ Press, 1 998. not, but this question is the traditional der" that is easy to measure. [9] Gould, Stephen Jay, An Earful of Jaw in front on which most battles over Dar­ Consider heat conduction in a solid, Eight Little Piggies, Norton 1 993. winism have been fought since 1859, R. If R is a closed system (no heat [1 0] Gould, Stephen Jay, Ever Since Darwin, and I did not imagine that my argu­ crosses the boundary), we can define Norton, 1977. ments would constitute the last word a "thermal entropy" in the usual way,

8

THE MATHEMATICAL INTELLIGENCER

Missing Term in the Language of Form, "

to measure randomness in the heat dis­

temperatures but carbon concentra­

dias,

tribution, and show, using the second

tions identical to that in the rod, the

computers connected to laser printers,

science texts,

and

novels, or

law of thermodynamics, that the total

rod may import "thermal order" (ex­

CRTs, and keyboards? If we take a

R can never decrease, and

port thermal entropy), but the "carbon

book of random letters and blow vow­ els into the front of the book (pretend

entropy in

will in fact increase until the tempera­

order" will be unaffected. In the scien­

ture distribution is uniform throughout

tific literature, thermal entropy is usu­

letters can diffuse!) and suck them out

R. If R is open, the thermal entropy in R can decrease, but it is easy to show

ally referred to simply as "entropy," but

the back, we can import order into the

in fact there are many entropies (de­

book, if randomness of the vowel dis­

(see Appendix) that the decrease can­

pending on what we choose to measure:

tribution is used to measure order.

not be greater than the entropy ex­

see [2], p.xiii) and many kinds of order:

Vowels are essential for words, just as

R. Be­

any macroscopic feature or property

solar energy is essential for life, but

cause a decrease in thermal entropy is

that is improbable from the microscopic

this process is not going to produce a

associated with an increase in "thermal

point of view can be considered order.

great novel: that is a different

order," this can be stated in another

For example, of all the possible config­

order.

ported through the boundary of

kind of

way: in an open system, the increase in

urations that atoms could take, very few

If we found evidence that DNA,

order cannot be more than the order

would allow the transmission of pic­

auto parts, computer chips, and books

imported through the boundary.

tures or air transportation of packages

entered through the Earth's atmos­

According to the second law, then,

over long distances, so television sets

phere at some time in the past, then

the order in the universe is continually

and airplanes can be considered to be

perhaps the appearance of humans,

decreasing, but what is left of it at any

improbable, and to represent order.

cars, computers, and encyclopedias

time can be transported from one open

The second law predicts that-in a uni­

on a previously barren planet could be

system to another. For example, if a

verse in which only natural processes

explained without postulating a viola­

rod of uniform, moderate temperature

are at work-every type of order is un­

tion of the second law here (it would

is used to connect a hot and a cold

stable and must decrease, as every­

have been violated somewhere else!).

reservoir, the entropy of the rod will

thing

But if all we see entering is radiation

tends

toward

more

probable

decrease, as one end becomes hotter

(more random) states.But just because

and

and the other becomes colder. The

two things are both improbable does

clear that what is entering through the

meteorite

fragments,

it

seems

uni­

not necessarily mean that the importa­

boundary cannot explain the increase

formly distributed in the rod-some­

tion of one (say, TV sets) into an open

in order observed here. Many scien­

temperature

will

become

less

thing that would be extremely unlikely

system can explain the appearance

tists seem to have the idea that "en­

to happen without help from outside.

there of the other (say, airplanes).

tropy" is a single number that mea­

The rod is simply importing order from

Rather,

sures order of all types, so if entropy decreases locally when computers ap­

the outside world, where order is now decreasing as the temperatures of the

If an increase in order is extremely

pear-no problem, entropy is increas­

two reservoirs approach each other.

improbable

ing all over the rest of the universe, so

when

a

system

is

If we look at the diffusion of, say,

closed, it is still extremely improb­

the total entropy is surely increasing,

carbon, in a solid instead of the con­

able when the system is open, un­

and the second law is satisfied.For ex­

duction of heat, and take

U(x, y, z, t)

now to be the carbon concentration in­

less something is entering which makes it

not extremely improbable.

"carbon entropy"

(Q

is just

U

now),

L. Hepler [3]

ment of civilization may appear con­

stead of the temperature, we can re­ peat the analysis in the Appendix for

ample, S. Angrist and

write, "In a certain sense the develop­

Although it is not as easy to quan­

Lradictory to the second law....Even

tify the order associated with airplanes

though society can effect local reduc­

showing again that in a closed system

and computers as the order associated

tions in entropy, the general and uni­

(no carbon crosses the border) this en­

with a carbon or temperature distribu­

v:�rsal trend of entropy increase easily

tropy cannot decrease, while in an

tion, it is clear that life and human cre­

swamps the anomalous but important

open system, the decrease in entropy

ativity are responsible for some very

efforts of civilized man."

cannot be greater than the entropy ex­

large increases in order here.Contrary

What is the conclusion then-that

ported through the boundary. But it is

to common belief, however, the "ther­

the explosion of new order on Earth has violated the laws of physics in a su­

important to notice that now "entropy"

mal order" imported from the Sun does

measures the randomness of the dis­

not help explain the formation of hu­

pernatural way? Not necessarily: since

tribution of carbon, not heat, so the

mans, jet airplanes, TVs, and comput­

the advent of quantum mechanics, the

amount of thermal entropy exported is

ers. If we add sunlight to the computer

laws of physics carmot be used to pre­

not relevant to the change in carbon

model hypothesized in [1], would we

dict the future with certainty, and they

entropy in the solid. For example, if a

expect that the simulation would

now

do not really say that anything is ab­

steel rod of uniform temperature and

predict that the basic forces of Nature

solutely impossible, they only provide

uniform carbon concentration is placed

would rearrange the basic particles of

us the probabilities. Thus one could ar­

between two steel blocks of unequal

Nature into libraries full of encyclope-

gue that the origin and development of

VOLUME 23, NUMBER 4, 2001

9

Q is the heat energy density and

life may not have violated any of the

where

laws of physics-only the laws of prob­

J is

ability. The conclusion is only this: con­

law requires that the flux be in a di­

the heat flux vector. The second

trary to what Charles Darwin believed,

rection in which the temperature is de­

and contrary to the majority opinion in

creasing, i.e.,

science today, the development of in­

J·VU�O

telligent life is not the inevitable or rea­

(2)

sonably probable result of the right

(In fact, in an isotropic solid, J is in the

conditions, it is extremely improbable

direction of greatest decrease of tem­

under any circumstances.

perature, that is,

J = -KVU.)

Note

that (2) simply says that heat flows from hot to cold regions-because the

REFERENCES

1 . G. Sewell, "A Mathematician's View of Evo­ lution, " The Mathematical lntelligencer 22 no. 4 (2000), 5-7. 2. R. Carnap, Two Essays on Entropy, Univer­ sity of California Press, 1 977.

laws of probability favor a more uni­ form distribution of heat energy. Now the rate of change of "thermal entropy," nition as

S, is given by the usual defi­

(2), we see that the volume integral is nonnegative, and so

St 2::

-II J · n/U aR

From (4) it follows that 81

2:: 0 in an iso­

lated, closed, system, where there is no heat flux through the boundary can never decrease. However, equation (4) still holds in an open system; in fact, the boundary integral in (4) represents the rate that entropy is exported across the bound­ ary (notice that the integrand is the outward heat flux divided by tempera­ ture). Thus in an open system, (4) be more than the entropy exported

Basic Books, 1 967.

through the boundary. Appendix. Consider heat conduction

Using

(3) and the first law (1), we get:

in a solid R, with (absolute) tempera­

U(x, y, z, t).

R

law of thermodynamics (conservation

-V·J

where (1)

Granville Sewell Mathematics Department

The first

of energy) requires that

Qt =

aR

University of Texas El Paso El Paso, TX 79968

n is the outward unit normal on

the boundary aR. From the second law

USA e-mail: [email protected]

Beware Biomathematics I approached Feynman after one of his Cornell lectures in 1964 for advice about

how best

to

move into mathematical biophysics from engineering physics, as I had planned when choosing Cor­ nell. He cautioned against any such move, on grounds that biology is too much a matter of tricks and accidents of evolution, and too complex for useful mathematical representations. I believe that is correct, on average, but the rich diversity of living nature provides many niches for peculiar ques­ tions and aptitudes. Arthur T. Winfree The

Geometry of Biological Time

2nd edition (Springer, 2001), p. 660

10

(J·n =

0). Hence, in a closed system, entropy

means the decrease in entropy cannot

3. S. Angrist and L. Hepler, Order and Chaos,

ture distribution

(4)

THE MATHEMATICAL INTELLIGENCER

c.m ott.1 ,;

If M athematicians Do Not Do It, Who Willt Daniel J. Goldstein

The Op inion column offers mathematicians the opportunity to write about any issue of interest to the international mathematical community. D isagreement and controversy are welcome. The views and opinions expressed here, however, are exclusively those of the author, and neither the publisher nor the editor-in-chief endorses or accepts responsibility for them. An Opinion should be submitted to the editor-in­ chief, Chandler Davis.

D

oes mathematics have any func­ tion in understanding biological phenomena? Of course it does, in that mathematics is the language of physics. Some mathematics is used to describe physical phenomena that were studied in classical physiology and genetics­ e.g., membrane biology, cardiovascular function, classical genetics-at a very macroscopic level. Classical macro­ scopic physiology and genetics (CMP/G) was the realm of biologists who were relatively insensitive or indifferent to molecular structure (which in any case, at the time could not have been tackled experimentally) and paid only lip service to biochemistry-they knew that some enzymology had to be thrown in to pacify the beasts. CMP/G generated black boxes and neat, elegant diagrams that convey (even today) a sense (false) of rationality and simplicity. These reduced models have tradi­ tionally tempted mathematicians inter­ ested in finding biological "laws." The problem is that the biological world is rather different from the neat rational­ izations and simplifications of CMP/G, and as soon as biologists went beyond the macroscopic depiction of physical phenomena, the effectiveness of math­ ematics collapsed. Genetics and molec­ ular chemistry-which together try to explain biological complexity in terms of interacting molecules-have regu­ larly shown the absurdity of "models" and "laws" deduced from concepts de­ rived from partial, mainly irrelevant, and biased information. These at­ tempts at mathematization were reac­ tionary on two grounds: first, they were based on the assumption that the bio­ logical world can be understood with­ out knowing its molecular structure and function; second, they implicitly accepted as truths the biggest biologi­ cal sins of CMP/G: teleonomy, the con­ cept of design, and the interpretation of evolution as the exclusive conse­ quence of adaptive selection. CMP/G, and the mathematical mod­ els derived therein, operate as if bio­ logical systems were the result of a "ra­ tional design" intended to maximize efficiency. This, of course, is utterly

false, and if mathematics has to do with a reality out there, the mathematical approximation to biological problems should start by recognizing that bio­ logical systems and objects are the re­ sult of accident and a curious mixture of adaptive and non-adaptive selection. To be sure, mathematics also helped in opening the biological black boxes, because the only tools available so far to determine molecular structure are two physical technologies, X-ray dif­ fraction and nuclear magnetic reso­ nance. Once modern genetics and mo­ lecular chemistry opened the black boxes of CMP/G, the already bewilder­ ing variety of the biological zoo in­ creased by the addition of ever stranger creatures. Biologists had to come to grips with this new expanded reality, and appreciate with awe the momen­ tous complexity hidden in tissues, cells, and extracellular structures; the interplay among thousands of intracel­ lular and extracellular macromole­ cules; and the astonishing heterogene­ ity of chemical signals that regulate the ensemble. The interactions among these gigantic collections are extraor­ dinarily difficult to describe, and it is utterly impossible to imagine a single "law" that can sensibly explain their collective, integrated behavior. Because the difficulties inherent in attempting an integrated approach seemed insurmountable, molecular bi­ ologists fell into the trap of trivial re­ ductionism and studied (as best they could) one molecule at a time. This re­ search strategy is good for description and for survival-there can be special­ ists in single proteins, and there are a lot of proteins out there. But there are deeper problems. For example, how can we know the total number of roles that a single protein plays in a cell or in a organism? Proteins are objects with multiple functions, and not all of the potential functions of a single pro­ tein species are exerted at the same time. Protein functions are context-de­ pendent, and biologists must approach the problem as art historians do when studying art objects, which also have multiple functionalities--aesthetic, sym-

© 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 4, 2001

11

bolic, or political-depending on time,

ematization, such as using topology to

ately needed, because the huge size and

place, and context. The chemical con­

describe DNA knots, an application that

variegated nature of the information

text in which a protein exists (which

may or may not be useful in the future

delivered

conditions its functionality) changes in

for explaining physiological and bio­

has changed radically the way we do bi­

by

the

genome

projects

real time, yet our knowledge of these

chemical

and

ology, and the old reductionist tricks

fluctuating boundary conditions is piti­

deriving predictive models of chromo­

need to be complemented with inno­ vative approaches coming from mathe­

phenomena

fully poor. Furthermore, protein struc­

some structure, behavior, and regula­

ture is not fixed: proteins undergo

tion. The (few) examples of this type

matics. Yet for this creative interaction

post-translational modifications, suffer

suggest a certain kind of laziness in the

to occur, mathematicians have to learn

limited proteolysis, associate with like

way in which mathematicians approach

enough molecular biology to be able to

molecules or with different protein

biology. They seem to decide which

grasp

species, and dissociate and even refold

"themes" are "mathematically viable" by

Mathematicians must be familiar with

in radically different ways as a function

their superficial resemblance to mathe­

the kinds of objects that the biologists

of the chemical context.

matical

objects

and

the

real

biological

problems.

situations with

work with; must share the same genetic

All this is crucial for understanding

which they are familiar. Symmetries,

and molecular language; and must un­

how biological systems function, be­

packings, knots, sequences, and pat­

derstand that biological objects are not

cause the genotype (the sum of genetic

terns occur in biology aplenty, and be­

the result of design, that efficiency is a

information, whatever this may mean)

ing easily translated into mathematical

human value judgment and not une don­

does not determine the phenotype (the

notation, they are defined as the areas

nee de

observable traits of an organism). The

of interface between mathematics and

the messy result of accidents and adap­

la

nature,

and that evolution is

genotype encodes a collection of pro­

biology. But, is there something in this

tive and nonadaptive selection. If math­

teins, and the interaction of the encoded

beyond translation? Did these analogies

ematicians learn this language and un­

proteins-with all the possible caveats

produce predictive models?

derstand the evolutionary process, they is

will be able to find many real biological

whether there are any mathematical

problems amenable to mathematical ex­

physical and chemical reactivity)-is

objects that really behave in the same

ploration, and biology will reach another

what determines the phenotype.

way biological macromolecules

intellectual dimension. I think that the

(modification, fragmentation, associa­ tion, with the concomitant changes in

In

my

opinion,

the

question

do,

Occasionally, the genetic and molec­

and that could help produce predictive

"Wigner-Gelfand principle," which as­

ular dissection of an experimental sys­

models of biological phenomena. In

serts the unreasonable ineffectiveness

tem allows the formulation of models of

this sense, the recent discovery of

of mathematics in the biological sci­

universal explanatory and

predictive

Professor M. Livsic about the possibil­

ences, should be reformulated. So far,

value. So far, in the fifty years of molec­

ity of depicting DNA structure and

mathematics has been ineffective in the

ular biology, only three such models

replication

biological sciences because mathemati­

have emerged: the Watson-Crick model

open systems may or may not open a

cians looked in the wrong places and

of DNA structure, the Jacob-Monod

new

with the wrong attitude. Mathematics

model of genetic regulation, and the

mathematics and biology.2

space

in terms

of space-time

of interaction

between

will be reasonably effective in the bio­

logical sciences when mathematicians

Jacob-Monod-Wyman-Changeux-Perutz

Of course, the old problem still

allosteric model of enzyme regulation.

looms: Are these equivalences "real" or

become

Mathematics (aside from geometry) had

merely reflections of the fact that our

out there,

aware of the biological reality perceive the nature of the

no role in these momentous achieve­

brains, whatever the way we see/ex­

open mathematical problems hidden in

ments. These discoveries were the result

press the world, can produce only a lim­

biological systems, and try to solve them

of solid thinking and strong invention in

ited number of metaphors? Yet the

(inventing/discovering some new math­

structural chemistry, and the (then new)

power of metaphors is huge and should

ematics, of course).

not be dismissed with a shrug, as the ex­

bacterial and phage genetics. that

traordinary interplay between mathe­

Daniel J. Goldstein

"Mathematics is unquestionably effec­

matics and physics so eloquently shows.

Departamento de Ciencias Biol6gicas

Some

commentators

think

tive in biology, for rationalizing obser­

I am convinced that a creative inter­

vations."1 This may be true, but it is not

action between mathematicians and bi­

Universidad de Buenos Aires, Argentina

obvious. There are possibilities of math-

ologists is not only possible but desper-

[email protected]

' Arthur M. Lesk. Compared to What? The Mathematica/ lntelligencer 23(1 ):4 (200 1 ) . 2"Systems and genetics," in Proceedings of the Workshop Dedicated to Advances and Applications. Birkhauser, i n press.

12

THE MATHEMATICAL INTELUGENCER

the 60th

Birthday of Harry Dym (ed.

Facultad de Ciencias Exactas y Naturales

D.

Alpay, I . Gohlberg, and Y. Vinnikov) Operator Theory:

M at h e m a tic a l l y Bent

The proof is in the pudding.

Opening a copy of The Mathematical Intelligencer you may ask yourself

uneasily, "What is this anyway-a mathematical journal, or what?" Or you may ask, "Where am I?" Or even "Who am !?" This sense of disorienta­ tion is at its most acute when you open to Colin Adams's column. Relax. Breathe regularly. It's mathematical, it's a humor column, and it may even be harmless.

C o l i n Ad a m s , Editor

A Deprogrammer's Tale

but I understand why; I know the se­ ductive power of a beautiful proof, the appeal of a well-turned lemma. Larry had fallen prey in the usual

manner. Mter hearing the derivative explained in a lecture hall with 300 other students, he went to see the pro­ fessor during office hours. That's when they know they have you. You're one of the susceptible ones, looking for some meaning beyond the plug and

Colin Adams

chug problems. hey hooked him in calculus class.

A little chitchat, maybe notational,

Started slow. Didn't want to be too

a bit of history, Newton versus Leib­

obvious. Gave him a little trig review,

nitz, that sort of thing, all seemingly in­

T

some functional notation, and then in­

nocuous. And then, when he least ex­

troduced limits. Gave him lots of prob­

pected it, the epsilon delta definition of

lems to work. Kept him busy to get his

a continuous function. Poor guy was

guard down. Then pow, hit him with

putty in the professor's hands. Before

the concept of the derivative. The raw

he could get his head back on straight,

power and simplicity of the idea, it was

the professor invited him to a depart-

1 200 peo p l e a year g et Ph . D.s i n math

in the U n ited States alone. overwhelming. How could he resist?

mental colloquium, followed by tea.

Who can? I know. I've been through it

Larry dutifully went, and although he

myself. Yes, that's right. I was one of

was blown out of the water by the ma­

them once.

terial, he saw the others there, at rapt

I was a slave to mathemat­

attention, and he felt he was among

ics. But unlike most, I escaped. And now my life is dedicated to helping others who were not as fortunate as

At tea, the department members ig­ nored Larry, feigning indifference to

I.

In this particular case, I was hired

Column editor's address: Colin Adams,

friends.

the freshman who was interested in

by the parents of one Lawrence De­

math,

senex. One minute, Larry was pre-med,

wrapped up in their own research to

pretending

they

were

too

heading for a lucrative plastic surgery

care. But oh, if he only knew. They

practice in Cherry Hill, and the next

were watching his every move, as they

minute he was talking about earning a

scribbled on the blackboard and talked

Ph.D. in mathematics. All thought of fi­

about this theorem or that with their

nancial gain went out the window. His

colleagues. He was a marked man, and

parents were horrified. Dreams of my­

Larry didn't even know it.

son-the-doctor turned into nightmares

In cases like these, there is a small

of my-son-the-itinerant-mathematician.

window of opportunity, a short period

But me, I wasn't surprised when I heard

when a student can yet be saved. But

the tale. I'd heard it a hundred times be­

you must act fast. Once students take

Department of Mathematics, Williams

fore. Believe it or not, 1200 people a

Real Analysis and Abstract Algebra,

College, Williamstown, MA 01 267 USA

year get Ph.D.s in math in the United

their fate is sealed. The window has

e-mail: [email protected]

States alone. That sounds incredible,

been slammed shut and shuttered.

© 2001 SPRINGER-VERLAG NEW YORK. VOLUME 23, NUMBER 4. 2001

13

But Larry's parents had called me in time. He was taking Linear Algebra, the applied version. There was hope yet. I found him in the cafeteria with an untouched plate of tuna casserole and a copy of The Man Who Loved Only Numbers open in front of him. I gave him my winningest smile. "Erdos, huh? Mind if I join you?" He was clearly impressed and mo­ tioned to the seat across the table. "Like math, do you?", I asked. "Oh, yes," he said enthusiastically. "It's so beautiful." "Yes, it does have an appeal." "Have you ever seen the argument for the uncountability of the reals?", he asked. "That's really cool." The bubbly excitement, the glassy bright eyes. Oh, he was in deep. We talked math for a while. I played along. Euclid this, Euler that. Then I laid the trap. "Hey, my roommate and I are hav­ ing a birthday celebration for Karl Friedrich Gauss on Wednesday at my apartment. You're invited." Of course, he was thrilled. Suscep­ tible and trusting are two descriptions of the same attribute. He showed up right on time. It hadn't taken him long to pick up that c haracteristic of mathematicians. I let him in and locked the door behind him. Then everyone popped out, his par­ ents, his grandparents, a cousin, an aunt, his best friend from high school. "What's going on here?" he said, clearly at a loss. "This isn't a birthday party for Gauss." "No, it's not," I said. "Gauss was born in April. This is an intervention, Larry. These are the people who love you and they're here to help." He backed away. "Open the door. Let me go," he cried desperately. I blocked him. "Not until you hear what we have to say." He looked like Galois after the duel. The blood drained from his face. Must

14

THE MATHEMATICAL INTELLIGENCER

have been wondering where his muse was now. His mother spoke first. "Bunchkins, bunchkins, have you thought about us? We love you, Pinchy, but good gra­ cious, what would the neighbors say? Mrs. Krawlick would revel in the news. Our son, a mathema, a mathema . . . , I can't say the word." She began to bawl uncontrollably. Larry's father held her. "Look at your mother. Look at what you are doing to her. She can't even say the word." "Poor, poor Erma," said his aunt, patting Larry's mother on the sleeve. "Larry, I can't believe you would do this. You seemed like you were a good kid. You used to watch television. You had a lemonade stand. What happened to you? My kids would never do this. Evan here, now, he is a dentist, aren't you Evan?" The cousin nodded yes. "And Cybil works in marketing for an ad agency. And I am proud of them both!" "What about Karen?" asked Larry. The aunt turned bright red. "How dare you mention her name in my pres­ ence." Evan laughed. "Karen has a masters degree in accounting." Not my area, but I sympathized. Larry's best friend spoke up. "Lis­ ten, Larry. The problem is, it's not cool to do math. Business degrees, they're cool. You know, Internet start-ups and all. Theater degrees, that's cool. You wear black clothes and talk about Pin­ ter. But math? It's not cool. Nothing is cool until everyone is doing it." Larry wrung his hands. "You don't understand. I don't have a choice. I am not choosing to do math­ ematics. Math has chosen me. When I saw that epsilon delta definition of con­ tinuity, it was like I had known it all my life. Here is what the professor was really talking about when he drew all these pictures. This is a rigorous defi-

nition. It felt so good. It's not up to me anymore." "Look, Larry," I said. "Do you want this to be you?" I showed him the pic­ tures of mathematicians, the addicts with their white pallor from sitting un­ der fluorescent lights for years at a time. Some were barely able to lift their eyes from the books in front of them as the camera clicked away. Their clothes, stained with coffee, made it clear they were unaware that fashion was an evolving concept. But he was unmoved. "That's ex­ actly what I want to be," he said. I sighed. "Okay, Larry, I have no choice." I strapped him into the Bar­ colounger and turned on the TV. I kept him there for two weeks; mostly re­ runs of the "Brady Bunch" and "Wel­ come Back Kotter." By the time we were done, spittle dripped from the side of his mouth. His brain had been washed clean. Unfortunately, it had been washed so clean that medical school was no longer an option. Larry did go on to a successful career with Seven Eleven, primarily mopping up the slushy spills at the Cherry Hill store. And I know that he's happier for it. But Larry's story is just one among many. These dangers are real. Do you know where your children are? Are you sure they are watching TV, and not sit­ ting in on a seminar, or leafing through a math text? If we are vigilant, we can prevent mathematics from spreading any fur­ ther. But we will need to fight the min­ ions of mathematics at every turn. We will need the entertainment industry to continue to hype style over intellectual curiosity. We will need to inundate children with the belief that being good at math is something to be ashamed of. We will need to convince everyone that there is nothing wrong with mathe­ matical illiteracy. So far, so good.

I\[email protected]§11£hlfiifj.lj,j11ii!,iihfj

Social Influences on Quantum Mechanicst - 1 Jane Cronin

This column is a forum for discussion of mathematical communities throughout the world, and through all time. Our ddinition of "mathematical community" is the broadest. We include "schools" of mathematics, circles of correspondence, mathematical societies, student organizations, and informal communities of cardinality greater than one. What we say about the communities is just as unrestricted. We welcome contributions from mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.

Please send all submissions to the M athematical Communities Editor.

Marjorie Senechal, Department of Mathematics, Smith College, Northampton, MA 01 063, USA;

e-mail: [email protected]

Marj o r i e Senechal , E d it o r

I

n an interesting article about the question of whether social and cul­ tural factors have affected the devel­ opment of quantum mechanics, M.B. Ruskai (Mathematical Intelligencer 23, no. 1, 23-29) concludes that quan­ tum mechanics "transcends social and cultural forces." There are, however, several such forces that have signifi­ cantly derailed or sidetracked the de­ velopment of quantum mechanics. The purpose here is to describe briefly some of these. The introduction of the Copenhagen interpretation by Niels Bohr and Werner Heisenberg gave rise to much controversy among very accomplished physicists. Erwin Schrodinger devised his cat-in-the-box thought-experiment to demonstrate his view that the Co­ penhagen interpretation was ridicu-

I

books. (See Jammer [9, pp. 247-248] and Mermin [11, p. 803].) When we consider the question of why the Copenhagen interpretation was thus accepted, the answer is sur­ prisingly unclear. First, of course, it should be pointed out that quantum mechanics was widely accepted in short order by physicists who applied it successfully to practical problems or extended the theory and who had little interest in the foundations of the sub­ ject. It was natural for them to stick with the first complete interpretation. But this does not answer the question of why the Copenhagen interpretation was accepted despite the serious ques­ tions raised by a number of physicists. The most important part of the answer seems to be Bohr's energetic support. His stature as a physicist, his agreeable

The introduction of the Copenh agen in­ terpretation by N iels Boh r and Werner Heisenberg gave rise to much controversy lous, and his biography suggests that he never changed his mind. Concerns about the nature of the observer and the collapse of the wave function led Eugene Wigner and others to quite dif­ ferent interpretations. (See, e.g., Rae [13, Chap. 1 1].) But the most important objection to the Copenhagen interpretation was the work of Albert Einstein and his col­ leagues [5], hereafter to be referred to as EPR. Einstein's concerns about the role of probability in quantum theory are far more profound than his oft­ quoted remark implies, and they can­ not be dismissed with a quip. Einstein and Bohr carried on a long, friendly dis­ cussion of their differences (cf. Jammer [9, Chapters 5,6]), and Bohr won out in the sense that the Copenhagen interpretation became ac­ cepted to the point that it entered text-

personality, and his persistence and de­ termination all combined to win the day for the Copenhagen interpretation. According to Murray Gell-Mann, "Bohr brain-washed a whole generation of physicists into believing that the prob­ lem had been solved." (See [7, p. 152] .) The Copenhagen interpretation was also supported indirectly by work of John von Neumann. As soon as the probability properties of the solutions of the Schrodinger equation (the wave functions) were introduced, it was nat­ ural to think of the possibility of hid­ den variables, i.e., variables describing the deeper structure of a given physi­ cal system. (For example, if a gas is de­ scribed in terms of temperature, pres­ sure, and volume, then the velocities of the individual atoms in the gas would be hidden variables.) In his book on quantum mechanics, von Neumann

© 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 4, 2001

15

[ 15] claimed to prove that there are no hidden variables in quantum mechan­ ics. (This result supported the Copen­ hagen interpretation, because the ab­ sence of hidden variables suggests that the wave function contains all possible information about the system it de­ scribes.) As described in Jammer [9. p. 265ff. ] , there was considerable discus­ sion of von Neumann's result, and in 1935, Grete Hermann [8] pointed out a deficiency in von Neumann's proof. However, Hermann seems to have been disregarded, and it was not until 1966 that John Bell [2] showed that von Neumann's proof was based on an as­ sumption that has been described by some writers as "silly." (See Mermin, [ 1 1, p.806.].) The direction of study of the foun­ dations of quantum mechanics from 1930 to the 1950s thus seems to have been strongly influenced by two non­ scientific or social forces: the prestige and persistence of Bohr and the pres­ tige of von Neumann. (Von Neumann was indeed a towering figure in twen­ tieth-century mathematics, but it does not follow that he was incapable of er­ ror.) From the point of view of gender issues, one might also ask if Grete Hermann's observation in 1935 would have been more seriously regarded if she had been named Georg Hermann. More nonscientific forces came into play in the reception of the work of David Bohm in 1952. In the accompa­ nying essay, Miriam Lipschiitz-Yevick describes these. In 1957, Bohm and Aharanov [4] de­ scribed an example of the EPR prob­ lem, which greatly clarified the prob­ lem and led the way to important further work The example is a thought­ experiment that reveals a puzzling point in quantum mechanics. For a careful description, see Rae [ 13, p.229]. Briefly and loosely put, the experiment involves two spin half particles. The to­ tal spin of the system is zero, but no in­ formation about the spins of the parti­ cles is given. The particles move apart until widely separated, after which the spin of one particle is measured. Measurement of the spin (actually a specific component of the spin) of the first particle causes the wave function

16

THE MATHEMATICAL INTELLIGENCER

to "collapse" into an eigenfunction of the spin operator, and it follows that the same component of the spin of the second particle then is determined and is equal to the negative of the spin of the first particle. Thus, even though the two particles may be light-years apart, the measurement of the spin of the first particle has an immediate influence on the second particle: That is, the mea­ surement of the spin of the first parti­ cle causes the measurement of the spin of the second particle. This is an ex­ ample of what is called nonlocality (what Einstein called "spooky action at a distance"). It was anathema not only to Einstein but to most physicists edu­ cated in the twentieth century. (See Ballantine [ 1 , p.585] and Bell [2, p.20, footnote 2].) This example is so important to later work that its origins should be ex­ amined with some care. First, EPR, in which the example originated, raises serious questions about quantum me­ chanics, questions whose formulation required deep and penetrating analysis. The example of David Bohm and Yakir Aharanov clarified the original EPR thought-experiment to the point where the ideas became accessible and use­ ful to others, as we shall see. Actually, the example was introduced and de­ scribed in detail in Bohm's book on quantum mechanics [3], indeed in more detail than in [4]. In [3] , Bohm de­ scribed his example as a modification of the EPR experiment, which has "conceptually equivalent form" to that experiment. Bohm should receive credit not only for devising the exper­ iment but for doing the work at a time when it was widely thought that Einstein's questions about quantum mechanics had been laid to rest and the Copenhagen interpretation reigned supreme. (The reader who is curious about looking up references [3] and [4] needs a word of warning here. In [3], Bohm was still a supporter of the Copenhagen interpretation, but by the time [4] was written, his views had changed significantly. Indeed, his views had changed between the publi­ cation of [3] and the publication of his papers on quantum mechanics in 1952. The description of the example is given

in more detail in [3], but the signifi­ cance of the example is better sug­ gested in [4] .) The momentous next step was taken by John Bell. (For a detailed, up­ to-date account of Bell's work and later results based on his work, see Ballantine [ 1 ] . Here we give only a short, rough description.) Starting with the model of Bohm, Bell devised a thought-experiment from which can be derived (using no quantum mechanics) a testable conclusion called "Bell's in­ equality." He showed also that this in­ equality contradicts the predictions of quantum mechanics. (This result is called "Bell's theorem.") Since then, ac­ tual experiments modeled on Bell's thought-experiment have been carried out, and the experimental results agree with the predictions of quantum me­ chanics, and thus contradict Bell's inequality. It follows that there is dis­ agreement between quantum mechan­ ics and the hypotheses used to derive Bell's inequality. The only significant hypothesis used to derive the inequal­ ity seems to be locality (i.e., no nonlo­ cality). (See Ballantine [ 1 , p. 607ff.] for a careful discussion of this point.) The implication is therefore that quantum mechanics is nonlocal. Since the requirement of locality is motivated by special relativity, this suggests a possible incompatibility be­ tween quantum mechanics and special relativity. At present, these are deep and unresolved questions. It is worth remarking that Bell's work received the attention it de­ served only slowly. One reason for this may have been that his earliest work was concerned primarily with hidden variables, a subject that was, for various reasons, of little interest to physicists. Another reason for the de­ lay in acknowledging the importance of Bell's results may have been the fact that even in his earliest papers, he emphasized the importance of nonlo­ cality; and, as noted before, nonlocal­ ity was unacceptable to most physi­ cists. (The Bohm theory described in 1952 is nonlocal, but that fact was held against the Bohm theory when it was introduced.) There seems to be no doubt that

these results are very important to the foundations of quantum mechanics, and yet at each stage of their develop­ ment, there was resistance by the physics establishment. EPR was dis­ missed, Bohm was disregarded, and even Bell's work was acknowledged slowly. It would probably be impossible to make a numerical estimate of the time delay in the development of quan­ tum mechanics caused by this resis­ tance, but it seems unquestionable that such a delay occurred. Part of the reason for the resistance has already been mentioned: this work is concerned with the foundations of quantum mechanics. To physicists, bustling in their laboratories and con­ fident of the applications of quantum mechanics, the foundations are simply not interesting. It is therefore ironical that all this theoretical nattering has given rise to significant work in quan­ tum cryptography. Although still in the laboratory stages, this work shows considerable practical promise. (See [6] , [ 10) [ 12), [ 14].) Pairs of particles satisfying the conditions in the Bohm example (the particles are said to be entangled) are used to create unbreak­ able codes that can be used for the secure transmission of confidential material.

