What are the meaning and the purpose of numbers? (Q2911593)

What is a number? The Pythagoreans had a clear answer: a number is a proper multiple of the unit. This was the basis of their number theory as laid down in Book VII of Euclid's \textit{Elements}. Since the early philosophers regarded the unit as indivisible, the Greeks did not develop a theory of fractions, although their astronomers (and, later, Diophantus) freely used them.NEWLINENEWLINEIt was also clear to the Greeks that lengths of line segments and the areas of polygons can be measured quantitatively. They stopped short of attaching ``numbers'' to geometric objects, and instead considered ratios of ``magnitudes of the same kind''. Examples of such ratios were the ratio of the diagonal and the side of a square, or that of the circumference and the diameter of a circle. It is not at all trivial to compare such ratios: the theory of ratios was developed by Eudoxus and is presented in Book V of Euclid's \textit{Elements}.NEWLINENEWLINEComparing certain propositions in Book V (on ratios of magnitudes) and Book VII (on numbers and their proportions) it becomes clear that the Greeks realized that ratios do behave a lot like proportions of numbers. It took many centuries of working with these objects until the ratio of the diagonal and the side of the square (or \(\pi\), the ratio of the circumference and the diameter of a circle) was accepted as some kind of ``number''.NEWLINENEWLINEThis vague concept of ``number'' sufficed for developing calculus. But then, Abel, Dirichlet, Cauchy, Weierstraß and others finally started putting calculus on a firm foundation by replacing arguments based on geometric intuition (a continuous function changing signs must have a root) by proofs. It was soon realized that such fundamental theorems depended crucially on the ``completeness of numbers'' (a notion which even Dedekind still called ``continuity'' (Stetigkeit)). Instead of accepting the completeness of the reals as a geometric ``axiom'', many mathematicians developed this result by extending the natural numbers step by step to the rational and then the real numbers. Perhaps the most familiar method of constructing the reals is the one used by Cantor, who considered Cauchy sequences of rational numbers.NEWLINENEWLINEDedekind's tool for constructing the real numbers out of the rationals is the ``Dedekind cut'', which he explained in his book [Stetigkeit und irrationale Zahlen. Braunschweig: F. Vieweg u. Sohn (1892; JFM 24.0248.02)]. The basic idea is that every real number \(\alpha\) defines a decomposition of the rational numbers into two disjoint subsets, those smaller than \(\alpha\) and those larger than \(\alpha\). Conversely, the real numbers may be constructed by looking at such subsets of the rationals. The core idea of Dedekind's theory is the same as that by Eudoxus in Book V of Euclid's \textit{Elements}, as Dedekind himself remarked in the preface of the book at hand: ``It is exactly this age-old conviction which is certainly the source of my theory [\(\ldots\)]''. In this respect, see [\textit{J. L.~Gardies}, ``Eudoxe et Dedekind'', Rev. Hist. Sci. 37, 111--125 (1984; Zbl 0571.01021)].NEWLINENEWLINEWith the construction of the real numbers from the rationals in place, Dedekind looked at the foundations of arithmetic, namely the natural numbers. This is what \textit{R. Dedekind}'s memoir [Was sind und was sollen die Zahlen? Braunschweig: Vieweg \& Sohn (1888; JFM 20.0049.05)] is all about. Dedekind introduced numbers using sets and functions in what was an extremely abstract approach at that time; Hilbert, upon his arrival in Berlin on a journey through various German university towns in 1888, recalled that ``in Berlin, I heard in all mathematical circles young and old talk -- mostly in a hostile sense -- about Dedekind's work `Was sind und was sollen die Zahlen?', which had just appeared''. This quotation from Hilbert's defense of the ``tertium non datur'' in his talk in Hamburg in 1930 [\textit{D. Hilbert}, ``Die Grundlegung der elementaren Zahlentheorie'', Math. Ann. 104, 485--494 (1931; Zbl 0001.26001; JFM 57.0054.04)] (see also ``Dedekind's analysis of number: systems and axioms'' by \textit{W.~Sieg} and \textit{D.~Schlimm} [Synthese 147, No. 1, 121--170 (2005; Zbl 1084.01008)]) is one out of many that show the attitude of Dedekind's contemporaries towards his abstract mathematics in general, and, in particular, towards the exposition found in ``Was sind und was sollen die Zahlen?''. In this respect see also Dedekind's correspondence with Keferstein published by \textit{M. A.~Sinaceur} [``L'infini et les nombres. Commentaires de R.~Dedekind à `Zahlen'. La correspondance avec Keferstein'', Rev. Hist. Sci. 27, 251--278 (1974; Zbl 0293.01030)].NEWLINENEWLINEThe negative reaction to Dedekind's memoir is also reflected in Dedekind's own remarks in the preface of the second edition of his booklet, which is reprinted here. In fact, to finitists like Kronecker it must have been an act of blasphemy that Dedekind defined the natural numbers with the help of infinite sets, whose existence he ``proved'' in a way that modern readers will find rather strange (see Satz 66). Perhaps Dedekind had the audacity to perform such an act because he regarded numbers as ``free creations of the human mind'' and not, as Kronecker did, as creations of God.NEWLINENEWLINEToday, Peano's approach to natural numbers is much better known than Dedekind's, and Dedekind's clarification of the notion of a number is interesting mainly to historians. I do not know how many of them read German (there exist English translations, in particular in [Essays on the theory of numbers. I. Continuity and irrational numbers. II. The nature and meaning of numbers. Chicago: The Open Court Publishing Co. London: Kegan Paul and Co. (1901; JFM 32.0185.01); Chicago: The Open Court Publishing Co. (1909; JFM 40.0220.02); New York: Dover Publications, Inc. (1963; Zbl 0112.28101); What are numbers and what should they be? RIM Monographs in Mathematics. Orono, ME: Research Institute for Mathematics (1995; Zbl 0826.01037)], and there are even translations into Russian [Kasan Ges. (2) 15, Nr. 2, 25--105 (1905; JFM 36.0087.04)] and Spanish [Madrid: Alianza Editorial. (1998; Zbl 1041.01012)]); but there must be even fewer who can read the present reissue, which is typeset in Gothic print. Perhaps Dedekind's ground-breaking book would have deserved more than a simple reprint.