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The logic employed is merely that underlying usual language (Gardies, 1997, pp. 49–75), with a large use of reductio ad absurdum. There are no reasoning errors, but an analysis using propositional calculus shows that the author appears to be unaware of some logical rules such as the equivalence between a statement and its contraposition (pp. 49-75). There is some evidence that Euclid does not separate in his proofs “what is pure logic and the part which is not” (pp. 49-75). We can also notice that Hilbert’s “foundations of geometry” follow the same way, using a logic not made explicit, underlying current language.
The two poles are not independent: the rules that a proof must fulfil arise from an agreement between mathematicians. It is a fact that there exist debates about some mathematical proofs, but in fact, they fall into two categories: • does this particular proof fulfil the rules mathematicians? • this proof uses new rules, can they be accepted? usually admitted P. ), Theorems in schools: From history epistemology and cognition to classroom practice, 27–42. © 2007 Sense Publishers. All rights reserved.
Actually, there exist in geometry some cases in which one can meet “naturally,” that is to say, without presupposing it, an infinite anthyphareisis, and, by an irony of fate, pointed out by K. Von Fritz, this case actually occurs when studying the emblematic figure of Pythagoreans: the star with five branches, that is to say the star-shaped regular pentagon. Using notations of Figure 2 below, applying anthyphairesis to the pair (AD,AE) leads to replace it by the pair (B′E′,B′A′), using only one intermediary (B′E,EA′), and mathematical knowledge attributed to that age is enough to perform the two stages; in fact we can consider that it is a matter of proof by visual evidence of the same kind as that of Meno.