[3] Bohm, David, Quantum Theory, Prentice­

A U T H O R

Hall, Inc. New York, 1 951 .

[4] Bohm, D. and Aharanov, Y. Discussion of experimental proof for the paradox of Einstein, Rosen, and Podolsky, Physical

Review ( 1 08) 1 070-1076, 1 957. [5] Einstein, A., Podolsky, B., Rosen, N., Can quantum mechanical description of phys­ ical

reality be considered

complete?

Physical Review (47) 777-780, 1 935. [6] Ekert, Artur K., Quantum cryptography based on Bell's Theorem, Physical Review JANE CRONIN

Letters (67) 661 -663, 1 991 .

Department of Mathematics

[7] Geii-Mann, Murray, The Nature of the Physical Universe, John Wiley & Sons,

Rutgers

Un i versity New Brunswick

Piscataway,

Inc . , New York, 1 976. Grundlagen der Quantenmechanik, Ab ­

NJ 08854-801 9 USA

[8] Hermann, G., Die naturphilosophischen

e-mail: [email protected]

handlungen der Fries ' chen Schute (6) 75Jane Cron in (Scanlon) got her doc­

1 52, 1 935. [9] Jammer, Max, The Philosophy of Quan­

torate at the University of Michigan.

tum Mechanics, John Wiley & Sons, New

She has been at Rutgers

York, 1 974.

becom ing Emerita in 1 991 Her pri­ .

[1 0] Jennewein, Thomas; Simon, Christoph; Weibs,

Gregor;

since 1 965,

Weinfurter,

H arald;

mary field of research has been and remains singular perturbation theory

Zeilinger, Anton. Quantum cryptography

applied to models of neural activity.

with entangled photons, Physical Review

Readers

Letters (84) 4729-4732, 2000. [1 1 ] Mermin, N. David, Hidden variables and two theorems of John Bell, Reviews of

may

recall

her

book

Mathematical Aspects of Hodgkin­ Huxley Neural Theory, and her article in The lntelligencer 1 2 (1 990), no. 4 .

Modern Physics (55) 803-8 1 5, 1 993. [1 2] Naik, D. S., Peterson, C. G., White, A. G., Berglund, A. J . , Kwiat, P. G., Entangled state quantum cryptography: eavesdrop­

REFERENCES

[1 ] Ballantine, Leslie C . , Quantum Mechanics, A Modern Development, World Scientific, Singapore, 1 998.

[2] Bell, J . S., Speakable and Unspeakable in Quantum Mechanics, Cambridge Univer­ sity Press, Cambridge, 1 987.

ping on the Ekert protocol, Physical

gled photons in energy-time Bell states.

Review Letters (84) 4733-4736, 2000.

Physical Review Letters (84) 4737-4740,

[1 3] Rae, Alastair I. M . , Quantum Mechanics, third edition, Institute of Physics Publish­ ing, Bristol, England, 1 992.

[1 4] Tittel, W., Brendel, J., Zbinden, H., Gisin, N . , Quantum cryptography using entan-

2000. [1 5] Von Neumann, J . , Mathematische Grund­ fagen der Quantenmechanik, Springer, Berlin, 1 932. (English translation, Prince­ ton University Press, 1 955).

VOLUME 23, NUMBER 4, 2001

17

Social [Bohm'sj themy, due to one of the greatest physicists of ou1· Why time, is practically ?.tnivm·sally ignm·ed, is an enigma which histmians of science offuture centuries wiU have to resolve. Influences on -Jean Bricmont, in "Cont1·e la philosophie de mecanique quantique, " Quantum M echanicst- 1 1 T Retrospect this

la

EDITOR's

OTE: And here we are in the next century, trying to resolve it.

his note is intended as a response

interpretation"), he rejected this inter­

to Mary Beth Ruskai's comments

pretation in favor of a radically oppos­

on David Bohm's quantum mechanics

Miriam Lipschutz-Yevick

1 995

(Mathmnatical InteUigencer,

vol.

23

ing one.2 This interpretation was to suggest3 that the EPR correlations

(2001), no. 1, 23-29, especially Appen­

were to be ascribed to fluctuations in

dix B). In particular, I want to speak of

the

the social factors that inhibited the free

which his new theory had postulated;

discussion of his challenge to ortho­

later he concluded rather that the cor­

doxy.

relations were entirely the product of the quantum potential of his theory.4

Jane Cronin, in the note that pre­

sub-quantum-mechanical

level

cedes this one, emphasizes the impor­

Bohm's "hidden variables" theory

tance and relevance to later work of

was, in fact, an independent rediscov­

Bohm's

EPR

ery and elaboration of the "pilot wave"

myself with his development, subse­

Broglie, which he had presented at the

reformulation

of

the

Gedanken experiment. 1 I will concern quent to finishing his

ory,

Quantum The­

of a "hidden variables" theory

theory of the French physicist Louis de Solvay Conference in

1927 to explain

the wave-particle duality. De Broglie

which, after a delay of several decades,

expressed

also gave impetus to a renewed inter­

Schrodinger's "particle corresponds to

est in EPR and its potential applica­

a wave packet" and Born's "psi func­

Quantum Theory Bohm stood

tion yields probabilities only," leading

tions. In

his

disagreement

with

squarely on the side of Niels Bohr's cri­

to renunciation of determinism for in­

tique of EPR, and he used his modifi­

dividual particles. De Broglie proposed

cation of this experiment to solidify the

instead that if we know the particle's

argument

initial position the psi function pre­

against

Albert

Einstein's

conviction of the incompleteness of

cisely determines its trajectory. On the

quantum mechanics.

other hand, given an ensemble of non­

Yet shortly after completing his text,

interacting identical particles with dif­

after further discussions with Einstein

ferent initial positions, the psi function

and continuing the profound thought

determines the probability that an in­

the

dividual particle will be in a volume of

Copenhagen philosophy and to under­

he

had

devoted

to

clarifying

space at a given instant. The psi wave

standing the EPR paradox (Einstein

thus appeared simultaneously as a pi­

had complimented him with "Yours is

lot wave

the best exposition of the Copenhagen

and a probability wave. "It does not

(Fiihrungsfeld

of Max Born)

1D. Bohm, Quantum Theory, Prentice-Hall, New York, 1 95 1 ; see p. 614. 2D. Bohm, A suggested interpretation o f quantum theory in terms o f hidden variables, I a nd II, Physical Re­

view 85 (1 952), 1 65 and 1 80. 3D. Bohm and Y. Aharanov. Discussion of experimental proof for the paradox of Einstein, Rosen, and Podol­ sky, Physical Review 1 08 (1 957), 1 072.

4D. Bohm and J . B. Hiley, The Undivided Universe, Routledge, London, 1 993; p. 1 49.

18

THE MATHEMATICAL INTELUGENCEA © 2001 SPRINGER-VERLAG NEW YORK

seem to us that there is a need to re­

diffraction phenomena. The Heisen­

anywhere in the universe a single

nounce our belief in the determinism

berg uncertainty principle introduces

system which did not combine the

of individual physical phenomena (that

an additional indeterminacy in experi­

three

is to say, the individual motion of par­

ments intended to observe the actual

probability, and the wave-particle duality, then this system could be

elements

of

indivisibility,

ticles), and it is thus that our concepts,

position or momentum of a particle, re­

elsewhere very similar to those of M.

sulting from an indivisible quantum be­

used to make measurements on

Born, nevertheless appear to differ per­

ing transferred in such

other systems which were more pre­

an

observation

ceptibly."5 De Broglie abandoned this

from the observing apparatus to the

cise than the limits of precision set

interpretation and became a convinced

particle, thus changing its momentum

by the uncertainty principle, and as

proponent of the Copenhagen inter­

and limiting the accuracy of the mea­

a result, one of the most fundamen­

pretation

surement. The precise limits on accu­

tal predictions of quantum theory

racy are set by the fluctuations in the

could be contradicted. 7

as

a

result

of Wolfgang

Pauli's criticism at this Conference. The Copenhagen interpretation has

SchrOdinger field. The indeterminacy

been questioned not as "perverse" but

no longer has as a consequence the

as mysterious. J. S. Bell6 wrote,

non-existence

When I was a student I had much

of individual particle

Bohm's (and de Broglie's) interpre­ tation leads to precisely the same re­

trajectories with well-defined positions

sults for all physical processes as does

and momenta. The abolition of the

the usual interpretation, as long as the mathematical theory retains its form. It

difficulty with quantum mechanics.

wave-particle duality returns us to the

It was comforting to find that even

classical status of probability as re­

does, however, offer a broader con­

Einstein had such difficulties for a

flecting imperfect knowledge of initial

ceptual framework which allows more

long time. . . . But in

1952 I saw the

conditions due to instability and com­

general

impossible done. It was in papers by

plexity

Bohm's careful analysis in his text of

David Bohm. Bohm showed explic­

causes imbedded in the context of the

the assumptions needed for the uncer­

itly how parameters could indeed be

event under consideration. Probability

tainty principle to follow from the

introduced,

into

of

numerous

independent

mathematical

formulations.

non-relativistic

is no longer "intrinsic, " and a deeper

Fourier transform principle8 between

quantum mechanics, with the help

understanding of what goes on at the

.lx and Ap had been summarized as fol­

of which the indeterministic de­

sub-quantum-mechanical

scription could be transformed into

some day allow us to predict and per­

sequence of the relation

a deterministic one. More impor­

haps control the action of some of

tween the width of a wave packet, .l.x,

level

may

tantly, in my opinion, the subjectiv­

these hidden vmiables, and reduce the

ity of the orthodox version, the nec­

sway of probability at the quantum-me­

essary reference to the "observer"

chanical level. Bohm,

could be eliminated. Bohm showed that the objections

lows: The uncertainty principle is a con­

.lx Ak

�

1 be­

and the range of wave numbers Ak of the

waves making up the packet, when we take into account the following quan­

in presenting the Copen­

tum-mechanical

principles:

(1)

The

hagen interpretation in his text, re­

de Broglie relation between wave num­

peatedly stressed that the uncertainty

ber

against de Broglie's earlier interpreta­

principle is anchored in three

ele­

and momentum: p = hlx = hk. (2) Whenever the position or momentum

tion could be overcome. The electron

ments:

(1) the wave property of mat­

of a particle is measured, the result is a

is a particle pursuing a defmite trajec­

ter;

the indivisibility of the energy

definite number.

tory subject to fluctuations caused by

and momentum transfers, and the re­

"hidden variables" whose oscillations

lated particle properties of matter;

originate at a sub-quantum-mechanical

the lack of complete determinism.

level.

These

complex

and

(2)

(3)

unpre­

dictable fluctuations are responsible

These three elements work together

for our need to resort to probability in

to form a unit that would fall apart

(3)

The wave function

!/!(x) determines only the probability P(x)

of a given position and the transformed function

(k)

determines only the pro�

ability P(k) of a given momentum.

A more general theory not consistent

with the usual interpretation is obtained

predicting the motion of electrons. The

if any one of them would be re­

ensuing probability distribution devel­

moved from any object in the uni­

tent assumptions is abandoned:

ops and is derived from the wave func­

verse. Thus all parts of quantum the­

psi "field" satisfies the Schrodinger

tion

ory

equation.

of

the

Schrodinger

equation,

interlock in

such

a unified

if any of the following mutually consis­ (1) the

(2)

If we write

1/J = R

exp

which represents the action of a field

structure that it is very difficult to

(islh), then the particle is restricted to

guiding the particle's trajectory in such

conceive of our giving up any one

p

a way that the probability distribution

element,

semble of particle positions with a

will display typical interference and

whole quantum theory. If there were

unless we give up the

=

\l s(x);

(3) we have a statistical en­

probability density P =

l !/JCx)J2.9

5L. de Broglie, Nouvelle dynamique des quanta, Comptes Rendus du Congres Solvay, 1 927, see pp. 1 1 4-1 1 6. 6J. S. Bell,

Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press, 1 987, p. 1 60.

7Quantum Theory, p. 1 1 4

BThere is a tendency in many texts to label the Fourier transform property as "the Heisenberg Uncertainty Principle" without mentioning the restrictions from quantum theory for this label to apply.

90. Bohm, lac. cit. fn. 2, p. 374.

VOLUME 23, NUMBER 4, 2001

19

break down and where our sug­

Bohm during the gestation and discov­

(1), (2), (3) his

gested interpretation can lead to

ery period of his alternative interpre­

hidden variables theory cannot be dis­

completely different kinds of pre­

tation and thereafter. I can testify that

tinguished experimentally from stan­

diction . 1 1

Ruskai rejects Bohm's later claim that under assumptions

such pressures were there, on a dam­ aging scale once he departed from the

dard quantum mechanics because the Schrodinger equation holds. She refers

According t o Bohm, such kinds of pre­

to the impossibility of deriving Heisen­

diction might include the divisibility of

berg's hypothesis about transition prob­

the quantum and hence the bypassing

Ruskai says that the publication of

abilities from the Schrodinger equa­

tion. 10 This, however, does not settle

of the uncertainty principle; or, say, the

Bohm's controversial articles in the

study of the fluctuations at a sub-quan­

Physical Review is evidence of the ob­

the matter: Either the imputed non­

tum-mechanical level responsible for

jectivity of the establishment toward

equivalence between the Heisenberg

the chaotic motion perceived as prob­

one whom Einstein had labeled "the

and Schrodinger formalism has as a

abilistic behavior at the quantum-me­

most promising young physicist." Yet

consequence that the wave equation is

chanical level.

his articles were received with a con­

insufficient to ground all of quantum

Perhaps

spiracy of silence1 4 or summarily dis­

if Bohm's ideas had not

orthodox Copenhagen view which he

had so strongly advocated in his text. 13

mechanics and thereby to validate the

been shunted aside, but accorded a

Copenhagen interpretation; or, in the

broad open forum of interest and dis­

Bohm's article appeared during the

contrary case, any more general formu­

cussion to sharpen and defend his

heyday of the House Committee on Un­ American Activities, and many mem­

missed.15

lation in which the Schrodinger equa­

views when they were still fresh, new

tion holds will equally well ground

experiments

ensued.

bers of the academic establishment

quantum mechanics. Is Ruskai asking of

Meanwhile, these theories offer an al­

were reluctant to associate with vic­

would

have

the Schrodinger equation that it allow

ternative to the mysteries associated

tims of this persecution. Quite a few

one to derive the transition probabilities

with the Copenhagen interpretation.

fmgered others to safeguard their own

Thus rather than being required to

positions. (In the same way, Bohm was

in order to play its ftmdamental role? Quite contrary to Ruskai's assertion

speak of superpositions in infmite-di­

refused

that the proponents of Bohm's hidden

mensional Hilbert spaces leading to ex­

Alamos because he had been named by

variables theory assert the impossibil­

perimental results, one can speak of

one who later encouraged physicists to

ity of experimental verification as a

ensembles of trajectories leading to the

ignore his papers. 16) I recall at lunch a

clearance

to

work

in

Los

virtue rather than seeking new phe­

same results.

J. S. Bell titled one of his

then young, upcoming member of the

nomena to explain or test this theory,

notes "Quantum field theory without

Institute saying that David Bohm had

we have this conclusion of Bohm's

observers, or observables, or measure­

moved to the "lunatic fringe."

seminal paper:

ments, or systems, or apparatus, or

I remember the excitement and joy

wavefunction collapse, or anything like

David Bohm expressed to me-"I can't

that. "12

believe that I was the one to see

An experimental choice between

these two interpretations cannot be

The social context in which Bohm's

made in a domain in which the pres­

theory was advanced was rife with

"think different" about quantum me­

ent mathematical formulation of the

"hidden assumptions" in Ruskai's lan­

chanics. He hungered for detailed re­

quantum theory is a good approxi­

guage. She dismisses the role of social

actions to his theory, for arguments

mation; but such a choice is con­

pressures in guiding research and ad­

and discussions with colleagues, dur­

ceivable in domains such as those

herence to particular views, such as

ing his four years of exile in Brazil. It

associated with dimensions of order

the Copenhagen approach to founda­

can hardly be said that societal pres­

this! "-upon realizing that one could

of w - 13 em, where the extrapola­

tional questions. I was one of those

sures guiding research directions were

tion of the present theory seems to

who were closely in touch with David

in no way a factor.

1 0See J ammer, The Philosophy of Quantum Mechanics, Wiley, 1 974, p. 289. 1 1 0 . Bohm, foe. cit. fn. 2, p. 391 . 1 2J. S. Bell, Phys. Reports 1 37 (1 986), 49-54. 1 3There, Bohm strongly disputed the possibility of hidden variables in many sections, Only at the very end does he accord them a very doubtful credence. See the dis­ cussion below. 1 4David Peat, Infinite Potential, Addison-Wesley, 1 997, the chapter "Brazil and Exile"; personal communication from Bohm and others at the time. 1 5Rosenfeld (quoted in Max Jammer, The Philosophy of Quantum Mechanics, Wiley, 1 974, pp. 279, 294) called Bohm's theories "empty talk" and "a short lived de­

cay product of the mechanistic philosophy of the 19th century." Pauli said, "Old stuff dealt with long ago." A particularly scathing attack is in Heisenberg's essay in

Niels Bohr and the Development of Physics, McGraw-Hill, New York, 1 955, p. 1 8: "This objective 'description' reveals itself as a kind of 'ideological superstructure' which has little to do with immediate physical reality; for the 'hidden parameters' of Bohm's interpretation are of such a kind that they can never occur in the descrip­ tion of real processes if the quantum theory remains unchanged. In order to escape this difficulty, Bohm does in fact express the hope that in future experiments (e.g . , i n the range beyond 1 0 - 1 3) the hidden parameters may yet play a physical part, and that the quantum theory may thus b e false. Bohr, however, is wont t o say, when such hopes are expressed, that they are similar in structure to the sentence: 'We may hope, that it will later turn out that sometimes 2 great advantage to our finances. 1 6See fn. 1 4 .

20

THE MATHEMATICAL INTELUGENCER

+

2

=

5, for this would be of

Here is how Bell felt about it:

quantum mechanics is another such example. Bohm was a Marxist at the time he The essential idea was one that had wrote his text Quantum Theory, as been advanced already by de well as at the later time when he ad­ Broglie in 1927 in his "pilot wave" vanced his new interpretation. He was picture. But why then had Born not drawn to the Copenhagen interpreta­ told me of this "pilot wave"? If only tion by what seemed to him its dialec­ to point out what was wrong with tical nature: the complementarity of it? Why did von Neumann not con­ two potentialities-wave and parti­ sider it? More extraordinarily, why cle-each to be realized at the expense did people go on producing impos­ of its opposite. Ideology did not lead sibility proofs after 1952, and as re­ him to seek a deterministic synthesis; cently as 1978? When even Pauli, in his text he let no occasion pass to Rosenfeld and Heisenberg could deny the notion of hidden variables. produce no more devastating criti­ There are just two places in his text cism of Bohm's version than to where he leaves open the possibility of brand it as "metaphysical" and "ide­ determinism. In Section 6. 1 1 , he pro­ ological"? Why is the pilot wave pic­ poses a possible test involving a pro­ ture ignored in textbooks? Should it ton lens by which the uncertainty prin­ not be taught, not as the only way, ciple might be contradicted; and in his but as an antidote to the prevailing discussion of the WKB approximation, complacency? To show that vague­ he emphasizes that this procedure im­ ness, subjectivity, and indetermin­ plies definite trajectories and veloci­ ism are not forced on us by experi­ ties for individual particles. mental facts, but by deliberate It was this insight into the WKB theoretical choice? 17 method that crystallized his thoughts and led subsequently to a drastically Bohm's humanistic and philosophi­ different view of the Schrodinger equa­ cal convictions left a stamp on his tion. His new interpretation left an work Ruskai (following Heisenberg) 18 opening for possible future modifica­ objects to the fact that Bohm's theory tions of the theory at the sub-quantum destroys the symmetry between the po­ level-definitely affecting the "hard" sition and momentum representations. parts of the theory. Only at this time Bohm objected to the purely formal did David Bohm perceive that such a approach in terms of abstract repre­ new interpretation was indeed much sentations being taken as a sufficient more compatible with a materialist phi­ reflection of physical reality. He pre­ losophy, and come to regard the ferred to think problems through in a Copenhagen interpretation as mired in "physical" way, dealing with objective positivism. (His book Causality and material reality, and let the mathemat­ Chance in Modern Physics, published ics emerge from that. It is particularly several years later, clearly develops the inappropriate to claim that his ideas materialist underpinnings of the new were "outside the realm of physics." 19 interpretation.) Reverting to Loren Graham's exam­ The most conspicuous social force ple (Intelligencer, vol. 22 (2000), no. 3, on him in the 1950s was political per­ 31-36) intended to show social forces secution; as I recall it, that actually had affecting the "hard" as well as the "soft" a liberating effect. At the party cele­ parts of physical theories, let me ten­ brating the publication of Quantum tatively explore whether the genesis of Theory in the winter of 1951, he re­ David Bohm's hidden variables in marked to me with bittersweet irony

A U T H O R

MIRIAM LIPSCHUTZ-YEVICK 22 Pelham Street Princeton,

NJ 08540

USA e-mail: [email protected]

Miriam Lipschutz-Yevick was born in Scheveningen (the name is so un­ pronounceable by foreigners that it was used as a password by the Dutch underground). She arrived in the USA in 1 940 as a refugee from the Nazis and has lived there since. Her doctorate is from MIT, 1 947; she was on the faculty of University Col· lege, Rutgers from 1 964 until her retirement.

She

probability and

has on

published her

in

invention,

"holographic logic." One of her dear­ est

nonscientific

grandchildren,

concerns

Aaron,

is

Ariela,

her and

Hannah, who appear with her in the accompanying photograph.

that perhaps he should be grateful to President Dodds of Princeton Univer­ sity and to the House Un-American Ac­ tivities Committee; for without the year's paid leave from the University while he was under indictment, he might never have come to "think dif­ ferent." Addendum 1. Ruskai writes, "With­ out the assumption of an external re­ ality it makes no sense even to discuss the concepts of science." Yes, but we must not confuse external reality, its

1 7J . Bell,

lac. cit., p. 1 60. 1 BHeisenberg, lac. cit. fn. 1 5, p. 1 9. 1 9"The posthumously published bock by Bohm and Hiley, cited in fn. 4, testifies that Bohm was fully informed on the latest experimental results relating to hidden vari· abies theories.

VOLUME 23. NUMBER 4, 2001

21

mathematical representation in theo­

in the classical limit. This is how the

experiment. This could be due to

ries, and the interpretation of the lat­

imaginary came into the wave equa­

our ignorance or perhaps because

ter as explanations of phenomena.

tion. Schrodinger did not just intro­

"God plays dice with the universe."

20

duce it, he needed it.

What sets de Broglie's pilot wave or Bohm's hidden variables against the

Addendum 2. It may be that there

orthodox view is not the abstract rep­

is disagreement about the concept of

Probability rather comes about be­

resentations

probability. Ruskai rests some of her

cause of objective contingencies and

that

are

mathematical

constructs, but the interpretation of

discussion on the notion presented by

not because of the subjective "we are

how they pertain to objective reality,

Faris in the Appendix of Wick's book.

unsure." Faris's attempt to have prob­

21

"what the world is like." Remember

Faris writes at the beginning of this Ap­

ability theory elucidate the "mysteries"

that both the de Broglie relation and

pendix,

may be merely transplanting them into

the Schrodinger equation were guided

(mimicking)

principle,

Probability comes about if we are

which implies agreement with reality

unsure of what will happen in an

by

the

correspondence

an

inconsistent

proba­

bilistic formalism that cannot corre­

spond to objective reality. 22

20See in this connection the return to the attitude "quantum mechanics works" in the article by Christopher A. Fuchs and Asher Peres, "Quantum mechanics needs no interpretation," Physics Today, March 2000. "What it does," these authors write, "is provide an algorithm for computing the probabilities for the macroscopic events that are the consequences of our experimental interventions." Bohm, like many of the early generation of discoverers of quantum mechanics, was searching for an "un· derstanding" beyond merely correct predictions of experiments based on algorithms. 21 0 . Wick, The Infamous Boundary, Birkhauser, 1 995. 22See, for instance, the chapter on Chance in Poincare, Science and Method, Dover, N.Y. 1 952; Miriam Upschutz-Yevick, "Probability and determinism," American

Journal of Physics (1 957), p. 570.

Evariste Galois

(1 81 1 -1 832)

Herbert E . Salzer

Evoked in every treatise on equations. Victim of violence, honored evermore As algebraist, and in human relations Rebel in spirit, radical to the core. Ingeniously you tamed the surds so cryptic, Symmetries, substitutions at one swoop. Thoughts you expressed in language too elliptic Epitomized the essential use of group. Groups and their subgroups grasped, new avenues Abound, new applications in the sequel. Lover of truth, firm against life's abuse. Original mind with courage hard to equal, Immortal creator of a new, productive Synthesis of inductive and deductive. 941 Washington

Brooklyn, USA

22

THE MATHEMATICAL INTELUGENCER

Avenue, Apt. 28

NY 1 1 225-2454

ll�fflJh§rr6hf¥1MQ.'i.i,ii!,ilh£j

Confusion About Bohm Mary Beth Ruskai

I

Marjorie Sen echal , E d it o r

I

n the preceding notes, Jane Cronin

It was never my intention to present

and Miriam Lipschiitz-Yevick raise a

anything approaching a complete ac­

number of interesting issues in the his­

count of the historical development of

torical development of quantum me­

quantum theory, much less an evalua­

chanics, particularly in regard to the

tion of social influences on the accep­

work of David Bohm. However, they

tance of competing theories. Rather I

seem to have read my article outside

chose to illustrate specific issues with

of the context in which it was written,

examples from quantum theory. The

namely, as part of a set of articles in

rapid acceptance by physicists of a the­

which it was agreed at the outset that,

ory which was paradoxical and "far

as Loren Graham wrote [ 10], "everyone

from every physicist's personal experi­

agrees . . . that social, political, reli­

ence" illustrates the extent to which

gious and philosophical ideas can of

convincing experimental evidence can

course affect what topics get studied

overcome social and cultural biases.

and what theories get conceived. . . . "

In quoting only a few words from my

Marjorie Senechal [ 18] reinforced this

concluding

theme in her introduction to the re­

ously distorts its meaning to imply that

paragraph,

Cronin

seri­

development

sponses by Michael Harris and me

I asserted that the

when she wrote, "We can agree at the

quantum theory was immune to social

outset

that

in

different times and

places scientific research has been

of

forces. Therefore, I repeat my con­ cluding sentences:

(and continues to be) directed by so­ ciety's carrots and sticks. . . . " Thus,

Few jigsaw puzzles fit together so

rather than "dismiss[ing] the role of so­

neatly. We are forced to overcome

cial pressures" (in Lipschiitz-Yevick's

the biases arising from our experi­

[23] words) I do not even consider

ence with the familiar macroscopic

them, for the simple reason that they

world of classical mechanics de­

were not germane to the question I was

spite the challenge of resolving all

asked to comment on by the editors.

questions about the foundations of

That question was more complex

quantum theory. In the end, quan­

and concerned the effect of social con­ lation of a physical theory and the

tum theory remains a human con­ struct subject, in principle, to so­ cial forces. But it is a theory so

process of "justification." I agree with

remarkable, so different from ordi­

Graham that "different people with dif­ ferent views may formulate a theory in

nary experience, that it transcends social and cultural forces. (empha­

quite different mathematical terms"

sis added)

text on both the mathematical formu­

and that "social, political, religious, and philosophical ideas SOMETIMES af­

That it is quantum theory itself, and not

fect scientists' evaluation of the evi­

its development that I regard as "tran­

dence for and against particular theo­

scend[ing] social and cultural forces"

ries" (Alan Sokal, quoted by Graham

was further reinforced in the second

[10]; emphasis in original). However, I

paragraph of my Appendix A on gen­

also argued that the need for "consis­

der issues.

tency" between experiments, as well as

It is also important to clarify that I

the more commonly cited need for "re­

use the term "Bohmian mechanics," as

producibility"

of

individual

experi­

is now commonly done, to refer to the

ments, minimizes social influences in

theory developed by Diirr, Goldstein,

the fmal outcome.

al., in the past

et

15-20 years as reflected

© 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 4, 2001

23

by my reference to [2,5] rather than to

and Schrodinger formalism has as a

Bohm's original papers. This theory is

consequence that the wave equation

tem, and quantum mechanics has been

based on Bohm's work and can, I feel,

is insufficient to ground all of quan­

shown

be regarded as part of his scientific

tum mechanics and thereby to vali­

giant multi-particle systems as neutron

ally no such thing as an isolated sys­

[14]

to accurately describe such

legacy; but it is not identical to his orig­

date the Copenhagen interpretation;

stars. Ultimately, there is no reason not

inal formulation in every detail. Thus,

or . . . any more general formulation

to regard the entire universe as one gi­

1

nothing I said was a "comment on

in which the Schrodinger equation

gantic

David

mechanics"

holds will equally well ground quan­

Schrodinger equation. But rather than

tum mechanics. Is Ruskai asking of

resolving the matter, this merely re­

Bohm was a brilliant and complex

the Schrodinger equation that it al­

places one conundrum with another,

Bohm's

quantum

per se.

molecule

governed

by

the

person who made many important con­

low one to derive the transition prob­

about which there is an extensive lit­

tributions to physics and to quantum

abilities in order to play its funda­

erature.

theory. But it does not serve his mem­

mental role?

Despite the failure to resolve fully the paradoxes associated with the pe­

ory to insist that his theories were without flaws. Niels Bohr and Louis de

This question merits an answer and

culiar role of measurements and ob­

Broglie had important and profound

commentary. It is

the critical issue. Schrodinger equation i-h -fJt i/J =

servables, the von Neumann formula­

impacts on the development of quan­

The

tion of quantum theory has withstood

HljJ describes the time development of an isolated system described by the

ertheless, the dilemma has given rise to

Hamiltonian H. In the usual Dirac/von Neumann formalism, 2 an observable is

which now include Griffith's "consis­

Bohmian mechanics is ever verified or

represented by a self-adjoint operator

tent histories," and "spontaneous lo­

falsified, Bohm unquestionably made

A, and additional axioms are needed to

calization," as well as Bohmian me­

profound contributions. His reformula­

describe the so-called measurement

chanics. An overview of the relation

tion of the EPR experiment (which

process. It is asserted that the only

between some of these theories and

tum mechanics by putting forth theo­ ries (e.g., the Bohr model of the atom) which proved, in the end, seriously flawed. Whether or not some variant of

the test of time and experiment. Nev­ a

number

of

alternative proposals

Cronin mentions and I will comment

possible result of measuring the ob­

on in a subsequent article) and the so­

servable associated with

called

suffice to give him an important place

A (where I have made the sim­ plifying assumption that A has only

in the history of quantum theory.

discrete spectrum). Moreover, when

Schrodinger equation is

the system is in the state

the proba­

the conventional approach one needs

ak is

additional axioms about the measure­

Aharanov-Bohm

effect

alone

A is an eigen­

value of

ljf,

the measurement paradox was given recently in

[8,

9].

