Hume / (English Men of Letters Series)

by Thomas Henry Huxley

The distinction here drawn between the truths of geometry and other kinds of truth is far less sharply indicated in the Treatise, but as Hume expressly disowns any opinions on these matters but such as are expressed in the Inquiry, we may confine ourselves to the latter; and it is needful to look narrowly into the propositions here laid down, as much stress has been laid upon Hume's admission that the truths of mathematics are intuitively and demonstratively certain; in other words, that they are necessary and, in that respect, differ from all other kinds of belief.

What is meant by the assertion that "propositions of this kind are discoverable by the mere operation of thought without dependence on what is anywhere existent in the universe"?

Suppose that there were no such things as impressions of sight and touch anywhere in the universe, what idea could we have even of a straight line, much less of a triangle and of the relations between its sides? The fundamental proposition of all Hume's philosophy is that ideas are copied from impressions; and, therefore, if there were no impressions of straight lines and triangles there could be no ideas of straight lines and triangles. But what we mean by the universe is the sum of our actual and possible impressions.

So, again, whether our conception of number is derived from relations of impressions in space or in time, the impressions must exist in nature, that is, in experience, before their relations can be perceived. Form and number are mere names for certain relations between matters of fact; unless a man had seen or felt the difference between a straight line and a crooked one, straight and crooked would have no more meaning to him, than red and blue to the blind.

The axiom, that things which are equal to the same are equal to one another, is only a particular case of the predication of similarity; if there were no impressions, it is obvious that there could be no predicates. But what is an existence in the universe but an impression?

If what are called necessary truths are rigidly analysed, they will be found to be of two kinds. Either they depend on the convention which underlies the possibility of intelligible speech, that terms shall always have the same meaning; or they are propositions the negation of which implies the dissolution of some association in memory or expectation, which is in fact indissoluble; or the denial of some fact of immediate consciousness.

The "necessary truth" A = A means that the perception which is called A shall always be called A. The "necessary truth" that "two straight lines cannot inclose a space," means that we have no memory, and can form no expectation of their so doing. The denial of the "necessary truth" that the thought now in my mind exists, involves the denial of consciousness.

To the assertion that the evidence of matter of fact, is not so strong as that of relations of ideas, it may be justly replied, that a great number of matters of fact are nothing but relations of ideas. If I say that red is unlike blue, I make an assertion concerning a relation of ideas; but it is also matter of fact, and the contrary proposition is inconceivable. If I remember[26] something that happened five minutes ago, that is matter of fact; and, at the same time, it expresses a relation between the event remembered and the present time. It is wholly inconceivable to me that the event did not happen, so that my assurance respecting it is as strong as that which I have respecting any other necessary truth. In fact, the man is either very wise or very virtuous, or very lucky, perhaps all three, who has gone through life without accumulating a store of such necessary beliefs which he would give a good deal to be able to disbelieve.

It would be beside the mark to discuss the matter further on the present occasion. It is sufficient to point out that, whatever may be the differences, between mathematical and other truths, they do not justify Hume's statement. And it is, at any rate, impossible to prove, that the cogency of mathematical first principles is due to anything more than these circumstances; that the experiences with which they are concerned are among the first which arise in the mind; that they are so incessantly repeated as to justify us, according to the ordinary laws of ideation, in expecting that the associations which they form will be of extreme tenacity; while the fact, that the expectations based upon them are always verified, finishes the process of welding them together.

Thus, if the axioms of mathematics are innate, nature would seem to have taken unnecessary trouble; since the ordinary process of association appears to be amply sufficient to confer upon them all the universality and necessity which they actually possess.