On the Philosophy of Discovery, Chapters Historical and Critical

by William Whewell

2. But leaving the Absolute and the Unconditioned, as notions which cannot be dealt with by our reason without being something entirely different from their definitions, we may turn for a moment to another notion which is combined with them by the expositors of whom I speak, and which has some bearing upon our positive science, because it enters into the reasonings of mathematics: I mean the notion of Infinite. Some of those who hold that we can know nothing concerning the Absolute and the Unconditioned, (which they pretend to prove, though concerning such words I do not conceive that anything can be true or false,) hold also that the Infinite is in the same condition;—that we can know nothing concerning what is Infinite;—therefore, I presume, nothing concerning infinite space, infinite time, infinite number, or infinite degrees.

To disprove this doctrine, it might be sufficient to point out that there is a vast mass of mathematical science which includes the notion of infinites, and leads to a great body of propositions concerning Infinites. The whole of the infinitesimal calculus depends upon conceiving finite magnitudes divided into an infinite number of parts: these parts are infinitely small, and of these parts there are other infinitesimal parts infinitely smaller still, and so on, as far as we please to go. And even those methods which shun the term infinite, as Newton's method of Ultimate Ratios, the method of Indivisibles, and the method of Exhaustions of the ancient geometers, do really involve the notion of infinite; for they imply a process continued without limit.

3. But perhaps it will be more useful to point out the fallacies of the pretended proofs that we can know nothing concerning Infinity and infinite things.

The argument offered is, that of infinity we have no notion but the negation of a limit, and that from this negative notion no positive result can be deduced.

But to this I reply: It is not at all true that our notion of what is infinite is merely that it is that which has no limit. We must ask further that what? that space? that time? that number?—And if that space, that what kind of space? That line? that surface? that solid space?—And if that line, that line bounded at one end, or not? If that surface, that surface bounded on one, or on two, or on three sides? or on none? However any of these questions are answered, we may still have an infinite space. Till they are answered, we can assert nothing about the space; not because we can assert nothing about infinites; but because we are not told what kind of infinite we are talking of.

In reality the definition of an Infinite Quantity is not negative merely, but contains a positive part as well. We assume a quantity of a certain kind which may be augmented by carrying onward its limits in one or more directions: this is a finite quantity of a given kind. We then—when we have thus positively determined the kind of the quantity—suppose the limit in one or more directions to be annihilated, and thus we have an infinite quantity. But in this infinite quantity there remain the positive properties from which we began, as well as the negative property, the negation of a limit; and the positive properties joined with the negative property may and do supply grounds of reasoning respecting the infinite quantity.

4. This is lore so elementary to mathematicians that it appears almost puerile to dwell upon it; but this seems to have been overlooked, in the proof that we can have no knowledge concerning infinites. In such proof it is assumed as quite evident, that all infinites are equal. Yet, as we have seen, infinites may differ infinitely among themselves, both in quantity and in kind. A German writer is quoted[301] for an "ingenious" proof of this kind. In his writings, the opponent is supposed to urge that a line BAC may be made infinite by carrying the extremity C infinitely to the right, and again infinite by carrying the extremity B infinitely to the left; and thus the line infinitely extended both ways would be double of the line infinite on one side only. The supposed reply to this is, that it cannot be so, because one infinite is equal to another: and moreover that what is bounded at one end A, cannot be infinite: both which assumptions are without the smallest ground. That one infinite quantity may be double of another, is just as clear and certain as that one finite quantity may. For instance, if one leaf of the book which the reader has before him were produced infinitely upwards it would be an infinite space, though bounded at the bottom and at both sides. If the other leaf were in like manner produced infinitely upwards it would in like manner be infinite; and the two together, though each infinite, would be double of either of them.

5. As I have said, infinite quantities are conceived by conceiving finite quantities increased by the transfer of a certain limit, and then by negativing this limit altogether. And thus an infinite number is conceived by assuming the series 1, 2, 3, 4, and so on, up to a limit, and then removing this limit altogether. And this shows the baselessness of another argument quoted from Werenfels. The opponent asks, Are there in the infinite line an infinite number of feet? Then in the double line there must be twice as many; and thus the former infinite number did not contain all the (possible) unities; (numerus infinitus non omnes habet unitates, sed præter eum concipi possunt totidem unitates, quibus ille careat, eique possunt addi). To which I reply, that the definition of an infinite number is not that it contains all possible unities: but this—that the progress of numeration being begun according to a certain law, goes on without limit. And accordingly it is easy to conceive how one infinite number may be larger than another infinite number, in any proportion. If, for instance, we take, instead of the progression of the natural numbers 1, 2, 3, 4, &c. and the progression of the square numbers 1, 4, 9, 16, &c. any term of the latter series will be greater than the corresponding term of the other series in a ratio constantly increasing, and the infinite term of the one, infinitely greater than the corresponding infinite term of the other.

6. In the same manner we form a conception of infinite time, by supposing time to begin now, and to go on, after the nature of time, without limit; or by going back in thought from the present to a past time, and by continuing this retrogression without limit. And thus we have time infinite a parte ante and a parte post, as the phrase used to run; and time infinite both ways includes both, and is the most complete notion of eternity.

7. Perhaps those who thus maintain that we cannot conceive anything infinite, mean that we cannot form to ourselves a definite image of anything infinite. And this of course is true. We cannot form to ourselves an image of anything of which one of the characteristics is that it is, in a certain way, unlimited. But this impossibility does not prevent our reasoning about infinite quantities; combining as elements of our reasoning, the absence of a limit with other positive characters.