What are numbers?Dedekind’s introduction of natural numbers in Was sind und

was sollen die Zahlen? (1888) as opposed to Frege’s in

Grundlagen der Arithmetik (1884).

Lorenzo Azzano

July 2014

Contents1 Introduction 2

2 Logicism and the rigorization of analysis 3

3 Some preliminary differences between Frege and Dedekind 11

4 Two approaches to natural numbers 174.1 Frege on natural numbers . . . . . . . . . . . . . . . . . . . . 174.2 Dedekind on natural numbers . . . . . . . . . . . . . . . . . . 24

5 Problematics in Dedekind’s account of naturals 28

6 Dedekind as a structuralist 356.1 Set-theoretical structuralism . . . . . . . . . . . . . . . . . . . 356.2 Eliminativist structuralism . . . . . . . . . . . . . . . . . . . . 376.3 Ante rem structuralism . . . . . . . . . . . . . . . . . . . . . . 386.4 Reck’s logical structuralism . . . . . . . . . . . . . . . . . . . 41

7 Conclusions 447.1 The best possible Dedekind . . . . . . . . . . . . . . . . . . . 457.2 Final comparison with Frege . . . . . . . . . . . . . . . . . . . 47

8 Bibliography 50

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1 IntroductionIn this paper I will discuss the philosophical implications of Dedekind’s in-troduction of natural numbers in the central section of his 1888 foundationalwriting Was sind und was sollen die Zahlen?, particularly as opposed to theapproach to numbers to be found in Gottlob Frege’s writings (above all, theGrundlagen der Arithmetik (1884) and both volumes of the Grundgesetzeder Arithmetik (1893, 1903)). This comparison will be crucial not only tohighlight Dedekind’s value as a philosopher, but also to discuss crucial issuesregarding the introduction of new mathematical objects, about their natureand our access to them.From a philosophical point of view, there has been a great deal of inter-est, during the XX century, towards Frege’s theory of natural numbers,whereas Dedekind’s approach was generally dismissed as a naïve and moremathematically-driven approach. Only recently philosophers of structuralistsympathies1 invoked Dedekind as their predecessor, but mentions of his phi-losophy are usually brief and arguable. Yet it can be claimed that Dedekind,far from being the naïve mathematician often depicted by philosophicalmythology, was a subtle thinker. And, unlike Frege, he was a mathematician:perhaps a charitable reconstruction of Dedekind’s ’philosophical’ approachtowards natural numbers may allow us to discuss philosophical issues aboutthe introduction and the nature of mathematical objects in a way that ismore close to inner-mathematical methods and concerns.

I now explain the structure of this paper. The next section will be devotedto a brief introduction of logicism, in order to understand the foundationalperspective about arithmetic both Frege and Dedekind shared: it will be ex-plained why it is so important to talk about natural numbers both from aphilosophical and mathematical viewpoint. In section 3 I will highlight themost important differences between their approaches. I will explore their re-spective accounts, and, more specifically, their introduction of naturals insection 4. In section 5 I will outline the usual philosophical criticisms toDedekind’s theory, and in section 6 I will try to make sense of it by in-vestigating its relation with recent structuralist theories. In section 7 I willsummarize my conclusions.

1Hellman (1989), Shapiro (1997).

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2 Logicism and the rigorization of analysisThe term “logicism” is usually associated with the claim that mathematics,or at least arithmetic, is a part of logic, or is derivable from logic: mathemat-ical concepts are to be accounted for through logical ones. In the strongestpossible interpretation, logicism states that mathematics (arithmetic) is justa notational variant of logic. More generally, it is argued that the logicalanalysis of mathematical concepts should ultimately allow us to prove allmathematical theorems on extremely general and simple logical grounds. Inthe words of Russell: “all pure mathematics deals exclusively with conceptsdefinable in terms of a very small number of fundamental logical concepts,and [...] all its propositions are deducible from a very small number of fun-damental logical principles”.2The philosophical reasons behind this account lied, originally, in the opposi-tion to empiricism and Kantian influences in arithmetic; both these positions,in one way or another, stress out the importance of sensible experience andintuition when it comes to mathematical knowledge and justification, whereaslogicism usually claims that such knowledge is pure a priori knowledge, and,furthermore has more to do with the laws and mechanisms of thought ratherthan with the outside physical world.3 Although some logicist thought ofhis positions in terms of the venerable analytic/synthetic opposition, and,more precisely, as stating that arithmetic, contrarily to what Kant says inthe Transcendental Aesthetic section of the Critique of Pure Reason, analytica priori, I will avoid this lexicon, as the meaning of the term “analytic” isknown to fluctuate from one author to another; for our current interest isnot important to determine whether or not logicism can be considered as the

2Russell (1903, pp. xvi).3One should not take this claim, however, as overlapping with the claim that math-

ematical statements, being logical, are formal in nature without any substantial contentabout facts or states of affairs. This issue has to do with a crucial point regarding what wethink about the object of logic. Nowadays we think of classical quantificational logic as be-ing merely formal: the only objects mentioned in a logical statement are those designatedby constants or quantified upon in a given interpretation; but the laws of logic are not perse about them, and the logical lexicon is not thought as having a designating value.Things were different for the logicians in the last decades of the XIX century. Frege, usuallyconsidered the father of modern logic by philosophers, was quite happy with second-orderlogic, which involved as a crucial feature quantification over entities known as “concepts”;other logicians considered set theory as a part of logic, such that logic involved referenceto special elements called “sets”. In general, the widespread idea of that time was thatlogic had a proper subject matter and it dealt with specific objects and their features. Forexample, for Frege arithmetic truths were about facts in a non-subjective but non-physicalrealm.

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thesis that arithmetic is analytic.4 Logicism, on the other hand, is primarilyconcerned with the idea that the kind of justification mathematical knowl-edge relies upon has nothing to do with any intuition, both a priori or aposteriori. “What is essential to the logicism of history (if not the logicism oflegend) is its opposition to the incursion of Kantian intuition into the contentof arithmetical theorems or their justification: the thesis that all our authorsshare is the contention that the basic truths of arithmetic are susceptible ofa justification that shows them to be more general than any truth securedon the basis of an intuition given a priori ”.5Logicism is usually associated with three names: Gottlob Frege, BertrandRussell and Rudolf Carnap. Frege is considered the fore-father, the one whoprovided the conceptual tools needed for the job through the introductionof relational and quantificational logic in the Begriffsschift (1879); he artic-ulated his logicist project first informally, in the Grundlagen der Arithmetik(1884), and later on, with significant differences, in what he considered to behis masterpiece, the Grundgesetze der Arithmetik (1893, 1903). According toFrege, natural numbers are extensions of second-order concepts regarding aspecial equivalence relation between first-order ones (more on this in section4). Out of all Frege’s writings about the foundation of arithmetic, the Grund-lagen is the only one where philosophical questions are explicitly asked andsystematically answered. Russell was one of the first to recognize the valueof Frege’s work; his first relevant contribution was however to express in asimple and straightforward way serious concerns about the consistency of settheory (what we now call naïve set theory) a fairly new mathematical theorythat, with some differences, Frege used as well (he employed the derivativeidea of extension of a concept instead of the basic idea of set); this went un-der the label of ’Russell Paradox’. After he discovered that his own personal“way out” of the paradox did not work, in 1906, Frege lost interest in logicismand reconsidered a Kantian position about arithmetic.6 On the contrary, theintellectual energies of the young Russell were to be exploited in the pursuitof a consistent variant of logicism. In particular, his primary objective wasto restore the logical basis of Frege’s foundational project through the use oftypes, first simple, then ramified, which stratified functions and argumentsin such a way that the self-reference which engendered the contradiction wasimpossible to obtain. Frege’s influence notwithstanding, the outcome of al-most ten years of work, together with the mathematician Alfred Whitehead,the Principia Mathematica (1910-13) was a completely original writing which

4Bertrand Russell, a prominent logicist, thought of arithmetic as synthetic.5Demopoulous and Clark, in Shapiro (2005, pp. 130).6Dummett (1991a, pp. 4-7).

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disagreed with Frege on many important respects (e.g., the lack of platon-ism). Although the ramified theory of types, with its highly controversialaxioms, failed to command belief as to how save set theory from the contra-diction, the Principia Mathematica were usually considered the best logicismhad to offer in its defence. This was the case in Rudolf Carnap’s presentationof logicism in the famous 1930 Königsberg conference; in his words “logicismis the thesis that mathematics is reducible to logic, hence nothing but a partof logic”.7 Carnap stressed the importance of the analyticity of arithmetic, asFrege did before him; so when the analytic/synthetic opposition, as Carnapunderstood it, went under the attack of Quine’s Two Dogmas of Empiricism(1951), it was considered as additional evidence that logicism was not anoption worth exploring, and it disappeared from the debate until the early’80s.

