Heuristic, Methodology Or Logic of Discovery? Lakatos on Patterns of Thinking
Total Page：16
File Type：pdf, Size：1020Kb
Heuristic, Methodology or Logic of Discovery? Lakatos on Patterns of Thinking Olga Kiss Budapest Corvinus University Heuristic is a central concept of Lakatos’ philosophy both in his early works and in his later work, the methodology of scientiªc research programs (MSRP). The term itself, however, went through signiªcant change of mean- ing. In this paper I study this change and the ‘metaphysical’ commitments be- hind it. In order to do so, I turn to his mathematical heuristic elaborated in Proofs and Refutations. I aim to show the dialogical character of mathemat- ical knowledge in his account, which can open a door to hermeneutic studies of mathematical practice. Heuristic or Methodology? The subtitle of Lakatos’ Proofs and Refutations is: The Logic of Mathematical Discovery (Lakatos, 1976). Although in the original version published in 1963 in the British Journal for the Philosophy of Science there was no such subtitle, it was not the invention of the editors. The unpublished 1961 Cambridge Ph.D. thesis of Lakatos has the same title (“Essays in the Logic of Mathematical Discovery”), and it was an early version of Proofs and Refu- tations.—But was it really a logic of mathematical discovery? And if it was, in what sense? There is a footnote in “History of science and its rational reconstruc- tions” (1970), a paper Lakatos wrote almost ten years later, in which he made the distinction explicit: heuristic means the rules of discovery, while logic of discovery or methodology makes the rules for appraising already An earlier version of a part of this paper was presented at the Conference of the Interna- tional Society for Hermeneutics and Science at Stony Brook, New York in 1994. I would like to thank the participants for the discussion and the OTKA Foundation for the scholar- ship to our studies with some of my colleagues on “Perspectives of historiography and phi- losophy of science” (T 046261). Perspectives on Science 2006, vol. 14, no. 3 ©2006 by The Massachusetts Institute of Technology 302 Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc.2006.14.3.302 by guest on 02 October 2021 Perspectives on Science 303 existing results of science (Lakatos 1978a, p. 103). As Teun Koetsier pointed out, there was a change in Lakatos’ use of the term ‘heuristic’. While heuristic in the Proofs and Refutations was a set of rules to guide problem solving for the individual scientist, “MSRP [Methodology of Scientiªc Research Programmes] does not give any ªrm heuristic advice to in- dividual scientists, but it deªnitely yields recommendations for a rational scientiªc community on the way it should act” (Koetsier 1991, p. 16). Al- though ‘heuristic’ is a central concept of the “Methodology of Scientiªc Research Programmes,” there are very few explicit hints in MSRP what heuristic means exactly. Lakatos makes vague statements, such as, heuris- tic “tells scientists” things to do or not to do, but these statements remain very obscure—he does not show a real, elaborated heuristic. This is why, I claim, this late work cannot be fully understood without his early 1963– 64 papers; his intentions become clearer, especially after we scrutinize his Proofs and Refutations. 1. Dialogue in the “classroom” In his Proofs and Refutations Lakatos reconstructs a vivid debate among mathematicians. The genre of the paper is moral drama written with an excellent sense of humor. The actors of Lakatos’ play—a teacher and his students—are in an imaginary classroom. In their talk, a ªctional dialogue of mathematicians of different eras unfolds. Students do not represent par- ticular mathematicians, but rather customs and typical behaviors of scien- tists faced with the coexistence of alternative proofs and refutations to the same theorem: Which one is wrong, and what is the problem with it? The actors are emotional, sometimes even passionate, enthusiastic, suspicious or desperate, but always witty. We can also see typical changes of their habits. The students’ names are Greek letters, to express their cross-cultural individuality. Each time they speak as someone in the ‘real’ history of mathematics, we can ªnd an exact reference to the primary sources in the footnotes. In this way one can say, the ‘real history’ runs in the footnotes, while the dialogue in the classroom is its ‘rational reconstruction’. Science begins with problems.1 The students in Lakatos’ “classroom” become interested in the problem at hand: is there a relation between the number of edges, vertices and faces of a polyhedron? They ªnd a conjec- ture (today it is known as the Euler theorem), and the teacher proves it for 1. At least in the Popperian logic of scientiªc discovery. Lakatos shows in this work the applicability of a ‘sophisticated version’ of Popperian methodology applied to mathemat- ics. I borrow the term of ‘sophisticated falsiªcationism’ from his later papers (see Lakatos 1978a). Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc.2006.14.3.302 by guest on 02 October 2021 304 Heuristic, Methodology or Logic of Discovery? Figure 1. Nested Cube, twin-tetrahedra, “urchin” that is star polyhedron, “pic- ture frame.” all polyhedra. The story could stop here, if the students were not advanced enough to ask questions. Is the proof correct? Is the theorem really valid? The ªrst meta-problem—how to criticize mathematical arguments— already arises by these steps. Suspicion alone is not criticism, but enough to make the Teacher modify his proof slightly. In order to make the criti- cism much tougher, the Teacher suggests that the students start looking for counterexamples. Some of them attack the conjecture (which can be seen as a theorem after having a proof), while others attack some steps of the proof. What to do next? The ªrst counterexample, given by Gamma, makes the Teacher modify his proof, after which he says “Criticism is not necessarily destruction”! (Lakatos 1976, p. 10) He ªnds the guilty lemma that is falsiªed by this counterexample, and reformulates it in order to correct the proof. The sec- ond counterexample, found by Alpha, is a cube-within-a-cube; creating a cavity (Figure 1a—“nested cube”) undermines not only the proof but also the theorem itself. The Teacher would have to surrender, but he does not. At this point the debate expands. Gamma rejects the defense of the proof, while Beta rejects the counterexamples. According to him ‘nested cube’ is not a counterexample, because it is not a polyhedron at all. The students start to create deªnitions for the term “polyhedron” (which was originally treated as self evident) in order to defend the conjecture by ex- cluding such objects from the ªeld of polyhedra. In a very short time we ªnd ourselves in a jungle of different deªnitions and crazy, perverted, and foolish counterexamples. Are these really polyhedra? They are rejected as monsters by Delta and Eta in defense of the proven theorem. More and more new counterex- amples arise to the increasing number of deªnitions and then again, new monster-barring deªnitions. Beta accepts all the counterexamples as poly- hedra, he simply adjusts the theorem to them by excluding them as excep- tions. When the theorem has too many counterexamples, he restricts its validity to a safe territory of convex polyhedra. Convex polyhedra must be Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/posc.2006.14.3.302 by guest on 02 October 2021 Perspectives on Science 305 Euler-like. But how can we know?—Beta did not do anything with the proof. Does it prove the new formulation of the theorem? Who knows? At this point Rho does something interesting: she does not change the deªnition of the polyhedra, but she re-interprets the counterexamples. She changes our way of seeing them. According to her, “urchin” is not a star- polyhedron with star-pentagonal faces but a simple, boring (although con- cave) polyhedron with triangular faces. “Monsters don’t exist, only mon- strous interpretations” (Lakatos 1976, p. 31). Lakatos refers here to the Skeptic criticism of the Stoic claim for the ability distinguishing ‘phantasia’ ([mental] representation) from ‘phantasia kataleptike’ (apprehen- sive [mental] representation) (Lakatos 1976, p. 32 n. 2), but Alpha consid- ers this monster-adjusting attitude as simply brainwashing. All of these struggles, however, seem to be ‘much ado about nothing’. (And, of course, we can follow the bouts of real mathematicians in the footnotes.) The question, whether the conjecture is true or not, remained unanswered. I claim that the quarrel and the counterexamples had noth- ing to do with the theorem and its proof. This is the point when the Teacher introduces us to the method of proof and refutations. Faced with the counterexamples, he carefully inspects the challenged proof, because he now thinks that some of the lemmas must have been false. After making a thorough analysis, he incorporates these lemmas into the theorem as pre- suppositions. In this way the theorem is not about ‘all polyhedra’, but about ‘simple polyhedra’ with ‘simply connected faces’. He now deªnes the terms ‘simple polyhedron’ and ‘simply connected faces’ in order to se- cure the previously attacked steps of the proof. These deªnitions cover theoretical concepts stemming from the proof-analysis, the ªnal source of which was the debate about the counterexamples. This method of lemma- incorporation shows the “intrinsic unity between the ‘logic of discovery’ and the ‘logic of justiªcation’ . .” (Lakatos 1976, p. 37) This “classroom dialog” could be seen as a socio-psychological account of the development of mathematics, but it is not so. All of the controver- sies become relevant to such a ‘rational reconstruction’, when we start to see their cognitive relevance, and the resulting concepts and trains-of- thought “frozen” in the papers, letters and textbooks.