aa r X i v : . [ m a t h . HO ] J a n RICHARD DEDEKIND: STYLE AND INFLUENCE
STEFAN M ¨ULLER-STACH
Abstract.
This text is based on an invited talk at the Dedekind memorialconference at Braunschweig in October 2016. It summarizes views from [7]. Intro
The monograph [7] contains a commented edition of the two books “Stetigkeit undIrrationale Zahlen” [3] (1872) and “Was sind und was sollen die Zahlen?” [4] (1888)by Richard Dedekind. Both books were conceived much earlier, the first in the late1850’s as a preparation for lectures in Z¨urich and the second in the 1870’s, ashandwritten manuscripts in the G¨ottingen archives show.Hilbert reported – after discussing with other mathematicians – about the hostile(“gegnerische”) reception of Dedekind’s 1888 book [7, Vorwort]:“Im Jahre 1888 machte ich als junger Privatdozent von K¨onigsbergaus eine Rundreise an die deutschen Universit¨aten. Auf meiner er-sten Station, in Berlin, h¨orte ich in allen mathematischen Kreisenbei jung und alt von der damals erschienenen Arbeit Dedekinds,,Was sind und was sollen die Zahlen?” sprechen – meist in geg-nerischem Sinne. Die Abhandlung ist neben der Untersuchung vonFrege der wichtigste erste tiefgreifende Versuch einer Begr¨undungder elementaren Zahlenlehre.”At the time, mathematicians had similar problems while digesting Dedekind’sequally influential and innovative work on Galois theory and algebraic numbertheory. Dedekind was so much ahead of his time in his research that some of hisachievements reach without any necessary “historical transformation” into the 20thcentury. The editors of Dedekind’s collected works (Fricke, Noether, Ore) confirmedthis view in 1932 [5, Nachwort, Vol. 3]:“Es ist ein Zeichen, wie Dedekind seiner Zeit voraus war, daß seineWerke noch heute lebendig sind, ja daß sie vielleicht erst heute ganzlebendig geworden sind.”Dedekind’s style has many implications for the reception of his work, as we will tryto explain here. 2.
Character and style
Dedekind had a very close relationship with his whole family. He lived with hismother and his sister Julie in their family’s home until Julie’s death in 1914. Weimagine the siblings spending hours together reading and talking. The family owneda weekend retreat in Harzburg where they used to meet with other friends andwent for hiking tours. Richard was close to the family of his brother Adolf aswell and supported them during a period of health problems of his nephew Atta.
These strong ties made Dedekind never leave his hometown Braunschweig for otheruniversities after he had returned from his first position in Z¨urich.Dedekind had an impressive and unique style in his works and showed a generouscharacter in his communication with colleagues. He characterized his own mathe-matical style in the following way [4, Preface]:“Was beweisbar ist, soll in der Wissenschaft nicht ohne Beweisgeglaubt werden ...Indem ich die Arithmetik (Algebra, Analysis) nur einen Teil derLogik nenne ...Die Zahlen sind freie Sch¨opfungen des menschlichen Geistes ...”The second sentence shows that he considered himself being a logicist (logicismwas originated by Frege as a program to reduce mathematics to logic). Dedekind’swords sound quite dramatic in a way. Indeed, the times in which he lived weretimes of major changes in science. Epple called the underlying phenomenon “DasEnde der Gr¨oßenlehre” [6]. The invention of set-theory by Cantor and Dedekindwas a scientific revolution, comparable to the invention of Riemannian manifoldsor Galois theory.The occurence of the word