In this reconstruction of the early history of logicism, Richard Dedekindis nowhere to be found. But why should he be there? After all, Dedekind wasa mathematician, and logicism as I defined it (an alternative to empiricismand Kantian positions) is a purely philosophical view. The short answer isthat both statements are false: Dedekind was not merely a mathematiciandevoid of philosophical interests and logicism was not a purely philosophicaltheory without roots in mathematics itself. I believe that this ’philosophi-cal side’ of Dedekind and this ’mathematical side’ of logicism converge untoa specific element, a crucial point of logicism, and the ultimate basis of amathematician’s knowledge according to Dedekind: the clarification of theconcept of natural numbers (viz., positive integers) in a non-mathematicalfashion. According to Gillies (1982) logicism essentially involves a twofoldtask:

• to define numerical concepts (natural numbers, since logicism dealsprimarily with arithmetic) through logical ones

• to characterize mathematical induction, the so-called passage from nto n+1, as a logical inference

I earlier on gave a brief philosophical introduction of logicism and its repre-sentatives; I would like now to introduce Richard Dedekind and the mathe-matical environment he moved in; in this way it will be extremely clear notonly that Dedekind was a logicist, but also that logicism itself was not just aphilosophical position, but the ultimate spearhead of a inner-mathematicalmovement called rigorization (or arithmetization) of analysis. This makes

7Carnap (1931, pp. 91).

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Dedekind not just a logicist “in passing”, but one of the purest kind.

Let’s now briefly introduce Dedekind; Richard Dedekind was a XIX centurymathematician who taught in Göttingen, Zürich and his hometown Braun-schweig. His record as a mathematician is outstanding. He was one of the firstto recognize the importance of groups in algebra, and his definition of ideals(basically a set-theoretical generalization of ideal numbers as inadequatelycharacterized by E.E. Kummer) was of crucial importance for modern ringtheory. He introduced and carefully defined many mathematical structures.More impressively, one has to keep in mind that the theory of sets was nota mere tool he used in his research; it was a theory he positively invented,together with his colleague and friend Georg Cantor: his research on sets, orhow he prefers to call them, Systems, goes hand in hand with his study on aset-theoretical concept that he thought to be of basic importance for the restof algebra and number theory, the concept of Körper (body); that is to say,a set of numbers closed under basic arithmetical operations. In 1856-58, he isone of the first to teach Galois’ group theory at the University of Göttingen(at the time, he was still a young unsalaried lecturer). He also worked aseditor and commentator of the work of Dirichlet, who had a great impact onDedekind’s formation as a mathematician.Abstract algebra was not however his only interest; Dedekind was one of theleading thinkers in the so-called rigorization (or arithmetization) of analysis,or foundations of real numbers.The mathematical problem Dedekind tried to tackle from 1858 to 1888 is notan easy one; in fact, it relates to a dichotomy that is as old as mathematicsitself. The discovery of irrational numbers is probably to be set around thefifth century BC, and it is due to Hippasus’ ex absurdo proof the discoverythat there are numbers that cannot be expressed as a ratio of integers (Hip-pasus’ proof basically shows that the alleged ratio corresponding to

√2, if

expressed as a fraction, could not be reduced to the lowest terms). The ex-istence of irrational numbers has a fairly simple geometrical representation:although all rational numbers n can be put in correspondence to points ofa line (the point that, together with a given point determines a segment of|n| length, once a unit is chosen), it is the case that not all points of a linecorrespond to rational numbers, as there are lengths that do not correspondto any of those quantities (numbers). This ultimately led mathematics to adichotomy, it being both the study of the discrete and the continuous. Whenthe calculus was invented in the XVII century, “infinitesimal” quantities wereoften introduced through resort to geometrical representations and intuitions,as I did before.The growing suspicion of many mathematicians in the XIX century, some

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of which were in Göttingen, was that, as heuristically useful this practicewas, the foundation of analysis could not really lie in geometry; mathematicsneeded a rigorous foundation, which for the “conceptual school” of Göttingenmeant that the basic concepts of analysis needed to be clarified in a non-circular and non-geometrical way. From an epistemological point of view, asstated by Carl Friedrich Gauss (a leading mathematician of the Göttingenschool) in a 1830 lecture, the problem was that the theory of numbers, beinga priori, needs a different epistemological basis than geometry, which onlyprovides us empirical knowledge; the point could also be that of a method-ologically pure explanation: analysis needed to be purified of external or“foreign” notions. Karl Weierstrass, one of the fathers of modern analysis,thus stated in a 1874 lecure that “the geometric representation of the com-plex magnitudes is regarded by many mathematicians not as an explanation,but only as a sensory representation, while the arithmetical representation isa real explanation of the complex magnitudes.”8Dedekind felt the need of a more rigorous foundation for analysis as soon ashe started to teach calculus in 1858, at the Polytechnic of Zürich. In his ownwords:

“As professor in the Polytechnic school in Zürich I found myself for the first timeobliged to lecture upon the elements of the differential calculus and felt more keenlythan ever before the lack of a really scientific foundation for arithmetic. In dis-cussing the notion of the approach of a variable magnitude to a fixed limiting value[...], I had recourse to geometric evidences. Even now such resort to geometric in-tuition in a first presentation of the differential calculus, I regard as exceedinglyuseful, from the didactic standpoint, and indeed indispensable, if one does not wishto lose too much time. But that this form of introduction into the differential cal-culus can make no claim to being scientific, no one will deny”.9

At the time, part of the work was already done; following Gauss, complexnumbers were reduced to pairs of real numbers, whereas integers were easilyunderstood through naturals, and rationals through integers. The problemwas rather how to reduce real numbers themselves, so, in other words, howto understand what an irrational number is only in terms of rationals (so,ultimately, naturals). This was no easy task, but the feeling that this laststep in the arithmetization of analysis could be taken was asserted with asignificant degree of confidence in Göttingen. For instance, Dedekind likes toremember Dirichlet, a mathematician in Göttingen who heavily influenced

8For this reference and the earlier one for Gauss, see Ferreirós (2007).9From the preface of Stetigkeit und irrationale Zahlen.

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him, saying that “every theorem of algebra and higher analysis, no matterhow remote, can be expressed as a theorem about natural numbers”.10 Thisis where Dedekind’s revolutionary set-theoretical treatment of real numberscomes into the picture, in the form of the so-called “cuts” of rationals. Thetreatment of real numbers in Stetigkeit und irrationale Zahlen, is not theargument of this paper; it is more interesting to note that Dedekind, unlikemany other mathematicians,11 did not stop in his foundational effort to thereduction of all mathematics to arithmetic, as he thought that arithmetic it-self, that is to say, both the numerical concepts and the rules of the discipline,stood in the need of clarification -through logic (set theory); this is where thetransition takes place from Stetigkeit und irrationale Zahlen (1872), whosesubject are real numbers, to Was sind und was sollen die Zahlen? (1888),which, on the other hand, deals with natural numbers. As our current con-cern are natural numbers, I will discuss primarily this second foundationalwriting, which was conceived by the author as providing the ultimate basisfor all of mathematics.This is to show that logicism, at least in the case of Dedekind, was not amere philosophical point, used by those that, for some reason or another didnot like what Kant had to say about mathematics in the Transcendental Aes-thetic: there is another side of the matter, one that is more closely related tomathematical practice itself. Logicism, as a pure non-mathematical analysisof numerical notions, together with the epistemological intention of gettingrid of a posteriori geometrical intuitions as genuine source of knowledge, wasto be the decisive spearhead of rigorization, one that has been clearly andstraightforwardly brought forward by Dedekind with his two foundationalwritings. In the words of Gillies (1982, pp. 8): “[...] There is a natural tran-sition from the arithmetization of analysis in the 1870’s to interest in thefoundations of arithmetic in the 1880’s”. Furthermore, it goes to show thatthe investigation on numerical concepts in arithmetic, the answer to the ques-tion “what are numbers?” is the key issue of logicism in more ways than one.That the history of logicism is somewhat more complicated than usuallyacknowledged by philosophers who ignore Dedekind’s contributions is sup-ported by the fact that Frege and Dedekind knew each other and recognizedthe value of each other’s work.12 Keep in mind that Dedekind’s last foun-dational writing, the one dealing with natural numbers and arithmetic, Wassind und was sollen die Zahlen? was published four years later than the

10From the preface to Was sind und was sollen die Zahlen?.11Such as his rival Leopold Kronecker, who took a primitivist and finitary stance towards

arithmetic which was diametrically different from Dedekind’s.12Reck recognizes similar “more mathematical” reasons for logicism in Frege’s early

writings as well. See Reck (2013, pp. 251-254).

