Letter | Published:

The Commutative Law of Addition, and Infinity

Naturevolume 81page69 (1909) | Download Citation



REFERRING to the review of Hilbert's “Grundlagen der Geometric,” on p. 394 of No. 2066 of NATURE (June 3), may I point out that the commutative law of addition can be proved without the help of any axioms at all, other than those of general logic? The method, indeed, used by Peano in 1889 (“Arithmetices Principia …,” Turin, 1889, p. 4), which is only based on axioms of a general nature (such as the principle of mathematical induction), and not on such special laws as the distributive ones, appears in so far superior to Hilbert's; and, since all Peano's axioms were proved in Mr. Russell's “Principles of Mathematics” of 1903, Hilbert's proof seems quite superseded. Further, the difficulties arising out of Dedekind's proof of the existence of infinite systems can be avoided without the introduction of “metaphysical” arguments about time and Consciousness (see Russell, Hibbert Journal, July, 1904, pp. 809–12), as, indeed, your reviewer seems to think possible. But the connection of the fact that the existence of an infinity of thoughts (which must be in time) with Hamilton's idea that algebra was interpretable especially in the time-manifold, just as geometry is in the spacemanifold, is not obvious.

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