Sch¨opfungen (i.e., creations ) is remarkable. There arethree areas where Dedekind employed this technique. First, in an unpublishedmanuscript “Die Sch¨opfung der Null und der negativen ganzen Zahlen” from 1872,in which he constructs Z and Q from N using equivalence relations. Then, in the1872 book “Stetigkeit und Irrationale Zahlen” [3], where he invented Dedekind cuts,and finally in his creation of “ideal numbers” through ideals in number rings. Inall three cases, new numbers are defined as sets. Divisibility and other conceptsarise from set-theoretic operations. Dedekind saw a step of abstraction in definingthe new numbers as single objects, therefore he used the phrase “Sch¨opfung”. Hedefended this viewpoint in a 1888 letter to H. Weber [8, p. 276].Dedekind’s mathematical style was characterized by other people as “axiomatique,formel et abstrait” (Sinaceur 1974) and as “built on traditions, abstract und Bour-baki style” (Scharlau 1981). Similarities to the Bourbaki group and eventually toGrothendieck are quite important. Dedekind prefered to study mathematics in amore formal way by using mappings instead of element-wise considerations andtheoretical concepts instead of examples. In this sense, he was quite different frommathematicians of his time.I would like to add that his style was very elegant, always concise and well-readablefor a modern reader, so also in this respect Dedekind was far ahead of his time. This“modernness” can also be expressed by saying that there is almost no “historicaltransformation” necessary in order to translate his ideas into our times. Dedekindpublished only relatively few papers after a rather long time of thinking. He wouldprobably hate the “modern” rapid publication style with its enormous output.A programmatic document for Dedekind’s approach to mathematics is his remark-able Habilitationsvortrag from 1854 [5, Vol. 3, p. 428]. In it he describes first somebasics of scientific theories in general and then proceeds to set up his program forthe creation of new objects and theories in mathematics.Some years later, Dedekind clearly distinguishes his own views from those of othersin the preface of his 1888 book [4]:“Das Erscheinen dieser Abhandlungen [von Helmholtz und Kro-necker] ist die Veranlassung, die mich bewogen hat, nun auch mit
ICHARD DEDEKIND: STYLE AND INFLUENCE 3 meiner, in mancher Beziehung ¨ahnlichen, aber durch ihre Begr¨un-dung doch wesentlich verschiedenen Auffassung hervorzutreten, dieich mir seit vielen Jahren und ohne jede Beeinflussung gebildethabe.”Dedekind’s personal style is present in his correspondence and relationships. Theletters to Cantor and others [1, 8] show – in their friendly and open way of sharingcommon knowledge – many positive features of the personality of Dedekind. Manyletters to his family were published by Ilse Dedekind [2]. They prove that the oldercolleagues Gauß, Riemann and Lejeune Dirichlet were Dedekind’s “Vorbilder” inG¨ottingen.In particular, Dedekind was close to Dirichlet, which eventually led to his edition ofDirichlet’s number theory lectures with the famous appendix (the “supplement”)on algebraic number theory. Dirichlet seems to have influenced Dedekind also infoundational matters in the way he prefered conceptual mathematics to examples(sometimes called the “second Dirichlet principle”) and in the idea of reducingmathematics to the theory of natural numbers [7, § intuitionism and constructivism in mathematics.After Cantor’s unthoughtful publication of some common ideas about the count-ability of algebraic numbers without mentioning Dedekind, thus violating scientificstandards boldly, Dedekind did not react anymore to Cantor’s letters for a certainperiod from 1874 on, most likely because of this instance. A letter from Dedekind’snephew Atta to Julie from 1887 [2, p. 223] proves that this was a well-known issuein the entire Dedekind family. In 1882 both met again, but even after this, as the1888 letter to H. Weber [8, p. 276] shows, Dedekind was not happy with Cantor’sattitude. In a letter to Hilbert from 1899 [7, § Axioms for N and recursion theory Dedekind found appropriate “axioms” for the natural numbers N (without callingthem axioms) and constructed models of N using some portion of his newly inventedelementary set-theory.Dedekind considered an infinite set X together with an injective self-map (“succes-sor map”) S : X → X and a distinguished element 0 ∈ X . We write ( X, , S ) for atriple with these properties. A chain