Men of Mathematics

by E. T. Bell

  In later life Dedekind regretted that the mathematical instruction available during his student years at Göttingen, while adequate for the rather low requirements for a state teacher’s certificate, was inconsiderable as a preparation for a mathematical career. Subjects of living interest were not touched upon, and Dedekind had to spend two years of hard labor after taking his degree to get up by himself elliptic functions, modern geometry, higher algebra, and mathematical physics—all of which at the time were being brilliantly expounded at Berlin by Jacobi, Steiner, and Dirichlet. In 1852 Dedekind got his doctor’s degree (at the age of twenty one) from Gauss for a short dissertation on Eulerian integrals. There is no need to explain what this was: the dissertation was a useful, independent piece of work, but it betrayed no such genius as is evident on every page of many of Dedekind’s later works. Gauss’ verdict on the dissertation will be of interest: “The memoir prepared by Herr Dedekind is concerned with a research in the integral calculus, which is by no means commonplace. The author evinces not only a very good knowledge of the relevant field, but also such an independence as augurs favorably for his future achievement. As a test essay for admission to the examination I find the memoir completely satisfying.” Gauss evidently saw more in the dissertation than some later critics have detected; possibly his close contact with the young author enabled him to read between the lines. However, the report, even as it stands, is more or less the usual perfunctory politeness customary in accepting a passable dissertation, and we do not know whether Gauss really foresaw Dedekind’s penetrating originality.

  In 1854 Dedekind was appointed lecturer (Privatdozent) at Göttingen, a position which he held for four years. On the death of Gauss in 1855 Dirichlet moved from Berlin to Göttingen. For the remaining three years of his stay at Göttingen, Dedekind attended Dirichlet’s most important lectures. Later he was to edit Dirichlet’s famous treatise on the theory of numbers and add to it the epoch-making “Eleventh Supplement” containing an outline of his own theory of algebraic numbers. He also became a friend of the great Riemann, then beginning his career. Dedekind’s university lectures were for the most part elementary, but in 1857-8 he gave a course (to two students, Selling and Auwers) on the Galois theory of equations. This was probably the first time that the Galois theory had appeared formally in a university course. Dedekind was one of the first to appreciate the fundamental importance of the concept of a group in algebra and arithmetic. In this early work Dedekind already exhibited two of the leading characteristics of his later thought, abstractness and generality. Instead of regarding a finite group from the standpoint offered by its representation in terms of substitutions (see chapters on Galois and Cauchy), Dedekind defined groups by means of their postulates (substantially as described in Chapter 15) and sought to derive their properties from this distillation of their essence. This is in the modern manner: abstractness and therefore generality. The second characteristic, generality, is, as just implied, a consequence of the first.

  At the age of twenty six Dedekind was appointed (in 1857) ordinary professor at the Zurich polytechnic, where he stayed five years, returning in 1862 to Brunswick as professor at the technical high school. There he stuck for half a century. The most important task for Dedekind’s official biographer—provided one is unearthed—will be to explain (not explain away) the singular fact that Dedekind occupied a relatively obscure position for fifty years while men who were not fit to lace his shoes filled important and influential university chairs. To say that Dedekind preferred obscurity is one explanation. Those who believe it should leave the stock market severely alone, for as surely as God made little lambs they will be fleeced.

  Till his death (1916) in his eighty fifth year Dedekind remained fresh of mind and robust of body. He never married, but lived with his sister Julie, remembered as a novelist, till her death in 1914. His other sister, Mathilde, died in 1860; his brother became a distinguished jurist.

  Such are the bare facts of any importance in Dedekind’s material career. He lived so long that although some of his work (his theory of irrational numbers, described presently) had been familiar to all students of analysis for a generation before his death, he himself had become almost a legend and many classed him with the shadowy dead. Twelve years before his death, Teubner’s Calendar for Mathematicians listed Dedekind as having died on September 4, 1899, much to Dedekind’s amusement. The day, September 4, might possibly prove to be correct, he wrote to the editor, but the year certainly was wrong. “According to my own memorandum I passed this day in perfect health and enjoyed a very stimulating conversation on ’system and theory’ with my luncheon guest and honored friend Georg Cantor of Halle.”