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Grundlagen of Arithmetik, but was developed independently. We know thatFrege knew right away of Dedekind’s contribution, as it taught a seminaron it in the academic year 1889/1890; when later on the first volume in theGrundgesetze was published, in 1893, he explicitly said that Dedekind’s 1888writing was the best work on the foundation of arithmetic he came acrossand that they agreed that the theory of numbers is a part of logic.13 It wasunusual for Frege to teach a seminar on another’s author book, especially ona subject so close to his personal interests.14 The praise was also unusual,coming from a man, like Frege, with a great deal of polemic attitude; how-ever this attitude finds its way in the passage of the Grundgesetze where theauthor presents his “fellow logicist” three criticisms; yet the overall tone ispositive.15

What about Dedekind? Did he knew of Frege’s contributions? Yes, he did. Ina very interesting letter, dated 1890 to H. Keferstein, a professor from Ham-burg who positively reviewed Was sind und was sollen die Zahlen?, Dedekindstated that he came across the Begriffsschrift and the Grundlagen der Arith-metik in 1889, when his work on arithmetic was already published;16 heseemed however pleased to point out that, despite Frege’s “inconvenient ter-minology”, the latter’s way of treating mathematical induction was substan-tially the same as his own. In 1893 (the year of publication of the first volumeof Frege’s Grundgesetze), in the preface of the second edition of his secondbooklet, he explicitly acknowledged the value of Frege’s work and made otherpositive remarks: he stated that their way of treating the passage from n ton+1 was the same, but they disagreed, to some extent, on the “essence onnumber”. If we look at Gillies’ list about what logicism ultimately is about(concept of numbers and mathematical induction), we can see that both ofthem were interested in both issues: they agreed on the solution to give tothe second, but disagreed on the first. The goal of this paper is, primarily, toexplore this disagreement.Both Frege’s theory of extensions of functions in theGrundgesetze and Dedekind’stheory of systems in Was sind und was sollen die Zahlen? made an implicit

13Frege (1893, pp. 196)14See Kreiser (2001) for more informations on classes taught by Frege during his career.15There is also a fourth criticism from Frege to Dedekind to take into account: the one

put forward in the second volume (third part) of the Grundgesetze, published in 1903. HereFrege’s attitude towards Dedekind was not as positive as before, but we should keep in mindwhat Dummett (1991a, pp. 243) wrote about this section “The Frege who wrote VolumeII of the Grundgesetze was [...] en embittered man whose concern to give a convincingexposition of his theory of the foundations of analysis was repeatedly overpowered by hisdesire of revenge on those who had ignored or failed to understand his work.” HoweverDedekind did neither.

16See Dedekind (1932).

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appeal to some kind of principle of unrestricted comprehension, roughly put,the principle according to which every condition defines a set. Thus, bothlogical systems were inconsistent, as a contradiction could be derived fromthem. Dedekind discovered this from Georg Cantor earlier on, in 1899; quiteunderstandably, he was devastated by this discovery, and he was reported tosay, being his theory of sets allegedly based on the basic rules of thought,that he wasn’t even sure human reasoning was rational after all. Frege wasequally shocked in learning of the paradox from a 1902 letter from BertrandRussell; yet, reading the famous second appendix of the second volume of theGrundgesetze, where he addresses this problem in a last-minute correction tohis last draft of the book, one may have the feeling of some kind of perversepleasure from his part when he points out that everyone else using a theory ofconcepts, classes or sets, is in his same position. Dedekind is clearly the firstone he is thinking about, as he explicitly mentions him in the correspondingfootnote. Thus their theories shared the same fate.

There is something ironic in what history had in store for these two im-portant thinkers. Dedekind, being a high-value mathematician, was knownas a “logicist” -as the father of logicism!- during his life. Ernst Schröder, themain German member of the Boolean school in logic, wrote in the Vorlesun-gen über die Algebra der Logik (1890/1895) about “those who, like Dedekind,consider arithmetic a branch of logic”. C.S. Peirce, who was very interestedin logic, recognized in a 1911 writing the value of Dedekind’s set theory andfoundational project as a significant development in the study of the relationbetween arithmetic and logic, which he thought himself to be very close. Fi-nally, David Hilbert, in a lecture on geometry in 1899, advices his audienceto read Was sind und was sollen die Zahlen? in order to learn more aboutthe relation between logic and arithmetic.17

On the contrary, before Russell’s passionate advertising, Frege’s work waspretty much ignored. However, things were about to change quite rapidly.While Dedekind’s value as a mathematician was never put into doubt, hisfoundational interests were somehow forgotten.18 It is not worthless to pointout that the α-δ conditions defining a simply infinite system in the Definition71 of Was sind und was sollen die Zahlen? which according to Dedekind rep-resented the true subject matter of arithmetic, became known as the “Peanoaxioms”, although Giuseppe Peano published very similar results only a yearlater. One can now find the “Peano-Dedekind axioms” label as well.

17See Ferreirós (1999, 2009) for references.18Altough, as I will explain later on, his influence on set theory was deep and lasting,

mostly on Zermelo.

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A better fate was expecting Frege, even if he died before he could see it. Al-though his logicist project was almost universally considered a failure becauseof the contradiction, interest for Frege as a philosopher started to grow in theearly ’50s. He was quickly to be recognized as the “first analytic philosopher”.Even if the Grundgesetze were never translated into English, in this periodOxford philosopher John Austin translated the Grundlagen, which, from aphilosophical point of view, became the main source to look for Frege’s viewson the foundations of arithmetic. Michael Dummett was probably the mostinterested in keeping the sparkle of interest alive, and after the publicationin 1973 of his book on Frege’s philosophy of language he worked for almosttwenty years on a book about Frege’s philosophy of mathematics. The crucialyear is however 1983, with the publication of Crispin Wright’s Frege’s Concep-tion of Numbers as Objects, which marks the birth of the so-called neo-logicistprogram. The basic idea of this project is that it is possible to isolate healthyparts of Frege’s logical system from inconsistency, by extruding Basic LawV, namely, the principle giving identity conditions to extensions of functionsand used to introduced them. It is sometimes called “Frege’s Theorem” theclaim that Peano’s axioms can be derived from weaker assumptions: Hume’sPrinciple (an abstraction principle giving identity conditions to numbers ofconcepts; more on that in section 4) and second-order logic; this is not anhistorically accurate reconstruction of how Frege tried to solve the problem,as he never acknowledged the possibility of a fall-back position to Hume’sPrinciple, which he did not take to be sufficient because it failed to deliverexplicit definitions for numbers. But as for now, “neo-logicism” is just anotherword for “neo-Fregeanism”.The conclusion is therefore that logicism is now quickly associated with a(mostly) philosophical position held first by Frege; the historical reconstruc-tion I gave in the beginning of this section (the one were Dedekind doesnot appear at all) then follows quite straightforwardly. On the contrary,during the last decades of the XIX century, logicism was championed byDedekind and was more closely related to inner-mathematical issues regard-ing the arithmetization of analysis.

3 Some preliminary differences between Fregeand Dedekind

Before presenting Frege’s and Dedekind’s introduction of natural numbers,it will be useful to point out some preliminary differences between theirapproaches to arithmetic and logic. There are four of them: the formal vs

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informal method, the difference between their choices about the primitivenotions of logic (concepts vs sets), the issue about psychologism, and finallythe cardinal vs ordinal conception of numbers. Let’s see them, in this order.

• Formal vs informal approach The first point I would like to high-light is not much of content, but rather of method. Whereas the Fregeof the Grundgesetze spent a significant amount of time presenting afull-blown axiomatic systems in order to show in details the proofshe was able to obtain with it, Dedekind’s exposition in Was sind undwas sollen die Zahlen? is much more informal. Although he starts thebooklet by presenting the theory he is going to adopt, and althoughhe utilizes a small amount of formal machinery, there is nothing likeFrege’s degree of precision and explicitness. More specifically, no ex-plicit list of axioms or rules of inference is given to the reader; in thewords of Gillies (1991, pp. 51): “Dedekind, though a logicist, has not thesame over-ruling passion to demonstrate his position conclusively, andis content with the usual informal mathematical standard of rigour.”As we will see, the Frege of the Grundgesetze will consider Dedekind’sinformal approach as a flaw of his system, as it does not allow but asketchy presentation of each proof and leaves open the possibility forsome hidden assumption the be smuggled in the demonstration. Forinstance (although this was not noted by Frege at the time) the axiomof choice is implicitly assumed in Theorem 159 to prove that if a set isDedekind-infinite, then it is countably infinite and vice versa.

• Concepts vs sets Both Frege’s and Dedekind’s “logic” (I use the quo-tation marks, as I find it quite difficult to understand, especially in thecase of Dedekind, what counts as logic and whether it is the same forthe two of them) involves some kind of notion of sets or classes; yetthere is a basic difference between them. For Dedekind, sets are logi-cal objects and their corresponding notion is a fundamental concept oflogic; the same does not apply for Frege, as he thinks of “concepts” asbasic logical notions and sets, that is to say, extensions of concepts, asmerely derivative. Let’s see both positions, and their relations.For Dedekind’s sets (System)19 are things (Ding) in the sense of ob-ject of our thoughts. They are the result of the exercise of a humanfaculty and, as such, they are the proper object of logic qua science ofthe rules of thought. He defines the most basic set-theoretical conceptsin the second paragraph of Was sind und was sollen die Zahlen? (as

19When discussing Dedekind’s set theory I will use the English terms “set” and “system”interchangeably.

