In the article above referred to, Schubert claims that in his System of Arithmetic, Potsdam, 1885, he "was the first to work out the idea referred to fully and logically, and in a form comprehensible for beginners;" although it had been previously expressed by Grassman, Hankel, E. Schroeder, and Kronecker. Such is the bibliography of this special presentation of the subject, so far as I am aware, not to mention Dr. Halsted's Number, Discrete and Continuous,* whose title promises a treatment of this subject, but which remains a fragment, dealing only with discrete number what I have called Primary Number. Should I be able to spur Dr. Halsted to a completion of this work I shall not have written in vain. Of course, Hankel's principle must be expounded in his Theorie der complexen Zahlansysteme, Leipzig, 1867; but I am yet ignorant of the specific publications of the other authors named, except that in Zeller's jubilee work the matter is referred to in an essay by Kronecker. On the other hand, the theory here advocated must not be deemed retrogressive, and referred to such writers as Frend,† who, though he very philosophically maintains that, since algebra has its origin and termination in arithmetic, it cannot be considered independent, and fairly enough regards algebra as "the science which teaches the general properties and relations of numbers," yet ends by practically throwing the greater part of the science of number overboard, in rejecting all algebraic forms which do not agree with his undeveloped concept of number. My theme may be regarded as the underlying harmony * Preface and four chapters (22 pp.) in Scientiae Baccalaureus, June, 1891. † Algebra, 1796. of the great makers of analytical mathematics, and my purpose, as an attempt to present to beginners fundamental theory commonly left for the speculations of the most advanced. Number is such a perfect and typical abstraction that it is difficult to see how a man who has, to use Newton's phrase, "in philosophical matters a competent faculty of thinking," could ever associate the terms concrete and number; nevertheless this confusion muddles many popular text-books. The question hardly requires or admits of argument. Since it is a vicious habit rather than an illogical deduction which is to be combated, good-tempered ridicule is perhaps the only fit rejoinder. In this spirit may I be permitted to relate an anecdote? Some years ago at the University of Virginia the Professor of Mathematics assigned several problems to be worked upon the blackboards by members of the Junior Class. To one he gave a problem concerning the number of oranges in a pyramidal pile of stated proportions. After expounding the error or propriety of the solutions of some of the other problems, the turn of the orange problem came. The student stood proudly beside his mechanically correct solution. "Well, Mr. Blank," exclaimed the Professor, "how many apples did you find?" A look of consternation overspread the youth's countenance. With a gesture of impatient annoyance he swept the erasing brush over the figures his chalk pencil had traced: "Oh," said he, "I thought you said oranges s! In all seriousness, the text-books we have all been abused by, expounding "concrete numbers," solemnly cautioning against confusion of multiplicand and multiplier, divisor and quotient, and unallowable combinations of the terms of a numerical proportion, are quite as ridiculous as our hero of the oranges. He displayed at least one virtue, - consistency. Such questions, however, though of great practical importance to the efficiency of our elementary schools, present no real difficulties. A little knowledge of psychology and mathematics will, if attention be called to the question, correct mistaken, and develop inchoate concepts of Primary Number. A far more difficult matter remains to attain for ourselves, and to lead our pupils to attain, a rational concept of number as continuous, a concept absolutely essential to modern mathematics, and now universally assumed as a fact, — implicitly so assumed, even when explicitly denied. It is also necessary to pass beyond the great step already made by Newton, who discerns the continuity of number, but leaves it only "triplex": "Estque (Numerus) triplex; integer, fractus, et surdus: Integer quem unitas metitur, Fractus quem unitatis pars submultiplex metitur, et Surdus cui unitas est incommensurabilis "*, with the implied limits zero and infinity. Newton also recognized qualitative distinctions, positive and negative, but the consequent neomonic (so-called "imaginary ") and complex numbers remain to be assimilated. I must return, however, to notice a uniquely erroneous view of primary number presented in the last issue of the INTERNATIONAL EDUCATION SERIES, The Psychology of Number, by James A. McLellan, A.M., LL.D., Principal of the Ontario School of Pedagogy, Toronto, and John Dewey, Ph.D., Head Professor of Philosophy in the University of Chicago, edited like all of the series by W. T. Harris, U. S. Commissioner of Education. The astounding thesis is maintained that number is not a * Arithmetica universalis: quoted from Halsted's Number, Discrete and Continuous. Of course, magnitude, does not possess quantity at all, and that "no number can be multiplied or divided into parts."* The authors vehemently assert that we might as well talk of any absurdity "as to talk of multiplying a number."† It is much to be regretted that a work of such prestige should merely shift the misconception of concreteness from numbers to the subjects of calculation, which we are told to believe are never numbers at all. Number is most emphatically shown to be "purely abstract," yet multiplication is claimed to be only of concretes. It is nonsense, we are told, to think of multiplying six by four; you can only multiply six inches, six oranges, by four. that numbers are multiplied is a fact, a fact that psychology may explain, but can in no wise question. After repeatedly insisting upon "the absurdity of multiplying pure number or dividing it into parts," § the authors admit without comment, and in seeming hesitation, "of course, in all mathematical calculations we ultimately operate with pure symbols." || What are these "pure symbols"? What can they be in arithmetic but the pure numbers themselves? It would be an error, shared by many algebraists, to conceive algebra as lacking specific content as operating with "pure symbols," whatever that may mean. The chapter on the Psychical Nature of Number is admirable, and I gratefully invoke its corroboration of what will be found in my syllabus on the subject; but that upon the Origin of Number, though very acute in tracing the dependence of measurement upon "adjustment of activity," seems to me mistaken in finding the origin of number in * Psychology of Number, p. 70. § Ib., p. 71, foot-note. measurement. Measurement is not the source of the concept of number, but a stimulation to clarify and develop the concept; and this is what the facts cited really show. The primary concept of number, as so correctly defined in the preceding chapter, is prerequisite to any attempt at measurement. The savage referred to needs the concept that the length of his arrow is some number of handbreadths before he can attempt to discover how many. And long before this he has learned to recognize a small group of objects as a "vague whole," and to "discriminate the distinct individuals,” i.e., in the very terms employed to define number, the concept originated before any measurement became possible. Nor, in truth, does number originate in counting, as so commonly asserted; and for a like reason, viz., the concept of some number must precede any device for naming or anywise specializing it. The position that number has its origin in measurement cannot seek strength from the procession toward the absolute of Hegel's ascending categories, quality, quantity (including number, as "quantum in its complete specialization"), measure (das Maass), essence; for Hegel's Maass, i.e., "qualitative quantity or measure," is a very different matter from Dr. Dewey's measurement, in fact, it seems very nearly the same as number according to the growing insight of modern mathematics. * Having mentioned Hegel, it is proper to remark that we are just now being reminded on every side - - Helmholtz not long ago admonished us that students of science are frequently driven by the very logic of their subjects into *The Logic of Hegel, translation, Wallace, p. 192 ("quantum, i.e., limited quantity," p. 190). † Ib., p. 200. |