### Lakatos as Dialectical Realist

#### Posted by David Corfield

Reading again *Proofs and Refutations* for my first Utrecht talk, I came across this part of the dialogue:

Omega: But I want to discover the secret of Eulerianness!

Zeta: I understand your resistance. You have fallen in love with the problem of finding out where God drew the boundary dividing Eulerian from non-Eulerian polyhedra. But there is no reason to believe that the term ‘Eulerian’ occurred in God’s blueprint of the universe at all. What if Eulerianness is merely an accidental property of some polyhedra? In this case it would be uninteresting or even impossible to find out the zig-zags of the demarcation line between Eulerian and non-Eulerian polyhedra. Such an admission however would leave rationalism unsullied, for Eulerianness is then not part of the rational design of the universe. So let us forget about it. One of the main points about critical rationalism is that one is always prepared to abandon one’s original problem in the course of the solution and replace it by another one. (pp. 67-68)

The students have come to the point of defining *Eulerianness* for a polyhedron as possessing the property that $V - E + F = 2$, for vertices, edges and faces. They were driven to devise this term having believed all polyhedra had this property, when counterexamples, such as the picture frame, emerged.

As it turned out, Eulerianness did pick out something worth demarcating – those polyhedra homeomorphic to a sphere. But, as Zeta remarks, the transformation of which problems and which properties are taken as central is no cause for concern for the rationalist. Indeed, if these changes are brought about for good reason, we should celebrate such changes.

My main problem with Lakatos is that he takes the vehicle for such transformation to be his method of proofs and refutations:

As far as naïve classification is concerned, nominalists are close to the truth when claiming that the only thing that polyhedra have in common is their name. But after a few centuries of proofs and refutations, as the theory of polyhedra develops, and theoretical classification replaces naïve classification, the balance changes in favour of the realist. (p. 92n)

The method takes the form:

(1) Primitive conjecture.

(2) Proof (a rough thought-experiment or argument, decomposing the primitive conjecture into subconjectures or lemmas).

(3) ‘Global’ counterexamples (counterexamples to the primitive conjecture) emerge.

(4) Proof re-examined: the ‘guilty lemma’ to which the global counterexample is a ‘local’ counterexample is spotted. This guilty lemma may have previously remained ‘hidden’ or may have been misidentified. Now it is made explicit, and built into the primitive conjecture as a condition. The theorem–the improved conjecture–supersedes the primitive conjecture with the new proof-generated concept as its paramount new feature.

(5) Proofs of other theorems are examined to see if the newly found lemma or the new proof-generated concept occurs in them: this concept may be found lying at the crossroads of different proofs, and thus emerge as of basic importance.

(6) The hitherto accepted consequences of the original and now refuted conjecture are checked.

(7) Counterexamples are turned into new examples - new fields of inquiry open up. (pp. 127-8)

Lakatos was right to point us in the direction of a dialogue, but it would do no justice at all to the discussions we have had on, say, generalized smooth spaces to interpret them as applications of the method of proofs and refutations.

He also usefully points us in the direction of natural kinds in mathematics.

## Re: Lakatos as Dialectical Realist

Is the method of proofs and refutations supposed to be prescriptive or descriptive? Could you remind us?