  * * *

  Dedekind’s mathematical activity impinged almost wholly on the domain of number in its widest sense. We have space for only two of his greatest achievements and we shall describe first his fundamental contribution, that of the “Dedekind cut,” to the theory of irrational numbers and hence to the foundations of analysis. This being of the very first importance we may recall briefly the nature of the matter. If a, b are common whole numbers, the fraction a/b is called a rational number; if no whole numbers m, n exist such that a certain “number” N is expressible as m/n,y then N is called an irrational number. Thus are irrational numbers. If an irrational number be expressed in the decimal notation the digits following the decimal point exhibit no regularities—there is no “period” which repeats, as in the decimal representations of a rational number, say 13/11, = 1.181818 . . . , where the “18” repeats indefinitely. How then, if the representation is entirely lawless, are decimals equivalent to irrationals to be defined, let alone manipulated? Have we even any clear conception of what an irrational number is? Eudoxus thought he had, and Dedekind’s definition of equality between numbers, rational or irrational, is identical with that of Eudoxus (see Chapter 2).

  If two rational numbers are equal, it is no doubt obvious that their square roots are equal. Thus 2X3 and 6 are equal; so also then are and But it is not obvious that and hence that The un-obviousness of this simple assumed equality, taken for granted in school arithmetic, is evident if we visualize what the equality implies: the “lawless” square roots of 2, 3, 6 are to be extracted, the first two of these are then to be multiplied together, and the result is to come out equal to the third. As not one of these three roots can be extracted exactly, no matter to how many decimal places the computation is carried, it is clear that the verification by multiplication as just described will never be complete. The whole human race toiling incessantly through all its existence could never prove in this way that Closer and closer approximations to equality would be attained as time went on, but finality would continue to recede. To make these concepts of “approximation” and “equality” precise, or to replace our first crude conceptions of irrationals by sharper descriptions which will obviate the difficulties indicated, was the task Dedekind set himself in the early 1870’s—his work on Continuity and Irrational Numbers was published in 1872.

  The heart of Dedekind’s theory of irrational numbers is his concept of the “cut” or “section” (Schnitt): a cut separates all rational numbers into two classes, so that each number in the first class is less than each number in the second class; every such cut which does not “correspond” to a rational number “defines” an irrational number. This bald statement needs elaboration, particularly as even an accurate exposition conceals certain subtle difficulties rooted in the theory of the mathematical infinite, which will reappear when we consider the life of Dedekind’s friend Cantor.

  Assume that some rule has been prescribed which separates all rational numbers into two classes, say an “upper” class and a “lower” class, such that each number in the lower class is less than every number in the upper class. (Such an assumption would not pass unchallenged today by all schools of mathematical philosophy. However, for the moment, it may be regarded as unobjectionable.) On this assumption one of three mutually exclusive situations
is possible.

  (A) There may be a number in the lower class which is greater than every other number in that class.

  (B) There may be a number in the upper class which is less than every other number in that class.

  (C) Neither of the numbers (greatest in [A], least in [B]) described in (A), (B) may exist.

  The possibility which leads to irrational numbers is (C). For, if (C) holds, the assumed rule “defines” a definite break or “cut” in the set of all rational numbers. The upper and lower classes strive, as it were, to meet. But in order for the classes to meet the cut must be filled with some “number,” and, by (C), no such filling is possible.

  Here we appeal to intuition. All the distances measured from any fixed point along a given straight line “correspond” to “numbers” which “measure” the distances. If the cut is to be left unfilled, we must picture the straight line, which we may conceive of as having been traced out by the continuous motion of a point, as now having an unbridgeable gap in it. This violates our intuitive notions, so we say, by definition, that each cut does define a number. The number thus defined is not rational, namely it is irrational. To provide a manageable scheme for operating with the irrationals thus defined by cuts (of the kind [C]) we now consider the lower class of rationals in (C) as being equivalent to the irrational which the cut defines.

