Page:Popular Science Monthly Volume 5.djvu/161

Rh answering, he gives me his opinion to the effect that the laws of motion are proved true by the truth of the "Principia" deduced from them. Of this hereafter. My present purpose is to show that Newton did not say this, and gave every indication of thinking the contrary. He does not call the laws of motion "hypotheses;" he calls them "axioms." He does not say that he assumes them to be true provisionally, and that the warrant for accepting them as actually true will be found in the astronomically-proved truth of the deductions. He lays them down just as mathematical axioms are laid down—posits them as truths to be accepted a priori, from which follow consequences which must therefore be accepted. And, though the reviewer thinks this an untenable position, I am quite content to range myself with Newton in thinking it a tenable one—if, indeed, I may say so without undervaluing the reviewer's judgement. But now, having shown that the reviewer evaded the issue 1 raised, which it was inconvenient for him to meet, I pass to the issue he substitutes for it. I will first deal with it after the methods of ordinary logic, before dealing with it after the methods of what may be called transcendental logic.

To establish the truth of a proposition postulated, by showing that the deductions from it are true, requires that the truth of the deductions shall be shown in some way that does not directly or indirectly assume the truth of the proposition postulated. If, setting out with the axioms of Euclid, we deduce the truths that "the angle in a semicircle is a right angle," and that "the opposite angles of any quadrilateral figure described in a circle are together equal to two right angles," and so forth; and, if, because these propositions are true, we say that the axioms are true, we are guilty of a petitio principii. I do not mean simply that, if these various propositions are taken as true on the strength of the demonstrations given, the reasoning is circular, because the demonstrations assume the axioms, but I mean more—I mean that any supposed experimental proof of these propositions, by measurement, itself assumes the axioms to be justified. For, even when the supposed experimental proof consists in showing that some two lines, demonstrated by reason to be equal, are equal when tested in perception, the axiom, that things which are equal to the same thing are equal to one another, is taken for granted. The equality of the two lines can be ascertained only by carrying from the one to the other some measure (either a movable marked line or the space between the points of compasses), and by assuming that the two lines are equal to one another, because they are severally equal to this measure. The ultimate truths of mathematics, then, cannot be established by any experimental proof that the deductions from them are true; since the supposed experimental proof takes them for granted. The same thing holds of ultimate physical truths. For the alleged a posteriori proof of these truths has a vice exactly analogous to the vice I have just indicated. Every evidence yielded by astronomy,