VII
Project Gutenberg EPUB3 edition
Poincaré, Henri and Halsted, George Bruce
Mathematicians, as I have said before, always endeavor to generalize the propositions they have obtained, and, to seek no other example, we have just proved the equality:
a + 1 = 1 + a
and afterwards used it to establish the equality
a + b = b + a
which is manifestly more general.
Mathematics can, therefore, like the other sciences, proceed from the particular to the general.
This is a fact which would have appeared incomprehensible to us at the outset of this study, but which is no longer mysterious to us, since we have ascertained the analogies between demonstration by recurrence and ordinary induction.
Without doubt recurrent reasoning in mathematics and inductive reasoning in physics rest on different foundations, but their march is parallel, they advance in the same sense, that is to say, from the particular to the general.
Let us examine the case a little more closely.
To demonstrate the equality
a + 2 = 2 + a
it suffices to twice apply the rule
(1)a + 1 = 1 + a
and write
(2)a + 2 = a + 1 + 1 = 1 + a + 1 = 1 + 1 + a = 2 + a.
The equality (2) thus deduced in purely analytic way from the equality (1) is, however, not simply a particular ease of it; it is something quite different.
We can not therefore even say that in the really analytic and deductive part of mathematical reasoning we proceed from the general to the particular in the ordinary sense of the word.
The two members of the equality (2) are simply combinations more complicated than the two members of the equality (1), and analysis only serves to separate the elements which enter into these combinations and to study their relations.
Mathematicians proceed therefore 'by construction,' they 'construct' combinations more and more complicated. Coming back then by the analysis of these combinations, of these aggregates, so to speak, to their primitive elements, they perceive the relations of these elements and from them deduce the relations of the aggregates themselves.
This is a purely analytical proceeding, but it is not, however, a proceeding from the general to the particular, because evidently the aggregates can not be regarded as more particular than their elements.
Great importance, and justly, has been attached to this procedure of 'construction,' and some have tried to see in it the necessary and sufficient condition for the progress of the exact sciences.
Necessary, without doubt; but sufficient, no.
For a construction to be useful and not a vain toil for the mind, that it may serve as stepping-stone to one wishing to mount, it must first of all possess a sort of unity enabling us to see in it something besides the juxtaposition of its elements.
Or, more exactly, there must be some advantage in considering the construction rather than its elements themselves.
What can this advantage be?
Why reason on a polygon, for instance, which is always decomposable into triangles, and not on the elementary triangles?
It is because there are properties appertaining to polygons of any number of sides and that may be immediately applied to any particular polygon.
Usually, on the contrary, it is only at the cost of the most prolonged exertions that they could be found by studying directly the relations of the elementary triangles. The knowledge of the general theorem spares us these efforts.
A construction, therefore, becomes interesting only when it can be ranged beside other analogous constructions, forming species of the same genus.
If the quadrilateral is something besides the juxtaposition of two triangles, this is because it belongs to the genus polygon.
Moreover, one must be able to demonstrate the properties of the genus without being forced to establish them successively for each of the species.
To attain that, we must necessarily mount from the particular to the general, ascending one or more steps.
The analytic procedure 'by construction' does not oblige us to descend, but it leaves us at the same level.
We can ascend only by mathematical induction, which alone can teach us something new. Without the aid of this induction, different in certain respects from physical induction, but quite as fertile, construction would be powerless to create science.
Observe finally that this induction is possible only if the same operation can be repeated indefinitely. That is why the theory of chess can never become a science, for the different moves of the same game do not resemble one another.