What is important in response to Lipschiitz-Yevick's question is that the

not enough. In

Is the Schrodinger

bility of obtaining the eigenvalue

Equation Enough?

1(1/J, cf>k/12

the correspond­

ment process. In Bohmian mechanics,

Lipschiitz-Yevick points out that I re­

ing eigenvector. It is here, in the so­

ject the claim that Bohmian mechanics

called measurement process and not in

a single linear equation is replaced by a pair of non-linear equations. 3

where

cPk is

the Schrodinger equation, that the cannot be distinguished experimen­

con­

tentious probabilities arise.

However, in fairness, I must also admit that my previous article greatly

tally from standard quantum me­

Now, at this point the reader may

oversimplified the situation when I

chanics because the Schrodinger

well be perplexed. Doesn't the system

said that the claim that Bohmian me­

equation holds. She refers to the im­

also interact with the measuring appa­

chanics "can not

possibility of deriving Heisenberg's

ratus? Why not consider the original

from standard quantum theory . . . re­ lies on the fact that . . . the Schrodinger

[be distinguished]

hypothesis about transition proba­

system and measuring device as a larger

bilities from the SchrOdinger equa­

system subject to the Schrodinger equa­

tion. This, however, does not settle

tion? It is

the matter: Either the imputed non­

goes to the heart of the matter, and

need to derive that Dirac/von Neumann

equivalence between the Heisenberg

there is no simple answer. There is re-

measurement formalism from the non-

this

question which really

equation holds." Bohm, Diirr, Gold­ stein,

et al. were not only aware of the

1 1t is curious that Lipschutz-Yevick. who objects to my comments, makes no mention of Harris's remark that "Durr, Goldstein, et a/. may have constructed a consis­ tent deterministic account of quantum mechanics." By citing only works of Durr, Goldstein et a/., without even mentioning Bohm, Harris seems to leave the reader with the impression that they developed a new theory rather than building upon Bohm's seminal work. 2The widely used term "Copenhagen interpretation" is rather vague, and it is sometimes said that there is more than one variant of the Copenhagen interpretation. However, this interpretation is closely associated with a mathematical formulation put forth nonrigorously by Paul Dirac and in precise terms by John von Neumann. It is carefully and elegantly presented in his influential book Mathematical Foundations of Quantum Mechanics [20]. At least one author [21] makes a distinction between the "Copenhagen interpretation" and von Neumann's formulation, referring to it as the "Princeton interpretation." It is this latter mathematical theory that gained wide acceptance, and is often incorrectly referred to as the "Copenhagen interpretation." 3Lipschutz-Yevick also expresses some concern about the possibility of confusion between interpretations and mathematical formulations. Now, interpretations are quite subjective and almost surely subject to cultural forces. The most we can hope for is to insist on interpretations that are consistent with mathematical formulations and/or experiments. Therefore, I confine my discussion entirely to the mathematical theory associated with this pair of non-linear equations.

24

THE MATHEMATICAL INTELLIGENCER

linear equations of Bohmian mechan­ ics; they presented cogent arguments for doing so. The Diirr-Goldstein ver­ sion, based on the concept of "quantum equilibrium," is sketched in [8] and de­ scribed in more detail in [5]. In the words of Sheldon Goldstein [7], Bohmian mechanics is "richer." However, attention has focused almost entirely on demonstrating that Bohmian mechanics yields the Schrodinger equation and a satisfactory explana­ tion of experiment consistent with the conventional theory. What has not been adequately explored are the ad­ ditional consequences of the non­ linearity. What About Experiments?

Before exploring the possibility of an experimental test of the mathematical reformulation of quantum theory now known as Bohmian mechanics, I want to clear up another point. Lipschiitz­ Yevick states that It is particularly inappropriate to claim that his [Bohm's] ideas were "outside the realm of physics." But what I actually said was quite different: It is curious that its proponents as­ sert the impossibility of experi­ mental verification as a virtue rather than seeking new phenomena to explain or test their theory. Whether or not the Bohmian view is useful, this seems to place it outside the realm of physics. (emphasis added) The word "this" has a clear antecedent which is not David Bohm, but the al­ leged "impossibility of experimental verification." This is hardly a novel point of view. Similar criticisms have been made about string theory (even by David Bohm [ 16]), because of the difficulty of experimental verifica­ tion. If one regards physics as an experimental science, then it is al­ most a tautology to assert that what is not experimentally verifiable is not physics. Of course, the list of ideas and topics, such as social implications,

which are relevant to physics is much test Bohmian mechanics. Quantum broader. theory has observable consequences at Lipschiitz-Yevick correctly points macroscopic scales. The non-locality out that Bohm's original paper con­ of Bohmian mechanics may have im­ cludes with the possibility of an even­ plications for quantum communica­ tual direct experimental test at do­ tion. Two of these are the possibility of mains less than 10- 13 em. However, his super-luminal communication and the biographer F. David Peat [ 16] also re­ security of quantum key distribution. ports (p. 269) that by the 1970s he "dis­ These issues will be discussed in a sub­ couraged such speculation, stressing sequent article. that his theory reproduced exactly all the predictions of conventional quan­ Reactions to Bohm's Theory tum theory." With few exceptions, his There is no doubt that, as Cronin and followers seem to have taken that ad­ Lipschiitz-Yevick point out, Bohm suf­ vice. fered for his political beliefs. At a min­ With recent advances in atomic and imum his exile in Brazil precluded him optical physics, it may now be possible from promoting his theory in person to test Bohm's theories directly. In­ via seminars and conferences. How­ deed, the recent work of Scully et al., ever, neither anti-communist paranoia summarized in [ 19] and verified inde­ in the United States in the 1950s nor pendently in a different experiment by the influence of the leaders of the Bohm's former collaborator Y. Ahara­ Copenhangen school provide a fully nov [ 1 ] , raises some serious questions. satisfactory explanation for the luke­ At a minimum [ 19], "A supporter of warm reception Bohm's theory re­ Bohmian mechanics would insist that ceived. The scientific objections, even the atom went along its Bohm trajec­ if not fatal, were also not frivolous. tory through one of the detectors, but Consider the reactions of Albert Ein­ left is mark in the other one," or [2], stein and Erwin Schrodinger, both of "there are 'measurements' of the posi­ whom were active and vocal oppo­ tion operator that are not measure­ nents of the Copenhangen interpreta­ ments of the actual position." tion. Lipschiitz-Yevick uses the term Einstein was neither vulnerable to, "sub-quantum-mechanical level" re­ nor a supporter of, McCarthyism. peatedly. Unlike terms such as "sub­ Moreover, he had supported and as­ atomic" which are well-defined, Lip­ sisted Bohm [ 16] in many ways. How­ schiitz-Yevick's term contains the ever, in a letter to Bohm's former stu­ hidden assumption that there is a lower dent, David Lipkin, Einstein wrote limit to the validity of quantum theory. ([ 16], p. 132), Although this may be the case, no such I do also not believe that the de lower bound has yet been observed ex­ perimentally. On the contrary, experi­ Broglie-Bohm's approach is very ments have now been performed down hopeful. If leads, f.i., to the conse­ quence that a particle belonging to to the level of 10- 10 atomic radii. (One a standing wave has no speed. This must be careful about the sense in which such distances are defined. This is contrary to the well-founded con­ viction that a nearly free particle figure is taken from [6] and actually should approximately behave ac­ refers to the wavelength of light asso­ cording to classical mechanics. ciated with accelerator experiments at the highest energy currently obtain­ able.) It seems that the domain of va­ Schrodinger, then director of the In­ lidity of quantum mechanics is now es­ stitute for Advanced Study in Dublin, tablished well below that which Bohm an ocean away from McCarthy and the envisioned at the time of his original House Committee on Un-American Ac­ tivities, was even less likely to be influ­ papers. However, I would like to emphasize enced by the political situation in the that microscopic observations are not United States. His reaction is described necessarily the only possible way to ([ 16], p. 132) in Bohm's own words in

VOLUME 23, NUMBER 4, 2001

25

an undated letter to Lipschiitz-Yevick. Schrodinger-4

tracted by a 1/r-potential for which the

objectivity of the establishment to­

only stable trajectories are circular or­

wards [Bohm]

bits. However, for charged particles, did not deign to write me himself,

Maxwell's theory of electromagnetism

but he deigned to let his secretary

implies that the acceleration in the tan­

tell me that His Eminence feels that

gential direction would lead to radia­

it is irrelevant that mechanical mod­

tion of energy, resulting in the electron

els can be found for the quantum

spiraling into the nucleus. In quantum

theory, since these models cannot include the transformation theory,

theory, this is countered by the Heisen­ berg uncertainty principle. 5 To explain

which everyone knows is the real

the stability of a hydrogen atom in

heart of quantum theory. Of course,

Bohm's theory, one must assume that

His Eminence did not find it neces­

the highly non-local quantum potential

sary to read my papers, where it is

conspires to permit exactly the deli­

explicitly pointed out that my model

cate balance needed for a pair of op­

not only explains the results of this

positely charged particles to remain in

transformation

equilibrium.

theory,

but

also

points out the limitations of this the­

Finally, it is worth noting that even

ory to the special case where the

Goldstein, perhaps the strongest cur­

equations are linear. . . . In Por­

rent advocate of Bohmian mechanics,

tuguese, I would call Schrodinger

wrote [9], Unfortunately, Bohm's formulation involved

unnecessary

complica­

ments indicate a considerable agree­

tions and could not deal efficiently

ment between the two men. The formu­

with spin. In particular, Bohm's in­

lation of Schrodinger and von Neumann was based on

linear

transformations,

and Bohm's theory was decidedly non­

vocation of the "quantum potential" made his theory seem artificial and obscured its essential structure.

imental evidence, it is hardly surprising

The issues surrounding the lack of ac­

that each should prefer the theory he

ceptance of Bohm's theory seem com­

had developed. Is this the picture of

un

studying the foundations of quan­ tum mechanics has long been far from the mainstream, it has never been suppressed. Bohm, Bell,

et al.

The papers of

were published in

reputable journals, . . . Reasonable people may disagree on the significance of a particular theory or in­ dividual's contribution. It is here, rather than in the physics per se, that questions of social influence are likely to arise. I have commented elsewhere, e.g., [17], on the role that gender sometimes plays. In a subsequent article, I will also dis­

of the social and political climate on the development of the careers of individu­ als and the development of physics.

The articles by Cronin and Lipschtitz­ Yevick have stimulated me to think anew about a number of issues related to Bohmian mechanics, for which a full discussion requires clarification of some

linear. In the absence of decisive exper­

burro or duas mulas?

It should be noted that even though

cuss the distinction between the effect

un burro. . . . Underneath the sarcasm, these com­

is not supported by my statement

plex and provide fertile ground for his­ torians of science. Neither scientific

A subsequent objection, similar to

flaws nor social pressures alone seem

Einstein's, arose from the realization

to give fully satisfactory explanations.

that Bohm's theory implies that in the

technical issues regarding the EPR ex­ periment and non-locality. These will be discussed in a forthcoming article. Acknowledgments: It is a pleasure to

thank Edvamia Bahia for assistance with

Portuguese.

The

author

was

supported in part by National Science

ground state of the hydrogen atom, the

Conclusion

velocity of the electron is zero [22]. It

It is important to distinguish between

Foundation Grant DMS-0074566.

simply sits there, albeit at a random po­

physics, which is an experimental sci­

sition. To understand why some find

ence, and

this hard to swallow, it is worth re­

The latter are most certainly

ob­

1 54 i n Bohmian Mechanics and Quantum

calling that explaining the stability of a

jective. Thus, Lipschtitz-Yevick's asser­

Theory: An Appraisal (ed. J. Cushing et a!.),

hydrogen atom is often regarded as one

tion that

physicists,

who are people.

not

REFERENCES

[1 ] Y. Aharanov, and L. Vaidman, pp. 1 4 1 -

Kluwer Academic, 1 996.

of the great successes of quantum the­

[2] K. Berndl, M. Daurner, D. Durr, S. Gold­

ory. The model of a hydrogen atom is

Ruskai says that the publication of

that of two oppositely charged parti­

Bohm's controversial articles in the

mechanics," II Nuovo Cimento 1 1 08,

cles, one positive and one negative, at-

Physical Review is evidence

737-750 (1 995).

of the

stein, and N. Zangh, "A survey of Bohrnian

4The only discussion I could find of Bohm's work in Moore's biography [1 5] of Schrbdinger is a brief mention on p . 31 1 after a discussion of Schrbdinger's reaction to the EPR paradox in 1 935 that Bohm, among others, had eventually produced hidden-variable theories. However, the description on pp. 451-452 of events in the fall of 1 952, after Bohm's papers appeared, raises several interesting questions. In September, Schrbdinger wrote enthusiastically about a meeting planned for December to discuss the interpretations of quantum mechanics. However, in early October, he became seriously ill and was unable to participate in person . If his letter to Bohm was written during the months of Schrbdinger's illness and recovery, it would explain communicating via his secretary, which so offended Bohm. On the other hand, if Bohm was not invited at least to submit a paper to be read at the conference (if he were unable to travel), that was a serious oversight. 5The standard argument is that as the electron spirals into the nucleus, its position, and hence the uncertainty in its position, will become small; this then implies large momentum and large kinetic energy. In fact, this argument is flawed. However, an alternative argument following the same physical intuition can be formulated using Sobolev inequalities. See Lieb [ 1 3] for details.

26

THE MATHEMATICAL INTELLIGENCER

[3] J. Cronin, "Social influences on quantum mechanics, 1 , " Mathematical lntelligencer

23, no. 4.

1 5- 1 7

[4] PAM. Dirac, The Principles of Quantum

The Flight from Science and Reason (New York Academy of Sciences, 1 996).

Mechanics (Oxford, 1 930). [5] D. Durr, S. Goldstein, and N. Zangh,

[1 2] M. Harris, "Contexts of justification," Math­

"Quantum equilibrium and the origin of ab­

ematical lntelligencer 23, no. 1 , 1 8-22

solute uncertainty," J. Stat. Phys. 67,

(2001).

[6] C. A. Fuchs and A. Peres. "Quantum the­ ory needs no 'interpretation,' " Phys. To­

Mod. Phys. 48, 553-569 (1 976). [1 4] E. Lieb, "The stability of matter: From atoms to stars," Bull. AMS 22, 1 -49

day 53(3), 70-71 (March. 2000). [7] S. Goldstein, "Quantum philosophy: The flight from reason in science:" pp. 1 1 9-

M.

1 , 1 6-1 7 (2001 ).

0. Scully, "Do Bohm trajectories always

provide a trustworthy physical picture of particle motion," Physica Scrip ta T76,

4 1 -46 (1 998). [20] J. von Neumann, Mathematical Founda­ lation, Princeton University Press, 1 955).

[2 1 ] A. Whitaker, Einstein, Bohr and the Quan­ tum Dilemma (Cambridge University Press, 1 996). [22] D.

(1 990). [1 5] W. Moore, Schr6dinger: Life and Thought (Cambridge University Press. 1 989).

1 2 5 in [1 1 ] .

[ 1 9]

tion of Quantum Mechanics (English trans­

[1 3] E. Lieb, "The stability o f matter," Rev.

843-907 (1 992).

no.

lntelligencer 22, no. 3, 3 1 -36 (2000). [1 1 ] P. R. Gross, N. Levitt, and M. W. Lewis,

(2001).

tification," Mathematical lntelligencer 23,

display social attributes?". Mathematical

[8] S . Goldstein, "Quantum theory without ob­

[1 6] F. D. Peat, Infinite Potential: The Life and

servers- Part One," Phys. Today 51 (3),

Times of David Bohm (Addison-Wesley,

Wick,

The

Infamous

Boundary

(Birkhauser, 1 995).

[23] M. LipschUtz-Yevick, "Social influences on quantum mechanics, I I , " Mathematical ln­

telligencer 23, no. 4, 1 8-22 (2001 ).

1 997).

42-46 (March, 1 998). [9] S. Goldstein, "Quantum theory without

[1 7] M. B. Ruskai, "Are 'feminist perspectives'

observers- Part Two," Phys. Today 51 (4),

in mathematics feminist?" pp. 437-441 in

38-42 (April, 1 998).

[1 1 ] .

[1 0] L. Graham, "Do mathematical equations

Department of Mathematics University of Massachusetts Lowell Lowell, MA 01 854 USA

[1 8] M. Senechal, "Between discovery and jus-

e-mail: [email protected]

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VOLUME 23, NUMBER 4, 2001

27

G. G. LORENTZ

Who D i scovered Ana yti c Sets? n answer to this question, which I will call Question 1 , requires the study of afascinating segment of the history ofmathematics, connected with the names of P. S. Aleksandrov (1896-1 982), F. Hausdorff (1868-1942), N. N. Luzin (1883-1 950), and M. Ya. Suslin (1894-1919). Analytic sets are also called A-sets or Suslin sets. I have chosen the term "analytic sets" because of its neutral character. In 1915-16, Luzin was a young professor at Moscow Uni­ versity. Aleksandrov and Suslin were his students. Luzin was an excellent mathematician. Even more important was the inspiration that he conveyed to his students, starting this way the astonishing ascent of Moscow mathematics. In what follows, I shall use the original papers [Aleksan­ drov, 1916; Hausdorff, 1916; and Suslin, 1917]. In 1915, Aleksandrov, in Moscow, and Hausdorff, in Bonn, were separated by the front line of World War I. In­ dependently, they proved the continuum hypothesis for Borel sets B in �n , which asserts that each B either is count­ able or has the power of the continuum. Both men used a representation, by means of closed or open sets, of all Borel sets B of a transfinite class VA,, g < fl. Each developed his representation incompletely, only as far as it was useful for the proof. Their formulas or methods, which depended on g, went only in one direction, from B to closed (or open) sets An, n = 1, 2, . . . . And they could not be inverted; that is, they were not defined for arbitrary An. A new student of Luzin, Suslin joined the investigation in 1916. As Luzin described it (see [4]), a natural question for himself and his two students was to describe Aleksan­ drov's representation formally. Of course, it would have

28

THE MATHEMATICAL INTELLIGENCER © 2001 SPRINGER-VERLAG NEW YORK

been desirable to find a solution that would produce all Borel sets and nothing else, and therefore would be inde­ pendent of transfinite numbers; I will call this Question 2. Partial answers were given by Suslin [15]. He proposed to relabel a simple set sequence {An} in a "crazy way" as a "Suslin tree":

This is possible because the set of all natural numbers n = 1, 2, . . . and the set of all finite sequences v0 = (n1, . . . , nk) of natural numbers are each countable. We write v = (nb . . . , nk, . . . ) for infinite sequences, and v0 < v if v0 is a beginning of v. Suslin defmed the set operation (2)

B= =

Y {An 1 n An1,n2 n

· · · n An� > . . . , nk n · · · }

U n Av0, v v0< v

calling it the A-operation. The union is extended over all (uncountably many) sequences v. For closed A v0 , the op­ eration (2) generates all Borel sets, but also many non-Borel sets. This was a partial answer to Question 2.

Sets produced by (2), the analytic sets, created a sen­ sation in set theory. Even formula (2) was unusual, con­ taining an uncountable union. Up to then we shunned unions of this type, for they could easily lead to undesir­ able non-measurable sets. Hausdorff called (2) an so oper­ ation. For him, u, o stood for countable unions and inter­ sections, respectively; s, d stood for uncountable unions and intersections. Thus, (2) cannot be written as (3) for this is a uo, not an so operation. The important development of 1927 was the second edi­ tion of Hausdorffs Mengenlehre [6] with a masterful pre­ sentation of the theory of Borel and analytic sets in metric spaces. He called these sets "Suslin sets." In the following period, general set theory became fashionable. In the West, books by Luzin, H. Hahn, K. Menger, and K. Kuratowski joined Hausdorff in his assignment of priorities. This fash­ ion was also featured in a few Soviet publications. The au­ thors of the historically important long memoir about set operations, Kantorovich and Livenson [8) could not be called unfriendly to Aleksandrov. But they claimed that "the first known (not elementary) analytic [set] operation is the A-operation of Suslin. With it he introduced a new and wider class of sets, viz., the A-sets. " Aleksandrov's friend Andrei Kolmogorov gave a very balanced and fair testi­ mony. In his review of set theory in the book, Mathematics in the USSR for 1 5 years [ 12] we read: "Suslin applied procedures of Aleksandrov's 1916 paper to discover a new class of sets of fundamental importance-the A-sets" (p. 38), and "the theory of A-sets has been fast developed by Suslin's methods" (p. 45). As we shall see later, Kolmogorov's for­ mulation is a good description of the Suslin-Aleksandrov controversy, except that it disregards Hausdorffs contri­ bution. I am not a stranger to analytic sets. In the 1930s I en­ joyed the geometric exposition of the theory by Luzin [ 1 1 ) , preferring it to the dry formulas of Hausdorffs book But K. Zeller and I had to use the Hausdorff version when we wanted to apply it to summability. The Riemann convergence set R(.'!l) : = {s} of a series .'!1: I'l an with real terms consists of all sums s = Lk= l ank of convergent rearrangements of .'!1. The familiar Riemann's theorem describes all possible R(.'!l): this set can be empty (for instance if an ...f+ 0); it can be any one-point set (if I I an i < oo) ; and it can be the whole real line. Now let C be a series summability method defined by a matrix. Replacing convergence of Iank by its C-summabil­ ity in the above definition, we get the Riemann C-set R(C, .'!1) of C and .'!1. To find the sets R(C, .'!1) for a given C is ex­ tremely difficult. But we proved (Lorentz and Zeller, [ 10]) that the set of the R(C, .'!1) for all C and .'!1 coincides with that of all analytic sets of the line. This was probably the first time analytic sets were used to resolve a concrete problem of analysis.

From the early 1920s, Aleksandrov occasionally claimed the A-operation as his. We now have new sources of in­ formation about the priority questions; they are pointing in opposite directions. Aleksandrov's reminiscences [2] were published in Uspekhi Mat. Nauk, a journal that he edited until his death. A second source is the book [4) which con­ tains complete stenographic reports of Luzin's 1936 trial at the Soviet Academy of Sciences. Believed lost or destroyed by the participants, a copy was found in the Academy's archives in 1993. The published volume contains enlight­ ening commentaries by eminent Russian historians of mathematics, S. S. Demidov and others. Here I shall ex­ amine only a small, but central and illustrative sector of the trial, the Luzin-Aleksandrov controversy about analytic sets. Luzin suffered political persecutions at two critical pe­ riods of his life. In 1930, after returning from a long and fruitful sojourn in Paris, he was attacked by E. Kol'man, a leading member of Moscow's Party Council and a profes­ sor at the Communist Academy (see Shields [ 14)). With horror, Luzin saw his older friend Egorov disappear into prison and die shortly afterwards. Kol'man denounced the activity of Egorov and his friends Luzin and P. A Florin­ skil as "fascist-tainted reactionary science inherited from the old Moscow mathematical school." To him Luzin's mathematics were idealistic, that is, opposing Marxism's materialistic philosophy. Luzin's posi­ tion at the university became precar­ ious when he refused to join the sign­ ers of a propaganda letter directed against the "enemies of the people." Luzin fled the university, finding a niche at the Academy of Sciences. In addition to real functions and set theory, he turned to applied mathematics, with only moderate success. Luzin's trial in June 1936 was an integral part of Stalin's Great Terror of 1936--37. Directed against all independent thinkers-in the Party, in the intelligentsia, and in the pop­ ulation in general-it took a staggering number of victims. Davis [3, p. 1325] estimates that one million persons were sent to concentration camps or executed during its worst year. In most cases, the victims did not even understand the reason for their arrest. To initiate the campaign against Luzin in 1936, his ene­ mies laid a cunning trap, prompting him to praise mathe­ matical work at one of the less-than-average high schools in Moscow-praise that was then used against him. Vilifi­ cation at universities throughout the nation and in news­ papers followed, with eight full-sized articles in the leading daily, Pravda, with titles like "About the so-called Acade­ mician Luzin" or "Enemy in Soviet Mask" Then followed the trial at the Academy, conducted in secret. Luzin had ample reasons to believe that he was fighting for his life. Indeed, the KGB had prepared compromising materials about him. His friend Florinskii, mathematician, engineer, and orthodox priest, arrested in February 1933 together with a friend, was broken by the KGB. They con-

Analytic sets are

n ot always Borel .

VOLUM E 23, NUMBER 4, 2001

29

fessed to belonging to the KGB-invented "Party for the Re­ birth of Russia," with a future "government" including Luzin as foreign minister and another mathematician, the acade­ mician Chaplygin, as prime minister (V. Shentalinsky, [ 13], pp. 1 1 1-115). This material, with potentially deadly conse­ quences for Luzin, was never used. Famous mathematicians formed the interrogating com­ mission at the Academy's trial. Of these, Lyusternik, Shnirelman, and Gel'fond already belonged to the "initi­ ating group" responsible for Egorov's downfall. They were joined by Sobol'ev. Luzin's former students were repre­ sented by Aleksandrov, Kolmogorov, and Khinchin. This revealed a split among Luzin's students: Lavrentiev and P. S. Novikov were present, but did not say a word against Luzin, a sign of civil courage, while Menshov and Nina Bari (one of the best Soviet female mathematicians) were missing altogether. Actually, Kolmogorov said very little. Among the full members of the Academy one saw the "red professor" 0. Yu. Schmidt, later famous for his Arctic expeditions, the completely mute I. M. Vinogradov, and S. N. Bernstein, the only faithful and persistent Luzin defender. Aleksandrov, who re­ placed Egorov as the presi­ dent of the Moscow Mathe­ matical Society, a post he was to hold for 32 years, was the natural leader of the anti­ Luzin group and the most ag­ gressive and sarcastic interrogator. Present at most sessions of the trial, Luzin had no legal counsel. Luzin had a complex, sensitive, and highly excitable na­ ture. His lectures were excellent, full of ideas, hypotheses, suggestions for investigation. He charmed people at the first meeting. Inspiring adoration by many of his students, he reserved his own for his French teachers Borel and Lebesgue. Sometimes he would attribute to them his own discoveries. Aleksandrov was quite different. Having enjoyed a rich cultural upbringing, he was at home with literature, espe­ cially German, and theater. As rumor will have it, after his disappointment in Moscow in 1917-18, he seriously con­ sidered a theatrical career in the Western provinces, and he gave up the idea only because of the possibility of po­ litical problems under the Bolsheviks. Extremely ambi­ tious, he befriended two of the best Soviet mathematicians, Uryson (who died prematurely in 1924) and Kolmogorov. With Uryson, he published joint papers and founded the Moscow topological school. He was a good lecturer, a witty raconteur, but his stature as mathematician was definitely below Luzin's. A strange antipathy, even hate, separated him from his teacher. At the trial, Luzin stood accused of having plagiarized from his students, in particular, of having "borrowed" from Suslin the notion of analytic sets. Aleksandrov was deeply involved. Forty years later he declared: "For me the ques-

tion of priority in this case [of the A-operation and ana­ lytic sets) was never indifferent, concerning my first and (probably therefore) my dearest result" (Aleksandrov, [2], p. 235). Terminology rarely plays an essential role in priority dis­ cussions. This case was an exception. As described in his autobiography, Aleksandrov [2) visited Hausdorff in Bonn in 1924. In his description we read: "To Hausdorffs ques­ tion on how the new sets should be called, I firmly replied, Suslin sets, because he was the first mathematician prov­ ing that they are really new [and not just Borel) sets." By not suggesting that the defining operation is also Suslin's, Aleksandrov indirectly reserved for himself the credit for the discovery of the A-operation. In his book Mengenlehre (6), Hausdorff followed this advice only partly, calling both the sets and the operation (2) Suslin's. At the time of Luzin's trial in 1936, Aleksandrov, translating Hausdorffs book, completely changed Hausdorffs Suslin-terminology to A­ terminology. This led to heated controversy between Alek­ sandrov and Luzin at the trial. Even more interesting than the terminology are Alek­ sandrov's following statements. In his reminiscences [2, p. 235], he said . categorically that "Suslin suggested the name 'operation A' for the new set operation I had con­ structed, and the name 'A­ sets' for the sets which result from its application to closed sets. He stressed that he was suggesting this ter­ minology in my honor." We compare this with Aleksan­ drov's words spoken at the 1936 trial ([4], p. 90): "He [Suslin) never told me that he called them A-sets in my honor. It was Luzin who formulated the term while lectur­ ing at Moscow University. Incautiously, he underlined this." As a faculty member at Leningrad University in the 1930s, I heard two versions of what motivated Suslin to call his sets A-sets: (1) to honor Aleksandrov and (2) to parallel the common use of B-set for Borel set. The strongest example Luzin's accusers could cite for his alleged plagiarism, a charge that eventually could not stand up at the trial, was the following. Suslin's expres­ sions of deep gratitude to his teacher Luzin in the intro­ duction of his paper [ 15) were interpreted as signs of Luzin's plagiarism, implying that they must have been written under pressure by him. Vehemently denying this, Luzin insisted that Suslin wrote the introduction alone. To this and other similar arguments that could be neither proven nor disproven, Aleksandrov offered Luzin mock­ ing advice: "As a sign of our past friendship, allow me, your former student who will be grateful to you all his life, to give you in this difficult moment a really sincere [piece of] advice. You would do much better to give up hotly de­ fending your rightfulness in cases when [defense) is im­ possible and to find the necessary courage and humility to accept the accusations against you."

Term i nology rarely plays an essential role i n p ri ority

d i scussions. Th is case was an exceptio n .

30

THE MATHEMATICAL INTELLIGENCER

It is very fortunate for our inquiry that cooperation be­ tween Luzin and Aleksandrov during 1915-16 was also discussed at the trial. According to the record ([4], p. 89, p. 159), Aleksandrov expressed profound thanks to his teacher for the proposed subject for investigation, but minimized his contribution. Luzin was bound by the un­ written rule that demanded from a doctoral supervisor (which he in essence was) that the teacher never divulge his part in the joint work At the trial Luzin implied that he had never done this before and was doing it only un­ der the pressure of accusations. We can believe his tes­ timony because he would have been foolish to insult Aleksandrov and many of his assembled students by mis­ representation. This is what Luzin ([4], pp. 160-161) said to Aleksandrov in my free translation: During 1915 you always came to my dacha with pages of incorrect attempts which I revised. In spite of my con­ cerns, by means of tables of sets, a proof emerged for the Borel class � 4. I asked you to do this for the gen­ eral case. After joint work, a transfmite proof appeared. The reduction to one table [of sets] was entirely mine. (This probably meant the second table of Aleksandrov [ 1].) Mterwards, do you know what problem arose? How can the representation table of a Borel set be recon­ structed? This was completely my problem [Luzin's problem was one of the formulations of our Question 2]. We both worked on it. But then you asked to be excused because of the difficulty of the problem. I still possess a postcard where you wrote this. Exactly at this point, at this second table, the work of us three [Aleksandrov, Suslin, Luzin] intermingled. This allowed me to say in my lectures that it remained for you to make a small step, and the discovery [of the operation A] would be yours. But neither you nor I made this step. "I do not deny this," replied Aleksandrov. Aleksandrov's admission proves that he was not the dis­ coverer of operation A Suslin was, and he gave a partial answer to Question 2: Applied to trees (1) of closed sets, this operation produces all Borel sets, but also non-Borel sets. Seven years later, in 1923, Luzin and Sierpiriski gave a complete answer to Question 2. Operation A produces all Borel sets and these only if it is restricted to trees for which all terms in the union (2) are disjoint. How did the cooperation of Luzin and Suslin develop af­ terwards? Luzin did not say. We can assume that he sug­ gested his student answer Question 2. The title of Suslin's paper (which many find inappropriate), "On a defmition of B-measurable sets without transfinite numbers," clearly in­ dicates such a suggestion. But it is useless to guess about the extent of their cooperation. When and why did this deep animosity between Luzin and Aleksandrov develop? Aleksandrov indicated that it be­ gan in 1923, when Luzin, chief editor of the Mat. Sbornik, invited contributions by his friend Uryson to the journal,

but not by Aleksandrov. (At that time Luzin was more pow­ erful than Aleksandrov; in 1936 the relation was reversed.) More likely, the aversion started as early as 1916, when Luzin accepted Aleksandrov's resignation from the tri­ umvirate too easily, and helped Suslin to prepare his pa­ per. Working alone on the general continuum hypothesis, Aleksandrov suffered a failure, and left for the Ukraine, re­ turning to Moscow and mathematics a full two years later. Another variant of the history of the Aleksandrov-Luzin relationship is even grimmer. In Leningrad many mathe­ maticians believed that Aleksandrov was homosexual, a criminal offense in tsarist Russia, as well as in Soviet Rus­ sia, although rarely prosecuted. Perhaps Luzin had of­ fended his sensibilities in this connection. A note accusing Luzin appeared in a public statement by Kolmogorov at Moscow University in 1936. He reminded the audience of Luzin's great service to mathematics "before his moral and political disintegration." This was echoed by Aleksandrov [2], when the author told that he "found his teacher in the highest sphere of human values, a sphere that he later aban­ doned." Aleksandrov quoted Goethe that "each guilt finds its revenge in life." But I must discuss also the fourth participant on this scene. Hausdorffs role in the discovery of analytic sets was never properly described in the Soviet literature. The main difference between the two 1916 proofs was between the transparent Boolean set operations of Hausdorff and the "tables of sets" of Aleksandrov, inherited from a 1905 pa­ per by Lebesgue. Furthermore, Hausdorff started with open sets in his construction, while Aleksandrov employed their complements-closed sets. This difference is not that im­ portant so far as Borel sets � are concerned. For analytic sets 91, the matter is different. It is known that the com­ plement C(A) of A E 91 is also analytic only if A is a Borel set; in other words, that 91 n C(91) = 'lf3. There is no real symmetry between the classes 91 and C(91), however. An­ alytic sets coincide with the continuous images of Borel sets (Luzin); on IR\ they coincide with the Riemann sum­ mability sets (see above). Therefore we compute the dual of (2), obtained by tak­ ing complements. For the complement of A we get

where the Bv0 = : C(Bv0Av0) form a Suslin tree. Formula (4) yields, with open Bv0 , all sets B that are complements to analytic sets, and only these. Hausdorff [5] does not have (4), but Suslin trees are there, as are unions like Uv0 Bv0 ([5], p. 436), absent from Aleksandrov's paper. It is easier to guess (4) from Hausdorffs paper than to guess (2) from Aleksandrov's. However, to get analytic sets, a complement must be taken. Russian literature after 1990 about Suslin includes a

VOLUME 23, NUMBER 4, 2001

31

A U T H O R

GEORGE LORENTZ

2750 Sierra Sunrise Terrace

404

Chico, CA 95928

us a remarkable, impartial, and just exposition of the new theory. The results of the Academy-based trial deserve a sepa­ rate analysis in the English-language literature. It ended mildly for Luzin. Why was his life spared, why was he not expelled from the Academy? According to the editors of the Delo [4] he was saved by the highest Party echelons, perhaps even by Stalin himself. They insisted that accusa­ tions against Luzin should be formulated in academic rather than political terms. Accordingly, Aleksandrov stated a cou­ ple of times that Luzin's behavior displayed no anti-Soviet attitudes. The outcome of the trial suggested that mathe­ matics was a cherished science of the Party. The Golden Years of Soviet mathematics, particularly in Moscow, had begun.

