I printed a paper on the Logic of Number in 1866, and it was not made up out of the first thoughts that came into my head about it, by any means, either.  But I was not satisfied with what I had done and studied over the matter a great deal until in 1882, or thereabouts, I printed another paper on the same subject, dealing with it in a wholly different way.  Still I was not satisfied, and after many years more study, I determined to write a book expounding the logic of algebra and geometry.  This I did in two years of solid work at my lonely country-place where I could and did labor day and night upon it uninterruptedly.  Messrs. Ginn accepted the book for publication;  but I was not satisfied with it and rewrote it entirely, putting another year’s labor into it.  I am not altogether satisfied yet;  but still, as far as the part relating to number goes, I think the theory of it is as well as I can do.

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In the first place, it is necessary to understand the general nature of mathematics and its reasoning.  Mathematics, speaking broadly, is historically the earliest of the sciences.  Unless a collection of absurd medical prescriptions be counted for science, the earliest scientific treatise which has come down to us is on mathematics.  Pythagoras was a true mathematician, ages before there was any true physics or [psychology] or philosophy.  Astronomy became scientific very early;  but it used mathematics from the outset.

The morphologistic biologists tell us that the development of the individual is an epitome of the previous history of the development of the race.  Some great pedagogists make this principle the chief guide to a true system of education.  Even if this exaggerates its importance (as I humbly opine it does) yet there is something in it;  and Dr. Thomas Hill was no doubt right that the study of mathematics should antecede puberty.  A child is better fit by far to understand mathematics than anything else except mechanics;  and it is almost the only study which will remain a valuable accomplishment though life.

All other sciences without exception depend upon the principles of mathematics;  and mathematics borrows nothing from them but hints.

Mathematics also heads the list of sciences in the sense of being the most abstract.  It is more abstract than metaphysics or even than logic itself.  For mathematics is the only science which asserts nothing as a fact.  It does nothing but make hypotheses and deduce their consequences.  (See my article “The Regenerated Logic” in the Monist for October 1896, p. 23.)  That is the first thing which must be clearly apprehended in order to understand number.

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My father’s definition “mathematics is the science which draws necessary conclusions” at least implies the truth.  Modern logic shows that all necessary inference is really mathematical; and no inference could be necessary if it related to anything more than a hypothesis.

But the best definition is “mathematics is the science of hypotheses,” or of precise hypotheses.  For one important part of the mathematician’s business is to frame his hypothesis and to generalize it.  The drawing of conclusions about it is not all.