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such, they are not primitive concepts, as in Zermelo’s set theory), andthinks of them extensionally. Overall, his understanding of sets is quiteprimitive: he implicitly assumes unrestricted comprehension and someversion of the axiom of choice, and, perhaps lured by the possibilityof thinking about sets as mereological aggregates of some kind, failsto distinguish between membership and containment (and, relatedly,between an element and its singleton), and fails to consider the emptyset as well. Although it is true that in the 1888 foundational writinghe used the same symbol ambiguously for both relations, he cannotbe accused of explicitly holding a view about sets as aggregates, and,by his own admission, the empty set can be supplemented if the needarises.A very advanced mathematician, he also introduced the notion of map-ping or representation between sets (Abbildung, based on another hu-man faculty) as a notion as basic as that of set itself; he additionallystudied its properties and applications on its theory; it is used to de-fine interesting properties of sets; for instance, we now call “Dedekind-infinite” a set that can be put in a bijection with one of its propersubsets (parts, in his terminology), one of the many definitions of in-finite that Dedekind introduces in Definition 64 of Was sind und wassollen die Zahlen?.20 Even the “infamous” pseudo-proof of the existenceof (Dedekind-)infinite sets in Theorem 66 had its weight, at least if weconsider that Axiom VII (of Infinity) of Zermelo’s set theory, in Zer-melo’s own words “is essentially due to Dedekind”21, presumably in thesense that Dedekind’s unsatisfactory proof demonstrated that the exis-tence of infinite setd has to be postulated. For an extensive overview ofDedekind’s and later set theory (especially Zermelo’s set theory), seeGillies (1982, chapter 8).Things are quite different for Frege, who thinks of concepts, instead ofsets, as basic notions of logic, with the notion of extension deriving fromit (“sets” are conceived as the extension of concepts). Keep in mind thatfor Frege concepts are not objects, as sets were for Dedekind; objectsand concepts are the two categories of Frege’s ontology; understandingthe nature of this bipartition is a very difficult task that I’m not going

20Although Cantor did ponder a similar definition of infinity since 1877 (when it wasworking to the paper “Ein Beitrag zur Mannigfaltigkeitslehre”) it was Dedekind himselfwho reassured him of its validity; in a letter to Weber (1888), he states that he is notlooking for a dispute about priority with his colleague and friend Cantor, although hemakes it very clear that Cantor himself felt unsure about this definition until Dedekindconvinced him he was indeed on the right tracks.

21Zermelo (1908, pp. 204).

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to tackle in this paper. As concepts are what predicate letters stand for,this choice means, from a logical point of view, that Frege’s theory doesnot essentially involve any set theory, but second-order logic, that is tosay, it involves the use of predicate variables. In the Grundlagen §45 heexplicitly rejects the possibility of taking natural numbers as relativeto sets rather than concepts; but given the arguments he suggests in1895,22 it seems that his dislike for sets was based on an understandingof the latter as mereological aggregates.There is another interesting point to be made here. An explicit theoriza-tion and axiomatization of his theory of extensions was available onlyin the Grundgesetze, whereas in the Grundlagen, he simply assumed,in §68, that a number (of a concept) is the extension of a second-orderconcept without telling the reader what an extension is. This can bequite frustrating for the reader of the Grundlagen, as Frege seems toassume that everyone knows what the “extension of a concept” is (how-ever keep in mind that, in the logic taught in his time, the distinctionbetween intension and comprehension was commonly drawn). In thecorresponding footnote of §68 he merely states that the extension of aconcept and the concept itself are two different things, for two reasons.They are very simple: extensions (as sets) are objects, while conceptsare not. Secondly, concepts are what we now call intensional entitieswhile their extensions are merely determined and identified by theirelements (sets are extensional entities).At some point between 1884 and 1893 Frege decided that some kindof theory of classes, or sets or extensions was what he needed to makesense of his claim of numbers being “said of concepts”. This was also theperiod Frege read and pondered Was sind und was sollen die Zahlen?,where set theory is explicitly and systematically used for the same foun-dational goals. Sundholm (2001) suggests that this is no coincidence,and that Dedekind’s theory had an heavy impact on the definitive ver-sion of Frege’s logicism. This issue seems to be crucial since this last op-eration (laying down a theory of extensions) is what condemned Frege’stheory to inconsistency.

• Psychologism. One of the philosophical issues Frege was most pas-sionate about in the Grundlagen was psychologism. Crude psycholo-gism about arithmetic is actually nothing more than Frege’s polemicaltarget according to which arithmetic is a subjective branch of knowl-

22“A Critical Elucidation of Some points in E. Schröder’s Vorlesungen über die Algebrader Logik ”. Schröder explicitly thinks of sets as aggregates; and, as I mentioned earlier,Shröder carefully read Was sind und was sollen die Zahlen? as soon as it was published.

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edge stemming from psychological activities, and numbers are mentalitems constructed by the mind. Frege’s position is that arithmetic can-not be subjective, that is to say, dependent on psychological humanactivity. And, likewise, its objects cannot be items created by the mind,as that would make them subjective too, with implausible consequences(there is a different 2 for everyone that thinks about the number 2? Dothese “2s” come and go together with the minds of those who thinkthem?).23 From a first reading of Was sind und was sollen die Zahlen?,Dedekind’s position could not be more different. He explicitly statesthat numbers are free creations of the human mind (Definition 73) andthat logical notions correspond to basic psychological faculties. Set the-ory, as a part of logic, describes the rules of thought; but it seems that“thought”, here, far from being an objectively rational kind of reason-ing, or even some kind of Platonic entity as Frege would later think,is simply the human faculty of thought. For instance, he refers to thefaculty of considering things together under a certain respect (System)and that of pairing one thing to another (Abbildung) as he was merelydescribing mental processes we all undergo in our everyday life.24

When we deal with natural numbers, the problem of psychologismamounts to the following question: if (again) numbers are objects, arenumbers created by the human mind? A positive answer to this questionwould be highly controversial, and without due qualifications (qualifica-tions that however Dedekind never does) is bound to make his theory ofnatural numbers simply unacceptable. The problem is that this seemsto be exactly what Dedekind had in mind; not only the passage whereit defines natural numbers (Definition 73) is awfully explicit about it,but he reiterates the same point in correspondence with his collabo-rator Heinrich Weber: “We are of divine species and possess, withoutdoubt, creative powers not just concerning material things (railroads,telegraphs), but especially concerning mental things”.25

This is one side of the problem that we are going to face in the inter-pretation of Dedekind’s theory of natural numbers.

• Cardinals vs ordinals Natural numbers such as 1, 2, 3 and so forth23Grundlagen, §26 and §27.24Dedekind seems to be completely insensitive to the philosophical issue of objectivity;

in this case, the problems amounts to the fact that if logic has to have a normative value,if it imposes a good kind of reasoning as opposed to a bad kind of reasoning, there mustbe some kind of objective canon the thoughts of various people are measured with. Thisseems to be the ultimate point Frege is trying to make against psychologists when it comesto logic, yet Dedekind is not bothered by it at all.

25Dedekind (1932, pp. 490).

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have many uses. First of all, they can be used to count elements andplace them in a succession; in such cases they are known as ordinals(which correspond to Enligsh expressions such as first, second, third...and so forth). They can also be used for cardinality purposes, that is tosay, loosely speaking, to count how many elements of some kind thereare: one cat, two dogs, three horses, and so forth. In such case, theyare known as cardinals. Yet, presumably, if there is such a thing asthe number 5, as both Frege and Dedekind seem to agree, the samenumber can be used for both purposes; it’s not like there is the number5-ordinal and the number 5-cardinal, two different entities which arein turn different from any fifth element in a succession. The number 5can be used as an ordinal or as a cardinal. This dichotomy generatesseveral difficulties for our two logicists which are interested in clari-fying their nature, as each one of them seem to exclusively considerone of these two sides as essential for the definition of the number 5.For Frege, naturals are essentially cardinals; for Dedekind, they areessentially ordinals. When discussing numerical concepts and assign-ing properties to the corresponding number-objects, this difference willbe crucial, as the exclusive consideration for either ordinals and car-dinals brings to incompatibilities between their accounts. As we willsee, Frege’s conception of positive integers as cardinals stems primar-ily by his philosophical conclusion that statements about numbers areultimately statements about concepts, while Dedekind’s interests innumbers were not as broad, and more closely related to arithmetic it-self: numbers are merely elements of a progression generated by theoperation of succession.26

The choice between cardinals and ordinals in one’s clarification of nu-merical concepts of course depends on what number statements onetakes to be worthy of analysis. Frege took such statements as “thereare 3 cats”, or “there are 5 horses”, in which numbers are applied forcardinality purposes relatively to concepts; yet none of those statements

26It was not, obviously, an idea Dedekind invented. Frege himself uses it to define num-bers through the notion of immediate successor (see next section). Their shared idea offormalizing arithmetic through the operation of succession plus some logical inference fromn to n+1 had roots in the Leibnizian idea (from the Nouveaux Essais, IV, §7) of definingnumbers in terms of the operation of succession and thus prove logically the operationof addition. Before Frege and Dedekind, this idea was also shared by Hermann Grassmanand Hermann Hankel; see Grassmann’s Lehrbuch der Mathematik fur höhere Lehrenstalten(1860). In the words of Frege (Grundlagen, §6, pp. 8): “Every number [...] is to be definedin terms of its predecessor. And actually I do not see how a number like 437986 could begiven to us more aptly than in the way Leibniz does it.” See section 4 and 7 to verify theagreement on this matter.

