Proof and ontology in Euclidean mathematics. (English) Zbl 1071.00004

Kjeldsen, Tinne Hoff (ed.) et al., New trends in the history and philosophy of mathematics. Papers from the conference, University of Roskilde, Roskilde, Denmark, August 6–8, 1998. Odense: University Press of Southern Denmark (ISBN 87-7838-606-3/pbk). University of Southern Denmark Studies in Philosophy 19, 117-133 (2004).
The author gives an interpretation of Euclidean axiomatics from the position of “epistemic naturalism”. He claims that “mathematical practice doesn’t require justifying an epistemic route to mathematical objects, and so mathematical objects are ontologically stipulated” (p. 118). This seems to solve the challenge mathematical ontology poses for epistemic naturalism. But it entails the new challenge of solving the problem to explain how theorem-proving forces mathematicians to believe that certain mathematical objects have certain properties (pp. 118–119). The author thinks that the stipulationist can explain the importance of ontological positing for mathematics, but the stipulationist also “needs to show that the ontologizing drive in that subject is never motivated by sensitivity to the presence of anything ontologically independent of us that mathematical terms refer to,” in order to keep compatible to epistemic naturalism (p. 120).
The author illustrates this with an interpretation of Book I of Euclid’s “Elements”. He conjectures that this book contains a pictoral proof system in which diagrams are an essential part of the proofs, and a language based proof system in which diagrams are of merely heuristic value (p. 123). The pictoral system gives the rules for the construction of admissible diagrams, and comprises meta-diagrammatic considerations on properties of the figures of the diagrams. Meta-diagrammatic reasoning points at pictoral facts in the diagram. The author’s examples are the uniqueness considerations in Postulates 1 and 2, considerations on intersections of two straight-lines, two circles, and a circle and a straight-line, finally the proof of Proposition 6.
There are reasons to go beyond a pictoral proof system. Pictoral proof systems do not generalize well, they furthermore multiply the cases needed to be shown in a proof. Language-based proof systems are much more convenient. In Euclid both approaches can be found.
For the entire collection see [Zbl 1054.01001].


00A30 Philosophy of mathematics