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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.)}

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You can certainly create a Hilbert system with

– Alex Nelson – 2016-05-19T15:25:46.180modus ponensas the only rule of inference, and describe quite a bit of math within it (certain automated proof checkers do this). Butallof mathematics within one formal system...easier said than done. Certainly proofs involving commutative diagrams might be a bit inelegant...As you say, the traditional syllogistic logic is modernly translated into monadic predicate logic which is a fragment of first-order logic. To "generate" math, of course, you need - in addition to logical language and the "inference mechanism" provided by logical axioms and rules - also mathematical specific

– Mauro ALLEGRANZA – 2016-05-19T15:56:40.970axioms.What exactly do you mean by "reduce"? In your question you seem to talk about translation to formal language, and not reduction (e.g. in the sense of logicism). – Eliran – 2016-05-19T16:52:56.740

@EliranH corrected. thanks. Yes, I mean "translate." (Yes, "reduction" is a separate logical term, not what I meant.) – Geremia – 2016-05-19T17:30:58.977

2all numbers greater than some number are greater than all numbers less than that number. 3 > 2, 1 < 2, therefore 3 > 1. The problem with traditional syllogism is quantification and anaphora (bound vars). we can't give a precise referent for "that number", since "some number is indeterminate". by contrast a modern version of same would bind x y and z by saying "for all x, y, z, x>y and y>z implies x>z", which covers the case of 3, 2, 1. you cannot translate traditional syllogism into modern FOL without changing it, since it lacks key conceptual innovations of the latter. – None – 2016-05-19T19:08:16.313

1ps. what counts as "mathematical reasoning" changes. mathematicians were perfectly content with Aristotlean logic, until they weren't. – None – 2016-05-19T19:11:36.613