is a subset C ⊂ X such that C is stable under S , i.e., S ( C ) ⊂ C . A chain containing 0 is called simply-infinite if it is the chain of STEFAN M¨ULLER-STACH all successors of 0, i.e., the intersection of all chains containing 0: \ C chain , ∈ C C = { , S (0) , S ( S (0)) , . . . } . Simply-infinite chains are determined uniquely by the triple ( X, , S ). For Dedekind,a model N for the natural numbers is a simply-infinite chain arising from a triple( X, , S ) where the set X may be arbitrarily large. Let us denote the restrictionof the map S to the simply-infinite set by ( N , , S ). Dedekind thus obtained thefollowing axiomatic characterization of the natural numbers: Dedekind’s Axioms (Nr. 71 in Dedekind 1888 [4])The triple ( N , , S ) satisfies:(D1) 0 is not contained in the image of S .(D2) S is injective.(D3) (Induction) The set N is simply-infinite, i.e., N = \ C chain , ∈ C C = { , S (0) , S ( S (0)) , . . . } . The following equivalent version is used by most authors [7, §
2] :
Dedekind-Peano Axioms
The triple ( N , , S ) satisfies:(DP1) 0 is not contained in the image of S .(DP2) S is injective.(DP3) (Induction) For any subset M ⊂ N such that(i) 0 ∈ M .(ii) n ∈ M ⇒ S ( n ) ∈ M one has M = N .Both descriptions above use quantification over subsets of N , hence are formu-lated using a second-order logic framework. Of course, Dedekind did not knowabout model theory in the modern sense, and the distinction between syntax andsemantics was not common then, but some of his remarks indicate that he tookcare to rule out possible non-standard models with undesired properties once someaxioms would be relaxed. A posteriori, assuming a first-order logic framework, weknow that (many) non-standard models indeed exist.The most important result in Dedekind’s book [4] is the famous recursion theorem .It implies the categoricity and semantical completeness of N . These two notionsmean that all models of N are isomorphic and any statement which holds for onemodel holds for any other model too. In modern form, the recursion theorem states: Recursion Theorem (Nr. 126 in Dedekind 1888 [4])Let Ω be a set together with a self-map θ : Ω → Ω und ω ∈ Ω. Then, there is one
ICHARD DEDEKIND: STYLE AND INFLUENCE 5 and only one map Ψ : N → Ω, such that the diagram N S (cid:15) (cid:15) Ψ / / Ω θ (cid:15) (cid:15) N Ψ / / Ωcommutes, i.e., such that Ψ(0) = ω, Ψ( S ( x )) = θ (Ψ( x )) . There are more general versions, where the function θ depends on two variables andthe rules are Ψ(0) = ω and Ψ( S ( x )) = θ ( x, Ψ( x )) [7, § §
14 in the proof of Satz 159.It leads naturally to the definition of “new” functions f = rec( g, h ) : N n +1 → N by primitive recursion [7, § f (0 , y , . . . , y n ) = g ( y , . . . , y n ) f ( S ( x ) , y , . . . , y n ) = h ( x, f ( x, y , . . . , y n ) , y , . . . , y n )from given functions g, h . General recursion theory (alias the theory of computation )needs one additional concept, called the (unbounded) search operator µ , in orderto obtain all computable (alias partial recursive ) functions.The invention of recursion theory is often attributed to G¨odel or Skolem. How-ever, it was Dedekind who first brought the recursive thinking to its full powerby proving the recursion theorem. Modern recursion theory owes to the workof Church, G¨odel, Herbrand, Hilbert (with Ackermann, Bernays), Grzegorczyk,K´alm´ar, Kleene, P´eter, Post, Skolem, Turing et al. [7, § Reception
