Can all mathematical reasoning be translated into traditional (Aristotelian, syllogistic) logic?
It would seem not ∵ one cannot syllogistically establish the validity of the reasoning in the following argument:
- 3 is greater than 2.
- 2 is greater than 1.
- ∴, 3 is greater than 1.
This doesn't work because "greater than 2" ≠ "2", or 2 ≯ 2.
The form of the following syllogism is valid, but it shows how a false mathematical premise can lead to a true conclusion:
- All multiples of 5 are even.
- 80 is a multiple of 5.
- ∴, 80 is even.
Thus, it doesn't seem traditional logic can handle mathematical reasoning. Didn't Aristotle, the medieval logicians, et al. realize this?
Poincaré thought that mathematical induction consisted in an ∞ number of syllogisms. Is that true?
(cf. Pierre Duhem's article contra Poincaré: "The Nature of Mathematical Reasoning" from "La nature du raisonnement mathématique," Revue de philosophie 21 (1912): 531-543.)