  One example will suffice. The irrational square root of 2 is defined by the cut whose upper class contains all the positive rational numbers whose squares are greater than 2, and whose lower class contains all other rational numbers.

  If the somewhat elusive concept of cuts is distasteful two remedies may be suggested: devise a definition of irrationals which is less mystical than Dedekind’s and fully as usable; follow Kronecker and, denying that irrational numbers exist, reconstruct mathematics without them. In the present state of mathematics some theory of irrationals is convenient. But, from the very nature of an irrational number, it would seem to be necessary to understand the mathematical infinite thoroughly before an adequate theory of irrationals is possible. The appeal to infinite classes is obvious in Dedekind’s definition of a cut. Such classes lead to serious logical difficulties.

  It depends upon the individual mathematician’s level of sophistication whether he regards these difficulties as relevant or of no consequence for the consistent development of mathematics. The courageous analyst goes boldly ahead, piling one Babel on top of another and trusting that no outraged god of reason will confound him and all his works, while the critical logician, peering cynically at the foundations of his brother’s imposing skyscraper, makes a rapid mental calculation predicting the date of collapse. In the meantime all are busy and all seem to be enjoying themselves. But one conclusion appears to be inescapable: without a consistent theory of the mathematical infinite there is no theory of irrationals; without a theory of irrationals there is no mathematical analysis in any form even remotely resembling what we now have; and finally, without analysis the major part of mathematics—including geometry and most of applied mathematics—as it now exists would cease to exist.

  The most important task confronting mathematicians would therefore seem to be the construction of a satisfactory theory of the infinite. Cantor attempted this, with what success will be seen later. As for the Dedekind theory of irrationals, its author seems to have had some qualms, for he hesitated over two years before venturing to publish it. If the reader will glance back at Eudoxus’ definition of “same ratio” (Chapter 2) he will see that “infinite difficulties” occur there too, specifically in the phrase “any whatever equimultiples.” Nevertheless some progress has been made since Eudoxus wrote; we are at least beginning to understand the nature of our difficulties.

  * * *

  The other outstanding contribution which Dedekind made to the concept of “number” was in the direction of algebraic numbers. For the nature of the fundamental problem concerned we must refer to what was said in the chapter on Kronecker concerning algebraic number fields and the resolution of algebraic integers into their prime factors. The crux of the matter is that in some such fields resolution into prime factors is not unique as it is in common arithmetic; Dedekind restored this highly desirable uniqueness by the invention of what he called ideals. An ideal is not a number, but an infinite class of numbers, so again Dedekind overcame his difficulties by taking refuge in the infinite.

  The concept of an ideal is not hard to grasp, although there is one twist—the more inclusive class divides the less inclusive, as will be explained in a moment—which shocks common sense. However, common sense was made to be shocked; had we nothing less dentable than shock-proof common sense we should be a race of mongoloid imbeciles. An ideal must do at least two things: it must leave common (rational) arithmetic substantially as it is, and it must force the recalcitrant algebraic integers to obey that fundamental law of arithmetic—unique decomposition into primes—which they defy.

  The point about a more inclusive class dividing a less inclusive refers to the following phenomenon (and its generalization, as stated presently). Consider the fact that 2 divides 4—arithmetically, that is, without remainder. Instead of this obvious fact, which leads nowhere if followed into algebraic number fields, we replace 2 by the class of all its integer multiples, . . . , −8, −6, −4, −2, 0, 2, 4, 6, 8, . . . As a matter of convenience we denote this class by (2). In the same way (4) denotes the class of all integer multiples of 4. Some of the numbers in (4) are . . ., −16, −12, .−8, −4, 0, 8, 12, 16, . . . It is now obvious that (2) is the more inclusive class; in fact (2) contains all the numbers in (4) and in addition (to mention only two) −6 and 6. The fact that (2) contains (4) is symbolized by writing (2) |(4). It can be seen quite easily that if m, n are any common whole numbers then (m) |(n) when, and only when, m divides n.

