Most of the mathematicians who during the last two generations have
treated the differential calculus have been of the opinion that an
infinitesimal quantity is an absurdity; although, with their habitual
caution, they have often added “or, at any rate, the conception of an
infinitesimal is so difficult, that we practically cannot reason about it
with confidence and security.” Accordingly, the doctrine of limits has
been invented to evade the difficulty, or, as some say, to explain the
signification of the word “infinitesimal.” This doctrine, in one form or
another, is taught in all the text-books, though in some of them only as
an alternative view of the matter; it answers well enough the purposes of
calculation, though even in that application it has its difficulties.
The illumination of the subject by a strict notation for the logic
of relatives had shown me clearly and evidently that the idea of an
infinitesimal involves no contradiction, before I became acquainted with
the writings of Dr. Georg Cantor (though many of these had already
appeared in the _Mathematische Annalen_ and in _Borchardt’s Journal_,
if not yet in the _Acta Mathematica_, all mathematical journals of the
first distinction), in which the same view is defended with extraordinary
genius and penetrating logic.
The prevalent opinion is that finite numbers are the only ones that we
can reason about, at least, in any ordinary mode of reasoning, or, as
some authors express it, they are the only numbers that can be reasoned
about mathematically. But this is an irrational prejudice. I long ago
showed that finite collections are distinguished from infinite ones
only by one circumstance and its consequences, namely, that to them
is applicable a peculiar and unusual mode of reasoning called by its
discoverer, DeMorgan, the “syllogism of transposed quantity.”
Balzac, in the introduction of his _Physiologie du mariage_, remarks that
every young Frenchman boasts of having seduced some Frenchwoman. Now, as
a woman can only be seduced once, and there are no more Frenchwomen than
Frenchmen, it follows, if these boasts are true, that no French women
escape seduction. If their number be finite, the reasoning holds. But
since the population is continually increasing, and the seduced are on
the average younger than the seducers, the conclusion need not be true.
In like manner, DeMorgan, as an actuary, might have argued that if an
insurance company pays to its insured on an average more than they have
ever paid it, including interest, it must lose money. But every modern
actuary would see a fallacy in that, since the business is continually
on the increase. But should war, or other cataclysm, cause the class
of insured to be a finite one, the conclusion would turn out painfully
correct, after all. The above two reasonings are examples of the
syllogism of transposed quantity.
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 _xⁿy⁰_ 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_xᵖy𐞥_. Then, this term with the two
following will be
=Transcriber’s Note:= Unicode has no subscript q character, so the Greek
subscript phi character ᵩ is used in these formulæ to represent it.
Italics have been removed for readability.