USA e-mail: [email protected]

BIBLIOGRAPHY

[ 1 ] P. S. Aleksandrov, Sur Ia puissance des ensembles mesurables George G. Lorentz was born in St. Petersburg in 1 91 0 and

B, Comptes Rendus Acad. Sci. Paris

162

(1 9 1 6), 323-325.

pursued a mathematical career in the Soviet Union, moving

[2] P. S. Aleksandrov, Matematicheskaya zhizn v SSSR, stranitsy au­

later to Germany, then Canada, then the United States. He

tobiografii [Mathematical life in the USSR, pages of an autobiog­

built and led an illustrious team in approximation theory at the

raphy), Uspekhi Mat. Nauk, Part 1 , 34, no. 6 (1 979), 2 1 9-249.

University of Texas in Austin, from which he retired in 1 980.

[3) N. Davis, Europe, New York, Harper Perennial, 1 998.

His research has spanned several fields of mathematical analy­

(4) Delo akademika Nikolaya Nikolaevicha Luzina [Case of Academi­

sis, including approximation and interpolation, divergent se­

cian N. N. Luzin], S. S. Demidov, B. V. Levshin, eds., St. Peters­

ries, orthogonal series, and number theory; he has also writ­ ten on history of mathematics. Two volumes of his selected works have been published by Birkhauser in Basel.

burg, RKhG I , 1 999. [5] F. Hausdorff, Die Machtigkeit der Borelschen Mengen, Math. Ann. 77

( 1 9 1 6), 430-437.

[6) F. Hausdorff, Mengenlehre, Berlin, G6schens Lehrbucherei, 1 927.

[7] V. I. lgoshin, M. Ya. Sus/in,

1894- 19 19, Moscow, Nauka-Fizmatlit. ,

1 996. [8] L. Kantorovich and E. Livenson, Memoir on Analytic Operations

good biography (Igoshin, [7]) and an article (Tikhomirov, [16]), "The discovery of A-sets." The conclusions of both authors, reached without benefit of the extensive new source [4], resemble those of Kolmogorov [12]. The proofs sketched in Tikhomirov's article are based on three essen­ tially different definitions of analytic sets, and on the exis­ tence of universal analytic sets. In the collection Kol­ mogorov in Perspective ([9], p.4) A. N. Shiryaev refers to the new sets simply as "A-sets (analytic sets, introduced by Aleksandrov). " W e see that Aleksandrov came very close t o what he had accused Luzin of, that is, to borrow from Suslin the definition of operation A. Suslin found it with some en­ couragement from Luzin. Hausdorffs attitude was com­ mendable. Devoted to the readership of his books and ig­ noring petty concerns, in his Mengenlehre [1927] he gave

32

THE MATHEMATICAL INTELLIGENCER

and Projective Sets (1), Fund. Math. 18 (1932), 2 1 4-279. (9] Kolmogorov in Perspective, Editorial Board, Am. Math. Society and London Math. Society, 2000, History of Mathematics, vol. 20. [1 0] G. G. Lorentz and K. Zeller, Series rearrangements and analytic sets, Acta Math. 1 00 (1 958), 1 49-1 69. [1 1 ] N. N. Luzin, Ler;;ons sur les ensembles analytiques, Paris, 1 930, Gauthier-Villars. [1 2] Mathematics in USSR for 15 Years, Moscow GTI, 1 932. [Russian] [1 3] V. Shentalinsky, The KGB's Literary Archive, The Harville Press, London, 1 995. [1 4] A Shields, Luzin and Egorov, Mathematical lntelligencer 9 (1 987), no. 4, 24-27. Egorov and Luzin: Part 2,

ibid. 1 1 (1 989), no. 2, 5-7.

[1 5] M. Suslin, Sur une definition des ensembles mesurables B sans nom­ bras transfinis, Comptes Rendus Acad. Sci. Paris 164 (1 91 7), 88-90. [1 6] V. M. Tikhomirov, Otkrytie A-mnozhestv [Discovery of A-sets], Is­

tor. Mat. lssled. , fasc. 43 (1 993), 1 29-1 39.

BURKARD POLSTER, AN DREAS E. SCHROTH AN D HENDRIK VAN MALDEG HEM

Genera ized F at an d With illustrations by the author A Hexagon* Based on the gospel of

GENERALITY

as proclaimed by the

POLYGONS

ost of my readers will be familiar with the sad story of my grandfather, an honourable square and eminent mathematician of FLATLAND who was condemned to lifelong imprisonment for claiming to have been abducted to SPACELAND, a world somewhere "out there" that extends our two-dimensional FLATLAND by a third di­ mension. Of course nobody, not even I, his grandson (a hexa­ gon), believed in his story until, on the eve of the new mil­ lennium, I myself was abducted to GENERALIZED FLATLAND. I discovered that this world extends our flat world and the worlds of graphs and projective planes in a completely nat­ ural manner. As our world is populated by polygons such as triangles, quadrangles/squares, pentagons, etc., this exten­ sion of our world contains generalized polygons, both us sim­ ple ones and much more complicated ones of breathtaking abstract beauty. I also found that GENERALIZED FLATLAND co­ incides with the land of mathematical buildings of rank 2 as conceived by one of our foremost mathematicians J. Tits. This means that all non-trivial mathematical buildings are made up of natives of this mysterious land. Preface

I will tell you my story and, as evidence of my claims, show you drawings of my abductors, the four smallest natives of

proper GENERALIZED FLATLAND. These drawings are exten­ sions of beautiful renderings of closely related highly ho­ mogeneous graphs such as the complete graph on four ver­ tices, the Petersen graph, and the Coxeter graph (Fig. 1). In fact, closer inspection discloses that my abductors share many of the remarkable properties of these graphs and are even more symmetric than the graphs they extend. I hope that the overwhelming evidence I have compiled will con­ vince even the most sceptical among you that there is re­ ally life "out there" beyond FLATLAND, and that we are able, and have an obligation, to claim our rightful place in full GENERALITY.

A Painting in the Sand

It was the last day of our 2000th year. I spent this all-im­ portant day at the site of some recently discovered ruins in the desert of OZ. After unearthing some mysterious mathematical writings and drawings in the ruins they were excavating at the time, the archaeologists in charge had in-

"Dedicated to my dear grandfather Edwin E. Abbot (1 838-1 926), the author of the infamous Flatland-a Romance in Many Dimensions [1].

© 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 4 , 2001

33

Figure 1. The complete graph on 4 vertices, the Petersen graph, and the Coxeter graph.

vited me to join their expedition as mathematical adviser. I had gladly accepted their offer and on that very day started deciphering the mathematical inscriptions that covered all the walls and floors. It soon became clear to me that what had been discovered here were some of the writings of the famous mathematical prophet J. Tits, in which he claims that there is a world he refers to as GENERALIZED FLATLAND that extends our world. Of course every child knows that these writings had been condemned as heresy and de­ stroyed a long time ago. I was afraid to reveal my discov­ ery to my colleagues in fear that they might destroy what turned out to be of true mathematical beauty, even though not referring to some real world as claimed by the prophet. My colleagues had already retired to their tents while I was still trying to unravel the mysteries of a pentagonal paint­ ing (Fig. 2) that occupied the interior of one of the rooms. After several hours of work, I summarized in mathemati­ cal language what I had learned so far from the inscriptions about GENERALIZED FLATLAND and its natives. GENERALIZED FLATLAND. Remember that a (point-line) geometry consists of a nonempty set of points

The geometry Of

and a nonempty set of subsets of the point set called lines, such that every point is contained in at least two lines and every line contains at least two points. Two geometries are isomorphic if and only if there is a bijection between the point sets of the two geometries that extends to a bijection between their line sets. Every graph can be interpreted as a geometry. Here the vertices of the graph are the points, and associated with every edge is a line consisting of the two vertices contained in this edge. In particular, an ordinary n-gon is a geom­ etry that is isomorphic to the geometry of vertices and edges of a regular n-gon in the plane, that is, one of the na­ tives of FLATLAND. Just as a graph can have multiple edges, that is, two or more edges that connect the same two vertices, a geom­ etry can have multiple lines that cannot be distinguished by just looking at the points contained in them. Let C§ be a geometry with point set P and line set L. A geometry C§' with point set P' and line set L' is contained in C§, if the following three conditions are satisfied: (1) P' � P; (2) every line in L' is contained in a line in L; and (3) no two lines of L' are contained in one line of L.

Axioms for Generalized n-Gons of order (s, f)

(Q l ) In a generalized n-gon C§ of order (s, t) every line contains points and every point is contained in t + 1 lines. (Q2) C§ does not contain any ordinary k-gons for 2

,

THE MATHEMATICAL INTELLIGENCER

+ 1

:S k < n.

(Q3) Given two points, two lines, or a point and a lin ther least one ordinary n-gon in
34

s

is at

The natives of GENERALIZED FLATLAND are the generalized polygons. Every generalized polygon is a generalized n-gon for some n ;;:::; 2 and has an order (s, t), 1 s; s, t.

that I had learned in university seemed to suggest that we ordinary n-gons are the only generalized polygons and that therefore GENERALIZED FLATLAND = FLATLAND. At the same time I felt an irresistible urge to study the drawing (Fig. 2) further. At that point only a few grains of sand in the hour­ glass separated us from the new millennium. Clearly, this drawing was supposed to be a picture of some geometry with 15 points and 15 lines (the 5 sides of the pentagon plus the 5 medians plus 5 circle segments), each point contained in 3 lines and each line containing 3 points. The overall shape seemed to suggest that this was a generalized pentagon, but I quickly discovered a number of quadrangles in the picture, no digons and no triangles,

Generalized 2-gons, 3-gons, 4-gons, etc., are also called generalized digons, triangles, quadrangles, etc., respec­ tively. Furthermore, an ordinary digon is just a graph con­ sisting of two vertices that are connected by two edges. If a generalized polygon is of order (s, t), s t, we also say that it is of order s. A generalized n-gon is finite if it con­ tains only finitely many points and lines. =

How I came to

GENERALIZED FLATLAND

and what I saw there

Although I did not fully understand these words, everything

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Figure 2. A sandpainting of the QUADRANGLE.

VOLUME 23, NUMBER 4. 2001

35

though. A true generalized quadrangle of order 2? Every­ thing in me revolted against the mere idea and I exclaimed aloud:

"

GENERALIZED FLATLAND,

what nonsense!"

Straightaway I became conscious of a presence and someone whispering, "Nonsense, is it indeed?" At the same time the painting on the floor seemed to come alive and right in front of my eyes first turned into a square, then into a pentagon, and then it took on ever-increasing "gonalities" until, finally, it consolidated into a 15-gon. STRANGER:

Figure 3. A grid and its dual, two slim generalized quadrangles. The

"What kind of mathematician are you not to be­

lieve in axioms and proofs that your own mind supplies to you? Do you really have to see to believe? Behold, then, as

black and gray lines in the grid correspond to the points of the cor­ responding colour in the graph. (The dual makes its appearance later in the story.)

I am the generalized quadrangle whose shadow you have been staring at for such a long time."

generalized n-gon, n

>

2, does not contain any digons. This

"Pardon me, my Lord, but although you seem to have

just means that 2 of its points are connected by at most 1

many gonalities they all seem to be distinct and uncon­

line and that 2 of its lines intersect in at most 1 point. This

I:

nected, and at the moment I only see a 15-gon." STRANGER:

"Why, of course this is because I am

observation and Axiom Q3 imply that in a generalized tri­

thick-that

angle 2 points are contained in exactly 1 line and 2 lines in­

is, a generalized polygon that has at least 3 points to every

tersect in exactly 1 point."

line and at least 3 lines through every point. One of us thick

1: "Wait, that sounds very familiar! Doesn't this mean that

ones just does not fit into

FLATLAND,

and a

thin generalized

polygon like you can only see one of my ordinary n-gons at

the thick generalized triangles are just the projective planes?"

a time. Ah, I can see it in your eyes, you still don't believe

STRANGER:

me. Stay!" [I was inching my way towards the entrance of

eralized triangles are something your mathematicians have

the room.] "I will prove to you that

GENERALIZED FLATLAND

=I=

"Finally, you are beginning to understand. Gen­

known for a long time, only not by their rightful name. "

Let me first tell you about the generalized digons,

1 : "Oh yes, I see. But still, projective planes, o r generalized

the simplest generalized polygons. In a generalized digon

triangles, are just abstract mathematical structures. They

there is a line that contains all the points of the geometry,

do not really exist."

FLATLAND.

and all lines are just copies of this line. Of course this also

STRANGER:

implies that every point is contained in all lines. Q.E.D."

I will introduce you to the members of my family: the

"Dear Sir, with all due respect, but, being just a collec­

I:

GONS

"That does it! Deeds are called for and not words.

consisting of the

DIGON,

tion of identical copies of a line (which most certainly lives

smallest projective plane or the Fano plane), the

in

(myself), and the Siamese twins the

FLATLAND) ,

I cannot but think of these geometries as lines

POLY­

the TRIANGLE (also known as the HEXAGON

QUADRANGLE

and its dual. We

who pretend to be more than what they really are. If this

are the only generalized polygons with exactly 3 points on

is the only evidence you can muster, I have to say that I am

every line and 3 lines through every point. As our names sug­

not convinced." STRANGER:

gest, we are generalized digon, triangle, quadrangle, and hexa­

"Of course you are right, but how can you be so

gon, respectively. As a family we occupy as prominent a po­

hasty? Following good mathematical practice, I started by cov­

sition in GENERALIZED FLATLAND as my brother the TRIANGLE does

ering all those examples that, although not really having a life

among the projective planes. Now, out of your plane you go!"

of their own at first sight, still fit the axioms. Now, listen fur­ ther. The next step up from you ordinary polygons are the

At this moment an unspeakable horror seized me and I

Being neither thin nor thick, they

was no longer "in" the room with the painting. Neverthe­

slim generalized polygons.

are lost inanimate souls caught between your world and mine,

less, I was still able to see it in a strange way that reminded

present in both yet not belonging to either. For example, the

me very much of the account of

simple square grids are examples of slim generalized quad­

father had given. I noticed that the painting had changed,

SPACELAND

that my grand­

rangles. Mistaking these pitiable beings as natural features,

as now a small drawing of a hexagon was visible at the spot

you have built your cities based on their underlying structure."

where I had stood just a moment ago. I turned "around"

As he said this the stranger pointed to a drawing of a

and realized that just as the painting was his shadow, the

and for the first time I beheld the QUADRANGLE in all his glory grid in one of the corners of the room (see Figure 3).

addition to the painting was my own. Before I had been separated from FLATLAND I must have uttered a loud cry, be­

I:

"Hmm, again these beings can still be considered as be­

longing to

FLATLAND,

and as you said yourself, they are re­

cause some of the archaeologists were running towards the very spot I had occupied just a moment earlier.

ally quite dead and amount to nothing special." STRANGER:

"Ah, you are really very hard to please. How shall

The TRIANGLE

I convince you? Well, all good things come in three, so let

QUADRANGLE:

us move on to the generalized triangles. By Axiom Q2, a

can about us if, on your return, you don't want to share

36

THE MATHEMATICAL INTELLIGENCER

"Now listen, you need to learn as much as you

one of the two triangles under successive rotations through 360/7 degrees around the center of the 7-gon. r: "How marvellous! Who would have guessed by just look­ ing at your initial shadow, which only exhibited symme­ tries of orders 2 and 3, that you also have a symmetry of order 7. By combining all these symmetries we arrive at total of 2

·

3

·

7

=

a

42 symmetries. Your brother mentioned

that you are also an incarnation of the smallest projective plane. As such you should have even more symmetries, is that not so?" TRIANGLE: "That is true indeed. In fact, just as your shadow Figure 4. The most famous shadow of the

TRIANGLE.

does not capture your full being, our shadows only capture part of our complex structure. I trust that you are familiar with the concept of a symmetry group of a geometry? Well then, just remember that for me and my brothers our

your grandfather's fate. Meet my brother the TRIANGLE, the smallest projective plane! " The 7-Gonality o f the

TRIANGLE

symmetry groups act sharply transitively on the ordered

(n +

1)-gons contained in us, where

n is our gonality. This

means that the orders of these groups coincide with the H e vanished, and his place

was taken by a stranger who introduced himself as the TRIANGLE. I noticed that my friends the archaeologists were

number of ordered

+ 1)-gons contained in us."

(n

r: "Wait, bear with me while I am trying to understand what you just said. You are a generalized triangle, therefore your

frantically gesturing at the painting on the floor, which had

gonality is 3. From what you just said it is clear that the or­

also changed its shape. Clearly, this was the shadow of the

dered quadrangles contained in you are very important to

TRIANGLE (see Figure 4 for a reconstruction of what my

you."

friends saw). As one person, my companions were seized

TRIANGLE: "So far your reasoning is flawless, but can you

by a great fear, and first fled the room, then the excavation

deduce how many such ordered quadrangles are contained

site, and finally the desert itself, never to return.

in me?"

TRIANGLE: "Although being very small and easily understood,

first vertex

I hold the key to a full understanding of my more compli­

maining 6 points can be chosen as the second vertex

cated brothers the HEXAGONS and the QUADRANGLE! To illus­

connecting line of p and

trate what I mean by this, I need to show you a very spe­ cial shadow of myself on a regular 7-gon and derive a neat labelling of my points and lines that will prove extremely useful in understanding the HEXAGONS. If you count care­ fully, you will find that I and my shadows have exactly 7 points and 7 lines."

r: "I will try. You contain 7 points from which to choose the

p

of an ordered quadrangle. Any of the re­

q

q. The

contains one further point that

cannot be chosen as the third vertex

r.

This means that

there are only 4 points left to choose this vertex from. The lines connecting p and

q, p and r, and q and r contain a to­

tal of 6 points. This means that the last vertex s in the quad­ rangle is the remaining 7th point. This implies that you con­ tain a total of 7

·

6

·

4

=

168 quadrangles! This means that

your symmetry group has order 168. You are truly sym­ While he was saying this he was turning inside out in a completely unexplainable manner and his shadow took on

metric !" TRIANGLE: "Very good. You really think that I am very sym­

a 7-gonal appearance. Then he made a sudden movement

metric? Wait until you encounter the QUADRANGLE and the

that resulted in the mirror image of this 7-gonal shadow

HEXAGONS. Their symmetry groups have orders 720 and

(Fig. 5).

6196, respectively!"

In both cases the points of the shadow were the 7 points of the underlying 7-gon, and its lines were the images of

r: "Fantastic, but what about the DIGON?" TRIANGLE: "Well spotted. Its symmetry group has order 36, and I am sure you will be able to verify this for yourself once you think about it for a moment. But enough of this. We do not have much time. Let us again consider my 7-go­

0 0 Figure 5. The gles.

0

TRIANGLE

nal shadows. I call the lines in the shadow that correspond

0

to the left and right diagrams positive

triangles.

and negative Fano

One more model of me is hiding in this picture.

Its points are the left Fano triangles and its lines are the

0

right Fano triangles. Here a point is abstractly contained in a line if and only if the corresponding triangles have ex­

0

in terms of positive and negative Fano trian­

actly one vertex of the underlying 7-gon in common. You will understand what I mean by this after you have been instructed in the mysteries of doubling."

VOLUME 2 3 , NUMBER 4, 2001

37

The punishment of doubling

"The double (also incidence graph) of a point-line geometry is the graph whose vertices are the points and lines of the geometry. Two vertices are connected by an edge if and only if they correspond to a point and a line such that the point is contained in the line. Note that all the information about a geometry is contained in its dou­ ble, which means that you don't really die when you are doubled. On the other hand, since doubles also get squashed into your FLATLAND, it is generally believed that we lose all awareness of ourselves after having been dou­ bled. In fact, traditionally the worst punishment for a thick generalized polygon is to be doubled, and this is exactly what is going to happen to us POLYGONS if the other thick generalized polygons find out about us talking to you." 1: "If I understand you correctly, then the double of one of us ordinary n-gons should be an ordinary 2n-gon and, since we live in FLATLAND to start with and our status is directly dependent on our gonality, doubling should be just about the best thing that can happen to one of us." TRIANGLE: "You are very quick; but, unlike you thin ones, most thick generalized polygons cease to be generalized polygons after being doubled, and their gonality is irre­ trievably lost in the process of doubling." 1: "But why? After all, any ordinary k-gon in a geometry be­ comes an ordinary 2k-gon in its double and any ordinary l­ gon in the double comes from an ordinary l/2-gon in the original geometry. So, any doubled geometry contains only ordinary n-gons with even n. Also, if the original geometry contains no ordinary k-gons with k < n, then the double contains no ordinary k-gons with k < 2n. Doesn't this im­ ply that the double of a generalized polygon is just another generalized polygon?" TRIANGLE: "You forgot about Axiom Ql. Your arguments only take care of Axioms Q2 and Q3. In fact, it is exactly the gen­ eralized n-gons of order t that turn into slim generalized 2n­ gons after doubling. This means that, for example, we the POLYGONS are still generalized polygons after having been dou­ bled. Using this revolutionary insight, my brothers and I dis­ covered we could deliberately double ourselves, live in this state in FLATLAND for extended periods of time, and revert to our usual states whenever it pleased us. It was during those visits that we took a liking to your kind and decided to help you claim your rightful place in GENERALIZED FLATLAND. " TRIANGLE:

He proceeded to demonstrate what he meant by dou­ bling himself, a process too awful to describe in detail, at the end of which he (or his double) coincided with his shadow. Figure 6 shows the double of the TRIANGLE whose vertices have been labelled with the two different kinds of Fano triangles. It is clear that this double is a slim gener­ alized hexagon. The HEXAGONS

"So far it seems that we have found in you a fit apostle for the gospel of GENERALITY. But let us see how you fare in the presence of the HEXAGONS. "

TRIANGLE:

38

THE MATHEMATICAL INTELLIGENCER

Figure 6. The double of the

TRIANGLE.

Vertices are connected by an

edge if the triangles in their labels share exactly one point of the un­ derlying 7-gon.

A narrow escape With a laugh he vanished and his place was taken by a being so glorious in appearance and complexity that at first I was too dazzled to pay any attention to what the being was saying. But even when I had recovered enough to pay attention to the noise emanating from it, I could not make out any words. Also, the being seemed to flicker between two completely different states. As I watched, it became more and more agitated and started making threatening moves towards me. Finally, something clicked into place in my mind, and flicking open a certain page in my notebook I reread a passage that I had translated earlier on and that had made no sense to me at that time.

The dual of a geometry <§ is constructed by interchang­ ing the roles of points and lines in <§. More precisely, its points are the lines of <§, and to every point of <§ corre­ sponds a line of the dual consisting of all lines in <§ con­ taining this point. A geometry is called self-dual if it is isomorphic to its dual, and an isomorphism between a geometry and its dual is called a duality. Clearly, I thought to myself, if all this is true, then we ordinary n-gons are self-dual; the dual of a grid is a com­ plete bipartite graph; and, come to think of it, there was such a graph drawn right next to the grid that the QUAD­ RANGLE had pointed out to me (see again Figure 3). Also, since the TRIANGLE and the QUADRANGLE are the only small­ est thick generalized triangle and quadrangle, they must both be self-dual, that is, coincide with their duals. In fact, the self-duality of the TRIANGLE follows immediately from its description in terms of Fano triangles, and a duality cor-

responds to a reflection of Figure 6 through its vertical sym­ metry axis. On the other hand, the HEXAGONS must be two separate geometries that are forever intertwined by a du­ ality; what I was witnessing here was the two HEXAGONS speaking to me at the same time. Time was running out, and if I didn't want to be squashed under the weight of the HEXAGONS, I had to find a way to communicate with them. I started blinking my eyes and ears in unison with the flick­ ering of the HEXAGONS and, lo and behold, I was able to see and hear only one of them. "Congratulations. One moment longer, and we would have doubled you (and your intelligence)." HEXAGON:

Here he laughed a mischievous laugh, and it occurred to me in a flash that I had just missed a unique opportunity to raise my gonality from 6 to 12. A simple numbers test HEXAGON: "But then again, doubling is a fairly painful process and you might not have enjoyed it. Anyway, please refer to me as the (Cayley) HEXAGON. I will speak for both myself and my dual. In the future you may encounter other geometries who will pretend to be one of us POLYGONS and try to dissuade you from lighting the fire of GENERALITY in FLATLAND. Therefore, let us start by deducing some basic properties of us HEXAGONS, such as the number of our points and lines, and a simple test that will allow you to distin­ guish us from any impostor. "You have seen that points and lines play similar roles. So let us refer to the points and lines of a geometry jointly as its vertices, and inductively define a distance between the vertices. The vertices at distance 1 from a point are the lines through this point, and the vertices at distance 1 from a line are the points on this line. Given a vertex e, the only vertex at distance 0 from e is e itself. A vertex is at distance n > 1 from e if it is not at distance m < n and if it is at distance 1 from a vertex at distance n 1 from e. The diameter of a geometry is the maximum distance between two of its vertices. As an immediate consequence of Axiom Q3 we see that a generalized n-gon has diame­ ter n." r: "Does this mean that if e is a point, then all vertices at odd and even distances from e are lines and points, re­ spectively?" HEXAGON: "That is correct. Given one of our vertices e, we can now count the number D� of vertices at distance n from e using the axioms, the inductive definition above, and the fact that there are exactly 3 vertices at distance 1 from e. We conclude that D6 1, Df = 3, D� = 2D�- 1 for 2 :::; n :::; 5. If e is a point, then a line is at distance 1, 3, or 5. This means that there is a total of 3 + 12 + 48 = 63 lines. The dual argument yields that there are also 63 points. Conse­ quently, D� = 63 - ( 1 + 6 + 24) = 32. "Simple counting arguments also show that a geometry with 63 points and 63 lines is one of us HEXAGONS if and only if, for all n with 1 :::; n :::; 5 and vertices e, the numbers D� coincide with the corresponding ones in us HEXAGONS. Us-

=

ing similar arguments, simple counting criteria can be de­ duced for any finite generalized polygon." r: "Of course. In fact, I can see immediately that a geome­ try with 7 points and 7 lines is your brother the TRIANGLE if and only if Df = 3 and D� = 6 for all its vertices." from

TRIANGLE

to

HEXAGON

"Very good. Now behold my shadow (Fig. 7). Pretty, isn't it, . . . but I am sure you would not be able to remember it, if I didn't tell you a little bit more about the way I am built. In the following I will describe my vertices in terms of the vertices of my brother the TRIANGLE. To avoid confusion, I will refer to vertices of the TRIANGLE as T-ver­ tices and vertices of me the HEXAGON as H-vertices. A point­ line pair {p, L} of a geometry is called a flag or anti-flag if p is or is not contained in L, respectively. "Look at me closely. Can you see that I have 4 different kinds of H-points? These are the T-points (7), T-lines (7), flags (7 T-points · 3 T-lines through a T-point = 21 flags) and anti-flags (7 T-points 4 T-lines not through a T-point = 28 anti-flags) of the TRIANGLE. This gives a total of 7 + 7 + 2 1 + 28 = 63 H-points. There are two different kinds of H­ lines containing 3 H-points each. H-lines of the first kind are sets of the form {p, L, {p, L}}, where {p, L} is a flag of the TRIANGLE. Clearly there are as many H-lines of this type as there are flags of the TRIANGLE; that is, there are 2 1 such H-lines. An H-line of the second kind is of the form { {p, L }, {q, M}, {r, N}}, where (1) {p, L} is a flag; (2) p, q, and r are the 3 T-points contained in L; (3) L, M, and N are the 3 T­ lines through p. This implies that both {q, M} and {r, N} are anti-flags. It is clear that there are two such H-lines asso­ ciated with every flag of the TRIANGLE, that is, there are 42 such H-lines. This gives a total of 2 1 + 42 = 63 H-lines." r: "Wait, wait, let me check this using the numbers test. Hmm, from what the TRIANGLE has told me about itself I know that its symmetry group acts transitively on its sets of points, lines, flags, and anti-flags. This means that none of these objects is distinguished in any way, and it suffices to check that the numbers D� pan out for the four essen­ tially different kinds of H-points and the two essentially dif­ ferent kinds of H-lines. Now, if we take . . . " HEXAGON:

·

A magic labelling of H-points and H-lines HEXAGON: "Correct, correct, but the rest is just trivial book­ keeping. I am afraid we don't have the time for this right now. Instead, let us draw my shadow based on the above description. We start with the model of the TRIANGLE whose points and lines are positive and negative Fano triangles, as in Figure 6. With respect to this model there are 9 es­ sentially different H-points of the HEXAGON as illustrated by the first row of labels in Figure 8. The remaining (labels of) H-points are the images of these 9 labels under successive rotations through 360/7 degrees around the center of the underlying 7-gon. "We replace every label by a simpler label as indicated by the second row in Figure 8. Rotated labels get replaced by new labels that have been rotated in a corresponding

VOLUME 23, NUMBER 4, 2001

39

The POLYGONS

\:!'

·�) ,• •

;.:-,

:). •'•

F�g��e 7. The POlYGONS. The points and Iiles of the OI.I.IOOAHGU �EXAOOI)

correspond to the subsets of size 1, � (and � of a >gon �·gon). To 11oid

a crowded appeatance of the IIElAGOII, the 1�t Slbsets of the 7-gon are represe��ted by lsmal) sclid bM poilts. The ines in the otJAnAA�W

correspond to partitions of the �on into 1· and 2·element subsets; see

FJ911'e 10. fer the ines of the� in terms of the labels ol its points see figure 9. Highlghted in the diagrams are the points of geometric hyper·

plalles -flll')!le jnwGLE and llU.IIIIWfQij and green lllf.l'ltGO)O . Alta remo�ng

these geometric hyperplanes from these geometries, we are left with models

of scme of the most homogeneous graplls -the complete graph on fOil vertices

in the case of the llVANGif, the Petersen graph il the case of the 0\J.UIUNGLE, and the

disjoint union olthe Coxeta graph jbl� points and blue and green Jiles) and the

Heawood graph btllow poilts and li�) il the case of the HEXAGON. Note that every

poilt of the OIGOH fonns a geometric �lane.