The reception of Dedekind’s work is partly a story of misunderstandings, whichhad both positive and negative effects.Keferstein’s critical, but mathematically not well-founded, comments on the 1888book led to a famous letter of Dedekind [7, 10] which explains Dedekind’s motiva-tions and some ideas very well, including the potential existence of non-standardmodels of the natural numbers.The “wrong” proof of Satz 66 in Dedekind’s 1888 book [4], in which he claimed theexistence of an infinite set by refering to non-mathematical objects (“thoughts”),was affected by the antinomies in set theory. Those eventually led to a crisis(“Grundlagenkrise”) in the foundations of mathematics after 1900. Dedekind alsoused the axiom of choice in his 1888 book. In contrast to the case of Frege, wedo not know in detail how Dedekind thought and felt about the antinomies afterhis book was criticized on their basis. The only witness for this is Felix Bernsteinwho remembered that Dedekind told him that he had almost started to doubtthe rationality of human thinking [5, Vol. 3, p. 449]. Nowadays, one postulatesthe existence of an infinite set and much of the foundational crisis seems to beexaggerated in retrospect. The antinomies had as an effect a lesser citation ofDedekind’s work, for example by Hilbert and others, even though his work was
STEFAN M¨ULLER-STACH silently incorporated in contemporary mathematics. In recent years, Dedekind’sposition in the foundations of mathematics is becoming readjusted again.The beginnings of set theory were quite “constructive” with rather “concrete” sets.There was no axiomatic method as in the later approach of Zermelo, who was a ded-icated follower of Dedekind and called the axiom of infinity after him. Dedekind’sletters with Cantor [1] had a strong influence on the notions of cardinality anddimension in early set theory, reaching as far as to Brouwer’s 1911 solution ofthe dimension problem for R n as conjectured by Dedekind, i.e., that there is nohomeomorphism between R n and R m unless m = n .There are also the letters to Lipschitz from 1876 [5, Band 3, p. 468-479] in whichDedekind explained very well, and way before Hilbert and Peano, the essence ofthe axiomatic method. In addition, he made clear that the axiom of continuity isnot part of Euklid’s considerations, but rather a property of the ground field inquestion. Frege, Hilbert and Peano used the axiomatic method in a more conse-quent way, sometimes even without any reference to concrete models, e.g. in thecase of Hilbert’s axioms for the real numbers. In contrast, Dedekind required theset-theoretic existence at the beginning of any new creation (“Sch¨opfung”) of amathematical structure.Most surprising to me was Skolem’s rejection of Dedekind’s achievements in recur-sion theory [7, § Influence in our times
Dedekind had a very strong influence on the development of algebraic numbertheory through the famous supplement [5, Vol. 3] to Dirichlet’s lectures on numbertheory, in which he developed algebraic number theory from scratch, using ideas ofGalois and Kummer. His presentation became a raw model for most number theorylectures and books to follow. Hilbert, in his “Zahlbericht”, promoted Dedekind’stheory even more. Subsequent developments include class field theory and latergeneralizations of the class number formula.There is a remarkable analogy between algebraic number theory and the theory offunction fields of one variable. The latter was developed by Dedekind and H. Weber[5, Vol. 1, p. 238] in 1882 as well as independently by Kronecker in his “Festschrift”(1882) for Kummer. Dedekind wrote a small unpublished paper “Bunte Bemerkun-gen” on his readings of Kronecker’s work. From a modern perspective, both ap-proaches are equivalent. The unification of algebraic number theory and the theoryof function fields, which was intended by both parties, reaches out into the 20thcentury to Grothendieck’s “marriage” of algebraic and arithmetic geometry via histheory of schemes and stacks. However, history has seen a lot of rivalry betweenfollowers of Dedekind and Kronecker. Weil mentions such a “partisan war” in his1954 ICM address and strengthens the role of Kronecker.
ICHARD DEDEKIND: STYLE AND INFLUENCE 7
Emmy Noether adored Dedekind. This is apparent from her numerous commentsin the collected works of Dedekind and her famous sentence “Es steht alles schonbei Dedekind”, as witnessed by her student van der Waerden. This shows that shemust have been on Dedekind’s side in the partisan issue.Dedekind’s 1888 book [4] set the path for the axiomatic treatment of arithmetic,nowadays called