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are proper to arithmetic. This difference is reflected in the quantity ofmathematics that is analysed in their foundational works: the Grundge-setze barely makes it to explain addition, whereas Was sind und wassollen die Zahlen? uses the revolutionary method of definition by in-duction to treat arithmetical operations up to exponentiation. Fregethought cardinality applications were fundamental in clarifying num-ber concepts, whereas Dedekind would have probably thought of themas foreign elements that do not have anything to say to the science ofnumbers itself. Frege does not justify its approach: it takes it as self-evident; in a footnote in §4 he introduces the numbers he is interested inas “positive whole numbers, which give the answer to the question “Howmany?”” Besides, Austin’s translation doesn’t help as he merely trans-late with “Numbers” (with a capital “N”) Frege’s preferred expressionAnzahl, which, unlike the more general Zahl, precisely means “cardinalnumbers”. Dummett (1991a, pp. 48) even suggests the presence of somemalevolence from Frege’s part: he seems to use the term Anzahl withgreat generality, as if it could be used to refer to numbers in general.The general problem is that, with such different ideas of what numbersare for, the ultimate answer to the question “what are they?” will bevery different as well. In particular, Dedekind’s minimal understand-ing of numbers as elements of an order will dictate a structuralist-likeflavour to his account of what numbers are.27

4 Two approaches to natural numbersIt’s now time to see, in turn, Frege’s and Dedekind’s introduction of naturalnumbers.

4.1 Frege on natural numbers

The constructive part the Grundlagen begins in section IV, §55, where Fregesummarizes his earlier achievements by saying: “we have learned that thecontent of a statement of number is an assertion about a concept”.28 Frege

27Whether or not Dedekind can be counted as a full-fledged structuralist, and in whichsense, is yet to be decided. More on this in sections 6 and 7.

28The main point of the earlier sections (together with the criticisms of the view ac-cording to which numbers are sets of units) was to establish that natural numbers are notproperties of external things experienced, as empiricists such as Mill allegedly thought,through acts of sensible perception. One may as well define Frege’s position, following histypical ontological bipartition, as the position according to which numbers are propertiesof concepts rather than objects; yet that would be inexact, being numbers objects and not

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takes into consideration statements about numbers as applied for cardinal-ity purposes, and his first attempt to logically define numerical statementsconsists in the introduction of what we may call numerical quantifiers. Forinstance:

There are 0 F =df∀x ¬F(x)There is 1 F=df∃x F(x) ∧ ∀y (F(y) → y=x)

The two previous definiens may be used to introduce new quantifiers, re-spectively ∃0 and ∃1. Although there is no method to produce them auto-matically, that is to say, by treating the indexes as elements in the successionof naturals, we can manually obtain the following ones:

∃2x... =df ∃x ... ∧ ∃y ... ∧ (∀z ... → (z=x ∨ z=y))∃3x... =df ∃x ... ∧ ∃y ... ∧ ∃z ... ∧ (∀w ... → (w=x ∨ w=y ∨ w=z))∃4x... =df etc etc etc...

And so forth. Keep in mind that although numbers figure as indexes of thequantifiers, the latter are introduced as notational variants of non-numericalstatements.Frege quickly discards this approach in §56: it is crucial to understand why.The reason has already been given earlier on, when I stated that there is nomethod to obtain one numerical quantifier from the previous one, treatingnumbers, à la Leibniz, one as the successor of the other. In fact, we mayhave defined quantifiers such as ∃1 and ∃2, but these numerical indexes haveno logical status whatsoever; in particular, the more general ∃n has not beendefined. Therefore, we have not yet fixed the meaning of any numerical ex-pression per se, but merely that of expressions such as “there is 1 F”. In thewords of Frege himself: “it is only an illusion that we have defined 0 and 1”.This is sometimes called the “Julius Caesar problem”, as the previous defini-tions, unlike a good definition of natural numbers, does not allow us to saywhether Julius Caesar (the Roman himself) was a natural number or not.That is clearly true. But in §56 Frege actually says something stronger that,without further justification, comes out as quite illegitimate. He states thatthe method of the numerical quantifiers is “unsatisfactory” as “we have noauthority to pick out 0 and 1 here as self-subsistent objects that can be recog-nized as the same again” (emphasis mine).29 But the problem is: why should

properties. The best way to express his position is thus the longer periphrasis: a statementabout number is ultimately a statement about concepts (and their cardinality).

29Sometimes the expression “Julius Caesar Problem” is used to describe both issues.

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we? Why shouldn’t it be possible to introduce (cardinal) natural numbers ina fully adjectival way through the definition of the more general ∃n quan-tifier? It might be true that every statement about a number reduces to astatement about a concept, but why should, for instance “the number belong-ing to the concept F is...” (where numerical expressions appear in substantiveform) be more primitive that “there are... Fs”? Frege offers no answer to thisquestion, aside from his philosophical prejudice that numbers are, and shouldbe treated as, self-subsistent objects. As thus, he excludes a priori the pos-sibility of an adjectival reconstruction of naturals. Dummett calls §56, whichhe considers one of the “weakest” passages in the Grundlagen, “Frege’s sleightof hand”; he writes: “Frege, impelled by his desire to establish that numbersare objects, seems to have been taken in by his own jargon. When ’Thereare 0 Fs’ and ’There is just 1 F’ are expressed as ’The number 0 belongs tothe concept F’ and ’The number 1 belongs to the concept F’ it looks moreplausible to complain that the definitions [of numerical quantifiers] do notentitle us to pick out 0 and 1 as self-subsistent objects; without the jargon,it would have been apparent that they were not meant to and did not needto”. 30

The problem therefore is how to clarify the meaning of numerical expres-sions in substantival form, given that we have no access to the objects theystand for; §62, where the constructive part of the Grundlagen truly begins,offer an answer to these questions. The idea is that to investigate numericalconcepts, we have to understand the meaning of statements where numericalexpressions occur in substantive form, preferably identity statements wherea single number is re-identified as the same: identity is a relation holdingbetween objects. Once we will be able to recognize a number as the sameagain, our job will be completed.31

In statements of the sort “the number of Fs is the same as the number of30The closest thing to an argument Frege presents against an adjectival strategy in §56

is that we need to prove that, for each concept, there is just one number belonging to it;substantival strategies, where numerical expressions are presented as referring to objectsseem suitable to express this fact through the relation, holding between objects, of iden-tity. But Frege does not explain why an adjectival strategy could not, in a different way,explain the same fact.Finally, see Gillies (1982, pp. 42) for a reason why an adjectival and non-Platonist recon-struction of cardinal numbers may be insufficient for the goals of arithmetic; for more onthis, see the first bullet point in the 7.2 section later on.

31Please note that through the adoption of this “molecular theory of meaning”, accordingto which numerical concepts must be investigated in the context of identity statements,many problems are solved at once: a semantical problem (how to define numerical con-cepts?), a metaphysical problem (what is the nature of numbers as self-subsistent objects?)and an epistemic one (how do we access such objects?).

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Gs iff...”, we are given identity statements for number of concepts. Such con-ditions are provided through a formula, introduced through a lengthy butinteresting geometrical analogy, that Frege attributes to Hume and, as such,has been called Hume’s Principle (HP):

(HP) the number of Fs is the same as the number of Gs iff F and G areequinumerous.

Equinumerosity, despite its name, has nothing to do with numbers: it simplyis a relation of equality according to which the objects falling under F andthose falling under G can be put in a one-to-one correspondence.32

HP is very important because of this: its two sides are semantically equiva-lent (obviously) but the left one is for Frege epistemically and semanticallyprior. It serves to clarify the meaning of the other side, the one with numbersin, without at the same allowing for numerical expressions to be reduced insuch a way that they no more refer to objects. This is an highly controversialpoint in Frege’s logicism that I’m not going to discuss: in his intention, theleft-right direction of HP was the correct pathway to numbers.As the geometrical analogy suggests from §66, however, HP cannot be theend of the story for numbers: we now know identity conditions for numbersof concepts, that is to say, we know when two of such numbers are the same,but we still don’t know what such a number is; we don’t have an explicit defi-nition that clarifies such expressions, and neither do we know what “number”,in general, means. In turn, this means that we do not have applicability con-ditions for numerical expressions such as “the number of...”. This is, onceagain, the Julius Caesar problem.This finally leads to the explicit definition of natural numbers in cardinalityapplications laid down informally in §68, according to which the number of Fis the extension of the second-order concept “equinumerous to F”; how doesFrege get to this conclusion? As noted, equinumerosity is an equivalence re-lation; therefore, there are equivalence classes33 of equal concepts (concepts,

32Again, Frege at the time did not work with a fully developed theory of sets or ex-tensions in mind, and the Grundlagen is only informally discussed; in §68 he uses theinvented word Gleichzablig, which Austin translates as “equal”. But he clearly had in mindsomething along the lines of a bijection; in a well-known passage in §70, he states: “If awaiter wishes to be certain of laying exactly as many knives on a table as plates, he has noneed to count either of them; all he has to do is to lay immediately to the right of everyplate a knife, taking care that every knife on the table lies immediately to the right of aplate. Plates and knives are thus correlated one to one”.

