The Mathematical Reasoning

The very possibility of mathematical science seems an insoluble contradiction.
If this science is only deductive in appearance, from whence is derived that
perfect rigour which is challenged by none? If, on the contrary, all the
propositions which it enunciates may be derived in order by the rules of
formal logic, how is it that mathematics is not reduced to a gigantic
tautology? The syllogism can teach us nothing essentially new, and if
everything must spring from the principle of identity, then everything should
be capable of being reduced to that principle. Are we then to admit that the
enunciations of all the theorems with which so many volumes are filled, are
only indirect 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 disappeared; it would have only been shirked.
Syllogistic reasoning remains incapable of adding anything to the data that
are given it; the data are reduced to axioms, and that is all we should find
in the conclusions.

No theorem can be new unless a new axiom intervenes in its demonstration;
reasoning can only give us immediately evident truths borrowed from direct
intuition; it would only be an intermediary parasite. Should 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 single glance perceive all its truths;
nay, one might even hope that someday a language would be invented simple
enough for these truths to be made evident to any person of ordinary
intelligence.