Description
Science and Hypothesis
PART I.
NUMBER AND MAGNITUDE.
CHAPTER I.
ON THE NATURE OF MATHEMATICAL REASONING.
I.
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.