33Again, Frege, in the Grundlagen, only speaks in term of “extensions” of concepts andobjects “falling under” concepts; but it is harmless to employ some set-theoretical notionat this point. I already discussed the peculiarities of Frege’s logic.

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we would say, with the same number of objects falling under them). If twoconcepts F and G are equinumerous, their number is the same; but also, theyare to be found in the same equivalence class, and therefore the extensionsof the concepts “equinumerous to F” and “equinumerous to G” are the same:this is a reason to believe that such extension is the original number we weresearching for (§69).HP is itself sufficient, together with Frege’s logic, to prove the validity of thePeano-Dedekind axioms for arithmetic; this is why it is held on such an highaccount by neo-Fregeans. Frege was not contented with HP, as it was notsufficient to answer the metaphysical question: what is a number? In Grund-lagen §73 he proved HP in the light of his definition of number of concept,and similarly he proved the axioms of arithmetic on the same basis in thefollowing passages.We must preliminarily note that the explicit definition of numbers as referredto concepts leads to other two definitions: the definition of (natural cardinal)number in general, in second-order logic, and the definition for the numbernought:

• (§72) n is a number =df there is a concept such that n is its number34

• (§74) 0 =df the number of the concept “self-different”

The point of the second definition is to pinpoint an equivalence class of con-cepts without objects falling under them (0).Before moving on to the succession of natural numbers, I would like to saysomething more about Frege’s treatment of numerical statements. We haveseen in the discussion of §56 that Frege refused a priori the possibility of afully adjectival strategy, that is to say an introduction of natural numbersthat primarily works with statements where numerical expressions figure inan adjectival position, as referred to concepts; he explicitly took expressionssuch as “there are 2 Fs” as dependent on “the number 2 belongs to the con-cept F”. This fact acquires a special meaning once we consider the differencebetween the two sides of HP: it is one thing to claim that there are as manyFs than there are Gs (a statement of first-order bijection which is surelycloser to numerically adjectival statements such as “there are 3 Fs”), anotheris to claim that the number of Fs is identical to the number of Gs. Fregeseems to consider those two formulation as equivalent, with the former (theadjectival one) being more fundamental than the latter. Nonetheless, purelyadjectival number statements such as “there are 3 Fs” follow from the explicit

34This definition, as Frege notes, is only apparently circular, for we have already definedthe notion of number of a concept.

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definition of substantive numerical expressions (e.g., “3 is the number of theconcept F”). Therefore, as Dummett (1991a, pp. 114-115) points out, we cansee Frege’s strategy as a mixed strategy between a fully adjectival strategyand a fully substantival one.

From §76 onwards, Frege discusses the succession of natural numbers, start-ing from the number zero so defined, and eventually proves the five Peano-Dedekind axioms of arithmetic.In §76 he introduces the notion of immediate successor (or predecessor):

n follows directly after m (in the series of naturals) iff there is a conceptF, and an object falling under it x, such that the number of the concept Fis n, and the number of the concept “falling under F but different from x” ism.

Put in an heuristically useful, but circular, way, m corresponds to the equiv-alence class of concepts with an additional object with respect to the onesin the equivalence class relative to n. In §77 Frege proves that there is anumber following after 0 and, according to the definitions in §74 and §76, isthe following:

1 =df the number of the concept “identical with 0”

The only object falling under that concept is the number nought itself; itis easy to see why, according to these definitions 1 follows after 0. There isno reason to stop: number 2 will be the number of the concept “identicalwith 0 or 1”, 3 the number of the concept “identical with 0 or 1 or 2”, and soforth; according to the previous definitions 2 is the immediate successor of 1,and 3 of 2. And so forth... In §78, additionally, Frege states that this relationof immediate successor/predecessor is one-to-one, which roughly correspondsto the statement according to which each number of concept has only onesuccessor. This is however not enough, if the goal is arithmetic: Frege onlyhas the general concept of (cardinal natural) number, and a list of numbers.There is no way he can generalize and speak about all numbers, or numbersarranged the series of natural; for this reason, there is still no way for himto prove any of the theorems of arithmetic. For instance, we may prove, onenumber after another, that each number of concept has a single successor,but we still cannot prove Peano’s axiom stating that every number has onlyone successor. This is what he now goes to discuss, and this is where hisapproach ultimately aligns with Dedekind.Consider a binary operation R over a set of elements. A concept F is saidto be hereditary in an R-series (R-hereditary) if and only if, each pair of R-

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related items is such that, if the first relatum has it, the second has it as well.In §79 Frege uses this definition to introduce the concept of ancestral of R.35

The concept, first introduced in the Begriffsschrift (Part III, Proposition 76),can be so introduced:

x comes before y in the R-series =df y falls under all R-hereditary conceptsunder which falls every object x is R-related.

We can state the point thus: the ancestral of R is the transitive closureover R. After some remarks about this definition in §81 Frege introduces thenotion of weak ancestral in §81:

y is a member of the R-series starting with x =df y either is in the an-cestral relation of R with x or either is identical to x.

The weak ancestral of R is thus the transitive and reflexive closure overR. If we take R to be the concept of immediately preceding as defined in§76, the ancestral operation of R corresponds to the numerical notion of “lessthan” (<), and weak ancestral corresponds to “less than or equal to” (≤).Through the notion of weak ancestral and 0, we can now define the conceptof natural number, not as relative to a concept, but as member of the series ofnaturals starting with 0. A natural number is simply whatever object that isgreater than 0 according to the definitions given (viz., if and only if it followsfrom the series of numbers starting with 0, that is to say, if and only if itstands in the weak ancestral relation (relative to the immediate predecessorrelation as given in §76) with 0). In the original words of Frege (§83): “n mustbe a member of the series of natural numbers beginning with 0”; this definesthe notion of number in the succession of naturals (ordinal number), based,as it appears in the definiens, on the notion of natural number as number ofa concept (cardinal number.)

This set of definitions allows Frege to prove the Peano-Dedekind axioms forarithmetic, a task he explicitly carries out in the Grundgesetze; it is howeverof no interest to go through the proof right now. We have all the elementswe need to make relevant comparisons with Dedekind’s theory of naturalnumbers. I now pass to Dedekind’s own theory.

35This term comes from Russell’s and Whitehead’s Principia Mathematica. In theGrundlagen der Arithmetik Frege uses the terms “following... in an R-series” and “comingbefore... in an R-series”.

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4.2 Dedekind on natural numbers

Was sind und was sollen die Zahlen? begins with in informal and somewhatincomplete introduction to set theory. The notion of set, part (subset) 36 andfunction are introduced and discussed. To understand Dedekind’s introduc-tion of natural numbers, we have to introduce four other notions: similarity,chain, infinite set and simply infinite set.

• Similarity (Definition 26 and 32) A function (Abbildung) on a systemis said to be a function of similarity (or a similar function) when the im-ages37 of distinct elements are distinct; or, in other terms, when identi-cal images are generated by identical elements: Definition 26 thus intro-duces what we now call injective functions. In Definition 32 Dedekindexpands the notion of similarity with the introduction of similarity be-tween two sets. Sets A and B are similar if there exists a similarityfunction φ(B)=A. This means, in modern terminology, that when twosets are similar when they can be put in a one-to-one correspondence.38

• Chain (Definition 37 and 44) Given a system S and a function φ: S→S, a set K ⊆ S is a chain [Kette] if and only if K’ ⊆ K. No restraint isgiven on the nature of the function φ in the definition of what a chainis: it could be any function. According to the text, being a chain is botha property of K and φ (the function that does the “chaining”).Many properties of chains are investigated, but in Definition 44 Dedekindadds a new crucial element, that of a chain of a set (relatively to a big-ger set). If A is a subset of S, we can call A0 the intersection of all thechains in S (defined as of Definition 37) for which A is a subset. Thisintersection, which is itself a chain, is the chain of A.

• Infinite set (Definition 64) A set is infinite if and only if it is similar toa proper part of itself. I already discussed the features of this definitionof infinity when discussing Dedekind’s theory of sets.

36Remember that, at this stage, Dedekind does not explicitly differentiate between sub-sets and elements of a set.

37Dedekind introduces the notion of image in the following way. According to Definition21, an image or transform of an element is first of all its output relatively to some function.So, for the element a and the function φ, we have φ(a), or, sometimes, the simpler a’.Images of sets (subsets of the domain) are only derivatively introduced from the imagesof their elements. So, given a function φ defined on the set A, the φ-image of A is eitherdenoted by A’ or φ(A). The term “codomain”, to define the larger sets all images belongto, appears only in section IV.

38He basically introduces bijection through an injection from the original codomain tothe original domain, which ensures that at least every element of the former is paired toone of the latter.

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• Simple infinity (Definition 71) A set N is a simple infinity, given atransformation φ on itself, if and only if it satisfies the following fourconditions

– α: N’ ⊆ N (N is a chain)

– β: N=10 (N is the chain of 1, where 1 is an element of N called“base-element”)

– γ: 1/∈ N’ 39

– δ: φ is a similar representation (an injection)

Those four conditions are sufficient to prove that N is an infinite setin the sense given in Definition 64 (a Dedekind-infinite set).40 Further-more, according to these conditions, the elements of N can be seen asordered in a succession by φ; this is what the succession of naturalnumbers ultimately amounts to.

