CHAPTER 2
Published online by Cambridge University Press: 05 June 2012
Summary
Editors' Introduction
Poincaré's proof of the Descartes–Euler conjecture is referred to above. In his doctoral thesis Lakatos introduced detailed consideration of this proof by a discussion of the arguments for and against the ‘Euclidean’ approach to mathematics. Parts of this discussion were incorporated by Lakatos into chapter i (see, e.g., pp. 50–6) and others were rewritten as parts of ‘Infinite Regress and the Foundations of Mathematics’ (Lakatos [1962]). We therefore omit this introductory discussion here.
The advocate of the Euclidean programme – the attempt to supply mathematics with indubitably true axioms couched in perfectly clear terms – has been Epsilon. Epsilon's philosophy is challenged, but the Teacher remarks that the most obvious and direct way to challenge Epsilon is to ask him to produce a proof of the Descartes–Euler conjecture which satisfies Euclidean standards. Epsilon takes up the challenge.
Translation of the Conjecture into the ‘Perfectly Known’ Terms of Vector–Algebra. The Problem of Translation
Epsilon: I accept the challenge. I shall prove that all simply-connected polyhedra with simply-connected faces are Eulerian.
Teacher: Yes, I stated this theorem in a previous lesson.
Epsilon: AS I have pointed out, I first have to find the truth in order to prove it. Now I have nothing against using your method of proofs and refutations as a method of discovering the truth, but where you stop, I start. Where you stop improving, I start proving.
Alpha: But this long theorem is full of stretchable concepts.
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- Information
- Proofs and RefutationsThe Logic of Mathematical Discovery, pp. 106 - 126Publisher: Cambridge University PressPrint publication year: 1976