  This might suggest that the notion of common arithmetical divisibility be replaced by that of class inclusion as just described. But this replacement would be futile if it failed to preserve the characteristic properties of arithmetical divisibility. That it does so preserve them can be seen in detail, but one instance must suffice. If m divides n, and n divides /, then m divides /—for example, 12 divides 24 and 24 divides 72, and 12 does in fact divide 72. Transferred to classes, as above, this becomes: if (m)|(n) and (n)|(l), then (m)|(l) or, in English, if the class (m) contains the class (n), and if the class (n) contains the class (l), then the class (m) contains the class (l)—which obviously is true. The upshot is that the replacement of numbers by their corresponding classes does what is required when we add the definition of “multiplication”: (m) × (n) is defined to be the class (mn); (2) × (6) = (12). Notice that the last is a definition; it is not meant to follow from the meanings of (m) and (n).

  Dedekind’s ideals for algebraic numbers are a generalization of what precedes. Following his usual custom Dedekind gave an abstract definition, that is, a definition based upon essential properties rather than one contingent upon some particular mode of representing, or picturing, the thing defined.

  Consider the set (or class) of all algebraic integers in a given algebraic number field. In this all-inclusive set will be subsets. A subset is called an ideal if it has the two following properties.

  A. The sum and difference of any two integers in the subset are also in the subset.

  B. If any integer in the subset be multiplied by any integer in the all-inclusive set, the resulting integer is in the subset.

  An ideal is thus an infinite class of integers. It will be seen readily that (m), (n), . . . , previously defined, are ideals according to A, B. As before, if one ideal contains another, the first is said to divide the second.

  It can be proved that every ideal is a class of integers all of which are of the form

  x1a1 + x2a2 + . . . + snan,

  where a1, a2, . . . , an are fixed integers of the field of degree n concerned, and each of x1 x2, . . . , xn may be any integer whatever in the field. This being so, it is convenient to symbolize
an ideal by exhibiting only the fixed integers a1, a2, . . . , an, and this is done by writing (a1, a2, . . . , an) as the symbol of the ideal. The order in which a1 a2, . . ., an are written in the symbol is immaterial.

  “Multiplication” of ideals must now be defined: the product of the two ideals (a1, . . . , an), (b1 . . . , bn) is the ideal whose symbol is (a1b1, . . . , a1 bn, . . . , a1bn), in which all possible products a1b1, etc., obtained by multiplying an integer in the first symbol by an integer in the second occur. For example, the product of (a1, a2) and (b1, b2) is (a1b1, a1b2, a2b1, a2b2). It is always possible to reduce any such product-symbol (for a field of degree n) to a symbol containing at most n integers.

  One final short remark completes the synopsis of the story. An ideal whose symbol contains but one integer, such as (a1), is called a principal ideal. Using as before the notation (a1)|(b1) to signify that (a1) contains (b1), we can see without difficulty that (a1)|(b1) when, and only when, the integer a1 divides the integer b1. As before, then, the concept of arithmetical divisibility is here—for algebraic integers—completely equivalent to that of class inclusion. A prime ideal is one which is not “divisible by”—included in—any ideal except the all-inclusive ideal which consists of all the algebraic integers in the given field. Algebraic integers being now replaced by their corresponding principal ideals, it is proved that a given ideal is a product of prime ideals in one way only, precisely as in the “fundamental theorem of arithmetic” a rational integer is the product of primes in one way only. By the above equivalence of arithmetical divisibility for algebraic integers and class inclusion, the fundamental theorem of arithmetic has been restored to integers in algebraic number fields.