Synthesized answer
Poincaré views classifying axioms as "à priori synthetic views" as "merely giving it a name" because it doesn't resolve the underlying difficulty of how mathematical truths are derived. He argues that even if the nature of synthetic views were understood, the contradiction would persist; it would simply be avoided rather than solved [1]. This critique reinforces his argument about the limitations of syllogistic reasoning by suggesting that such classifications don't empower syllogisms to do more than what they are inherently capable of.
He contends that syllogistic reasoning is incapable of generating new knowledge beyond the data it is given, which in the case of mathematics, are axioms [1]. If axioms are considered purely as data, then conclusions derived solely through syllogism can only contain what was already present in those axioms [1]. Poincaré questions whether the "syllogistic apparatus" merely serves to obscure this borrowing, implying it doesn't possess a "creative virtue" distinct from the syllogism [2].
Synthesized from the book passages below. Chat with the book on Feynman for follow-up.
From the book
direct ways of saying that A is A? No doubt we may refer back to axioms which are at the source of all these reasonings. If it is felt that they cannot be reduced to the principle of contradiction, if we decline to see in them any more than experimental facts which have no part or lot in mathematical necessity, there is still one resource left to us: we may class them among à priori synthetic views. But this is no solution of the difficulty it is merely giving it a name; and even if the nature of the synthetic views had no longer for us any mystery, the contradiction would not have…
ould we not therefore have reason for asking if the syllogistic apparatus serves only to disguise what we have borrowed? The contradiction will strike us the more if we open any book on mathematics; on every page the author announces his intention of generalising some proposition already known. Does the mathematical method proceed from the particular to the general, and, if so, how can it be called deductive? Finally, if the science of number were merely analytical, or could be analytically derived from a few synthetic intuitions, it seems that a sufficiently powerful mind could with a…
he geometrical axioms are therefore neither synthetic à priori intuitions nor experimental facts . They are conventions. Our choice among all possible conventions is guided by experimental facts; but it remains free , and is only limited by the necessity of avoiding every contradiction, and thus it is that postulates may remain rigorously true even when the experimental laws which have determined their adoption are only approximate. In other words, the axioms of geometry (I do not speak of those of arithmetic) are only definitions in disguise . What, then, are we to think of the question: Is…
tains as particular cases all the majors. This unending series of syllogisms is thus reduced to a phrase of a few lines. It is now easy to understand why every particular consequence of a theorem may, as I have above explained, be verified by purely analytical processes. If, instead of proving that our theorem is true for all numbers, we only wish to show that it is true for the number 6 for instance, it will be enough to establish the first five syllogisms in our cascade. We shall require 9 if we wish to prove it for the number 10; for a greater number we shall require more still; but…
efore, purely analytical, and would not lend themselves to proofs analogous to those of Euclid. On the Nature of Axioms. —Most mathematicians regard Lobatschewsky's geometry as a mere logical curiosity. Some of them have, however, gone further. If several geometries are possible, they say, is it certain that our geometry is the one that is true? Experiment no doubt teaches us that the sum of the angles of a triangle is equal to two right angles, but this is because the triangles we deal with are too small. According to Lobatschewsky, the difference is proportional to the area of the triangle,…
More questions about this book
- Poincaré posits a fundamental paradox concerning mathematical science. Explain, in your own words, the two contradictory possibilities he presents (deductive rigor vs. gigantic tautology) and why each, on its own, seems to invalidate the true nature of mathematics.
- What does Poincaré mean by suggesting that if mathematics relies solely on formal logic, it would be "reduced to a gigantic tautology"? Provide an example to illustrate how this perspective would strip mathematical theorems of their perceived novelty and insight.
- "The syllogism can teach us nothing essentially new," Poincaré declares. If this is true, what specific aspects of mathematical activity or discovery does this statement challenge, and how might one begin to explain the generation of genuinely new mathematical knowledge?
- If pure deduction cannot yield new knowledge, and merely naming axioms doesn't solve the problem, what *alternative* mechanisms or sources of mathematical truth might Poincaré be implicitly seeking to reconcile both the undeniable rigor and the perceived innovation within mathematics?