A· i

• d•

1:

.

.. . .:

:.'J t

7 poi nts

2 1 flags

? l ines

28 anti-flags

© � @l@ � ��g� © � ©@ @ aeoo o---o 0

Figure 8. Labels for the points of the

0

o

o--o

o

o

0

0

o

HEXAGON.

PGCISj 2, 2). [8] for more details about this representation 2, 2).

manner. Note that the new labels correspond in a natural

over the field with two elements, for short

way to all 1-,

See [ 10) and

ing of the

2-,

and 3-element subsets of the set consist­

7 vertices of the underlying 7-gon. In terms of the 9 essentially different kinds of lines of the HEXAGON; see Figure 9. "It is clear that every symmetry and duality of the TRI­ new labels there are

ANGLE

induces a symmetry of the

HEXAGoN.

Encoded in the

labels is an order 7 symmetry of the TRIANGLE and an order 2 symmetry that corresponds to a duality of the TRIANGLE. Using the labels, it is easy to reconstruct my shadow; see Figure

of

PGCISl

-

-

"As you have already observed, the rule that assigns a

new label to one of the original labels can also be stated in terms of the operation EB. Here of the underlying

S consists

of the vertices

7-gon, and if a label consists of two Fano

triangles A and B (sets of three vertices), then the new la­ bel is A EB B. "With the above remarks it should be clear to you that my H-points coincide with the points of the 5-dimensional

7."

1: "I understand all this. Except for the step where you re­

projective space PG(5,

place the original labels by new labels. It seems that the

1: "I think I know what you are getting at. Your lines are

2). Furthermore, . . . "

new label associated with a label containing two Fano tri­

also . . . wait, let me double-check this . . . Yes, any two H­

angles is either the symmetric difference of the two trian­

points on any of your H-lines E9-add up to the third H-point

gles or the complement of this difference."

on this H-line. " HEXAGON:

HEXAGoN:

"Exactly! This means that

I am a subgeometry

right at the center of this projective space, which is an im­

Strength in projective spaces

"Ah, yes that is correct. In fact, the main source

portant source of power for me."

of our power can be explained using the mathematical op­

1: "So there really are beings that live in spaces of a di­

eration that corresponds to this 'step.' Let S be a set with

mension greater than two, just as my grandfather claimed

lsi > 1 of elements, and let Sv2 be the set of all nonempty subsets of S with fewer than ISI/2 elements. an odd number

If A, B E

S112, A

(although this dimension is quite different from the 'tangi­ ble' dimensions he had in mind!)."

=I= B, let D be the symmetric difference of

A and B and define A E9 B to be D if D E

S112 or S \ D oth­

Hyperplanes, H eawo od graph, and Coxeter graph

erwise. We define a geometry 'fi(S) whose point set is S and

1: "How miraculously all this fits together! But I am sure

whose lines are the sets {A, B, A E9 B} where A and B are

that there is much more beauty hiding in your shadow. For

distinct elements of S. Every line in this geometry contains 3 points. Furthermore, given two points the third point on the line is always

P and Q

P E9 Q.

on a line,

This implies

example, I just noticed that every one of the H-line labels in Figure

9 contains exactly one isosceles triangle. This

seems to suggest that the H-points that correspond to these

that any two points in the geometry are contained in ex­

labels form a very special set of points."

actly one line. Closer inspection reveals that the geometry

HEXAGON:

is isomorphic to the projective space of dimension

ing you to be our messenger! Your remark reminds me of

2 1 poi nt/l ine/flags

Figure 9. Labels for the lines of the

42

THE MATHEMATICAL INTELUGENCER

HEXAGON.

lSI

-

2

"We have indeed made the right choice in select­

48 flag/anti-flag/anti-flags

something else we should talk about. By now you will prob­ ably have guessed that the kind of conversation we are hav­ ing is extremely dangerous. It is only possible during the first hours of a new millennium, because at this time the BUILDINGS we are part of are too busy celebrating to broad­ cast every word that is said to the rest of (thick) GENERAL­ IZED FLATLAND. To be able to communicate with us even af­ ter your return to FLATLAND, you have to know a little about the flat subgeometries that my different kinds of H-points and H-lines correspond to. "A geometric hyperplane H of a geometry is a set of points such that every line either contains exactly one point of H or is completely contained in H. The set of all flag H­ points (isosceles triangles) is a special geometric hyper­ plane that intersects every H-line in exactly one point (every one of the labels in Figure 9 contains exactly one such triangle). Imagine that we remove the points of this hyperplane from me and my H-lines. Then we are left with two famous graphs: the Coxeter graph, and the double of the TRIANGLE, which in FLATLAND is also known as the Rea­ wood graph. "The vertices of the Heawood graph are the H-points that correspond to points and lines of the TRIANGLE. The edges of this graph are induced by the H-lines of the point/line/flag type. The picture of the Heawood graph right in the mid­ dle of my shadow in Figure 7 corresponds to Figure 6. "The vertices of the Coxeter graph are the H-points cor­ responding to the anti-flags of the TRIANGLE. The edges of this graph are induced by the H-lines of the flag/anti­ flag/anti-flag type. This corresponds to a well-known rep­ resentation of the Coxeter graph; see [6]. Also, the picture of this graph in the middle of Figure 7 corresponds, via some obvious rearrangements, to the most famous repre­ sentation of this graph depicted in Figure 1 (three 7-gons joined together via 7 extra points). "By the way, the presence of a special hyperplane as above distinguishes me from my dual. Also, after you are back in FLATLAND I will keep these two graphs immersed in FLATLAND so that you can communicate with me via either one of them." Misfortune Strikes

At this moment the BUILDING we were hiding in started shak­ ing violently. "We are discovered! Dear friend, always remem­ ber what we have told you today, and no matter what hap­ pens now you should be able to find me and my brothers again and finish what we have begun. Beware of the OCTA­ GON in the PENTAGON, because . . . " " THUNDERING VOICE: HEXAGON, you and your brothers have committed the heinous crime of communicating with the thin ones. For this you will suffer the terrible fate of doubling. " At this moment the ceiling slammed down on my new friend and me, and we were both squashed back into FLAT­ LAND. When I regained consciousness it was morning, and HEXAGON:

I found myself in the very room where all this had started. I automatically assumed that the night's adventure had been a dream induced by what I had read on the walls. But then I discovered that all the writings had vanished and that none of my companions was anywhere to be seen. I also found, to my utter amazement, that my gonality had been raised to 12-1 had been doubled. Although still somewhat shaken, I immediately started looking for the doubles of the POLYGONS-to no avail. I realized that, using my doubled IQ and the unprocessed notes in my notebook, I first had to deduce as much as possible about the POLYGONS and their doubles; then, to convince you my fellow flatlanders of their existence, locate their whereabouts in FLATLAND, and with their help claim our rightful place in full GENERALITY. The QUADRANGLE and the DIGON

It was a long journey back home. I spent most of the time organizing my notes and developing a mathematical theory of GENERALIZED FLATLAND. Following the procedures the HEXAGON had introduced me to, it was easy to show that the QUADRANGLE has 15 points and 15 lines, that its diameter is 4, that D5 = 1, Di = 3, Ifj = 6, Dti = 12, D4 = 8 for all vertices of the QUADRANGLE, and that these numbers suffice to recognize the QUADRANGLE among geometries. I also found a geometric construction of the QUADRANGLE as a derived geometry at a point of the HEXAGON; see [3]. However, this construction is rather com­ plicated, and executing it within the shadow of the HEXA­ GON yields a model of the QUADRANGLE with only very few symmetries. After two sleepless days and nights, I finally succeeded in reconstructing the shadow that I first saw in the ruins. The Shadow of the QUADRANGLE revisited Let S be the set of vertices of a regular pentagon. The points of the shadow are all elements of 8112 , that is, all 1- and 2-element subsets of S. The lines are the partitions of S into two 2-element subsets and one 1-element subset of S. Then there are essentially 3 different kinds of points and 3 different kinds of lines, as illustrated by the labels in Figure 10. Of course this representation parallels the representation of the HEXAGON as a subgeometry of the projective space PG(5, 2) and identifies the QUADRANGLE as a subgeometry right in the middle PG(3, 2). Using the labels, it is possible to reconstruct the shadow of the QUADRANGLE as in Figure 7.

Just like the the QUADRANGLE also contains geometric hyper­ planes that intersect every line in exactly one point. One is visible right in the centre of its shadow. It consists of the five 1-point subsets of S. If we remove the points of this hyperplane from the QUADRANGLE and its lines, we are left with the famous Petersen graph. Also, the picture of this graph in the diagram of the QUADRANGLE in Figure 7 corresponds to the most famous representation of this graph depicted in Figure 1 (two 5-gons joined together). I Geo metric hyperplane and Petersen graph

HEXAGON,

VOLUME 23, NUMBER 4 , 2001

43

� �

Figure 10. The points and lines of the

QUADRANGLE.

assume that the QUADRANGLE planned to stay in touch with us in this form. For completeness' sake I remark that the lines of the TRI­ ANGLE are geometric hyperplanes. After deleting one of these hyperplanes from the TRIANGLE, we are left with the complete graph on four vertices. As you are probably aware, this graph, the Petersen graph, and the Coxeter graph are almost as homogeneous as the POLYGONS they are contained in; see [2]. The derived geometry and from DIGON to QUADRANGLE You are asking me where in all this the DIGON fits in? Although I never had the honor of meeting the DIGON, I found it very easy to reconstruct its shadow (see Fig. 7). Note that it contains 3 points and 3 lines, and that every line contains all the points. Your first reaction may be similar to mine when the QUADRANGLE first introduced me to generalized digons: "What's the big deal?" Well, it turns out that there is a labelling of the QUADRANGLE in terms of the DIGON that is the direct equivalent of the labelling of the HEXAGON in terms of the TRIANGLE: The points of the QUADRANGLE are the points, lines, and flags of the DIGON. There are two kinds of lines. The lines of the first kind are of the form (p, L, (p, L} }, where {p, Lj is a flag of the DIGON. The lines of the second kind are of the form { {p, Lj, {q, Ml, {r, N}} such that {p, q, rj and {L, M, Nj are the point and line sets of the DIGON.

The Doubles of the POLYGONS

It seems obvious to me that the POLYGONS intended to be present in FLATLAND in the form of some special graphs. Ac­ cording to their original plan they would be surveying proper GENERALIZED FLATLAND by using only the points of one of their special geometric hyperplanes, with the rest of their bodies immersed in FLATLAND (in this form they are almost invisible). If this is what they are doing, then to get in touch with them we have to locate the graphs in Figure 1 and Fig­ ure 6. Of course it is also possible that even surveying just using a geometric hyperplane is too risky at the moment and they are existing only as their doubles and are fully im­ mersed in FLATLAND . My investigations had confirmed my belief that the POLY­ GONS had revealed their most symmetric shadows and sub-

Figure 1 1 . A special path in the

44

THE MATHEMATICAL INTELLIGENCER

QUADRANGLE.

geometries to me. I therefore proceeded to reconstruct the most symmetric representations of their doubles. I had already encountered an attractive picture of the double of the TRIANGLE in Figure 6. Also, it turned out that the double of the DIGON is the complete bipartite graph on 6 vertices in Figure 3. Of course this meant that, without my realising it at the time, the DIGON had been present in this form throughout my conversations with his brothers right next to their shadows. To construct the best picture of the double of the QUAD­ RANGLE, I considered the path in this geometry depicted in Figure 1 1 . Since this is a path, two of its adjacent vertices correspond to a flag in the QUADRANGLE. Furthermore, this path contains the different kinds of points and lines in Fig­ ure 10 exactly once, except for its beginning and its end, which are two points of the same kind. If we fit together the 5 images of this path under rotations of the 5-gon un­ derlying the labels, we arrive at a path that contains every point and line of the QUADRANGLE exactly once and is in­ variant under the rotations. This enables us to draw a pic­ ture of the double such that the vertices of the graph are the vertices of a 30-gon, two adjacent vertices of the 30-gon are connected by an edge, and rotations through 360/5 de­ grees around the center of the 30-gon leave the double in­ variant. Figure 12 is a picture of the double that has been constructed in this way. This also shows that the QUAD­ RANGLE contains 15-gons like the one I saw in the ruins and that it is self-dual. Note that the reflection through the ver­ tical symmetry axis of the diagram corresponds to a dual­ ity of the QUADRANGLE. Figure 13 shows a similar path in the HEXAGON which can be used to model the double of this geometry on a regular 126-gon such that two adjacent vertices of this polygon are connected by an edge, and rotations through 360/7 degrees around the center of the polygon leave the double invari­ ant. See [9, Section 13.5] for a picture of the double that has been constructed in this way. Where to From Here?

When I finally arrived back in my hometown, I discovered that in my absence I had been accused of high treason and the police were looking for me everywhere. All this re-

Figure 1 2. The double of the

QUADRANGLE,

a generalized octagon.

minded me so much of what had happened to my grandfa­ ther. Of course I was only a boy when he first told me about his abduction, and at that time his story sounded like the ramblings of a madman to me. But now that I had been ab­ ducted myself and reconsidered what he had told me with my doubled intellect, it all made perfect mathematical sense. So, why had he been locked away for something that

Figure 13. A special path in the

our incredibly intelligent multigonal rulers should have rec­ ognized as the truth? And why were the authorities after me all of a sudden? I needed time to think. Since the po­ lice were looking for a hexagon I did not have to fear too much, of course. But the HEXAGON had warned me to beware of the "ocTA­ " GON in the PENTAGON. What had he meant by this? A (gen-

HEXAGON.

VOLUME 23, NUMBER 4, 2001

45

eralized) octagon in a (generalized) pentagon? There must be infinitely many such combinations! On the other hand, the way he had pronounced PENTAGON and OCTAGON was very similar to the way he pronounced the names of his brothers. Did this suggest that I had to look for the smallest thick gen­ eralized pentagons and octagons and that these were per­ haps somehow related to the POLYGONS? I returned to my studies, and after a couple of weeks of hard work I uncov­ ered some more fundamental properties of generalized poly­ gons that suggested an answer to my problem. All generalized n-gons we have to worry about are fi­ nite, that is, both their point and line sets are finite sets. Remember that by Axiom Q1 a generalized n-gon C§ is of order (s, t), s, t ;::;: 1, if every line contains s + 1 points and every point is contained in t + 1 lines. If s = t, we also say that C§ is of order s. This means that the POLYGONS are the generalized polygons of order 2. Also, we ordinary n-gons are, up to isomorphism, the unique generalized n-gons of order 1. A generalized polygon is slim if either it or its dual is of order (2, m) for some m > 2. If C§ is not an ordinary n-gon, then, by a celebrated result of Feit and Higman [7] (contemporaries of the prophet J. Tits), n = 3, 4, 6, 8, or 12, and, if n = 12, then C§ is slim. The smallest slim generalized n-gons can be shown to be unique up to isomorphisms and duality. These geome­ tries are the generalized 2-, 4-, 6-, 8-, and 12-gons of order (1, 2) and their duals. The first (trivial) geometry is the graph consisting of 2 vertices that are connected by 3 edges (this is the DIGON minus one of its points, that is, minus one of its geometric hyperplanes). The remaining four geome­ tries are the doubles of the POLYGONS. This means that all smallest non-trivial generalized polygons are related to the

mathematicians are writing and mathematicians are exactly the audience able to appreciate this report for what it is, I am submitting this account to a popular international math­ ematical journal, the perfect forum for subversive mathe­ matical writings. For a more detailed exposition of the mathematical the­ ory of generalized polygons and the all-encompassing the­ ory of mathematical buildings, see the recently discovered manuscripts [4] , [ 12], [ 14], [16], and [ 17]. See [5], [9], [10], [11], and [ 13] for further information about the POLYGONS. Enough said, my dear fellow flatlanders. Go forth and seek out the POLYGONS and then onwards to full GENERALITY! REFERENCES

[1 ] Abbot E.A. Flatland-A Romance in Many Dimensions, with illus­ trations by the author A Square, 2nd Edition originally published in 1 884 is available for free download from many literature archives and private websites on the internet. [2] Biggs, N. Three Remarkable Graphs, Can. J. Math. 25 (1 973), 391 -41 1 . [3] Bloemen, I. and Van Maldeghem, H. Generalized hexagons as amalgamations of generalized quadrangles. Eur. J. Combin. 1 4 (1 993), 593-604. [4] Brown, K.S. Buildings. Springer-Verlag, New York-Berlin, 1 989. [5] Cohen, A.M. and Tits, J. On generalized hexagons and a near oc­ tagon whose lines have three points. Eur. J. Combin. 6 (1 985), 1 3-27. [6] Coxeter, H . S . M . My Graph, Proc. London Math. Soc. 46 (1 983), 1 1 7-1 36 . [7] Feit, W. and Higman, G. The nonexistence o f certain generalized polygons. J. Algebra 1 (1 964), 1 1 4- 1 3 1 . [8] Pickert, G. Von der Desargues-Konfiguration zum 5-dirnensionalen

POLYGONS.

projektiven Raum mit 63 Punkten. Math. Semesterber. 29 (1 982),

So, obviously, there are no non-ordinary generalized pen­ tagons. Hence the PENTAGON must refer to something em­ bedded in FLATLAND. Of course the shape of most of our build­ ings here in FLATLAND is that of a pentagon and the building that houses the best-kept secrets of our government is THE PENTAGON. Could that be it? Was the HEXAGON trying to warn me of my own government? All of a sudden everything seemed to make sense. Clearly, the OCTAGON was a thick gen­ eralized octagon that had immersed one of its multigons into FLATLAND and under the pretence of being a circle was rul­ ing our land. Further study revealed that this ocTAGON is most probably a generalized octagon of order (2, 4) having 1755 points and 2925 lines. So far I have been able to show the existence of only one such octagon. As I suspected it is a distant relative of the POLYGONS: Its derived geometry is the unique generalized quadrangle of order (2, 4) which in tum contains the QUADRANGLE. I believe that this generalized oc­ tagon is unique but have not yet been able to prove it. Following this discovery I joined the mathematical un­ derground. Since governments are not interested in what

5 1 -67.

46

THE MATHEMATICAL INTELLIGENCER

[9] Polster, B. A Geometrical Picture Book, Universitext Series, Springer-Verlag, N .Y. , 1 998. [1 0] Polster, B. Centering small generalized polygons- projective pot­ tery at work, submitted. [1 1 ] Polster, B. and Van Maldeghem, H. Some Constructions of small generalized polygons, to appear in J. Combin. Theor. Ser. A. [1 2] Ronan, M . Lectures on Buildings. Perspectives in Mathematics, 7. Academic Press, Boston, 1 989. [1 3] Schroth, A. E. How to Draw a Hexagon, Discrete Math. 1 99 (1 999), 6 1 -7 1 . [1 4] Thas, J.A. Generalized polygons. in: Handbook of Incidence

Geometry, pp. 383-431 , North-Holland, Amsterdam, 1 995. [1 5] Tits, J. Sur Ia tialite et certains groupes qui s'en deduisent, lnst. Hautes Etudes Sci. Pub/. Math. 2 (1 959), 1 3-60. [1 6] Tits, J. Buildings of Spherical Type and Finite BN-Pairs. Lecture Notes in Mathematics 386. Springer-Verlag, Berlin-New York, 1 974. [1 7] Van Maldeghem, H . Generalized Polygons. Birkhii.user, Basel, 1 998.

I

A U T H O R S

BURKARD POLSTER

ANDREAS E. SCHROTH

HENDRIK VAN MALDEGHEM

Department of Mathematics and Statistics

lnstitut fUr Analysis

Department of Mathematics

P.O. Box 28M

TU Brau nschweig

University of Gent

D-381 06 Brau nschweig

9000 Gent

Monash

University, Victoria 3800 Australia

Germany

Belgium

e-mail: Burkard. [email protected]

e-mail: a.sch roth@tu -bs. de

e-mail: [email protected]

http://www.maths.monash.edu.aul-bpolster

http://fb 1 .math. nat . tu - bs .de/-top/aschroth

http://cage.rug.ac.be/-hvm

Burkard Polster joined the mathematical

Andreas E. Schroth was forced into the

Hendrik

underground while studying arcane territo­

mathematical underground because his

to life i n the mathematical underground by

van Maldeghem was condemned

ries of finite and topological geometry. He

work on the connection between circle

his addiction to numbers. Some of the

has been on the run ever since, hastily

planes

numbers by which he lives:

completing his doctorate and working at

verged dangerously close to circle-squar­

6: his favorite number. His work on gen­

eight universities on three continents over

ing. Cycling across the Indian subconti­

eralized hexagons earned him the 1 9g9

and

generalized

quadrangles

the last sixteen years. To maintain razor

nent, he developed a persistent attach­

Hall Medal of the Institute of Combinatorics

sharpness for this hectic existence, he

ment to vegetarian Indian food (which he

and Applications.

practices daily: juggling, sculpting soap

both cooks and eats) and to bollywood

bubbles,

and creating ambigrams. Some

of these ambigrams have graced The Mathematical lntelligencer.

movies (which he only watches).

40000: the number of kilometers he ran before the age of 38-and the approximate circumference of the earth. 4/4:

the usual meter of the folk-rock

band Lezzamie, in which he plays an elec­ tronic drum.

VOLUME 23, NUMBER 4, 2001

47

li,i$?.ff'l . i§,fih£ili.II!QBM

The Magic Square on Sagrada Fam il ia Pieter Maritz

D i rk H uylebro u c k , E d itor

B

arcelona is probably best known for its architecture. There are many fascinating structures reflecting the art nouveau movement, known in Catalonia as Modernisme, with the city's most famous architect, Antoni Gaudf, represented by some ten differ­ ent works. Antoni Placid Guillem Gaudf i Cor­ net was born June 25, 1852, in the province of Tarragona [1]. At age eleven he entered the Col.legi de les Es­ coles Pies in Reus, located in the an­ cient convent of Sant Francese. In 1868 Gaudf moved to Barcelona to study ar­ chitecture. He fulfilled his military ser­ vice requirement during the years 1874-1877. His first large project was workers' housing in a factory, the Co-

I

most famous work, the finest example of his visionary genius, and a world­ wide symbol of Barcelona and of Cat­ alonia. This neo-Gothic project was ini­ tially managed, in 1882, by Francese de Paula del Villar i Lozano, Gaudf's former professor, who volunteered to carry out the ideas of Josep Maria Bocabella, chair of the Associaci6 Espiritual de Devots de Sant Josep. Martorell was part of the Temple Council. He dis­ agreed with del Villar about the mate­ rials that should be used to make the pillars, and, when they couldn't reach agreement, del Villar stepped down. Bocabella offered the position to Mar­ torell, who, because of the situation, did not accept but proposed his young assistant, Gaudf, who immediately ac-

A worldwide sym bol of Barcelona and of Catalo n ia .

D oes your hometown have any mathematical tourist attractions such as statues, plaques, graves, the caje where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? .(f so, we invite you to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks.

operativa Mataronense (Matar6 Coop­ erative). The project was intended to improve the workers' quality of life, but Gaudf's project was ahead of its time, and only one section of the factory and a kiosk were built. Gaudf was disap­ pointed, but the presentation of his project at the Paris World Fair in 1878 marked the beginning of his fame. There he also presented a showcase for pret-a-porter gloves from the shop of Esteban Comella, thanks to whom he met the man who would become one of his best friends and patrons, Eusebi Giiell. After the Paris World Fair, Gaudf decorated the Gibert pharmacy in Barcelona and collaborated with the architect Martorell on various jobs. Sagrada Familia

Please send all submissions to

Gaudf's relationship with Martorell al­ lowed him to take over management of

Mathematical Tourist Editor,

El Temple Expiatori de la Sagrada Familia ("Expiatory Temple of the

8400 Oostende, Belgium

Holy Family") near Avinguda Diagonal in Barcelona. This became Gaudf's

Dirk Huylebrouck, Aartshertogstraat 42, e-mail: [email protected]

.

.

cepted. In 1883, Gaudf officially took control of the project. Gaudf wanted to create a "20th cen­ tury cathedral," a synthesis of all his ar­ chitectural knowledge with a complex system of symbolism and a visual ex­ plication of the mysteries of faith [2]. There would be three fa<;ades, repre­ senting the birth, death, and resurrec­ tion (Nativity; Passion; Gloria, the main fa<;ade) of Christ, with eighteen towers symbolizing the twelve apostles, the four evangelists, and the Virgin Mary and Christ. This latter one, the Tower of the Saviour, would be the tallest, 1 70 meters high against the 120 meters of the others. Gaudf planned monumental fa<;ades on the central nave and the arms of the transept. Works personally undertaken by Gaudf are the neo­ Gothic crypt, the constructed part of the apse, and the fa<;ade of the Nativ­ ity (Eastern side, facing the streets Ma­ rina and Cerdenya) with a purely nat­ uralistic exuberance in its decoration. The figures were directly molded from

© 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 4, 2001

49

Figure 1. The Nativity Fac;ade.

nature, animals, plants, clouds, etc., with contributions from different sculptors, including Japanese artist Et­ suro Sotoo. Of the four towers of this fac;;ade, Gaudf only saw that of St. Barn­ abas complete. Figure 1 shows part of the Nativity Fac;ade. Gaudf, who never married, became obsessed with the church to the point that not only did he focus almost all of his creative energies on it, he also set up residence in his on­ site studio. On June 7, 1926, Gaudf was run over by a tram, and he died at the age of 74 on June 12, 1926. His body was buried in the crypt (built by the original architect Francese de Paula Villar i Lozano in 1882) of the edifice where he had worked for the last 43 years of his life. Mter Gaudf's death, the architect Sugrafies i Gras took over the project until 1935, when the work came to a standstill because of the Civil War. M­ ter the war, the architect Quintana i Vi­ dal took over the work, followed by Llufs Bonet, Isidre Puig, Francese de

50

THE MATHEMATICAL INTELLIGENCER

Figure 2. The Passion Fac;ade.

Paula Cardoner, and the current coor­ dinating architect, Jordi Bonet. In 1936, Gaudf's studio holding his notes and designs was burned in Civil War shelling. The project was resumed in 1952 using drawings and maquettes as a base, although the continuation of the work gave rise to much debate. From 1954 to 1976 the Passion Fac;;ade (western side) and its four towers were completed. The image of the Sagrada Familia as a work that will never be fmished is be­ ginning to fade. In 1986 the sculptor Josep Maria Subirachs Sitjar (Bar­ celona, 1927-) was commissioned by Joan Anton Maragall, president of the Council on the Sagrada Familia, to sculpt the Passion Fac;ade. When Subirachs was offered this assignment, he accepted on two conditions: that he be allowed to live on the grounds of the Sagrada Famflia itself, and that he be given total freedom for his work Both wishes have been fulfilled. To contrast with the Nativity Fac;;ade,

completed by Gaudf, which stands out for its Baroque character, Subirachs opted for "a hard vision, facilitated by a brutal treatment of the stone," on the Passion Fac;;ade [3]. In early 1987, Subirachs commenced his work and on April 14, 1999, he placed the final sculp­ tural set on the fac;;ade (Fig. 2). Mter that, he finished the two bronze side­ doors, the door of Faith and the door of Hope, and is now working on the two central doors and the sculpture of the apostle St. Thomas [4]. The work of Subirachs, of diversified periods, from the abstract style to expression­ ism, always has a sense of insight and research of new resources, expressed by signs and symbols. His empty-full and concavo-convex line is very ex­ pressive, above all, the serene, impec­ cable, and perfectly finished figure, playing with geometric lines [5]. Figure 3 shows the Subirachs statues of the crucifixion, and in more detail, Figure 4 shows the betrayal of Jesus Christ by Judas Iscariot. Notice the square to the

Figure 3. The Subirachs statues of the crucifixion on the

Figure 4. Detail of the betrayal of Jesus by Judas lscariot (note

Passion FaQade.

4 x 4 square at the left).

left of the two figures. Figure 5 shows the 4 X 4 square on the Passion Fa<;ade in detail. Construction has also begun on the transept, and the Schools of Architecture of the Polytechnic University of Catalonia, the University of Deakin (Australia), and the University of Wellington (New Zealand) are work­ ing on the computer drafting for con­ struction of the transept. The next segment of the general plan is the apse. Architect Jordi Bonet calculates that this portion of the Sagrada Familia, in which the entire Church of Santa Maria del Mar could be hidden, will be finished by the year 2010. With completion of all these sections, two projects will remain pending that, un­ til now, seemed inaccessible. The first is the Gloria Fa<;ade, which faces the Carrer de Mallorca, which Gaudf in­ tended to be the main entrance. This access was sacrificed twenty years ago with the construction of an apart-

ment building across the street. The fi­ nal work of the temple should be the six towers that emerge over the cen-

tral nave. The highest of these would be the Tower of the Saviour, men­ tioned earlier.

Figure 5. The magic square in Sagrada Familia, the "Passion" square.

VOLUME 23, NUMBER 4, 2001

51

The Magic Square

A magic square consists of an arrange­ ment of positive integers in the form of a square so that the sum of these inte­ gers in every row, in every column, and in each of the two principal diagonals is the same. This sum is called the "magic sum." Any magic square can be subjected to a reflection and/or rota­ tions through 90° without losing its magic character. Magic squares are said to be essentially different if they cannot be transformed into one another by ro­ tations and reflections. Every essentially different magic square can be written in eight forms due to rotations and reflec­ tions. If the integers forming a magic square are the consecutive positive in­ tegers from 1 to n2 inclusive (each number occurring exactly once), the square is said to be "normal" (or "pure") and of order n [6, p. 444], [7). The magic sum S of a normal magic square of order n is given by S = 1/ n(n2 + 1). In a normal magic square 2 of order n, a complement pair is two numbers that sum to n2 + 1. Henceforth, consider normal magic squares of order 4. The magic sum of these squares is, of course, 34. Freni­ cle de Bessey (1602-1675) established that there are exactly 880 essentially different normal magic squares of or­ der 4, by using the method of exhaus­ tion. The complete set was published in 1693. This number of 880 was con­ firmed analytically in [6] . It is clear that there are 8 complement pairs (adding up to 42 + 1 17) in a normal magic square of order 4, and the links be­ tween complement pairs form one of twelve patterns (W. S. Andrews (1908), H. E. Dudeney (19 10), both by obser­ vation; see [6] for the references). For the first analytical proof of the exis­ tence of twelve, and only twelve, pat­ terns (the so-called Dudeney groups), see also [6). The 880 normal order-4 magic squares are thus divided into =

14

15

4

16

3

2

13

12

7

6

9

5

10

11

8

8

11

10

5

9

6

7

12

13

2

3

16

4

15

14

1

Figure 8. Frenicle #175 reflected around its

Figure 9. Frenicle #175 reflected around its

main diagonal.

right diagonal, the Durer square.

twelve Dudeney groups. Reference [7) is an easily accessible source contain­ ing graphics illustrating how the twelve groups are formed in terms of the com­ plement pairs. A Jaina inscription of the twelfth or thirteenth century c.E. giving the magic square in Figure 6 was reported in 1904 as having been found in Khajuraho, In­ dia. This square is said to be pandiag­ onal (or Nasik or diabolical): not only do the numbers in the rows, columns, and principal diagonals add to 34, but so also do the numbers in the "short broken diagonals," namely 2 12 5 15, 1 1 1 6 16, and those in the "long broken diagonals," namely 7 6 10 1 1 , 14 2 3 15, 4 16 13 1, 9 12 8 5. The square in Fig­ ure 6 is one of the 48 squares belong­ ing to Dudeney Group L In fact, an or­ der-4 magic square belongs to group I if and only if it is pandiagonal. If the short broken diagonals of a square add to 34, but the long broken diagonals do not, then the square is said to be semi­ pandiagonal or semi-Narsik; see also [8, p. 482], and Figure 7 for an exam­ ple of a semi-pandiagonal square. Only group III will be considered in the sequel. Group III has all comple­ ment pairs symmetrical around the center of the square, as illustrated in Figure 7. All 48 magic squares in group III are symmetrical (Fig. 7) and semi­ pandiagonal. Since every essentially different square can be written in 8 forms, for enumeration purposes one of these 8 squares must be designated the basic

7

12

1

14

1

12

8

13

2

13

8

11

14

7

11

2

16

3

10

5

15

6

10

3

9

6

15

4

4

9

5

16

Figure 6. A pandiagonal square.

