Let us, therefore, see the geometer at work and seek to catch his process.
The task is not without difficulty; it does not suffice to open a work at random and analyze any demonstration in it.
We must first exclude geometry, where the question is complicated by arduous problems relative to the rôle of the postulates, to the nature and the origin of the notion of space. For analogous reasons we can not turn to the infinitesimal analysis. We must seek mathematical thought where it has remained pure, that is, in arithmetic.
A choice still is necessary; in the higher parts of the theory of numbers, the primitive mathematical notions have already undergone an elaboration so profound that it becomes difficult to analyze them.
It is, therefore, at the beginning of arithmetic that we must expect to find the explanation we seek, but it happens that precisely in the demonstration of the most elementary theorems the authors of the classic treatises have shown the least precision and rigor. We must not impute this to them as a crime; they have yielded to a necessity; beginners are not prepared for real mathematical rigor; they would see in it only useless and irksome subtleties; it would be a waste of time to try prematurely to make them more exacting; they must pass over rapidly, but without skipping stations, the road traversed slowly by the founders of the science.
Why is so long a preparation necessary to become habituated to this perfect rigor, which, it would seem, should naturally impress itself upon all good minds? This is a logical and psychological problem well worthy of study.
But we shall not take it up; it is foreign to our purpose; all I wish to insist on is that, not to fail of our purpose, we must recast the demonstrations of the most elementary theorems and give them, not the crude form in which they are left, so as not to harass beginners, but the form that will satisfy a skilled geometer.
DEFINITION OF ADDITION.--I suppose already defined the operation _x_ + 1, which consists in adding the number 1 to a given number _x_.
This definition, whatever it be, does not enter into our subsequent reasoning.
We now have to define the operation _x_ + _a_, which consists in adding the number _a_ to a given number _x_.
Supposing we have defined the operation
_x_ + (_a_ - 1),
the operation _x_ + _a_ will be defined by the equality
(1) _x_ + _a_ = [_x_ + (_a_ - 1)] + 1.
We shall know then what _x + a_ is when we know what _x_ + (_a_ - 1) is, and as I have supposed that to start with we knew what _x_ + 1 is, we can define successively and 'by recurrence' the operations _x_ + 2, _x_ + 3, etc.
This definition deserves a moment's attention; it is of a particular nature which already distinguishes it from the purely logical definition; the equality (1) contains an infinity of distinct definitions, each having a meaning only when one knows the preceding.
PROPERTIES OF ADDITION.--_Associativity._--I say that
_a_ + (_b_ + _c_) = (_a_ + _b_) + _c_.
In fact the theorem is true for _c_ = 1; it is then written
_a_ + (_b_ + 1) = (_a_ + _b_) + 1,
which, apart from the difference of notation, is nothing but the equality (1), by which I have just defined addition.
Supposing the theorem true for _c_ = [gamma], I say it will be true for _c_ = [gamma] + 1.
In fact, supposing
(_a_ + _b_) + [gamma] = _a_ + (_b_ + [gamma]),
it follows that
[(_a_ + _b_) + [gamma]] + 1 = [_a_ + (_b_ + [gamma])] + 1
or by definition (1)
(_a_ + _b_) + ([gamma] + 1) = _a_ + (_b_ + [gamma] + 1) = _a_ + [_b_ + ([gamma] + 1)],
which shows, by a series of purely analytic deductions, that the theorem is true for [gamma] + 1.
Being true for _c_ = 1, we thus see successively that so it is for _c_ = 2, for _c_ = 3, etc.
_Commutativity._--1º I say that
_a_ + 1 = 1 + _a_.
The theorem is evidently true for _a_ = 1; we can _verify_ by purely analytic reasoning that if it is true for _a_ = [gamma] it will be true for _a_ = [gamma] + 1; for then
([gamma] + 1) + 1 = (1 + [gamma]) + 1 = 1 + ([gamma] + 1);
now it is true for _a_ = 1, therefore it will be true for _a_ = 2, for _a_ = 3, etc., which is expressed by saying that the enunciated proposition is demonstrated by recurrence.
2º I say that
_a_ + _b_ = _b_ + _a_.
The theorem has just been demonstrated for _b_ = 1; it can be verified analytically that if it is true for _b_ = [beta], it will be true for _b_ = [beta] + 1.
The proposition is therefore established by recurrence.
DEFINITION OF MULTIPLICATION.--We shall define multiplication by the equalities.
(1) _a_ × 1 = _a_.
(2) _a_ × _b_ = [_a_ × (_b_ - 1)] + _a_.
Like equality (1), equality (2) contains an infinity of definitions; having defined a × 1, it enables us to define successively: _a_ × 2, _a_ × 3, etc.
PROPERTIES OF MULTIPLICATION.--_Distributivity._--I say that
(_a_ + _b_) × _c_ = (_a_ × _c_) + (_b_ × _c_).
We verify analytically that the equality is true for _c_ = 1; then that if the theorem is true for _c_ = [gamma], it will be true for _c_ = [gamma] + 1.
The proposition is, therefore, demonstrated by recurrence.
_Commutativity._--1º I say that
_a_ × 1 = 1 × _a_.
The theorem is evident for _a_ = 1.
We verify analytically that if it is true for _a_ = [alpha], it will be true for _a_ = [alpha] + 1.
2º I say that
_a_ × _b_ = _b_ × _a_.
The theorem has just been proven for _b_ = 1. We could verify analytically that if it is true for _b_ = [beta], it will be true for _b_ = [beta] + 1.
