I think that deduction used in syllogistic systems, employing axioms and hence inferential conclusions i.e. theorems is rather weak, and cannot suffice for the robustness needed for proving mathematical theorems? Is there any point 'for' Euclid's approach?

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    Not clear at all; do you refer to syllogism in the "technical" sense ? If so, what about Euclid's syllogistic approach ? Very few (more or less none ) of Euclid's Elements theorems are proved with formal syllogisms. Dec 16, 2017 at 21:26
  • @MauroALLEGRANZA Hm, I was referring to his approach of finding theorems, where we have two premises or axioms, and we deductively form a theorem based on the premises. Dec 16, 2017 at 21:57

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You are correct that syllogistic, which corresponds to monadic predicate calculus in modern terms, is insufficient for doing mathematics. Modern formalisms use polyadic calculus. However, Euclid does not use syllogistic alone (in fact, he hardly uses it at all). Recent studies of Euclid's method, especially Manders's classic Euclidean Diagram, show that his use of synthetic construction and reading off of diagrams is irreducible to Hilbert style axiomatic reasoning and is the main tool of Euclid's demonstrations, see What caused or contributed to Euclid's Elements and Synthetic Geometry falling into disfavor?

Kant, like Locke before him, saw that analytic, i.e. derivable in syllogistic, consequences were insufficient to prove even theorems of Euclidean geometry, let alone calculus, so he introduced synthetic a priori constructions to explain how non-trivial mathematics was possible. But the motivating distinction between "logical" (analytic) and "geometric" (synthetic) arguments in Euclidean geometry predates even Euclid himself, and occurs already in Aristotle, who went as far as to say that all thinking requires building images (phantasma) in De Memoria et Reminiscentia:

"An account has already been given of imagination in the discussion of the soul, and it is not possible to think without an image. For the same effect occurs in thinking as in proving by means of diagrams. For in the latter case, though we do not make any use of the fact that the size of the triangle is determinate, we none the less draw it determinate with respect to size. [quoted from Euclid's Pseudaria by Acerbi.]

The difference in logical strength is discussed at length in Friedman's Kant's Theory of Geometry:

"Our distinction between pure and applied geometry goes hand in hand with our understanding of logic, and this understanding simply did not exist before 1879 when Frege's Begriffsschrift appeared... Euclid's axioms do not imply Euclid's theorems by logic alone. Moreover, once we remember that Euclid's axioms are not the axioms used in modern formulations... it is easy to see that the claim in question is perfectly correct. For our logic, unlike Kant's, is polyadic rather than monadic (syllogistic); and our axioms for Euclidean geometry are strikingly different from Euclid's in containing an explicit, and essentially polyadic, theory of order. The general point can be put as follows. A central difference between monadic logic and full polyadic logic is that the latter can generate an infinity of objects while the former cannot.

[...] Does this... show that Euclid's axiomatization is hopelessly "defective"? I think not. Rather, it underscores the fact that Euclid's system is not an axiomatic theory in our sense at all. Specifically, the existence of the necessary points is not logically deduced from appropriate existential axioms. Since the set of such points is of course infinite, this procedure could not possibly work in a monadic (syllogistic) context. Instead, Euclid generates the necessary points by a definite process of construction: the procedure of construction with straightedge and compass."

After the onset of modern logic the issue of analytic vs synthetic got reformatted. Frege and Peirce, for instance criticized Kant for holding that mathematics is synthetic or for defining "analytic" too narrowly, their notion of analytic was of course much stronger than his because of much stronger logic, and on it classical mathematics is indeed analytic. Interestingly enough, while Frege thought that this made construction wholly unnecessary according to Peirce modern logic simply codifies it:

"But neither Kant nor the scholastics provide for the fact that an indefinitely complicated proposition, very far from obvious, may often be deduced by mathematical reasoning, or necessary deduction, by the logic of relatives, from a definition of the utmost simplicity, without assuming any hypothesis whatever (indeed, such assumption could only render the proposition deduced simpler); and this may contain many notions not explicit in the definition.

[...] But Kant, owing to the slight development which formal logic had received in his time, and especially owing to his total ignorance of the logic of relatives, which throws a brilliant light upon the whole of logic, fell into error in supposing that mathematical and philosophical necessary reasoning are distinguished by the circumstance that the former uses constructions. This is not true. All necessary reasoning whatsoever proceeds by constructions; and the only difference between mathematical and philosophical necessary deductions is that the latter are so excessively simple that the construction attracts no attention and is overlooked.

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    A question, my IB book states that ''the model of (mathematical) reasoning developed by Euclid is known as formal system, and it has three key elements; axioms, deductive reasoning and theorems'' and then the next page mentions how he (Euclid) used ''syllogisms to derive his simple theorems e.g. lines perpendicular to the same line are parallel''. I am just wondering as to whether the author of my book is delusional or is it me, have I not got the gist of the argument? Dec 16, 2017 at 23:52
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    What is the difference between synthetic construction and syllogism? Dec 17, 2017 at 0:05
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    @SelenaCarlos What your book says first is true not of Euclid's Elements (c. 300 BC) but of Hilbert's Foundations of Geometry (1899 AD). Unfortunately, reading of Hilbert into Euclid is quite common, it saves textbook authors a lot of effort. As for "syllogisms", some trivial reasoning in geometry can be so framed (usually for deriving what Euclid calls "corollaries", not "theorems") although Euclid does not do it. For an analysis of Euclid's approach see Rodin's Doing and Showing, he touches also on the difference.
    – Conifold
    Dec 17, 2017 at 0:08
  • Well, well. In respect of the use of of logic it seems I have been confusing Hilbert with Euclid. Thanks for the correction.
    – user20253
    Feb 14, 2018 at 12:38

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