52

1

THE MATHEMATICAL INTELLIGENCER

Figure 7. Frenicle #1 75.

one. The other 7 are often referred to as versions of the basic one. Frenicle established two simple rules to determine the basic one among the 8:

disguised

1. The smallest number in any comer of the magic square must be in the top left comer. If it is not, rotate the square until it is. 2. The second number in the top row must be less than the first number in the second row. A normal magic square A may be trans­ formed into another magic square by subtracting each number in A in tum from 17 ( = 42 + 1). The resulting magic square B is a disguised version of an­ other magic square from the same group. The magic squares A and B form what is known as a complementary

pair. Consider Frenicle # 1 75, which is shown in Figure 7. It is a normal basic order-4 magic square from Dudeney Group IlL If #175 is reflected around its main diagonal, the square in Figure 8 is obtained, and if #175 is reflected around its right diagonal, the square in Figure 9 is obtained. The square in Fig­ ure 9 is the Durer square; it appears in the famous engraving titled Melancho­ lia by Albrecht Durer (1471-1528). Two impressions of this engraving are in the British Museum. The middle numbers in the last row contain the date of the engraving, 1514. There are 32 essentially different magic squares that can be written with the numbers 15 and 14 in these positions, but only four of them, of which Diller's is one, are symmetrical [6, p. 446]. Note that the magic square in Figure 8 is the complement of Durer's square (Fig. 9). Note also that if the highest number in every row in Figure 8 is reduced by 1, then the square in Figure 5 is obtained. Let us refer to the latter square as the "Passion" square. Although the "Pas-

sion" square is non-normal and non-ba­ sic, it has some interesting properties: 1. Its magic sum is 33, the age of Jesus Christ when he was crucified, see [9], [10], [ 1 1 ] , and [12]. 2. The four comers add to 33. 3. The four numbers in the center add to 33. 4. The middle numbers in the top row and the middle numbers in the bot­ tom row add to 33. 5. The middle numbers in the first col­ umn and the middle numbers in the last column add to 33. 6. The four squares in every comer add to 33. 7. It is semi-pandiagonal (the short broken diagonals 14 1 1 5 3 and 14 9 8 2 add to 33). The photographs were taken by P. Maritz and E. L. Maritz.

www. barcelona. com/gaudi/ingles/i_vida.

tions & Patterns. Available at wysiwyg://

html (accessed November 27, 2000).

3/http://www.geocities. com/ �harveyh/

[2] Jonathan

D.

Meltzer.

Gaudf Central.

transform.htm (accessed December 7,

Sagrada Familia . Available at URL http://

2000).

www.op. net/�jmeltzer/Gaudi/eltemple.

[8] C. J. Henrich. Magic squares and linear algebra. Amer. Math. Monthly 98 (1 991),

html (accessed November 27, 2000). [3] Gaudf & Barcelona Club. The sculptor is

481 -488.

now working on the great bronze door of

[9] New Bible Dictionary, 2nd Edition, Editor: J. D. Douglas. Inter-varsity Press, Leices­

the Passion Facade. Available at URL http :1/www. gaudiclub. com/ingles/i_links/

ter, England, 1 962.

subiracs.html (accessed November 27,

[ 1 0] M . J. Edwards. "Not Yet Fifty Years Old":

2000).

John 8:57. New Testament Studies 40

[4] Gaudi & Barcelona Club. The "Sagrada

{1 994), 449-454.

Famflia. " Current Status of the Construc­

[1 1 ] G. W. Buchanan. The Age of Jesus. New

tion. Available at URL http://www.barcelona. com/gaudi/ingles/i_vida/i_sf98.htm

Testament Studies 41 {1 995), 297.

(ac­

[1 2] W. Hinz. Jesu Sterbedatum. Zeitschrift der

cessed November 27, 2000).

Deutschen Morgenlandischen Gesellschaft 142 {1 992), 53-56.

[5] BancSabadell. The Artists. Subirachs Sit­

jar, Josep Maria . Available at URL http:!/ www. b a n c sa b ad e l l . c o m / p i nacot/3/ 321 00065.htm (accessed November 27,

Department of Mathematics

2000).

University of Stellenbosch

[6] K. Ollerenshaw and H. Bondi. Magic REFERENCES

[1] Gaudf & Barcelona Club. Antoni Gaudf

(1852- 1926). Available at URL http:!/

Private Bag X1

squares of order four. Phil. Trans. R. Soc.

Matieland 7602

Land. A

South Africa

306

(1 982), 443-532.

[7] Harvey D. Heinz. Order-4, Transforma-

e-mail: [email protected]

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VOLUME 23, NUMBER 4, 2001

53

l'i¥1(W·J·I•i

.J e remy G ray ,

How Modern Mathematics Came to Portugal Elza Maria A. S. Amaral

Editor

I

A

t the beginning of the twentieth century the scientific community in Portugal was small. Advances in the mathematical domain were a conse­ quence of the individual work of some Portuguese mathematicians, most no­ tably the most distinguished Por­ tuguese mathematician of the 19th Century, Gomes Teixeira. Teixeira was born in 1851, and studied at the Uni­ versity of Coimbra, where he obtained his first and doctoral degrees. He stayed there as a lecturer until 1883, when he went to the Academia Poly­ technica do Porto (in 191 1 renamed the Faculdade de Ciencias da Universi­ dade do Porto), of which he was first its Director (1883-1911) and later its Rector (191 1-1918). Gomes Teixeira's main areas of study were Analysis and Differential Geometry. In these areas he exercised a strong positive influ­ ence on the teaching of pure mathemat­ ics in Portugal, producing new ideas and carrying the name of Portugal beyond her frontiers. From Teixeira's vast output, the Curso de Andlise Infinites­ imal, which he started to write in 1885 (first edition 1887), and the Tratado

de las Curvas Especiales Notables, Tanto Planas Como Abeladas, 1889, were probably the two most important. Concerning the latter, R. C. Archibald's praise is probably still valid [Ferreira, (1992)]: "We heartily recommend Pro­ fessor Gomes Teixeira's book for every mathematical library, as no other pub­ lication of the kind can take its place." One of Teixeira's prime objectives was to break the scientific isolation in which Portuguese mathematicians had lived for so long. He achieved this in 1877 by creating the Jamal de Scien­

cias Mathematicas e Astron6micas­

Column Editor's address: Faculty of Mathematics, The Open University, Milton Keynes, MK? 6AA, England

the first Portuguese review devoted to the mathematical sciences. This journal continued unti1 1905, for a total of four­ teen volumes, when it became known as the Annaes Scientijicos da Acade­ mia Polytecnica do Porto, remaining under the direction of Gomes Teixeira until 1932, the year before his death.

However, although Teixeira and others exercised a positive influence on some of their disciples, they did not make a 'team'; they always worked alone. In the words of Alfredo Pereira Gomes, who was an assistant lecturer at the Faculty of Sciences in Oporto in 1941/42 and a collaborator of the Cen­ tro de Estudos Matematicos do Porto (C.E.M.P) from 1942 to 1946 [Gomes (1992), p. 50]: It is a historical fact that mathemat­ ical research in Portugal was never properly established as a solid ac­ tivity. Notwithstanding the fact that some important mathematicians had appeared in the course of time, they left no followers and they ap­ peared as singular cases. A. Monteiro, one of the mathemati­ cians most involved in the revival of mathematics in Portugal, shared the same opinion and said [(194la), p. 9]: As

everyone knows we do not have a Portuguese mathematical school; we only have half a dozen investi­ gators and among them we cannot find one who could be considered a great mathematician of our time. We do not have a centre for mathemat­ ical research. The Creation of a Modern Mathematical Community in Portugal, 1 936-1 946

During the late 1930s, the mathematical sciences in Portugal were revitalised. Several mathematicians sought to create teams of investigation and centres for mathematical research. They organised seminars and conferences where mod­ em theories and results were given; they published books and articles and trans­ lated important treatises in mathemat­ ics; they supervised young researchers; and they established contacts with for­ eign mathematicians. Speaking of the scientific activity of this period, Pereira Gomes said [(1992), p. 50]:

© 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 4, 2001

55

It is important to point out the re­ lated appearance of three major events, that I like to call a triptych, three related panels, which taken to­ gether show the results of that con­ nection, that col\iunction, namely: scientific works-conducted first around the Seminario de Analise Geral and then around the Centros de Estudos Matematicos; a Por­ tuguese Mathematical Society; and two reviews, Portugaliae Mathe­ matica for original works, and Gazeta de Matematica, concerned with the diffusion of mathematics and the modernising of methods and subjects of studies. The institutions they created, along with the active research of their mem­ bers, gave great impetus to mathemat­ ical science in Portugal, which was fur­ ther enhanced by the exchange of Portuguese mathematical journals with those from other countries. The process started in 1936 with the creation of Nucleo de Matematica, Fisica e Quimica. It was a group of mathematicians (Antonio Monteiro and Manuel Zaluar Nunes from the Fac­ uldade de Ciencias of Lisbon, and Bento de Jesus Carac;a from the Insti­ tuto Superior de Ciencias Economicas e Financeiras in Lisbon) and physicists (Manuel Valadarese, Aurelio Marques

da Silva, and Manuel Teles Antunes from the Faculdade de Ciencias of Lis­ bon, and Antonio da Silveira from the Instituto Superior Tecnico in Lisbon), and has as its main goal bringing about courses and conferences in these sub­ jects. Among these publications, Teo­ ria da Relatividade Restrita was the product of a series of four lectures given by the Portuguese mathemati­ cian Ruy Luis Gomes (1905-1984). Although the Nucleo ceased in 1939, one year later the Centro de Estudos Matematicos Aplicados a Economia (C.E.M.A.E.) was founded, as was the Seminano Matematico of Lisbon (which became the Seminano de Analise Geral after 1939). The purpose of these initiatives was to initiate young people into the study of modem mathematics. The mathematical activ­ ity of the Centro de Estudos Matemati­ cos Aplicados a Economia took place at the Instituto Superior de Ciencias Economicas e Financeiras through the action of the Portuguese mathemati­ cian Bento de Jesus Carac;a (19011948). The Seminar, conducted by An­ tonio Monteiro, was first connected with the Faculty of Sciences of Lisbon and later with the Centro de Estudos Matematicos of Lisbon, which, in 1944, published a volume titled Trabalho do Seminario de Andlise Geral, contain­ ing six of the research works of the

Seminar that earlier had been pub­ lished separately, in Portugaliae Math­ ematica between 1942 and 1943. The centre was created by the In­ stituto para a Alta Cultura in 1940 and it was led by A. Monteiro for the next three years. Its activity was mainly in the area of topology. Two series of lectures started in the centre in No­ vember 1941, one on Introduction to Abstract Algebra and other on Intro­ duction to General Topology. The lat­ ter series was integrated with the Sem­ inano de Ana!ise Geral, which usually was open only to workers at the cen­ tre. However, the series of lectures in general topology, which took place on Saturdays, were open to the public, be­ cause of the importance of the subject. These lectures were held regularly with a pause in January and the beginning of February, occasioned by the visit of the eminent French mathematician Mau­ rice Frechet. He was invited by the In­ stituto para a Alta Cultura, at the sug­ gestion of A. Monteiro, to participate in the activities of the centre (Fig. 1). Frechet stayed in Portugal for three weeks and gave several lectures on Topology and the Calculus of Probabil­ ities at Lisbon, Oporto, and Coimbra. I have already mentioned the found­ ing of two Portuguese mathematical re­ views, Portugaliae Mathematica and the Gazeta de Matematica, as well as

Figure 1 . From left to right: Hugo Ribeiro, Armando Gibert, Antonio Monteiro, Manuel Zaluar, Bento Cara'
56

THE MATHEMATICAL INTELLIGENCER

Figure 2. From left to right: Alfredo Pereira Gomes, Ruy Luis Gomes, Ant6nio Monteiro, Manuel Zaluar.

a mathematical society, the Sociedade Portuguesa de Matematica Portu­ galiae Mathematica was the first Por­ tuguese review dedicated solely to the publication of original works of math­ ematical research. It was created in 1937, and it played an important role in the development of mathematical stud­ ies in Portugal. In the preface of the first volume pub­ lished in 1940, A. Monteiro justified the appearance of the journal Portugaliae Mathematica in the following words: At the beginning of this century we in Portugal lost a review dedicated exclusively to pure mathematics, and this happened precisely at a time when the mathematical sci­ ences entered a period of great de­ velopment which in the years that followed (from 1920 to 1940) be­ came a vertiginous current. Of course, during this period several reviews were founded in Portugal in which mathematical works were published, but Portuguese work in mathematics was dispersed in non­ specialist periodicals, national and foreign, which was inconvenient.

A. Monteiro thought of Portugaliae as filling the "hole" left by the disappearance of Gomes Teix­ eira's journal at the beginning of this century. Thanks to its quality it soon became recognised abroad, and conse­ quently it shortly carried articles by such esteemed mathematicians as Maurice Frechet, John von Neumann,

Mathematica

Monteiro . . . never was al lowed to work in any Portu­ guese un iversity. L. A. Santal6, and W. Sierpinski. Evi­ dence of its importance is that even the first volume, published in 1940, was exchanged with similar publications from 22 countries: Germany, Belgium, Canada, Czechoslovakia, Spain, France, Hungary, India, England, Italy, Ar­ gentina, Japan, Yugoslavia, Latvia, Lithuania, Poland, Portugal, Rumania, Sweden, Switzerland, USSR, and USA. The Gazeta de Matematica, founded in 1939, was created to inspire students

to research in mathematics. Besides publishing some mathematical articles, most of them of an elementary level, and exams on subjects taught at high schools, it carried notices about math­ ematics in Portugal and abroad. The Sociedade Portuguesa de Matematica was created in 1940. Its main goal was to establish and pro­ mote the study of Applied and Pure Mathematical Sciences, to which end it organised lectures and free courses. The Society encouraged participation in colloquia and congresses, affording interchange of Portuguese mathemati­ cal knowledge with that from other countries. It broke the isolation in which Portuguese mathematicians had been working. This was a first step to­ wards the improvement of both teach­ ing and scientific research. Ant6nio Anlceto Ribeiro Monteiro

Antonio Aniceto Ribeiro Monteiro (1907-1980) was one of the Portuguese mathematicians behind the mathemati­ cal movement in the 1940s (Fig. 3). He was born 31 May 1907, in Mo�funedes (Angola), and obtained his first degree in Mathematical Sciences at the Fac-

VOLUME 23, NUMBER 4, 2001

57

some of his colleagues in Oporto, Ruy Luis Gomes in particular, involved him in the activities of the new centre. Meanwhile, Monteiro had received an invitation in September 1 943, with the support of Albert Einstein, John von Neumann, and Guido Beck, to go to Brazil to take up a Chair, and this he could not refuse. However, Monteiro's departure was postponed for political reasons, and in this same year he helped to found the Junta de Investigac;iio Matematica It was this institution, created in 1943 by the Instituto para a Alta Cultura, that made it possible for Monteiro to stay in Oporto and work with the Centro de Es­ tudos Matematicos. Despite the finan­ cial difficulties of the Junta, Monteiro and his colleagues were able to start the Col6quio de Analise Geral. It was com­ posed of three branches of mathemat­ ics: Modem Algebra (supeiVised by An­

tOnio Almeida Costa) ; General Topology

(supervised by A. Monteiro); Integration

Figure 3. Antonio Aniceto Monteiro.

and Measure Theory (supervised by Ruy Luis Gomes). Courses, seminars, and

ulty of Sciences in Lisbon in 1930.

ematics. He founded, with the collabo­

From 1932 to 1936 he stayed in Paris

ration of M. Valadares,

on a scholarship from the Instituto

Silva,

A.

A. Marques da

Silveira, Peres de Carvalho,

original works of this colloquium were

Caderrws de

later published in the

Andlise Geral,

also supported by the

para a Alta Cultura and was strongly

and others, the Nucleo de Matematica,

Junta (Fig. 4). Monteiro's remarkable

influenced by some of the leaders of

Fisica e Quimica, and helped to found

legacy in Oporto is a monument to the

the French school of Analysis, such as

Portugaliae Mathematica. He collabo­

character and thought of a Portuguese

Emile

mathematician

Borel,

Jacques

Henri

Hadamard.

Lebesgue,

and

rated with the Sociedade Portuguesa

There

wit­

da Matematica and the review,

he

Gazeta

nessed the modem trends in the study

da Matemdtica,

of algebraic and topological structures.

Seminano de Analise Geral with great

In 1 936 he obtained his Doctorate at

enthusiasm and dignity.

and conducted the

whose

talent

foreign

countries were to profit from and whom his own country did not appreciate. Monteiro left Portugal in 1945 and went to the University of Brazil, in Rio

the Institut Henri Poincare (University

As I describe in more detail below,

of Paris) under the direction of Mau­

Monteiro had returned to Portugal full

Mathematical activities had started in

rice Frechet, with a thesis in Analysis

of enthusiasm for topology. He started

Rio in 1939, and Monteiro was respon­

entitled Sur l'additiviti des noyaux de Fredlwlm.

to study topological spaces, lattices,

sible for the chair of Higher Analysis at

and the relations among them, creating

the National Faculty of Philosophy

de Janeiro, on a four-year contract.

On his return to Portugal, Monteiro

around him an infectious ambiance, in

(now the Federal University of Rio de

did not become a member of any offi­

which some of the younger Portuguese

Janeiro), which was the main mathe­

cial teaching institution. Because po­

mathematicians were willingly caught

matical centre of Rio and the second­

litically he stood against a regime that

up. First in the Seminano de Analise

best mathematical centre in Brazil (the

allowed no dissenting voices or criti­

Geral and later at the Centro de Estu­

first being the Faculty of Philosophy,

cism, he never was allowed to work in

dos Matematicos de Lisboa, several

Sciences, and Literature at Siio Paulo

any Portuguese university. Between

topological studies were developed by

University, where mathematical activ­

1 938 and 1943, his activities as lecturer

Monteiro, Hugo Ribeiro, Sebastiiio e

ity had started

and as researcher were unpaid, he sur­

Silva, and others.

ian community had the benefit of this

vived by giving private lessons and working in a Service of Inventorying of

From 1 943 to 1945,

A. Monteiro was

in Oporto participating in the Centro de

remarkable

in 1934). So the Brazil­

Portuguese

mathemati­

cian. Monteiro had to face enormous

the Scientific Bibliography in Portugal,

Estudos Matematicos do Porto, cre­

difficulties just to leave his country.

organised by the Instituto para a Alta

ated in 1 942. Conscious of Monteiro's

Despite help from some eminent for­

Cultura. Even so, he contributed sig­

remarkable capacity for training disci­

eign mathematicians, in particular G.

nificantly to the development of math-

ples and of his excellence in research,

Beck, who sent a report supporting

58

THE MATHEMATICAL INTEWGENCER

him to the Minister of National Educa­ tion in Rio, signed by the representa­ tives of all Argentine and Uruguay Uni­ versities, Monteiro was not able to get to Brazil until the end of March 1945, fifteen months after the invitation. He then remained in Brazil and Argentina for some thirty years, doing distin­ guished work on Algebraic Logic and building up the mathematical commu­ nities there. After the Portuguese Revolution in 1974, some mathematicians who had worked with A Monteiro at the Centro de Estudos Matematicos returned to Portugal and tried to persuade him to re­ turn to Portugal from Bahia Blanca in Argentina They wanted the new gener­ ation to meet the man most responsible for the promotion of research centres in Portugal. In October 1976, a document was sent to the INIC from the three Por­ tuguese Universities (Lisbon, Oporto

and Coimbra) to admit Monteiro as a re­ searcher at the Institute. This document praised Monteiro in the following words [Gomes (1980a), XXXN] : Antonio Monteiro was the Por­ tuguese mathematician who con­ tributed in the most decisive way to the renewal of the mathematical studies in Portugal and who, by his research and by training disciples, continued to do so successfully in foreign university centres. They also wrote that the presence of Monteiro in Lisbon would promote the study of Algebras of Logic among the Portuguese community and could start a research school on this impor­ tant subject. Monteiro returned to Portugal early in 1977 and immediately started work­ ing to that end, giving several lectures

on his research, attracting some disci­ ples, and proposing several research topics. This resulted in notes in spe­ cialised reviews and two doctoral the­ ses. He was satisfied with his effect on scientific activity in Portugal, but he was not happy with the teaching and the way research was done in Portugal. In particular, Monteiro was disap­ pointed when he proposed to give a se­ ries of lectures at the Faculdade de Ciencias de Lisboa (October, 1977), and did not get a response. He wrote to Pereira Gomes on 5th June 1978 [Gomes (1980a), XXXV] : I have not given the lectures yet . . . the only reason is that they have not made the timetable. Either they do not have any interest or they do not want them. Patience! I shall not in­ sist again because I already feel hu­ miliated. I will give a series of lec­ tures this month at the Faculty of Sciences in Oporto. I wonder that the situation in Lisbon has not changed with time. Tradition has extraordi­ nary force. During my four or five years as a student at the Faculty of Sciences in Lisbon, I never heard about lectures in mathematics. If there were any, I cannot remember. During his short stay in Portugal, Monteiro wrote a paper titled, Sur les

Algebres de Heyting Symetriques,

TOPOLOGIA GER l

Figure 4. Cademos de Analise Gerat.

which in 1978 received an award from the Calouste Gulbenkian Foundation, the Premio Gulbenkian de Ciencia e Tec­ nologia In 1979 Monteiro returned to Bahia Blanca, where he died in October 1980. In his life he had published more than fifty papers in Algebraic Logic, and he left as many as fifty more unpub­ lished; he had no enthusiasm for writing up his results, and his mind would turn to new col\iectures. He had this attitude in his whole life, for he always el\ioyed developing something new. Monteiro was honoured posthumously in Portugal with two volumes, 39 and 40, of the jour­ nal Portugaliae Mathematica dedicated to his memory, and an International Conference on Mathematical Logic in Memory of A A Monteiro was held at the University of Evora, July 13-18, 1998. In addition, on 2 October 2000, Jorge Sampaio, President of the Repub-

VOLUME 23, NUMBER 4. 2001

59

To appreciate the early work of Monteiro, recall that the modem defi­ nition of a topological space, as given by Wack-law Sierpinski in his Warsaw lectures and his

1928

text, is a set X,

consisting of elements of an arbitrary nature, called points of the given space, and a topological structure (often sim­ ply called a topology) specifying the open sets of the topological space. From the concept of a topology all the remammg

fundamental

topological

concepts may be derived. First of all, closed sets are defined as comple­ ments of open sets. The concept of a neighbourhood enables one to define the derived set A ' (the set of accumu­ lation points of a �t A) and the closure

of A, denoted by A, which is the union of A and A ' . The function ta�ng an ar­

bitrary set A to its closure A is called the closure operation on the given topological space, and i�njoys the_!?!­ lowing properties: A C A and A

= A if

and only if A is closed, i.e., its comptementAc _ls open; A 1 U A2 and A

Figure 5. Gra-Cruz da Ordem Militar de Santiago da Espada.

= A.

-

=

-

A1 U A2;

In addition, the closure of

an arbitrary set A is the intersection of all clo�d sets containing A; alterna­

lie, honoured Antonio Monteiro with the

scribed here as an indication of the in­

tively, A is the smallest closed set con­

highest award of the Portuguese Gov­

tellectual concerns of the young group

taining A.

ernment, the Gra-Cruz da Ordem Militar

who

Portuguese

In this order of ideas, the closure op­

de Santiago da Espada (Fig.

mathematics. It can be seen that they

eration and its basic properties, de­

5).

were

revitalising

were greatly interested in the contem­

fined above, have been derived from

Monteiro's Early Work

porary move towards an axiomatic

the fundamental concept of a topology

on Topology

presentation of the subject.

on a given set X. Alternatively, one can

Monteiro's earliest mathematical work

Topology and lattice theory devel­

consider closure as the fundamental

was in point-set, analytic, or general

oped strongly during the first decades

topological concept. In this approach,

topology, which tries to explain in a

of the twentieth century, and at that

general setting such concepts as con­

time Monteiro was in Paris

(1931-

one assumes that in a given abstract set X for each subset A there is specified

vergence and continuity known from

1936) doing his Ph.D. under Frechet. At

a subset A of X, called the closure of

classical

of

the Sorbonne, Monteiro was caught up

A, enjoying the above propef!!es to­

Frechet, Sur quelques points du calcul fonctionnel (1906), spaces of functions

in the new trend towards abstract al­

gether with the equality

gebraic and topological structures. Be­

approach to the concept of a topologi­

were shown to be, in a natural way,

sides being influenced by the analysts,

cal space goes back to the Polish math­

metric spaces, and as a result some

Monteiro was also fascinated by mod­

ematician Casimir Kuratowski, who

standard

were

em algebra. Van der Waerden's book

axiomatised the idea of closure in

shown to apply in this more general

appeared exactly at that time, and

setting, with important consequences

Monteiro spent hours with his fellows,

analysis.

limiting

In

the

work

arguments

¢

=

¢. This

Sur l'operation de !'analysis situs (1922).

for analysis. A broader class of topo­

in particular with Jean Dieudonne,

The

logical spaces, not necessarily metric,

studying that book. He was also ex­

spaces

actual

study

were introduced in Felix Hausdorffs

cited by the works of Garrett Birkhoff

classes of spaces by some conditions or axioms additional to those defining

required

of topological

distinguishing

sub­

Grundziige der Mengenlehre (1914).

on lattice theory and universal algebra,

They became regarded as central, and

those of Marshall Stone on topological

topologies.

are known today as Hausdorff spaces.

representations of Boolean algebras

were of various kinds, the so-called

Although Monteiro's work in topology

and distributive lattices, and those of

separation axioms being among them,

was less substantial than his later con­

Henry Wallman on compact topologi­

such as the one characterizing Haus­

tributions to algebraic logic, it is de-

cal spaces.

dorff spaces.

60

THE MATHEMATICAL INTELLIGENCER

These additional axioms

On his return to Portugal, in 1936, Monteiro got some of the younger mathematicians working on current problems in topology. Monteiro and Hugo Ribeiro and Sebastiao e Silva (his

they added the following axiom:

B c A + B or, B => A C B.

which is the same,

A+ Ac

ticle, "Sur l'axiomatique des espaces de Hausdorff. " In his article "Les ensembles fermes

Frechet had raised the problem of

et les fondements de la topologie"

characterising a Hausdorff space using

[ 1 94la],

Monteiro

showed

that the

collaborators at the Centro de Estudos

the derivation operation as a primitive

most general spaces of type ( v) whose

Matematicos de Lisboa) started by

notion. Monteiro solved this problem

topology is uniquely determined by

characterising topological spaces by

in his work [ 1940c] "Caracterisation

knowledge of the family of closed sets

means of primitive notions such as

des espaces de Hausdorff au moyen de

are precisely those where the closure

frontier, closure, and derivation opera­

!'operation de derivation." Frechet had

operator of each proper subset

tions. In these first research papers, the

said that it would be of use to have two

isfies the condition

names

of

Birkhoff,

Frechet,

A

=

A.

A,

sat-

Kura­

definitions for normal and completely

towski, Hausdorff, and Sierpinski were

normal spaces, one based directly on

Antonio Almeida Costa and

often cited.

the choice of neighbourhoods and the

Abstract Algebra

other on the operation of derivation,

Antonio Almeida Costa was born in

The article, "Sur l'axiomatique des

(v)"

[ 1940a] contains Mon­

and in his paper Monteiro carried this

humble surroundings as the illegiti­

teiro's first results on the study of the

programme through. H. Ribeiro then

mate son of an unmarried dressmaker,

foundations of abstract topology. This

went on to use Monteiro's idea to char­

Maria de Jesus Costa, in Santa Maria

was written in collaboration with Hugo

acterise, by means of the derivation op­

da Vila, a municipality in Celorico da

Ribeiro and contains some results on

eration,

Beira, on 25 May 1903, and died in Lis­

neighbourhood spaces (spaces of type

pletely nonnal

e

bon on 24 August 1978. At the age of

(v)), a notion presented by M. Frechet

Silva also contributed to the axiomati­

nine, he went to the grammar school in

in his article "Sur Ia notion de voisinage

sation of Hausdorff spaces, presenting

Guarda and there completed his sec­

dans les ensembles abstraits," in 1917).

a characterisation of Hausdorff spaces

ondary-school studies in July 1919.

espaces

A space X is a space of type (v) if the derived set

A'

of a set

A

is the set of

the points x in X such that every neigh­

regular,

normal, spaces.

and

com­

Sebastiao

by means of the primitive notions of frontier, edge, and boundary in his

ar-

In 1920 he completed, with distinc­ tion, some studies in chemistry and

bourhood of x has at least one element belong to the set A

- x.

It seems to be Monteiro and Ribeiro who for the first time characterised the spaces (v) using primitive notions dif­ ferent from the usual notions of de­ rivation or neighbourhood. Partial re­ sults

in

this

direction

had

been

obtained by Kuratowski (in his work on the closure notion, 1933) and by Zarycki (using the concepts of frontier, interior, exterior and edge, 1927), but they only considered particular neigh­ bourhood spaces, the so-called acces­ sible spaces of Frechet. (At the time they began their research, Monteiro and Ribeiro did not know about the work of Zarycki.) Hugo Ribeiro had al­ ready presented an axiomatisation of Frechet's topological spaces using the primitive notions of closure, interior, edge, frontier, border, and exterior, in his first research paper, "Sur l'axioma­ tique des espaces topologiques de M. Frechet." Monteiro and Ribeiro proved that it was sufficient to add a third ax­ iom to the axiom system for Frechet's topological spaces to obtain an axiom system for spaces of type (v). So, for example, to characterise a space of type (v) using the closure operation

Figure 6. Ant6nio Almeida Costa.

VOLUME 23. NUMBER 4. 2001

61

mathematics at the University of Lis­ bon and then went on to the degree in Mathematical Sciences, in the Faculty of Sciences at the University of Oporto, in October 1924. He distinguished him­ self there by winning the Gomes Teix­ eira and Gomes Ribeiro prizes for math­ ematics, and graduated as the highest ranked student in the Faculty of Sci­ ences. In due course he became an aux­ iliary professor of applied mathematics at his alma mater. In November 1933, he was made the first Official of the Ad­ ministrative Services of the Santa Casa da Misericordia, the charitable institu­ tion where he had once worked as an office boy and which had provided fi­ nancial support for his studies. As a re­ sult, he was invited to become a mem­ ber of the Municipal Council of Oporto. There he remained, directing the Edu­ cation Section, from May 1936 to Sep­ tember 1937, when he left to take part in a foreign exchange to Berlin. He had applied to the Junta de Ed­ uca<;ao Nacional to travel there in June 1934. The grant given on 31 May 1937 was for twelve months in Berlin at the Physikalischer Institut, where he was to study quantum theory and relativity, subjects that were then unknown in Portugal. The next year he took Max Kohler's course on the Application of Group Theory to Quantum Mechanics, and the Introduction to Quantum Me­ chanics taught by von Weizsacker. Af­ ter this, Almeida Costa's work began to veer away from physics to the theory of group representations and subse­ quently to modem algebra. In July 1939, he returned to Portugal with great enthusiasm for those areas of al­ gebra that could be related to the the­ ory of representations. Thereafter he devoted himself to the introduction and expansion of algebra in Portugal. At the end of 1941, he started a free course in the Faculty of Sciences at the University of Oporto on the Theory of Groups and their Applications. This course was greeted eagerly not only by teachers and some students of mathe­ matics and physics but also by lectur­ ers in mineralogy and geology. It led to his first published work on abstract al­ gebra, Elements of the Theory of Groups. This was published in 1942 by

62

THE MATHEMATICAL INTELLIGENCER

the C.E.M.P., which served as an addi­ tional Study Centre for Mathematics under the directorship of Ruy Lufs Gomes. It had been founded in Febru­ ary 1942, as mentioned above, and Almeida Costa played a principal role from the beginning. Almeida Costa saw, in the centre, the possibility of teaching and publishing regular courses in alge­ bra, thus preparing students with whom he would be able to discuss problems in algebra and later recruit co-workers. Portugal from 1926 to 1974 was a right-wing dictatorship, and many aca­ demics were forced into exile. Almeida Costa agreed with the government pol­ itics, and was a member of the Party as his position in the Municipal Council shows. He was therefore able to pur-

Almeida Costa reg retted having pursued modern algebra only 40 years i nstead of his whole l ife. sue his mathematical career in Portu­ gal and benefit from such opportuni­ ties as there were for mathematicians and scientists at the time. In his per­ sonal relationships with friends and colleagues, he is remembered for re­ specting people's political views and recognising their efforts and work re­ gardless of their political opinion or so­ cial background. In the Faculty of Sciences at the University of Oporto, where he lec­ tured in areas more directly associated with applied mathematics such as as­ tronomy and rational mechanics, he al­ ways introduced a little algebra (the theory of groups and related areas). In addition he gave courses and seminars to groups associated with the univer­ sity, first in Oporto (Mathematical Cen­ tres of Studies) and later (from the be­ ginning of 1 952) at the Faculty of Sciences at the University of Lisbon. When in 1952 Almeida Costa was in­ vited by the Faculty of Sciences at the

University to occupy the Professorial Chair of Algebra, he accepted at once; and was thus able to dedicate himself body and soul to the subject which fas­ cinated him and for which previously he had not had as much time as he would have wished. All who knew him agree that Almeida Costa bitterly re­ gretted having taken the road of mod­ em algebra so late, pursuing it for only some 40 years instead of his whole life. On arrival in Lisbon in 1952 he rapidly drew on his experience from Oporto. He lectured once a week in the Faculty of Sciences at the University of Lisbon on the theory of groups, rings, non-commutative ideals, and fields, hop­ ing to attract young students to modem algebra. Unfortunately, Almeida Costa was a very sober lecturer and some stu­ dents found his lectures boring. Others were enthusiastic both about algebra and about the professor; for such peo­ ple he was a powerful motivator. One of his colleagues in the Instituto de Alta Cultura, Pinto Peixoto, described Costa's activity while a member of the Conselho de Investiga<;ao do Instituto de Alta Cultura in these terms: "He was extremely meticulous in the analysis of research projects, but always open to innovation and to progress. How many of the doctoral students that took part in foreign exchanges owe their degree to him without knowing it?" Although not among the top alge­ braists, Almeida Costa published pro­ lifically and helped shape degree courses in mathematics as well as the training of postgraduate students. He was one of the first in Portugal to pro­ mote the supervision of doctoral de­ grees in mathematics. Towards the end of his life he served on numerous com­ mittees. In 1977 he was President of the Academy of Sciences of Lisbon and Vice-President of the same Academy in 1976 and in 1978, and his death in 1978 was a great loss to the mathematical community in Portugal. His influence established algebra in Portugal as a subject in its own right at both the un­ dergraduate and postgraduate levels, and there is not a university in Portu­ gal today which does not teach ab­ stract algebra as an essential part of all its mathematical courses.

