The proposition that finite and infinite collections are distinguished by the applicability to the former of the syllogism of transposed quantity ought to be regarded as the basal one of scientific arithmetic.
If a person does not know how to reason logically, and I must say that a great many fairly good mathematicians,—yea, distinguished ones,—fall under this category, but simply uses a rule of thumb in blindly drawing inferences like other inferences that have turned out well, he will, of course, be continually falling into error about infinite numbers. The truth is such people do not reason, at all. But for the few who do reason, reasoning about infinite numbers is easier than about finite numbers, because the complicated syllogism of transposed quantity is not called for. For example, that the whole is greater than its part is not an axiom, as that eminently bad reasoner, Euclid, made it to be. It is a theorem readily proved by means of a syllogism of transposed quantity, but not otherwise. Of finite collections it is true, of infinite collections false. Thus, a part of the whole numbers are even numbers. Yet the even numbers are no fewer than all the numbers; an evident proposition since if every number in the whole series of whole numbers be doubled, the result will be the series of even numbers.
1, 2, 3, 4, 5, 6, etc. 2, 4, 6, 8, 10, 12, etc.
So for every number there is a distinct even number. In fact, there are as many distinct doubles of numbers as there are of distinct numbers. But the doubles of numbers are all even numbers.
In truth, of infinite collections there are but two grades of magnitude, the _endless_ and the _innumerable_. Just as a finite collection is distinguished from an infinite one by the applicability to it of a special mode of reasoning, the syllogism of transposed quantity, so, as I showed in the paper last referred to, a numerable collection is distinguished from an innumerable one by the applicability to it of a certain mode of reasoning, the Fermatian inference, or, as it is sometimes improperly termed, “mathematical induction.”
As an example of this reasoning, Euler’s demonstration of the binomial theorem for integral powers may be given. The theorem is that (_x_ + _y_)^n, where _n_ is a whole number, may be expanded into the sum of a series of terms of which the first is _x_^n_y_^o and each of the others is derived from the next preceding by diminishing the exponent of _x_ by 1 and multiplying by that exponent and at the same time increasing the exponent of _y_ by 1 and dividing by that increased exponent. Now, suppose this proposition to be true for a certain exponent, _n_ = _M_, then it must also be true for _n_ = _M_ + 1. For let one of the terms in the expansion of (_x_ + _y_)^M be written A_x^p__y^q_. Then, this term with the two following will be