Note 136 — Chapter XXXVIII · THE METHODOLOGY OF SCIENCE (continued 2)

Of recent years certain philosophers known as logicians, Bertrand Russell in England, Couturat in France, among others, have stressed the logical aspect of mathematics. The question is whether they have not over-stressed it. That mathematical reasoning implies clear thinking and complies with the rules of logic has never been denied; nevertheless the assertion that all mathematical reasonings are of a purely deductive nature, and are reducible to the rules of logic, is an opinion which is by no means unanimous. Some of the greatest among the modern mathematicians, notably Poincaré and Klein, have protested vigorously against this view and have pointed out numerous cases of circularity in the arguments and lack of rigour in the definitions presented by the logicians. Over and above this aspect of the matter, they have maintained that the rules of a game are not everything in its make-up. To say that mathematics and logic are one and the same would be equivalent to maintaining that poetry was nothing but grammar, syntax and rules of versification, or that music was nothing but counterpoint and harmony. It is conceivable that we might acquire as thorough a knowledge of counterpoint and harmony as Beethoven may have possessed and yet be unable to compose a work rivalling any of his symphonies. We should have no hesitancy in granting that a Beethoven must obviously have been gifted with some mysterious faculty which had been denied us; and this faculty, whatever its essence, would relate to music, would be a part of music, since were all men lacking in it, there would be no great music. Under the circumstances, in spite of what might be called our logical knowledge of music, could we truthfully claim to have as thorough a knowledge of it as a Beethoven?

And it is exactly the same with pure mathematics. We know from experience that many persons, though possessing highly logical minds, are yet refractory to advanced mathematics. Were mathematics nothing but logic, this situation would seem extraordinary. Logistics, from a failure to see in mathematics anything but a series of rules and regulations with no creative faculty behind it, has been christened “thoughtless thinking” by its adversaries. But without wishing to take sides in a controversy for which the majority of persons evince but little interest, there is a point which the unprejudiced onlooker must perceive. With the sole exception of Hilbert, who, though opposing Russell’s views, defends opinions of a somewhat similar nature, none of the logicians have contributed to the constructive side of mathematics. This again appears somewhat strange when we recall that one of the earliest boasts of this school of thinkers was that logistics would give them wings. One cannot help but suspect that the logicians are lacking in some creative faculty of which they may not be conscious, and that as a result they perhaps occupy in mathematics a position analogous to that of the professor of counterpoint and harmony in music. Under the circumstances, it is questionable whether they possess a sufficient understanding of this difficult science to contribute any information of value.

Of course a charge of this sort cannot apply to Hilbert, whose great work in the creative regions of mathematics has proved him to be gifted with the creative faculty in addition to the purely formal dissecting faculty which all mathematicians, regardless of their tendencies, must necessarily possess. Owing to Hilbert’s attitude, the problem is generally regarded as controversial.