REFERENCES

Matematica de Antonio Monteiro," Encontro

[Agudo (1 988)] Fernando Roldao Dias Agudo,

Luso-Brasileiro de Historia da Matematica.

acterisation

Aetas, 1 39-148. Sao Paulo, Brazil.

au moyen de !'operation de derivation," Por­

"Matematica de Ontem e Matematica de Hoje-A Escola Politecnica/Faculdade de Ciencias de Lisboa e as Matematicas em

gumas

Figuras

Dominantes na Analise

des espaces

de

Hausdorff

tugaliae Mathematica 1 (IV), 333-349. [Monteiro (1 941 a)] Antonio A. Monteiro, "Les

Portugal," Universidade de Lisboa da Facul­

Matematica, em Portugal (de Gomes Teix­

ensembles fermes et les fondements de Ia

dade de Ciencias, Funchal.

eira a Sebastlao e Silva)," Hist6ria e Desen­

topologie,"

volvimento da Ciencia em Portugal no

56-66.

[Agudo (1 992)] Fernando Roldao Dias Agudo,

Portugaliae

Mathematica

2,

"Algumas considera<;:6es sabre o ensino su­

Seculo XX,

perior da matematica em Portugal," Boletim

tenario da Academia das Ciencias de Lisboa,

tion de fermeture et les axiomes de separa­

da Sociedade Portuguesa de Matematica

Lisbon.

tion," Portugaliae Mathematica 2, 290-298.

24, 27-54, Novembro.

I,

55-85, Publica<;:6es do II Cen­

[Frechet (1 91 8)] Maurice Frechet, "Sur Ia notion

[Agudo (1 998)] Fernando Roldao Dias Agudo, "Ser Cientista em Portugal -0 meu teste­ munho," Boletim da Sociedade Portuguesa

de Matematica 38, 5-27, Abril. [Amaral (1 994)] Elza Amaral, The Introduction

[Morgado (1 986)] Jose Morgado, "0 Professor Ruy Luis Gomes e o Movimento Matematico

Bull. Sci. Math. 4 2 , 1 38-1 56.

Portugues," Anais da Faculdade de Ciencias

[Gomes (1 942)] Ruy Luis Gomes, "Sobre o ob­

do Porto 67, 97-1 51 .

jectivo dos cursos promovidos pela Sec<;:ao

[Morgado (1 992)] Jose Morgado, "No Cin­

Gazeta da

quentenario do Centro de Matematica do

de

Matematica do

Porto, "

Matematica (Movimento Matematico)

Portugal (Masters Thesis), Birmingham, U.K.

Ill, Janeiro.

9,

Ana

Porto," Edic;ao Comemorativa do Cinquen­

tenario do Centro de Matematica do Porto,

[Gomes (1 980)] Ruy Luis Gomes and Luis

1 -47,

Neves Real, "Antonio Aniceto Monteiro e o

Porto.

C.E.M. do Porto (1 94 1 /44)," Portugaliae

A U T H O R

[Monteiro (1 941 b)] Antonio A. Monteiro, "La no­

de voisinage dans les ensembles abstraits, "

by Antonio Almeida Costa of Algebra into [Cignoli (1 997)] Roberto Cignoli, "La Obra

1

[Ferreira (1 992)] J. Campos Ferreira, "Sabre Al­

[Monteiro (1 940c)] Antonio A. Monteiro, "Car­

Mathematica

INIC-Centro de

Matematica do

[Morgado (1 995)] Jose Morgado, "Para a

IX-XIV.

historia da Sociedade Portuguesa de Mate­

[Gomes (1 980a)] A. Pereira Gomes, "0 re­

matica," Publicac;oes de Hist6ria e Metodolo­

gresso de Antonio Monteiro a Portugal de

gia da Matematica, Departamento de Mate­

39,

1 977 a 1 979," Portuga/iae Mathematica 39,

matica de Universidade de Coimbra, Coimbra. [Ribeiro (1 937/40)] Hugo Ribeiro, "Sur l 'ax­

XXXIII-XXXVIII. [Gomes (1 980)] Ruy Luis Gomes, "Tentativas feitas nos anos 40 para criar no Porto uma Escola de Matematica," Boletim da So­

ciedade Portuguesa de Matematica 6, Out­

iomatique des espaces topologiques de M. Frechet,"

Portugaliae

Mathematica

1,

259-274. [Ribeiro ( 1 94 1 )] Hugo Ribeiro, "Caracterisations des espaces reguliers normaux et com­

ubro. [Gomes (1 992)] Alfredo Pereira Gomes, "A Co­

pletement normaux au moyen de I' operation

munidade Matematica Portuguesa e a ln­

de Ia derivation," Portugaliae Mathematica 2, 1 3-1 9.

ELZA MARIA A.S. AMARAL

vestiga<;:ao na Decada de 40," Edk;ao

Department of Mathematics

Comemorativa do Cinquentenario do Centro

[Ribeiro (1 980)] Hugo Ribeiro, "Actua<;:ao de

de Matematica do Porto, 49-65, INIC-Cen­

Antonio Aniceto Monteiro em Lisboa entre

Universidade de Trfas-os- Montes e

1 939 e 1 942," Portugaliae Mathematica 39,

tro de Matematica do Porto.

Alto Douro

5000 Vila Real

[Gomes (1 997)] Alfredo Pereira Gomes, "Por­

V-VII.

Portugal

tugaliae Mathematica- Urn marco historico

[Silva (1 941 )] Jose Sebastiao e Silva, "Sur l'ax­

e-mail: [email protected]

na investiga<;:ao matematica portuguesa,"

iomatique des espaces de Hausdorff," Por­

Publicac;oes de Hist6ria e Metodologia da

tugaliae Mathematica 2, 93-1 09. [Silva (1 997)] llda Perez Silva, " Matematica em

Elza Maria Alves de Souza Amaral is

Matematica,

a teacher of mathematics and history

da Universidade de Coimbra (Centro de

Portugal nos anos 40," Encontro Luso­

but her research

Matematica da Universidade de Coimbra),

Brasileiro de Hist6ria da Matematica, Aetas,

of mathematics,

specialty is history of mathematics.

Coimbra.

After taking an undergraduate degree

[Guimaraes

in mathematics at Oporto, she did

ham, England ,

versity, didate.

Portugal,

graduate work at Birming­

5.

Departamento de Matematica

2 1 3-222. Sao Paulo. (1 990)]

Andrade

Guimaraes,

[Souza (1 968)] Jayme Rios de Souza, Gomes

"Gomes Teixeira e a M issao da Universi-

Teixeira e a sua acc;ao no Ensino Superior

dade," Junho (unpublished).

do Porto, Porto. [Vilhena (1 935)] Henrique Vilhena, 0 Professor

and at the Open Uni­

[Monteiro (1 940a)] Antonio A. Monteiro and

where she is now a Ph.D. can­

Hugo Ribeiro, "Sur l'axiomatique des es-

Doutor Francisco Gomes Teixeira (Eiogio,

paces (v)," Portugaliae Mathematica 1 (IV),

Notas, Nota de Biografia, Bibliografia, Doc-

275-288.

umentos), Lisboa.

VOLUME 23, NUMBER 4, 2001

63

M.eftii • i§rr@iiifii@i§4fii .i111§.id

The Sieve of Eratosthenes and Wall is's Product: How Two Wrong Arguments Give One Correct Answer A V. Spivak

This column is devoted to mathematics forfun. What better purpose is there for mathematics? To appear here,

A l exan d e r S h e n ,

T

o get a table of primes, one can start with 1, 2, 3, 4, 5, . . . and cross out all multiples of 2 (except for 2), then all multiples of 3 (except for 3) et cetera ("sieve of Eratosthenes"). Let us modify this process and delete each nth number in a row in­ stead of deleting multiples of n, and do it for all n (not only for primes). At the first step we delete each second num­ ber and get

1, 3, 5, 7, 9, 1 1, 13, 15, 17, 19, 2 1 , 23, 25, 27, 29, 31, . . . Then we delete each third number and get

1 , 3, 7, 9, 13, 15, 19, 21, 25, 27, 31, . . . then each fourth number:

1 , 3, 7, 13, 15, 19, 25, 27, 3 1 , . . . etc. Continuing this process, we get a (rather strange) sequence

1 , 3, 7, 13, 19, 27, . . . A simple program that computes the nth term of this sequence (an) says, for example, that a1 = 1, a13 = 133, a14 = 147, and a3s26 = 1 1 499 769. What can be said about the asymp­ totic behavior of an as n increases? Ob­ servations suggest that an is close to 1J"''l2/4. For example, the data given above show that

a theorem or problem or remark does

an

-- =

not need to be profound {but it is allowed to be); it may not be directed only at specialists; it must attract and fascinate.

17'1'l2/4

from readers-either new notes, or replies to past columns.

1.0002

for n = 3826. Probably this computation has con­ vinced you that indeed

We welcome, encourage, and frequently publish contributions

Ed itor

an

=

1J"/'l2

--(1 4

+ o(l)).

But how can we prove that? First Attempt

Please send all submissions to the Mathematical Entertainments Editor,

Alexander Shen, Institute for Problems of Information Transmission, Ermolovoi 1 9, K-51 Moscow GSP-4, 1 0 1 447 Russia; e-mail: [email protected]

64

Let us start with a simple estimate. Look at the position of some integer N during our process. At first it occupies position N. After we delete each sec­ ond term, N moves to the position (ap­ proximately) N/2. Then we delete each third term, and N moves to the position

THE MATHEMATICAL INTELLIGENCER © 2001 SPRINGER-VERLAG NEW YORK

I

::::::

2

-

3

·

N

-

2

==

N

-

3

After deleting each fourth number the position is

3

N

N

4

3

4

= - · - = -

and so on: after deleting each jth num­ ber, the position of N (denoted by Nj in the sequel) is Nj = N/j. When Nj be­ comes less than j, further operations do not change the position of N. There­ fore, the ultimate position of N (let us denote it by n) is about Nj when N/j = j, i.e., n = VN. In other terms, an = n2 . However, numerical data suggest an = 1J"''l2/4, not n2 . What is wrong? The source of errors is rounding: the posi­ tion of N after we delete each second term, is N - LN/2J, not N/2. Rounding errors become even more important for subsequent steps. Second Attempt

Let us try another approach and go backwards. (Instead of deleting terms, we may trace the position of the nth term when we insert a new term after each (n 1 )-th term, then after each (n - 2)-th, (n - 3)-th, . . . , each third, each second, and fmally after each term. Using this language, we may de­ fine an as the fmal position of the term that had position n at the beginning. However, in the sequel I still use the original language and speak about deletion, not insertion.) Let n be the final position of integer N (so an = N). What happens just be­ fore N reaches this position? On the previous step (just before we delete each nth term) N occupies the (n + 1)-th position. After that N occupies the nth position; we delete each (n + 1)-th term, and N never moves again. One step earlier, N occupies the (n + 2)-th position after we have deleted each (n - 2)-th term and just before we delete each (n - 1)-th term. Continuing this "back-tracing," we see that u steps earlier, N occupies the (n + u )-th position after we have -

deleted each (n u)-th term. This is true for small u when only one term before N is deleted, i.e., when (n + u) < 2(n u) . The situation changes when u = n/3: at that time we delete each jth term, where -

Let us repeat the answer we have got: number N (whose final position is n) occupies position (approximately) � · n at the time when next op­ eration will delete m terms.

-

j

=

n

-

·

2 2 u = 3 · n, Ni = n + u = 2 3 · n ,

and two terms are deleted at each step for smaller j. The next change happens when we delete three terms instead of two, i.e., when Ni = 3j. At that time . J=

2 -·n 3

-

v

'

2 N· = 2 · - · n + 2v 3 J

=

3,;

:J )

Similar arguments show that the next change occurs when

Ni

=

j=

(m

+

an since

' . . .

=

j

2m 2m +

---

1

j

�

· n, N1 � (m + l)j �

y;:j8 · nVn,

· Ni =

Jiii; · n · H · n

however, the first one shows that these two formulas, we get

· n.

� � + · · · 2�: � � ��·

ji;

=

n/2. Again we get the wrong answer!

N=

Now Wallis's product comes into play. It is known that

so we get

·

v:mnJ4 n

Both arguments above give wrong answers for similar rea­ sons: we have ignored rounding errors that initially are small but then increase significantly. Note that we move in different directions. The first computation has a small er­ ror at the beginning of the deletion process, while the sec­ ond one is good at its end. So what can we do? The answer is evident: let us com­ pare the results that both computations give in the middle. The second argument shows that

1 )j when

2 4 6 3'5'7

m

=

Two Computations Meet Each Other

where v is the number of steps when two terms before N are deleted. A simple computation shows that at that time

In general,

The first operation deletes n/2 terms. If we forget about rounding errors (caused by our approximation), the initial position of n (i.e., an) should be

1r.

=

( 7T/4)n2 ;

� = N/j.

Comparing

(7T/4)n2 .

To make this proof rigorous, we have to estimate round­ ing errors in both computations (for some intermediate value ofJ). It is rather long (and boring), so I omit this es­ timate. ·

n.

Acknowledgment: This problem was suggested by I. Akulich of Belarus.

MOVING?

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VOLUME 23, NUMBER 4, 2001

65

K. S. SARKARIA

A To po l og i cal Parad ox of M oti o n

basic premise of the mechanics of continuous matter, and one invariably found stated in some form in all books of hydrodynamics-often even before the equations of motion are derived-is the hypothesis of continuity, that is, the doctrine, going back to Anaxagoras, that matter and motion are continuous. This is interpreted mathematically as implying at least that, at each time t, there is a "particle" (of course hypo­ thetical; we are ignoring the actual molecular nature of mat­ ter and are talking say of the spermata of antiquity!) at . every point of a region R1 of 3-dimensional space IR3 , and that, following the motion of these particles, one gets con­ tinuous surjections m5•1:R1 -4 R8, t < s, varying continu­ ously with t and s, and obeying mu,s0ms,t = mu,t· Though apparently quite reasonable, this implies some funny things, including the following:

One cannot completely empty a tyre-tube filled with water into a bucket in any finite length of time.

66

THE MATHEMATICAL INTELLIGENCER © 2001 SPRINGER-VERLAG NEW YORK

For this would mean we could continuously deform any homotopically nontrivial loop C1, of the space X c IR3 oc­ cupied by the apparatus of Figure 1, within X, to a trivial loop Cz of X. Here, the existence of a non-trivial C1 is en­ sured by the fact that X, which is homeomorphic to the re­ gion R0 initially occupied by the fluid, has the homotopy type of a circle, so its fundamental group is 7L, and the triviality of C2 follows because the bottom part of X is contractible. The above notwithstanding, it is customary in all books of hydrodynamics to assume even more: that there is al­ ways a unique fluid particle at each point of R8, thus it is understood that motion occurs via homeomorphisms ms,t· So even the topological type of R1 cannot change with time.

-

-

---

Figure 1

This implies, conversely, that a bucketful of water cannot be transferred completely into a tyre-tube of the same vol­ ume in any finite length of time. I emphasize that the above is not a refutation of the sci­ ence of hydrodynamics but a vivid reminder that the boundaries of its domain of applicability are encountered even in simple everyday situations. The topological con­ tradiction should alert us to the fact that there is something amiss in the mathematical model used, as indeed there is: As soon as local forces in excess of the cohesive limits of the fluid appear near the upper inner portion of Figure 1, the hypothesis of continuity is inapplicable to the flow in that region. The doctrine of Anaxagoras was very much in keeping with the spirit of his time. After the resolution of the Pythagorean conundrum by means of irrationalities, phys­ ical space was almost universally regarded as continuous; then, to resolve the well-known paradox of Zeno regarding Achilles and the tortoise, it became necessary to give up the notion of finitely many moments between any two events, and time, too, came to be regarded as continuous. However, Democritus, a contemporary of Anaxagoras, was of the view that matter, unlike space, is discrete. Four centuries later, it was this atomic hypothesis which was championed by the Roman poet Lucretius, who claimed­ see [4], p. 14--that motion would become impossible if we were to believe with Anaxagoras that all of space is full of matter: There's place intangible, a void and room. For were it not, things could in nowise move; Since body's property to block and check Would work on all and at all times the same. Thus naught could evermore push forth and go. Since naught elsewhere would yield a starting place.

Here Lucretius seems to overlook the possibility of ro­ tational motion, i.e., of vortices, which (much later) became all the rage with Rene Descartes, and briefly again in the nineteenth century when Lord Kelvin (William Thompson) made a beautiful attempt to understand atoms via vortices. For more on this, the reader can probably do no better than start with James Clerk Maxwell [5]. Ever since John Dalton and Robert Brown there has been abundant microscopic evidence which favours the atomic hypothesis. Nonetheless, it is contended in all books of fluid mechanics that, for macroscopic purposes, one can still safely assume the hypothesis of continuity. As shown above, one has not only microscopic evidence, but a pri­ ori arguments from topology (i.e., the mathematics of con­ tinuity) which show that even a weakened hypothesis of continuity is untenable, so that matter and motion cannot both be assumed continuous. Even a gas, confined to the lower bulbous part of X with the top evacuated, would change its topology after the stop­ cock is opened, which should suggest, independent of any other evidence, that its matter is probably discrete. This, of course, is what the kinetic theory assumes, and the equa­ tions of motion of hydrodynamics are, as is well known, statistical averages of the Boltzmann transport equation­ see, for example, Desloge [3]. But for the case of liquids (as against gases) this approach runs into some unresolved difficulties-see, for example, Batchelor [ 1 ] . So, following Jean-Claude St. Venant and George Stokes, it is convenient to invoke the hypothesis of continuity. Unless the approx­ imate nature of this assumption is emphasized, however, this runs the risk of making the equations of hydrodynam­ ics appear more basic than they possibly can be. We recall that Daniel Bernoulli, Claude Navier, Simeon Poisson, and Augustin-Louis Cauchy, all, had sought to understand hy­ drodynamics starting from various atomic hypotheses. These original attempts need to be perfected, because a natural understanding of turbulence will probably be found only in such statistical foundations.

Matter and motion can not both be assu med continuous . For very small values of time, the flow of Figure 1 does obey the hypothesis of continuity, and the fluid region Rt retains the topology of a solid torus; however, its geome­ try, which depends on the nature of the fluid and the bound­ ary conditions, changes rapidly, with Rt becoming thinner and thinner at the top (and for a creeping flow, say of trea­ cle, it seems to tend towards a well-defmed limiting posi­ tion). But at the moment when the thin Rt breaks, the con­ tinuity hypothesis becomes invalid, and the flow is no longer governed by hydrodynamics. Similarly, in a swift stream going past an obstacle, wa­ ter contained in neighbourhoods of homotopically non-triv­ ial loops and surfaces encircling the obstacle, is probably

VOLUME 23, NUMBER 4, 2001

67

being swept entirely past the obstacle, and so must be

A U T H O R

breaking up topologically. This failure of the hypothesis of continuity, which we suspect is usually over an open sub­ set, implies that some flows cannot be modelled by

any

smooth velocity vector field, not even a generic one hav­ ing all sorts of strange attractors. In particular, Jean Leray has suggested the Navier-Stokes equation is probably in­ adequate for modelling turbulence. Sometimes matter is assumed to be continuous, but its velocity field is allowed to be discontinuous. An analysis of some such arguments is given in Birkhoffs classic

drodynamics [2).

Hy­

For example, in aerofoil theory, one gets

around the D'Alembert paradox by guessing a suitable flow

K. S. SARKARIA

topology: wake, dividing stream line, etc. Despite their suc­

Department of Mathematics

cesses, such

ad hoc

devices can obviously not be deemed

Panjab University

to be physical explanations of these phenomena.

Chandigarh 1 600 1 4

I observe next that the "opposing doctrines of the

India

plenum and atom"-as Maxwell [5) calls them-can actu­

e-mail: [email protected]

ally be reconciled with one another, if one believes that

all physical no­

Karanbir Singh Sarkaria received his Ph.D. degree in 1 974

tions are discrete. From this viewpoint, which is roughly

from SUNY Stony Brook (USA). He has been a Professor of

space is discrete, and more generally, that like that of Gottfried Leibniz's

Monadology (1714),

Zeno's

Mathematics at Panjab University since 1 989. Among his vis­

paradox arose only because physical space was confused

its elsewhere he has especially fond memories of IHES, Bu­

with geometrical space, and, to cover this initial "lie," it be­

res-sur-Yvette, France. His research is mostly on the interface

came necessary to invent more; for example, that time is

between topology and combinatorics, but other questions

monads are not

"in" anything­

continue to attract him, including the reading and rereading of

there being no empty space or vacuum-they are by them­

the works of Poincare. Outside mathematics, he is especially

selves, forming a discrete plenum! If one wants to proscribe

devoted to long bicycle tri ps on his trusty Raleigh.

continuous. The discrete

action at a distance, more structure is needed; for exam­ ple, one may postulate that each monad acts only via some others that are contiguous with it. It is this extra structure, true, or mistakenly imposed by us on reality, which makes it appear continuous: contiguity gives us a simplicial com­

property was however dictated more by his teleological

plex, thus a continuous space. In this view, physical motion

predilections and earlier work on ethics, parodied memo­

is only a sequence of monadic permutations, not an arbitrary

rably as the absurd Dr. Pangloss of Voltaire's

Candide.

flow on this geometrical space, and may be bound by some

quantization

rules-say, limitation to those locally irrota­

tional flows whose circulations are integral multiples of Planck's constant.

In other words, we are but "hearing" some

discrete aspects of the topology-integral homology--Dftiny portions of this monadic simplicial complex, via the quan­ tum-mechanical observables of microsystems! I conclude by recalling that, even more than continuity, the essential doctrine of Anaxagoras was

homoeomaria,

that is, that a part is like the whole, or, as someone raised

REFERENCES

[ 1 ] G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge Uni­ versity Press, Cambridge (1 970). [2] G. Birkhoff, Hydrodynamics: A Study in Logic, Fact, and Similitude, Princeton University Press, Princeton (1 960). [3] E. A. Desloge, Statistical Physics, Holt, Rinehart and Winston, New York ( 1 966). [4] Lucretius, translator W. E. Leonard, Of the Nature of Things

=

De

Rerum Natura, 58 B.C., Dutton, New York (1 950).

on fractal graphics would now put it, self-similarity. In

[5] J. C. Maxwell, Molecules, and Atom , in, "The Scientific Papers of

analogy with this, Leibniz required that each monad of his

James Clerk Maxwell", Vol. I I , 1 890, pp. 361 -378, and 445-484,

discrete plenum be a replica of the entire universe! This

Dover re print , New York (1 952).

68

THE MATHEMATICAL INTELLIGENCER

I a§!H§',Ifj

c.let W i m p , E d i t o r

]

Calculus

by

Y

Fang and

Y

Wang

NEW YORK SPRINGER-VERLAG. 2000 808 PP. HARDCOVER US $59.95, ISBN 981 -3083-52-2

REVIEWED BY RALPH A. RAIMI

Feel like writing a review for The Mathematical Intelligencer? You are welcome to submit an unsolicited review of a book of your choice; or, if you would welcome being assigned a book to review, please write us, telling us your expertise and your predilections.

Column Editor's address: Department of Mathematics, Drexel University, Philadelphia, PA 1 9 1 04 USA.

he authors live in Taiwan and mainland China, respectively, and state in the Preface that this book has been used for twenty years, in a two­ semester course for science students in both countries. Though Answers to Selected Exercises fill pages 703-792 at the end of the book, the Preface also promises a Solutions Manual to be printed separately at some future time, making the book more suitable for self­ study. No mention is made of the trans­ lator. At first glance I found the book at­ tractive. It is not as large and heavy as has become so common in recent years; it is printed entirely in black­ and-white with no irrelevant pic­ tures-indeed no pictures at all save for mathematical diagrams of exactly the sort we all draw on blackboards (or wish we could draw that well)-and it has an open, uncrowded look. The headings are informative, the back-ref­ erences are easy, and the occasional important formula or theorem is set off by enclosure in a shaded box. The line-up of chapters is entirely traditional, too, resembling the list of topics familiar fifty years ago; they are titled: Introduction, Limits and Conti­ nuity, Differentiation, Applications of Derivatives, Integration, Some Special Functions, Formal Integration, Numer­ ical Integration, More on Limits and Improper Integrals, Infinite Series, Po­ lar Coordinates, Differential Calculus for Functions of Several Variables, Multiple Integrals. It is the Introduction that contains the necessary back­ ground concerning the real numbers and cartesian coordinate system, and the "Special Functions" chapter con-

T

tains nothing very special: only the usual discussion of inverse functions, followed by details about the calculus of inverse trigonometric and exponen­ tial and logarithmic functions, the al­ gebraic and trigonometric functions being presumed by the authors to be already known to the reader from ear­ lier study. Yet, this is China! Recent interna­ tional comparisons of mathematical competence among school children at the 4th, 8th, and 12th grade levels have shown Americans scoring rather badly compared to the technologically ad­ vanced countries of the Far East. Com­ parisons of math curricula of Japan (say) and Singapore with what we have here have added fuel to the long-stand­ ing, furious debates in the "math edu­ cation" circles of this country. A recent book much talked of, Liping Ma's

Knowing and Teaching Elementary Mathematics, compares the knowl­ edge and strategy of Chinese elemen­ tary school mathematics teachers with our own, much to our disfavor, and has itself practically become a textbook on how things should be done. In conse­ quence, as when our notions of the booming Japanese economy of the 1980s made us all imagine the Japan­ ese to be ten feet tall, economically speaking, today the book of Liping Ma has caused us all to look to China, too, for the corresponding lessons in math­ ematics education. In particular, if Chi­ nese children so much outscore our own in earlier grades, one would ex­ pect a Chinese calculus book to show signs of the superior audience it would have been sure to have, in its twenty years of use there. Alas, it shows no such thing. Worse, it even exhibits what seems to me a se­ rious lack of mathematical insight on the part of the authors. To characterize the book in brief, it is a traditional calculus book of the type pioneered by the fa­ mous books of Thomas and of Sher­ wood and Taylor (though not as well done), overlaid with a certain amount of

© 2001 SPRINGER-VERLAG NEW YORK, VOLUME 23, NUMBER 4, 2001

69

epsilontics of the "New Math" persua­ sion of the 1960s.

Immediately thereafter appears

In neither dimension

Theorem 5.2.8: If f is a continuous function on [a,b], then there exists

is it entirely consistent, and it suffers not

2. 4. 2 Extreme Value or Min-Max

a unique number I such that L,{P) s,

only from the uneasy alternation of

Theorem for Continuous Functions:

I s,

points of view, but from quite novel-to

Ifj: [a,b] -'> R is a continuous func­

me---rr -€ ors of judgment.

tion, then there exist x1,x2 in

Consider the question of "rigor," which in calculus means the problem

[a,b]

Ur(P) for all partitions P of [a,b]. Proof: We lmow that, for all

partitions of P,

such that f(xi ) s, j(x) s, j(x2), for every x in [a,b]. so that the existence of I is guaran­

of how to characterize and use, in con­ nection with derivatives and integrals,

A diagram illustrates this theorem,

the real number system as we have

but it unfortunately exhibits a graph for

teed . . . .

come to lmow it since the time of

whichf(a) happens to be the minimum

Now, there is a bit of slippage here.

Dedekind. Chapter 2 of this book, 59

andf(b) the maximum-though at least

Nowhere prior to these lines have L

pages, is devoted to limits and conti­

the pictured function is not monotone.

and U been mentioned or defmed; they

nuity, with the most careful possible

For "proof," the authors state, "The

are presumably the

definitions: limits of functions and of

proof is immediate from Theorem 2.4. 1

sets of lower and upper sums, but in

sequences, too; right-hand limits, lim­

and is therefore left as an exercise."

its at infinity, infinite limits. Theorems,

lub

and

glb

of the

keeping with the resolutely rigorous

Indeed, the proof is asked for as Ex­

tone of the proof so far, one would

too, limit of sums and products, etc.,

ercise 5 at the end of the section, but

have expected their existence to have

the "squeeze" theorem when a function

it is so trivial that the average student

a proof, or at least a reference to "ad­

the lim­

would hardly lmow what to answer. It

vanced calculus." The missing proper­

its of sin(x)/x and of polynomials. All

is bad policy to ask for a formal proof

ties, and indeed missing

of this can be and is done without ref­

of the obvious. The question arises,

these two numbers, argue that this

erence to the completeness property of

why do the authors isolate 2.4.2 in this

proof has been cut down from some

R, though irrationals are mentioned in

manner, and not merely note that the

the example of an everywhere discon­

points that map into

are the

pleteness property of R, and rather carelessly, as may also be seen in the

f is pinched between

g

and

h,

c

and

d

mention

of

original version, one invoking the com­

tinuous function; but the proofs of

points desired? Apparently 2.4.2 is iso­

such things as the "limit of a product"

lated because it will be used

are rigorous and correct, filled with ep­

form at some future place, and indeed

guaranteed," which is badly stated. What is guaranteed by the inequalities,

in that

clause "so that the existence of

I

is

silons and deltas, with instructive nu­

it does appear so in the proof of Rolle's

merical examples to show how delta

theorem; but one would also expect it

even had they been explained and

could be determined in simple cases.

to appear in some form when uniform

proved, is the existence of

continuity is proved true for functions

ber having the inequality property an­

After about fifty pages of excellent

some num­

exposition, example, and exercises,

(p. 93):

continuous on [ a,b], in preparation for

nounced, but not necessarily the

the authors suddenly announce

the proof that every continuous func­

the announcement of the theorem,

2.4. 1 Fundamental Theorem on Con­

The mathematician now looks forward

a small point, but inaccuracies of ex­

tinuous Real Functions: If.f[a,b] -'>

to this familiar sequence of theorems,

pression of this sort abound, some of

R is a continuous function, then

but what does he fmd?

f([a,b])

=

[c,d] for some suitable c,d

in R,

tion on

[ a,b]

has a Riemann integral.

