Review: [Untitled] Reviewed Work(s): Proofs and Refutations: The Logic of Mathematical Discovery. by Imre Lakatos; John Worrall; Elie Zahar W. D. Hart Mind, New Series, Vol. 87, No. 346. (Apr., 1978), pp. 314-316. Stable URL: http://links.jstor.org/sici?sici=0026-4423%28197804%292%3A87%3A346%3C314%3APARTLO%3E2.0.CO%3B2-S Mind is currently published by Oxford University Press.
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thinks, 'subjective'. A naturalistic view o f evaluation, n o w becoming thinkable again, would make Il"\Toodfield's interpretation consistent w i t h taking the functional approach i n e.g. sociology seriously, b u t he shows n o sign o f thinking that view correct. Having gone into the nature o f b o t h goals and functions, and found t h e one t o have nothing directly t o d o w i t h t h e other (see p. 126) Woodfield seems surprised at t h e end o f his book t o find himself nevertheless able t o present a 'unified theory' o f teleology. Its essence, h e says, 'lies i n welding a causal element and an evaluative element t o yield an explanatory device. T h e causal clause identifies an actual or envisaged effect o f a certain event, t h e evaluative clause says that this e f f e c t is good f r o m some point o f view, and t h e whole thing saps that t h e combination o f these elements provides raison d'etre [sic] o f t h e event' ( p . 205). Woodfield's book is rich i n argument, valuable for providing a m a p o f recent controversies relating t o t h e problems o f teleology as well as for advancing n e w suggestions. O n t h e other hand, he adopts 'conceptual analysis' as t h e framework for his thinking, without fundamental argument, which lends it a period atmosphere, and his vernacular style is streamlined b y recourse t o t h e graceless jargon and wearing clichCs o f Oxford philosophy i n t h e sixties. Both these features make m e anxious t o b e able t o expose t h e book's superficiality. B u t , I hope because it hasn't any, I can't. UNIVERSITY O F LANCASTER
VERNON PRATT
Proofs and Refutations: The Logic of Mathellzatical Discovery. B y IMRE LAKATOS. Edited b y JOHN WORRALL and ELIE ZAHAR. Cambridge University Press, 1976. Pp. xii+ 174. A7.50, P.B. L1.95. Lakatos' four original articles under this title were at the time refreshing and exciting. Nowadays it is only with mixed feelings that one recommends t h e m t o students, for even reading between t h e lines it is hard t o see just what his thesis was. T o that e n d , the present volume is welcome. I shall concentrate here o n stating what I take t o b e Lakatos' view. His main interest t h e n was t h e epistemology o f mathematics, and his main purpose was t o defend what he calls its heuristic. T h i n k o f inquiry as t h e search for knowledge. ( B y knowledge I do not m e a n just propositional knowledge, b u t also knowing things; and I have i n mind particularly a special kind o f knowledge, understanding.) A s a search, inquiry may b e an activity with means as well as ends. Heuristic is the strategy o f deploying the means o f inquiry. It is a mistake o f at least emphasis t o think o f principles o f proof (like complete induction) as t h e means o f mathematical inquiry. Instead, because o f the way t h e y focus attention, concepts and definitions are t h e most interesting mathematical means o f inquiry. It is as a route t o understanding concept formation that Lakatos makes his strongest case for placing heuristic centrally within systematic epistemology. Here is his main example. Euler conjectured that for any polyhedron w i t h V vertices, E edges and F faces, F+V = E+2. Cauchy gave the
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following argument. Remove a face from the polyhedron. Each of that face's edges and vertices being shared with another face, E and V remain fixed while F is reduced by one. Imagine that the object, now holed, is rubber and stretch it out flat on a plane. Triangulate each remaining face by connecting one of its vertices by a diagonal to each of the rest of its vertices (except the neighbours of the vertex you chose). Each diagonal splits one old face into two, increasing F by one, while also adding one to E ; since it does not change V, the number V+F-E is fixed throughout triangulation. Remove the outside triangles one by one until only one triangle is left. Since an exterior triangle shares either one or two edges with other triangles, each extraction eliminates either two edges, one face and one vertex or else one edge, one face and no vertices. In neither case does extraction change V+F-E. Obviously in the last triangle V + F - E = I , so in the first flat figure V+ F- E = I as well; hence for the polyhedron V + F - E = 2, as Euler said. Cauchy's argument is a