Theorem 66 allegedly proves the existence of at least one infinite set. The-orem 72 proves that every infinite set contains as a part a simple infinity.Therefore, Theorem 66 and 72 jointly prove that there is at least a simpleinfinity. This is, for Dedekind, a point of great importance, as knowing thatthere is an infinite set or a simple infinity ensures that there is nothing inco-herent or ill-conceived in the notions themselves: Theorem 66, in particular,works as a proof of consistency.41

This leads straightforwardly to Definition 73, where Dedekind introduces the39It can be easily proved that 1 is the only element of N that is no image of any other

element. This helps representing the α-δ conditions as the Peano axioms for arithmetic.40Once again, Dedekind merely states that such a proof exists, but it’s easy to see it;

first of all, we have to prove that N and N’ are similar; we have to prove the existence ofa similarity function ψ(N’) = N, such that its inverse is ψ−1(N) = N’. ψ−1 can be clearlyidentified with φ itself, so ψ = φ−1; because of Definition 26, if φ is similar, so is φ−1;therefore, the ψ function between N’ and N is similar. Thus, N and N’ are similar systems.By α, N’ ⊆ N, and, by γ, N 6= N’. N is similar to one of its proper parts; therefore, is aninfinite set.

41The “infamous” Theorem 66, which was suggested to Dedekind by the Paradoxiendes Unendlichen from Bernard Bolzano (§13), can be disputed as being an actual math-ematical proof. First of all, by considering the set of “the totality [...] of all things, whichcan be objects of my thought” as its starting point, the proof is immediately subject to aCantorian-like paradox of cardinality. Zermelo (1908, pp. 204, footnote 8) made this remarkand, as I already suggested, this is probably why he decided to postulate the existence ofinfinite sets instead of trying to prove it. Furthermore, this alleged proof has philosophi-cally puzzling consequences: Dedekind speaks about “my own realm of thoughts” as beinginfinite, which is clearly absurd; in fact, no human mind can hold infinite thoughts all to-gether. Perhaps he is thinking about the set “of everything thinkable”, which is more likelyto be infinite. Zermelo, in the passage mentioned, uses this reformulation, but this is also

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notion of natural number. Let’s read the whole passage, in the Beman trans-lation of Was sind und was sollen die Zahlen? :“If in the consideration of a simply infinite system N set in order by a transforma-tion φ we entirely neglect the special character of the elements; simply retainingtheir distinguishability and taking into account only the relations to one anotherin which they are placed by the order-setting transformation φ, then are theseelements called natural numbers or ordinal numbers or simply numbers, and thebase-element 1 is called the base-number of the number-series N. With referenceto this freeing the elements from every other content (abstraction) we are justifiedin calling numbers a free creation of the human mind. The relations or laws whichare derived entirely from the conditions α, β, γ, δ in [71] and therefore are alwaysthe same in all ordered simply infinite systems, whatever names may happen to begiven to the individual elements [...], form the first object of the science of numbersor arithmetic.”

It seems from a first reading that Dedekind is trying to make two pointshere:

• arithmetic, the science of natural numbers, is primarily interested inthe α-δ conditions for a system; that is to say, it is primarily interestedin the notion of a simple infinity, and secondly

• numbers are characterized exclusively as being the elements of a simpleinfinity through some kind of abstraction process.

The first point is a point of what we may call methodological structuralism:what the mathematician (or, in this case, the arithmetician) is interested iswhat can be derived from the structural features of a system such as the α-δconditions, rather than whatever peculiarities the elements of a simple in-finity display. This point is reinforced by the model-theoretic considerationsput forward by Dedekind in the passages from 132 to 134. Simple infinitiesform an equivalence class under the similarity relation: they are all similar toall the others (Theorem 132). This allows Dedekind to state, in Remark 134,that “every theorem regarding numbers, i.e., regarding the elements n of thesimply infinite system N set in order by the transformation φ [...] possessesperfectly general validity for every other simply infinite system Ω set in orderby a transformation ψ and its elements ν [...].The same significance whichthe transformation φ possesses for the laws in the domain N, in so far as

suggested by the paraphrase Dedekind himself offers, the one I already quote above. Yetthis generates a departure from actual thoughts to possible thoughts (Platonic ideas?),which does not sounds like anything Dedekind would say: it actually sounds closer toFrege’s philosophical sensibilities. On this point, also Russell (1919, pp. 139).

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every element n is followed by a determinate element φ(n) = n’, is found [...]to belong to the transformation ψ for the same laws in the domain Ω [...] Bythese remarks, as I believe, the definition of the notion of numbers given in[73] is fully justified.”According to this passage, Definition 71 offers an adequate characterizationof natural numbers. In the same passage Dedekind states that the element νof Ω, the one that was put in correspondence through the function introducedin Theorem 132 to the n element in the number-series N, “can be called thenth element of Ω and accordingly the number n is itself the nth numberof the number-series N”. Nonetheless, I’m not entirely sure what the valueof this isomorphism is supposed to be when it comes to the introduction ofnaturals.Keep in mind that N, the number-series, appears to be the same arbitrarysimple infinity introduced in Definition 71, one of the sets in the equivalenceclass introduced in 132 and 133 (at least the notation is the same). Doesthis passage mean that numbers could be the elements of whatever simplyinfinite set in the class? Or that this equivalence relation should allow usto posit as the proper object of arithmetic as something “over and beyond”the successions we can find throughout space-time? Is perhaps the object ofarithmetic the structure itself that is preserved in all systems of the class? Aswe can see, answering to the question “what are numbers?” entails, when itcomes to Was sind und was sollen die Zahlen? answering another question:“was Dedekind a structuralist, and of what kind?”.The second point, which is more closely related to the nature of numbers,is also ambiguous. In particular, it is open to interpretation as exactly whatdoes the term “abstraction” stand for. Dedekind writes that, in order to getnumbers from the elements of a simply infinite set, what we need to do isto “neglect the special character of the elements; simply retaining [...] therelations to one another in which they are placed by the order-setting trans-formation”. This passage is highly ambiguous; more precisely, it is ambiguouswhether Dedekind’s abstraction is merely a subjective act of ignoring thatdoes not change the properties of its subject, or something more. Were theformer to be the case, numbers would simply be the elements of whateversimple infinity (whatever elements of an order). On the contrary, were ab-straction to be something more than a methodological act of neglet, numberswould be something new. This passage does not help us understand what“abstraction” precisely means in this context: the use of the expression “ne-glect” suggests the first alternative, while the following sentence, claimingthat numbers are “created” by the mind, points to the second one. Those are(some of) the problems of interpretation that I will discuss in the followingsections.

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5 Problematics in Dedekind’s account of natu-rals

Through the notions of, respectively, weak ancestral and chain, Frege andDedekind are able reconstruct in a very similar way the succession of nat-urals, and they almost obtain the same result when it comes to show thatmathematical induction is a special case of a logical inference. But they de-fine natural numbers in a very different way; we have already introducedseveral differences between their approaches in the third section (cardinalsvs ordinals, psychologism vs antipsychologism), but it is now time to see thephilosophical reaction to Dedekind’s introduction of natural numbers. Thecriticisms I’m going to discuss were offered against Dedekind with the inte-tion of being knock-out arguments against his position, and they are probablyresponsible for the lack of attention reserved to him in the recent past of thephilosophy of mathematics.As I said, Frege offered three criticisms to Was sind und was sollen dieZahlen? in the first volume of the Grundgesetze, but none of them wasdirectly related to Dedekind’s introduction of naturals. Furthermore, it iscrucial to notice that although Frege spent a significant amount of time inthe Grundlagen criticizing psychological abstraction in mathematics, he doesnot speak ill about the introduction of numbers by abstraction in Definition73 of Dedekind’s foundational writing. Admittedly, the kind of abstractionemployed by Dedekind is at least partially different.Psychological abstraction was a very common position held by philosophersand mathematicians in the XIX century according to which new objects, inthis case numbers, could be obtained through a creative psychological pro-cess of generalization. In the words of Dummett (1991a, pp. 50): “One of themental operations most frequently credited with creative powers was that ofabstracting from particular features of some object or system of objects, thatis, ceasing to take any account of them. It was virtually an orthodoxy [...]that the mind could, by this means, create an object or system of objectslacking the features abstracted from, but not possessing any others in theirplace”. In the case of cardinal numbers, the idea was that numbers couldbe generated as sets of units through psychological abstraction from the pe-culiarities of the elements of some set; what one (allegedly) obtains in theend of this procedure, is the number corresponding to the cardinality of theoriginal set. Husserl subscribed to this theory in his 1891’s book Philosophieder Arithmetik, despite an earlier and substantial discussion with Frege onthe matter.42 Also Georg Cantor held this theory, as well as other important

42See Dummett (1991a, chapter 12), and Dummett (1991b, chapter 2.)

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mathematicians such as Schröder, Thomae and Lipschitz.43

The idea of creating numbers as units of distinct but featureless units throughpsychological creation was fiercely opposed by Frege in the Grundlagen derArithmetik(§29 - §44); although his criticisms were exceptionally on the spot,hardly anyone gave up his support to the position because of them: their im-pact on the mathematical community was close to zero. They can be sosummarized:

• we cannot generate the objects of arithmetic through mental acts. Thisis, again, an anti-psychologist point on the objectivity of the discipline.Whether or not the mind can create something, that something cannotbe numbers.