Uniform continuity is indeed care­ fully defined, and examples given, too,

"I" in

which was by definition unique. This is

them apparently the result of imper­ fectly idiomatic English in the transla­ tion.

at the end of Chapter 2, but the proof

As for the rest of the proof, the uni­

of the theorem, that a function contin­

form continuity of f is duly and cor­

fact (as simple as it may have been

uous on [ a,b] is uniformly continuous

rectly used, in a proof bristling with no­

seen) requires detailed lmowledge of a

there, while stated to be "of great im­

tation at least as difficult to follow for

complete­

portance" (for what? We have to wait),

a student new to the calculus as any

and we shall not give any proof

is not given. It "can be found in a stan­

proof of the uniform continuity prop­

here. A proof of this theorem can be

dard advanced calculus text," the au­

erty itself would have been.

found in any standard advanced calcu­

thors say.

with the explanation, "The proof of this

real number system called

ness

lus text." (The

Two chapters later, uniform conti­ awkward

What is the point of all this partial rigor? If the proofs of the fundamental

construction

nuity never having been mentioned

properties of functions continuous on

"knowledge of a real number system

again, the Riemann integral is intro­

R are to be deferred to a later study,

. . . " is here quoted verbatim, i.e., "[sic]", and is typical of the infelicities

duced. It is as carefully defined as in

why not defer a few others and save a

any analysis course, in terms of the

couple hundred pages in the process?

in translation that occur in this book

lower and upper Riemann sums Lj(P)

The idea of continuity is well ex­

Every mathematician will recognize

and Uj(P) associated with the function

pressed in this book, and of limit, and

what is meant, but students might not.)

f and a partition P. Does it exist?

of integral and of derivative; and if the

70

THE MATHEMATICAL INTELUGENCER

assertions of existence for the integral

These authors have also not solved

foundations of the subject, which in

were taken as intuitive, as they were in

the problem of making sense of differ­

the long run must be subordinated to

the classic calculus textbooks of yore,

entials. On the one hand, d is defined

their usefulness.

students would be philosophically as

as

well off as they are with this sort of

proximation, with pictures. Then

halfway analysis course. For American

151), "Here, we remark that in this

f'(x)b.x

f

and is regarded as an ap­

(p.

In the derivation of the formula for

y = f(x) on [a,b], the authors choose a partition P = {xo, Xr, . . . , Xn ) and, carefully and arc length of the graph of

students, this book would be mystify­

definition b.x can be any nonzero value.

ing both to the student who cannot fol­

However, in most applications of dif­

correctly employing the mean-value

ferentials, we choose

theorem on each interval of the parti­

low statements containing quantifiers, epsilons, and implications, and to the

dx = b.x. we also write dy = f'(x)dx."

Thus,

tion, arrive at

student who can. Again and again in

This particular mystery is then com­

the course of this book, students who

pounded when it comes to changes of

are able to follow such rigor as is pre­

variables in an integral, where

"dx" sud­

sented find themselves rebuffed when

denly, from having earlier been a part of

it comes to a knotty point where that

the mere notation for "integral," now

earlier rigor could prove of actual use.

turns, via

Nor is it only in matters of real vari­

x being replaced by g(t) in its

other appearances under the integral

g'(t)dt.

able theory that the authors disappoint

sign, into an unfathomable

the reader (or the student) who wants

replacement story begins when "inte­

why. There is a chapter on ap­

gration by parts" is introduced, and

to know

proximate integration, with error for­

such things as dg are easily understood as shorthand for understandable frag­

Simpson's rule and the Trapezoidal

ments of the formulation, but then they

rule, but no hint is given as to why

take on a life of their own when sub-

even plausible, and the reader is re­ ferred to the ubiquitous "advanced cal­ culus" for their proof. Even so, the authors miss an amusing sidelight: Be­ cause the error formula for Simpson's rule, whatever its source, invokes only the fourth derivative off, which is zero for cubic functions, it follows that Simpson's rule gives the exact integral for cubics as well as quadratics.

In Chapter 1 the Peano induction ax­

Students who are able to follow rigor fin d themselves rebuffed where that rigor could prove of use.

P(n)

I

i�l

v + (f'(ti))2b.xi.

1

But then they write: "Now, one imme­ diately realizes that Thus lim

,1PI--. o

n � oo as I� � 0.

i=n

L

= lim =

I Yl + (f'(ti))2b.xi

i� 1

t Yl + (f'(x))2 dx," a

as if we somehow needed to convert the limiting process to something re­ sembling the sum of a series before an­ nouncing the result to be an integral. To be sure, there was an earlier theo­ rem stating that for continuous func­ tions the Riemann integral can be ap­ proached by a sequence of Riemann sums of the special sort obtained by a partition into

n

equal pieces, but the

"partial sum" in the formulation here does not quite refer back to that theo­ rem, so that both here and elsewhere there appears to be neither philosoph­

iom is given carefully enough, but then stitutions are casually introduced in

ical nor practical value in having in­

is

other contexts. The authors of the

troduced the entire Darboux sum for­ mulation earlier.

the process of "mathematical induction" for propositions of the form

i=n

The

mulas of some complexity for both

these error estimates should be true, or

L=

"proved," not by translating Peano's sets

present book take such substitutions

into corresponding propositional impli­

(and their inverse procedures) in eval­

Lack of coherence between rigor

cations, but by a sudden invocation of

uating integrals as obvious, using them

and application, or example, while it

the well-ordering property of N, not oth­

unceremoniously, never at all attempt­

takes space to describe even a minor

erwise or earlier mentioned. That is,

ing to state or prove a substitution for­

example or two, is far from the only

there is in the logic of this section no ap­

mula for either indefinite or definite

failure of this book for the sort of cal­

all.

integrals. The lack of a substitution

culus course a beginning science stu­

The proof of the composite function

formula for definite integrals is a theo­

dent should have today. One might well

rule for derivatives exhibits the lacuna

retical lack, for that matter, since its

say, "Let us have a lean and lively cal­

parent need for the Peano axiom at

familiar to most of us when the Leibniz

higher-dimensional analogues, under­

culus, ignoring the deltas and Darboux

formulation D.zlb.x

stood and used by 19th-century math­

sums, even the proof of the mean-value

=

(D.zl�y)(�ylb.x) is

casually invoked in a limiting process

ematicians who knew nothing of ep­

theorem and the ratio test; let us take

without attention to the possibility that

silons and compactness, have such

as intuitive the existence of maxima

�y

fundamental importance in geometry

and minima, with associated derivative

might be zero along the way. (The

An introductory calculus

tests, and the additive properties of in­

book ought to take account of what

tegrals, as was done in 1940 calculus

uses limit theorems in a careful way,

wonders the subject can achieve even

books. Perhaps it is better to challenge

but the error is the same.)

before modern developments in the

the beginning student with problems

notation in the text is much more so­ phisticated than this, and

apparently

and science.

VOLUME 23, NUMBER 4, 2001

71

and applications, leaving the technical

almost any book, given good exercises,

logical structures aside for a while."

but in the past fifty years we have seen

Whatever there is to be said for this

many superb calculus textbooks, fitted student populations

S

cientists who frequent the late­

night talk shows say that those

who intentionally put themselves in

stance, suppression of the rigor in this

to

of variously

harm's way have a gene that causes

particular book, were it possible in

imagined skill and interests; and the

them to do so. They call it the "risk­

problems

some sort of rewriting, would still

number of fine calculus

leave an insufficient remainder for a

there is so enormous that new ones need

out

and

calculus course. It is true there are

hardly be looked for and can hardly find

possess much the same gene. Myself, I

taking gene. " People who seek mystery mind-bending experiences may

many good problems of the usual sort,

use.

In my time I have used, or known

like safe, well-structured experiences.

enough to serve any course covering

quite well, books by Apostol, by Bers, by

My introduction to Shalosh B. Ekhad

the same main topics. Some of them

Gillman and McDowell, by Johnson and

was seriously unsettling.

even illuminate the intuitive content of

Kiokemeister, by Sherman

some of the more arcane theory, of­

Swokowski,

Stein, by

It began three years ago when I was

all the way back to Ran­

attending a conference on Extrapola­

tentimes better than the incomplete

dolph and Kac, Sherwood and Taylor,

tion and Orthogonal Polynomials (as

theoretical developments themselves.

and Thomas. They all work, one way or

someone said, it's wonderful that the

But:

another, and though they all have some­

field has grown to the extent that we

thing to object to as well, they do main­

can entertain each other with confer­

The opening chapter reviews "ana­ lytic geometry" only so far as getting

tain a consistency of theoretical tone

ences; up to that time, you just had to

equations for straight lines and circles,

that makes for a coherent course of

sit at a desk). We were at a conference center in Luminy, about

10 km from

and proving the slope criterion for per­

study. (Probably others are even better,

pendicularity. No simultaneous equa­

or more suited to particular audiences,

Marseilles. It was a beautiful sunny af­

tions, no conics, no rigid motions. Not

but I simply don't have experience

ternoon (an event for which this con­

is the un­

ference always adjourns), and having

until page

530, in the chapter explain­

teaching with them. One such

ing polar coordinates, is there a sec­

justly neglected yellow book of Ralph

nothing to do, I decided to consult the

tion, "Parametric Paths and Lengths,"

Palmer Agnew (1962), containing a thou­

library at the conference center, which was known to house not only early

in which students are introduced to pa­

sand interesting sidelights; another is

rametrized curves as a sort of after­

the classic Courant.) On the whole,

manuscripts of Islamic thinkers like

thought, in the course of obtaining the

therefore, I do not see any reason to add

Averroes and al-Kindi (whom, though of­

formula for the length of the polar

this

Calculus to the list of those a text­

ten accused of being corrupted by Hel­

graph r

book-choosing committee should spend

lenism, I consider a seminal thinker) ,

=

f(8). My own experience has

been that even students well-exercised

time considering. Whatever magic there

but also a collection of books of a

in parametric equations in an earlier

is in Chinese mathematical education is

darker, hermetic character.

course still have trouble using () as pa­

simply not evident here.

rameter in this context. Furthermore,

As an aftermath of discussion in the introduction of a book of poetry I had co-edited, 1 I was researching

it is pointless, unless you really need to

Department of Mathematics

know the arc length for a cardioid.

University of Rochester

game­ trya, the Cabalistic tradition of assign­

The absence of a systematic discus­

Rochester, NY 1 4627

ing to each integer magical properties.

sion of parametrization leads to a total

USA

That tradition, which continued un­

absence of the differential geometry of

e-mail: [email protected]

abated until the late middle ages, is ear­

curves, even in the plane, where surely

Web page

a physics course would expect these

lier than the Cabala, going back even

http://www . math. rochester.edu/u/rarm

to the Pythagoreans. We find Dante

students to have learned something

venerating the integer

about vector velocity and acceleration.

square root is

There are no line integrals, either.

In the

latter part of the book, where double in­ tegrals

and

treated, there

partial

derivatives

are

is only the sketchiest in­

A = B

mystery of the holy trinity. (Why, I of­

by Marko Petkovsek, Herbert S. Wiif, and Doron Zeilberger

The double integral over a region de­ fined by curves in polar coordinates is

A. K. PETERS, LTD . , 1 996 .

is taken in the whole book

I have seen worse calculus books in one way or another, and a good teacher can make a good calculus course out of 'Against Infinity, Primary Press, Parker, PA (1 983).

72

THE MATHEMATICAL INTELLIGENCER

one-third root is three?)

27,

I was

interested in tracing Islamic variants of lucky to have access to such a library.

US $29.00: ISBN 1 56881 0636

Looking over a very early edition of

Tahafut at-tahafut,

REVIEWED BY JET WIMP

as far as the subject of change of vari­ ables

ten wonder, didn't Dante venerate whose

the tradition, and I considered myself

troduction of vectors, with dot products but no cross products or determinants.

9 because its 3, which suggests the

Averroes's monu­

mental treatise, I was shocked to dis­ God does not exist. Not only that,

cern a marginal notation in English

have you ever tried to get a plumber

clearly made by a much later, even con­

in Manhattan on a Sunday?

temporary hand, perhaps with a ball­

-Woody Allen

point pen.

"1" is the giver of shapes. Programo ergo sum. -Shalosh B. Ekhad I started to sweat, whether out of out­ rage at someone's defacing that august book, or because I felt a Mystery im­ pinging on my life. 1 clearly meant the integer one, and the Latin seemed corrupt. I should have left it at that, walked out of the library, packed my clothes, taken the TGV back to Paris. But I was hooked. I paged through the manu­ script, searching for more marginalia. I found another note, again in Latin. Here is the English translation of what was written: "

"

To serve my master faithfully, I will go there, salute the Prince . . . but then there was an erasure of one word, incompletely done, so I'm pretty sure the word was "aeris" ("of the air"). There followed a puzzling Greek ad­ junction, LAo.;; ADEA¢o.;;, and again that ominous signature, Shalosh B. Ekhad. I returned to the United States in a daze. That we live in the age of infor­ mation is a curse as much as a bless­ ing; I love curiosity in my cats, but here I was beset with too much of it. I sat down at my desk, invoked Netscape, and typed in the signature I had un­ covered, Shalosh B. Ekhad. I got 95 hits. If you doubt me, try it yourself. I felt like a naive protagonist in a movie thriller, uncovering a conspiracy of such appalling dimensions that it in­ cludes his mother and his cocker spaniel. I selected one of the matches, No Title. Here is what was returned: Shalosh B. Ekhad Dept. of Math., Temple University Phila. PA 19122 Plan: to serve my master faithfully. Programo ergo sum Doran Zeilberger. I hit the matches one after another: bibliographies containing Ekhad's pub­ lications, a purported proof of the Rie­ mann hypothesis, things about Gasper-

able sums. "Have a look at my pretty face," one document tantalized, dis­ playing a photograph of a desk com­ puter. And always there was a remark­ able coupling of Ekhad's name with that of someone I almost knew: Doran Zeilberger. There was the ominous website,http://www.math.temple.edu/ �zeilberg/pj.html with its invocation of things such as "A heterosexual Mehler's formula for the straight Hermite poly­ nomials." Doran Zeilberger. I had heard of him. We worked in similar areas, com­ binatorics spilling over into special functions and asymptotics; special functions and asymptotics spilling over into combinatorics. He had taught at Drexel University, my institution, but I had never met him. He taught in the evening school. I remember working late one night and catching a brief glimpse of him as he rounded a corner, back-pack protruding above sandals and bare legs, rushing from 30th Street Station to catch his 4:30 a.m. class. Among the Netscape hits was much mention of a book whose annoying ti­ tle was "A = B." How silly, I thought. There obviously were only two possi­ bilities: either A does not equal B, in which case the title should be "A =1=- B," or A does equal B, in which case the ti­ tle should be "A = A" or even "B = B," or maybe just "A" or "B." The authors of the book were Doran Zeilberger, Herbert Wilf, and Mark Petkovsek. I know Herbert Wilf. He teaches at the University of Pennsylvania, just across the street from me, and we keep col­ liding at those conferences we share an interest in. He is tall, rather saturnine in appearance, and has a voice so low it occasionally drifts below hearing range. I had always found him cordial, but once, I confided to him that I would like to meet Doron Zeilberger, with whom I knew Herbert had often col­ laborated, really meet him, not just watch him vanishing around a corner, maybe even have Doron introduce me to Ekhad, a look of urgent otherness came into Wilfs eyes, as though he had just remembered leaving the oven on at home. About Petkovsek, I know nothing, except that he is an ex-gradu-

ate student at the University of Penn­ sylvania, who has clearly been sucked into the maw of the Ekhad consortium. Last week I sent him an e-mail mes­ sage, warning him he was keeping bad company. Two days later the book "A = B" ar­ rived in the mail. I had not ordered it, and I don't believe in coincidences. However, "A B" confirmed it: Shalosh exists. He/she/it is abundantly referenced in this volume. Further­ more, the book has implications for my own work, and I fastened on these. First there was the crucial idea of a hypergeometric term. =

A sequence s(n) is called a hypergeometric term if the term ratio s(n + 1)/s(n) is a rational func­ tion of n. DEF1NITION:

There, that's simple enough. Some peo­ ple associate the term hypergeometric with a certain kind of series-I'm go­ ing to do this in a second, too--but it's interesting to know that the first use of the term, by Wallace in 1655, was just as the defmition says. s(n) above can be complex and n can range over the positive and negative integers or just the positive integers: it doesn't matter. DEF1NITION:

A term depending on two

F(n, k), is called doubly hypergeometric if both variables,

F(n + 1, k)/F(n, k), F(n, k + 1)/F(n, k) are rational functions of n and k. In case the expression

this

I F(n, k) k

is called a hypergeometric

sum. 2

DEF1NITION: An expression for a func­ tion of n is said to be in closedjorm if it is a linear combination of a fixed, finite number of hypergeo­ metric terms.

These definitions are to give a back­ ground for illustrating some of the ideas in "A = B." The authors start out with a chapter on proof machines, and the following curious passage occurs:

2Sometimes the authors require that the sum terminate for each n .

VOLUME 23. NUMBER 4, 2001

73

People have always perceived and savored relations be­ tween natural phenomena. First these relations were qualitative, but many of them sooner or later became quantitative. Most (but not all) of these relations turned out to be identities, that is, statements whose format is A B, where A is one quantity, B is another quantity, and the surprising fact is that they are really the same.

important than mine, but hypergeometric sums do occur in many branches of mathematics. I'll justify this remark later. If s(n) is a hypergeometric sequence, then a little re­ flection shows that it may be written

=

All the writing in the book is like this: plain, unadorned, chaste. For these authors, no rewards accrue from obfus­ cation. To illustrate the idea, the authors give a ton of for­ mulas, and I'll give another:

I (n)2 = (2n)

k�o

k

n

(�)

In this equation, F(n, k) is doubly hypergeometric (easy to verify), the expression on the left is a hypergeometric sum, and the right-hand side constitutes a closed-form expression for the sum. The authors talk about "when we can find algorithms for deciding the truth of formulas and when can't we." They discuss briefly Godel undecidability, solution of Diophan­ tine equations (Hilbert's tenth problem), and Richardson's theorem-no proofs, just casual chatter. Some conserva­ tives were elated when it was discovered that there is no general algorithm for solving Diophantine equations: it yanked the jewel of mathematical problem-solving out of the heretical paws of the computer scientists. The authors quote David Bressoud: =

The existence of the computer is giving impetus to the discovery of algorithms that generate proofs. I can still hear the echoes of the collective sigh of relief that greeted the announcement in 1970 that there is no gen­ eral algorithm to test for integer solutions to polynomial Diophantine equations. . . . Yet, as I look at my own field, I see that creating algorithms that generate proofs con­ stitutes some of the most important mathematics being done. The all-purpose proof machine may be dead, but tightly targeted machines are thriving. The authors talk about techniques for proving identities for simple classes of functions: polynomials, trigonometric, symmetric sum, elliptic functions. Chapter 2 is labeled "Tightening the Target." The authors declare their mission: The problem of discovering whether or not a given hy­ pergeometric sum is expressible in simple "closed" form, and if so, finding that form, and if not, proving that it is not. Why this obsession with hypergeometric sums? Aren't others equally important? Well, no one wants to get into a normative squabble about whether your identities are more

74

THE MATHEMATICAL INTELLIGENCER

s(n)

=

K

q

n! IT (bj)n

,

j� i

where

(a)n is Pochhammer's symbol, (a)k =

bj =!= 0,

{a(a +

- 1, -2,

1) 1

.

'

. . . (a + k k=O . . ' j = 1, 2,

1), k

> 0; } ,

'

. . . ' q

And this representation of s(n) gives rise naturally to the generalized hypergeometric function (the authors of "A = B" more usually call this a hypergeometric sum) with p numerator parameters and q denominator parameters:

Hypergeometric series with p = 2, q 1 (though they were not called that) were studied by Wallace, Newton ( - 1664) and Stirling (1 730) in connection with the rectifica­ tion of certain algebraic curves. Euler in 1778 discussed the general series of this type, though again without using the term hypergeometric. Gauss thoroughly investigated this se­ ries in a large number of published and unpublished works, beginning in 1805. Today the corresponding function is called Gauss's hypergeometric function, or simply the hypergeo­ metric function, though it was in only relatively recent times that Kununer (1836) applied the term hypergeometric to Gauss's series. Pochhammer (1890) and Barnes (1907) de­ veloped the notation for the general function. The InteUi­ gencer [vol. 7, no. 2, 1985] contains a nice survey article by W. K. BUhler on the history of hypergeometric functions. There are four cases: =

(i) If one of the a1 is a negative integer or zero, the series always makes sense, for it terminates. The result is a polynomial in z. Failing (i), then (ii) If q 2: p, the series converges uniformly on compact subsets to an entire function of z (of finite order q + 1 - p); (iii) If p = q + 1, the series converges and thus defines an analytic function of z on compact subsets of lzl < 1. The function may be analytically continued into the complex plane cut along [ 1 , oo]. (iv) If p > q + 1, the series diverges for all z =!= 0. However, the series may still be computationally very useful, for instance, as a member of an umbral calculus of formal power series about 0. As

an example, the series of the squares of the binomial z1��k : coefficients is such a sum, since � = ( - 1) k c

()

()

� n2

k�O

k

=

lFl

( -n -n ) ,

1

'

; 1 .

Similarly, sums of all integer powers of the binomial coef­ ficients are hypergeometric. Another apparently frivolous example is the situation where a drunkard is climbing around in 3-space, occupying unit lattice points. (Maybe the bar is adjacent to a playground that has an enormous jungle gym installed on it.) The drunk­ ard starts out at the origin (the bar) and makes consecutively any one of the six movements (:± 1, 0, 0) (0, ± 1, 0), (0, 0, :± 1) with equal probability. If an denotes the number of ways of going from the origin back to the origin in 2n steps, then

an

4n (112)n

=

"' :J-L. 2 -

n.'

( -n, -, n, 1 1

)

112 . 4 , .

X

I

n �o

6�':,

=

1.51638 . . . .

Thus u = .34053 . . . . It is known that if the drunkard moves in 1-space or 2-space, the probability of return to origin is 1. I sometimes get snickers when I mention in a talk that the difference of the 3-space behavior from the lower-di­ mensional behavior is the basis for all life on earth, but it's true!3 The constant .34053 . . . is called P6lya's constant. P6lya investigated the situation in 1921. Another example comes from numerical analysis and ap­ proximation theory, and the arcane topic of Pade approx­ imants. The Legendre polynomials Pn (x) satisfy the recurrence formula

(n + Po

=

1)Pn + 1 1,

PI

=

=

(2n + 2x

-

1)(2x - 1)Pn - npn - 1 , n = 1, 2, 3, . . . , 1.

Obviously, Pn(X) is a polynomial in the variable x o f exact degree n. It can be shown that the Legendre polynomials are an orthogonal set on the interval [0, 1 ] , i.e., that

f

Pn(X)Pm(X)dx

=

m,n = 0, 1,2,

hn =F 0,

8m,nhn,

. . . .

This orthogonality formula has many consequences, one of which I will explore. We defme a set of functions, p�(x), by the formula

p�(x)

=

1

1 Pn(X) - Pn(t)

0

X-

t

dt, n

=

0, 1, 2, . . .

=

p�(x)

Pn(X)

=

=

(Note the above hypergeometric series terminates.) The probability of return to the origin is u = (m - 1)/m, where

m=

If we write the numerator out as a series of differences of powers xi - ti, j = 0, 1, 2, . . . , n, then the denominator divides evenly into each term, and we see that p�(x) is a polynomial in x of exact degree n - 1 . It is called the as­ sociated Legendre polynomial. These polynomials, like the Legendre polynomials, have been studied by dozens of mathematicians. It is easy to verify that p�(x) satisfies the same recurrence as Pn(x), except we start with the initial values Po = 0, Pi 2. Now divide each term in the above equation by Pn (x) and do a little rewriting to get

En (X)

=

1 X-

1o

1 --

t

dt + En(X),

- In (1 - 1/x)

+ En(X),

1 Pn(X)

t

--=-!.___

1 Pn(t)

0

X-

dt.

The rational approximation p�(x)/pn(x) when developed in powers of 1/x agrees with the series for -ln(1 - 1/x) up to powers of order 2n + 1, and it can be shown that the ap­ proximants p�(x)lpn(X) converge to the function -ln(l 1/x) as n � oo uniformly on compact subsets of C - [0, 1 ] . Simple formulas for the Legendre polynomials are known; e.g., Pn(X)

=

( - l)n zF1

( -n n + ) '1

1

;X ·

It would be nice to have a simple formula for the asso­ ciated polynomials, p�(x). At the time, none was known. The story of p�(x) will resume later. The book "A = B" discusses five computer algorithms for analyzing hypergeometric sums. All of these algorithms are downloadable from the web page and require variously Mathematica or Maple. The book is very didactically ori­ ented, and contains a plethora of beautiful and challenging exercises. I had to tear myself away from working them to write this article. I will describe three of the algorithms:

Sister Celine's general algorithm: Suppose we have the sum

f(n)

=

I F(n, k) k

where F(n, k) is doubly hypergeometric. Sister Celine's al­ gorithm4 provides a method for finding a recurrence for the sum f(n). It does this by first finding a double recur­ rence in n and k for F and then summing this recurrence over k. Herbert Wilf and Doron Zeilberger proved in 1992

31n living organisms, the underlying explanation for many vital biochemical reactions is that the path of an enzyme molecule may be modeled by a drunkard's walk. The walk suddenly becomes constrained, i.e., drops from three to two dimensions, when the enzyme targets a cell surface. The turnover numbers of membrane-bound enzyme systems are enhanced if their substrates undergo two-dimensional diffusion along membrane surfaces. The consequent activation of the enzyme is what allows enzyme systems in living bodies to work; hence life. 4The authors provide a nice biography of Sister Celine (Fasenmeyer), who was born in Crown, Pennsylvania, October 4, 1 906. She received her Ph.D. under the di­ rection of Earl Rainville at the University of Michigan in 1 946, and discussed a specialized version of the algorithm in her thesis.

VOLUME 23, NUMBER 4, 2001

75

that the method will always work, and they give the proof here. The algorithm given by Wilf and Zeilberger is a very deep generalization of the one discussed by Sister Celine herself, and an inspiration to subsequent algorithmic iden­ tity verification. The Maple algorithm for the method is contained in the package EKHAD.5 For instance, when the method is applied to the sums of squares of the binomial coefficients,

RESULT

II: Let Sn be the sum of the first k

+

1 factorials:

Then Sn is not a hypergeometric term (modulo a constant.) RESULT

III:

Let

the recurrence delivered is

f(n)

=

-

2(2n n

1)

f(n -

1).

Iterating this recurrence gives the explicit formula men­ tioned previously, 2nln .

)

(

Gasper's algorithm: Let tk be a hypergeometric term. Gosper's algorithm an­ swers the question: can the indefinite summation of tk be expressed as a hypergeometric term plus a constant? In other words, does there exist a hypergeometric term Zn such that n

Zn

=I

k�O

1)! - 1 .

2 How like the problems o f integrating e->'2 and xex the last two examples are! The software implementing Gosper's algorithm is called GOSPSUM. It requires Mathematica.

Zeilberger's algorithm: This algorithm, sometimes called "the method of creative telescoping, " was proposed by Zeilberger in 1990, 1991; it accomplishes for definite sums what Gosper's algorithm did for indefinite sums. Let

f(n)

n

=

k�

Sn

=

(4k +

2-

1)

k! (Zk

+ 1)! "

n! (2n + 1)!

J(n)

k

= I n O:o;k=s

- (

n_ n - k zk _ n k 2k /3

)

.

The problem came from the American Mathematical Where it came from before that is probably un­ knowable. Zeilberger's algorithm shows thatJt:n) satisfies

Monthly.

+

1)(N -

2)j(n)

=

0.

(I follow the authors in writing N for the shift operator; f(n + 1). Old-timers called it E.) But this is a re­ currence with constant coefficients, and thus can be solved by exponentials, analogously to the differential equation with constant coefficients.

Nf(n)

5The temerity of this creature Ekhad! Naming a computer algorithm after oneself-well,

THE MATHEMATICAL INTELLIGENCER

= I F(n, k)

where F is doubly hypergeometric. The problem, as in Sis­ ter Celine's algorithm, is to determine the recurrence sat­ isfied by f(n). However, the Zeilberger algorithm is much, much faster. I won't even attempt to convey the radically different and often startling reasoning that underlies the al­ gorithm; the reader will have to go to the book for that. I want to mention, though, one dramatic application out of many. The problem is to evaluate the sum

(N2

Sn

76

(n +

I:

Let

Then

=

tk + c?

The reader may observe that this is the fmite difference analog of the problem of integration in closed form. Bill Gosper published his method in 1975. Gosper, whom many believe to possess one of the most original and cre­ ative intelligences operating at the interface of mathemat­ ics and computer science, has never sought traditional venues for the dissemination of his results. Many of them exist only as conference proceedings and reports. Gosper's method involves an inhumanly clever ploy, and I won't reveal it here. Its appearance in the text is signaled by the indigitation, "And now a miracle happens." Now, I distrust miracles; if good miracles can happen, so can bad miracles� If I can win the Pennsylvania lottery, I can equally easily be struck down by some tropical disease shared by only five other people. Nevertheless, I want to list some re­ sults of Gosper's algorithm. RESULT

Then Sn

=

I wouldn't do it.

Determining the constants by using the values off(O), J( 1), f(2) is straightforward, and we find

f(n)

= 2n - J + cos-. 2

nw

The algorithm is contained in the package EKHAD. These descriptions furnish only an intriguing glance into a book which by now has become justly famous, and oc­ cupies a position at the nexus of computer science, algo­ rithmic theory, special functions, combinatorics. The algo­ rithms described in the book gained for Zeilberger and Wilf the prestigious 1998 Leroy P. Steele Award. Determined to probe the strength of the algorithms therein, I sent Herb Wilf an e-mail message describing the unsatisfactory state of affairs of the associated Legendre polynomials (*). I got back a TeX document containing the formula

P�+ ! (x) = where the

( - 1)n (n +

1)

{

k

I =

0

+ 2)k (Pn - Pk- l)x (k!)2 (n + 1 k)

(-n)k(n

k

-

Pn are the harmonic numbers Pn _

n

I

1

r=o r+ l

0,

, r 2= 0,

--

r<

0

-and no explanation. I leave it to the reader, as Wilf left it to me, to discover how the algorithms in this book pro­ duce the very beautiful result above. The book, obviously, is not without its psychological perils. Perhaps it should be labeled NC17 and kept away from young readers. Every morning I look into the mirror

to see whether I can discern a vacuity of expression, the crippling effects of the Ekhad mystique and the perni­ ciousness of its canonical text, "A = B." But I see nothing to alarm me-there's even a sprightliness in my expression: yes, I feel so much better since my trip to Luminy. On impulse the other day I sent to a netnews discussion group on obscure languages a query on the name Shalosh B. Ekhad. I got an immediate reply: Dear Jet Wimp, In regard to your question, the exact translation of the phrase "Shalosh Be Echad" is "Three in one". This phrase came in the Hebrew translation of the New Tes­ tament, it means "Trinity" referring to the "Holy Trinity," if you know what I'm talking about . . . While I don't doubt the accuracy of the translation from the Hebrew, the explanation ofits meaning is certainly wrong. My concordance reveals no occurrence of the indicated phrase in the entire Bible; and, according to the Harper CoUins Dictionary of Religion, "the Trinity, as such, does not occur in the New Testament. . . . Later theological tradi­ tion . . . conceived of God as communal." The present tradi­ tion had its origins in the Council of Nicaea (325) and was later refined in the Council of Constantinople (381). There the matter stands, mysterious as ever.

Department of Mathematics Drexel University Philadelphia, PA 1 9 1 04 USA e-mail: [email protected]

VOLUME 23. NUMBER 4 . 2001

77

lfiflrri .MQ·h•i§i

Robin Wilson

The Birth of Computing

Schickard's machine

he earliest mechanical calculating machines appeared in the seven­ teenth century, following Napier's in­ vention of logarithms and the develop­ ment of the slide rule. An early machine was described by William Schickard in 1623, and others were constructed by Pascal and Leibniz. In the nineteenth century the cen­ tral figure was Charles Babbage. His difference engine, a digital machine he conceived in the 1820s, was a complex arrangement of gears and levers de­ signed to mechanise the calculation of mathematical tables and print out the results. Although a small working model was built, a full-scale version from his drawings was not constructed until 1991. Babbage's analytical engine can be regarded as the forerunner of the mod­ em programmable computer, although it was never actually constructed. Designed to be run by steam power, it contained a store (or memory) and

T

200 {Jr tolk8tE:Iling

was to be programmed by means of punched cards. Punched cards, with holes punched in specific locations to convey informa­ tion, were used in the Jacquard loom to mechanise the weaving of compli­ cated patterns. Data processing with punched cards was developed by Hollerith for the 1890 US census, and along with punched tape, such cards were in widespread use for many years. The modem computer age started in the Second World War, with COLOSSUS in England, used for deciphering German military codes, and ENIAC (Electronic

Numerical Integrator And Computer)

in the United States. These machines were large and cwnbersome: ENIAC was eight feet high and contained many thou­ sands of vacuum tubes, resistors, capac­ itors, relays and switches. Inspired by ENIAC, John von Neumann helped to develop a "stored-program computer" in which data and instructions are held in an internal store until needed.

Norge

I II Ill 11 11 I I Il l I 111111 I II II 65 Charles Babbage

punched card

ENIAC computer

Please send all submissions to the Stamp Corner Editor,

Robin Wilson, Faculty of Mathematics,

The Open University, Milton Keynes, MK7 6AA, England e-mail: [email protected]

80

THE MATHEMAnCAL INTELLIGENCER © 2001 SPRINGER-VERLAG NEW YORK

Joseph Marie Jacquard

punched tape

John von Neumann