Dedekind-Peano arithmetic . Recursion is also used in Hilbert’s fini-tistic proof theory program as a metamathematical tool. Furthermore, Dedekind’srecursion theorem is related to G¨odel’s two incompleteness theorems via the useof primitive recursive functions and (via transfinite or higher type versions) to theconsistency proofs for arithmetic of Gentzen and G¨odel, hence to modern prooftheory in general [7, § Hilbert problems ,especially in Nr. 1 (continuum hypothesis), Nr. 2 (consistency of arithmetic), Nr.7 (trancendence theory), Nr. 9 (reciprocity laws), Nr. 10 (diophantine equations),Nr. 11 (quadratic forms), Nr. 12 (generalized Kronecker-Weber theorem) andNr. 24 (proof theory). Among the Millenium problems of the Clay Foundationthe problem “P versus NP” is very close to recursion theory and the Birch andSwinnerton-Dyer conjecture to algebraic number theory.Finally, the unpublished manuscript “Sch¨opfung der Null und der negativen ganzenZahlen” (1872) may be viewed as being pointing to K-theory and the theory of mo-tives of Grothendieck. Those theories are related to extensions of the class numberformula of Dirichlet and Dedekind, e.g. Beilinson’s conjectures and their variants,and are vital areas of research in modern arithmetic and algebraic geometry. Thetheory of automorphic L-functions and the Langlands program can be seen as avast extension of Dedekind’s ζ -functions and the theory of modular forms. References [1] Jean Cavaill`es, Emmy Noether (Hrsg.): Briefwechsel Cantor-Dedekind, Paris (1937).[2] Ilse Dedekind: Unter Glas und Rahmen, Appelhans Verlag, Braunschweig (2000).[3] Richard Dedekind: Stetigkeit und Irrationale Zahlen, 7. Auflage, Vieweg Verlag, Braun-schweig (1965). Erste Auflage 1872.[4] Richard Dedekind: Was sind und was sollen die Zahlen?, 10. Auflage, Vieweg Verlag, Braun-schweig (1965). Erste Auflage 1888.[5] Richard Dedekind: Gesammelte Werke (Fricke, Noether, Ore (Hrsg.)), 3 B¨ande, ViewegVerlag, Braunschweig (1930).[6] Moritz Epple: Das Ende der Gr¨oßenlehre: Grundlagen der Analysis 1860-1910, Kapitel 10in: Geschichte der Analysis (Jahnke (Hrsg.)), Spektrum Verlag, Heidelberg, 371-410 (1999).[7] Stefan M¨uller-Stach (Hrsg.): Richard Dedekind: ,,Stetigkeit und Irrationale Zahlen” (1872)and ,,Was sind und was sollen die Zahlen?” (1888), edition with commentaries, KlassischeTexte in der Mathematik, Springer Verlag (2017).[8] Katrin Scheel: Der Briefwechsel Richard Dedekind – Heinrich Weber, de Gruyter Verlag(2014).[9] Jens Erik Fenstad (ed.): Selected Works in Logic by Thoralf Skolem, UniversitetsforlagetOslo, 499-514 (1970).[10] Christian Tapp (Hrsg.): Richard Dedekind: Brief an Keferstein / Letter to Keferstein, seearXiv:1611.10065 (2016).[1] Jean Cavaill`es, Emmy Noether (Hrsg.): Briefwechsel Cantor-Dedekind, Paris (1937).[2] Ilse Dedekind: Unter Glas und Rahmen, Appelhans Verlag, Braunschweig (2000).[3] Richard Dedekind: Stetigkeit und Irrationale Zahlen, 7. Auflage, Vieweg Verlag, Braun-schweig (1965). Erste Auflage 1872.[4] Richard Dedekind: Was sind und was sollen die Zahlen?, 10. Auflage, Vieweg Verlag, Braun-schweig (1965). Erste Auflage 1888.[5] Richard Dedekind: Gesammelte Werke (Fricke, Noether, Ore (Hrsg.)), 3 B¨ande, ViewegVerlag, Braunschweig (1930).[6] Moritz Epple: Das Ende der Gr¨oßenlehre: Grundlagen der Analysis 1860-1910, Kapitel 10in: Geschichte der Analysis (Jahnke (Hrsg.)), Spektrum Verlag, Heidelberg, 371-410 (1999).[7] Stefan M¨uller-Stach (Hrsg.): Richard Dedekind: ,,Stetigkeit und Irrationale Zahlen” (1872)and ,,Was sind und was sollen die Zahlen?” (1888), edition with commentaries, KlassischeTexte in der Mathematik, Springer Verlag (2017).[8] Katrin Scheel: Der Briefwechsel Richard Dedekind – Heinrich Weber, de Gruyter Verlag(2014).[9] Jens Erik Fenstad (ed.): Selected Works in Logic by Thoralf Skolem, UniversitetsforlagetOslo, 499-514 (1970).[10] Christian Tapp (Hrsg.): Richard Dedekind: Brief an Keferstein / Letter to Keferstein, seearXiv:1611.10065 (2016).