masterpiece, partly for the cases where it fails. Punch a square tunnel straight through a cube. If you remove an unviolated face, you cannot flatten out the remainder; and you cannot triangulate a punched out face. Being able to flatten out the remainder is the same as being able to pump the original polyhedron up into a ball; the punched out cube pumps up into an American doughnut. And a diagonal wholly within one of the punched out faces splits it into one piece. Both sorts of peculiarity are at bottom the same and illustrate a property now called multiple connectivity. Diagnosing cases where Cauchy's argument fails brings that property to our attention, just as Cauchy's argument brings to our attention properties (like connectivity) invariant under pumping but not punching. This is concept formation; it was the origin of all topology. There is nothing wrong with organizing epistemology around justification and, in the present state of the art, taking logical deduction as a paradigm of justification. Note two obvious points about proof. One cannot both prove and refute something; and the logical principles deployed in refutation are the same as those in proof. The second point means that an epistemology fixated on proof must take its concepts (the substituents for its schematic predicate letters with their uses, extensions and intensions) as given. But paradoxically enough, and this is Lakatos' main heuristic discovery, a sufficiently nayve conjecture like Euler's can sometimes have both proofs as brilliant as Cauchy's and counter-examples as illuminating as the punched-out cube. The most interesting principle of mathematical heuristic which Lakatos broached, albeit very clumsily, is this: a natural way to try to solve such a paradox is to try to draw a generalizable distinction between exactly those cases where the proof is valid and just those cases which include counter-examples. If the problem is sufficiently deep, then drawing such a distinction is concept formation. Lakatos has isolated an important pattern of inquiry; unlike the logical stereotypes of an epistemology focused on proof, Lakatos' pattern treats proof and refutation together, and for that very reason does not take all its concepts as given. Surely a good epistemology should account for the rationales of notions as well as the reasons for beliefs;
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even i f stipulative definition is mere abbreviation, it can take genius t o notice what deserves abbreviation. It m u s t not b e thought that Lakatos' dialectical pattern i n conceptual dynamics is peculiar t o Cauchy's topological treatment o f Euler's conjecture, or even special t o mathematics. T h e editors include work b y Lakatos o n other mathematical examples (which are fascinating and help one t o start t o see t h e general point h e was after). Once you have t h e idea, t h e n if you are familiar with the history o f inquiry i n mathematics, physics or some other serious pursuit, then you can multiply examples yourself, look for t h e best general statement o f t h e pattern, and investigate t h e question o f its importance. Hence Lakatos' insistence o n t h e importance o f history o f science for philosophy o f science: you cannot formalize (codify or stereotype for systematic examination and understanding) what you do not know. (Lakatos m u s t take his o w n medicine. W h e n h e wrote, it was not fashionable t o d o philosophy systematically. But, t o steal a phrase f r o m t h e villiain o f that piece, you cannot command a clear view o f what you refuse t o organize.) After all, as Russell pointed out, a formalization is judged i n part b y h o w well it represents its historically given discipline. Lakatos' main a i m was t o secure heuristic a central place i n epistemology, especially that o f mathematics. T o that end h e wrote delightful and informative mathematics and history o f mathematics (though, again, it would have been so m u c h easier t o see what h e was o n about had h e written conventional, disciplined, organized and abstract philosophical prose m u c h o f t h e time instead o f his allusive and elusive dialogues). O n a fourth level, he takes occasional potshots at t h e exposition o f some mathematicians. H e does not practise what he preaches, and he is silliest w h e n he attacks one o f t h e masters o f that difficult art, Paul R . Halmos; for an antidote t o Lakatos' v e n o m , see Halmos, ' H o w t o iTTrite Mathematics', L'Enseignenzent mathLmatiqz~e,xvi (1970), 123-152. S u c h things are only minor flaws i n a minor masterpiece. T h e work o f Lakatos finally made easily available i n this handsome volume inaugurates a study which deserves t o flourish. UNIVERSITY COLLEGE, L O N D O N
W.
D. HART
