Science is that knowledge which can be gained by exact observation and correct thinking.  If science makes use of any methods but these it ceases to be itself.  Science has therefore nothing to do with morals:  it gives the suicide his pistol, the surgeon his life-saving lance, but neither admonishes nor judges them.  It has nothing to do with emotion:  it exposes the chemistry of a tear, the mechanism of laughter; but of sorrow and happiness it has naught to say.  It has nothing to do with beauty:  it traces the movements of the stars, and tells of their constitution; but the fact of their singing together, and that “such harmony is in immortal souls,” it leaves to poet and philosopher.  The timbre, loudness, pitch, of musical tones, is a concern of science; but for this a Beethoven symphony is no better than the latest ragtime air from the music halls.  In brief, science deals only with phenomena, and its gift to man is power over his material environment.

MATHEMATICS

The gift of pure mathematics, on the other hand, is primarily to the mind and spirit:  the fact that man uses it to get himself out of his physical predicaments is more or less by the way.  Consider for a moment this paradox.  Mathematics, the very thing common sense swears by and dotes on, contradicts common sense at every turn.  Common sense balks at the idea of less than nothing; yet the minus quantity, which in one sense is less than nothing in that something must be added to it to make it equal to nothing, is a concept without which algebra would have to come to a full stop.  Again, the science of quaternions, or more generally, a vector analysis in which the progress of electrical science is essentially involved, embraces (explicitly or implicitly) the extensive use of imaginary or impossible quantities of the earlier algebraists.  The very words “imaginary” and “impossible” are eloquent of the defeat of common sense in dealing with concepts with which it cannot practically dispense, for even the negative or imaginary solutions of imaginary quantities almost invariably have some physical significance.  A similar statement might also be made with regard to transcendental functions.

Mathematics, then, opens up ever new horizons, and its achievements during the past one hundred years give to thought the very freedom it seeks.  But if science is dispassionate, mathematics is even more austere and impersonal.  It cares not for teeming worlds and hearts insurgent, so long as in the pure clarity of space, relationships exist.  Indeed, it requires neither time nor space, number nor quantity.  As the mathematician approaches the limits already achieved by study, the colder and thinner becomes the air and the fewer the contacts with the affairs of every day.  The Promethean fire of pure mathematics is perhaps the greatest of all in man’s catalogue of gifts; but it is not most itself, but least so, when, immersed in the manifoldness of phenomenal life, it is made to serve purely utilitarian ends.