• even if that wasn’t the case, one cannot obtain through abstraction a setof distinguishable and featureless units from a set of a given cardinality.Take a set of four cats: if you abstract from their peculiar differences,what you obtain is perhaps the more general concept of “cat”, ratherthan another object called “unit”, with less properties than the originalcats. But even if that was the case, that unit-object would not berepeated four times, since the source of difference between the four cats(e.g., difference in weight, location and so forth) has been abstractedaway in the process. And since, of course, the original set of four catscannot be the number four, this means that the whole theory is notgoing to work. The same goes for any set of any given cardinality.

Does Dedekind’s abstraction fall prey to these objections? Only to some ex-tent. First of all, it does not fall prey to the second point, as Dedekind doesnot work with cardinal numbers and does not pretend to treat them as sets offeatureless units obtained through abstraction from a set of a given cardinal-ity. In particular, the elements obtained after the abstraction (the elementsof the simply infinite system) are themselves numbers, and are not feature-less; they have the properties given by conditions α-δ in Definition 71. Eachof them, furthermore, has a special and distinct role in the order given bythe function φ (e.g., “1”, the base-element, has different properties than itssuccessor, namely, its φ-image); they are not, and they are not meant to be,neither featureless or indistinguishable.The second issue about psychological abstraction relates directly to the issueabout psychologism in Dedekind that I introduced earlier on: it seems thatDedekind allowed for numbers to be created by the human mind; his notionof object (Ding) as object of thought, allows for that, and he does not seem

43For Cantor, see Beiträge zur Begründung der transfiniten Mengenlehre in Cantor(1932, pp. 282-356).

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to be bothered by the issue of objectivity as Frege was.

Altough Bertrand Russell in his Principles of Mathematics (1903) expresseswords of gratitude for Dedekind’ work on analysis, he has harsher wordswhen it comes to some philosophical aspects of his work, and, closer to ourinterest, about the introduction of numbers. This is both the case for thereal numbers in Stetigkeit und irrationale Zahlen, where the latter are intro-duced as something new and different from the set-theoretic constructionson rationals they correspond to, and for the natural numbers in Was sindund was sollen die Zahlen?. In the second case, as we have seen, naturals areintroduced by abstraction out of the elements of a simple infinity. But Rus-sell suspects that what Dedekind is doing is merely assuming their existence:in Russell’s famous expression (1919, pp. 71): “The method of ’postulating’what we want has many advantages; they are the same as the advantages oftheft over honest toil.” This remark is similar to a similar one, made by Fregeearlier on, in the second volume of the Grundgesetze; Frege’s point is thatDedekind’s principle of abstraction is not explicitly regulated by any rule andit is unclear whether there is any limit to it; in short, he is wondering whetherDedekind’s abstraction does not allow him to posit whatever entity he wants.In the words of Frege: “If there are logical objects at all—and the objects ofarithmetic are such objects—then there must be a means of apprehending, orrecognizing, them. This service is performed for us by the fundamental lawof logic that permits the transformation of an equality holding generally intoan equation [i.e., Basic Law V]. Without such a means a scientific foundationfor arithmetic would be impossible.”Here Frege’s point seems to be very close to Russell’s (that perhaps wasinspired by this earlier argument). The concept of number has to be prop-erly clarified through an explicit definition, or, when that is not possible,through the resort to what we may now call, a “contextual definition”, aprinciple assigning identity conditions for the new entities (as it was the casefor Basic Law V for extensions of concepts, numbers, in Frege’s final theory).In this way, we acquire a way to recognize and properly characterize whatwe introduce; this leads to the “Julius Caesar problem” I introduced earlieron: numerical predicates need applicability conditions in order to answer thequestion “what is a number?” Definition 73, and, in general Was sind und wassollen die Zahlen?, does not provide such answers, as, for instance, naturalsare not explicitly introduced through definitions.44

Russell’s second criticism towards Dedekind relates more closely to the no-tion of abstraction as free creation and Dedekind’s allegedly structuralistposition. In the Principles of Mathematics (pp. 248-249) he rightly statesthat Dedekind’s sees natural numbers as ordinals rather than cardinals; but

44I will however argue in section 7 that Dedekind’s naturals are properly characterized.

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then he says that ordinals are somewhat more “complicated“than cardinals.Why is that? Russell says:

“It is impossible that the ordinals should be, as Dedekind suggests, nothing butthe terms of such relations as constitute a progression. If they are to be anythingat all, they must be intrinsically something; they must differ from other entitiesas points from instants, or colours from sounds. [...] Dedekind does not show uswhat is it that all progressions have in common, nor give any reason for suppos-ing it to be the ordinal numbers. [...] What Dedekind presents to us is not thenumbers, but any progression: what he says is true of all progressions alike, andhis demonstrations nowhere [...] involve any property distinguishing numbers fromother progressions. No evidence is brought forward to show that numbers are priorto other progressions. We are told, indeed, that they are what all progressions havein common; but no reason is given for thinking that progressions have anything incommon beyond the properties assigned in the definition, which do not themselvesconstitute a new progression.“

Russell’s argument here seems to be that, when thinking about ordinals, we areonly thinking about specific progressions of elements (e.g., points, instants) whichare not themselves numbers. Nothing is given, in describing such progressions andtheir elements, that should allow us to postulate an additional progression of num-bers; there are two closely related reasons for that:

• Dedekind seems to hold that, through abstraction, the series of ordinals iswhat all progressions have in common; yet what all progressions have in com-mon are the α-δ conditions specified in Definition 71. This is unquestionablytrue.45

• numbers as elements of a progression obtained by abstraction would onlyhave structural properties, and this is unacceptable.

The second point is of crucial importance here. It seems that Dedekind’s ordinalnumbers have no other properties except for being elements of that progressions(unlike points, or instants, which have other, perhaps intrinsic, properties). Theproblem then is that numbers, qua objects, cannot be characterized in a structural-only way through the relations they display because of the φ function over thesimple infinity; and this is why thinking of naturals as ordinals is harmful. Forinstance, the natural number n cannot only be fully characterized as “being the

45There is however a way to make sense of this claim, which corresponds to what is nowknown as ante rem structuralism; if Dedekind is so interpreted, the subject of abstraction,what is common to all progressions is a Platonic structure-type exhausted by conditionsα, β, γ, δ; in this context, the ordinal numbers introduced in Definition 73, rather thanbeing full-fledged “objects” that need to have non-relational properties, are merely placesin the structure. See section 6.2 for this option; it is however highly questionable whetherDedekind held something even close to this.

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n-th element in an order”: there has to be something playing this role. In short,according to Russell, exactly because there cannot be objects only characterized re-lationally/structurally, numbers as elements of a “special progression” (presumably,the number-series N introduced in Definition 73) cannot be prior to the progres-sions we normally encounter, e.g., progressions of instants, points and so forth.46

Did Dedekind think that numbers could be characterized only by structural prop-erties? Possibly. At least, he doesn’t seem to take this conclusion as problematicanywhere in Was sind und was sollen die Zahlen?. The number 1, which he takesto be the first natural number, seems to be only characterized as being the base-element, e.g., the only element of N not contained in its image. To understandthis point we have to take into consideration the crucial role abstraction playsfor Dedekind; psychological or not, it provides a role of creation which presum-ably allows for such unusual objects to exist. Dummett (1991a, pp. 52), who takesDedekind to be a psychologist, puts the point cleverly, although in an highly sar-castic manner: “he [Dedekind] believed that the magical operation of abstractioncan provide us with specific objects having only structural properties: Russell didnot understand that belief because, very rightly, he had no faith in abstraction thusunderstood.”Therefore, the question as to whether numbers, as objects, can be characterizedonly through the order they are found in, depends on other questions: what is ab-straction, for Dedekind? What is an object?

What did Dummett himself believe of Dedekind? It is important to answer thisquestion since Dummett’s eight-pages-dismissal of Dedekind’s philosophy of math-ematics in the fifth chapter of his important book Frege: Philosophy of Mathematicswas probably the first serious discussion about Dedekind’s philosophy of mathe-matics since Russell. As seen before, he basically sides with Frege and Russell. Asmentioned in the earlier quotes, Dummett seems quite happy to criticize Dedekindon the grounds of him being a supporter of psychological abstraction; it shouldbe clear now, however, that it is not obvious whether Dedekind can be considereda psychologist, and, furthermore, the abstraction device employed in Definition73 of Was sind und was sollen die Zahlen? is not the one, used by many math-ematicians, Frege attentively demolished in the Grundlagen; Frege himself didn’toppose Dedekind on this grounds, but later wondered, in the second volume ofthe Grundgesetze, about the limits and conditions of Dedekind’ abstraction. YetDummett completely fails to take into consideration this fact.Dummett also agrees with Russell, complimenting his passage about Dedekind inthe Principles of Mathematics, about the possibility of number being structural-only objects; and, being highly critical about Dedekind’s abstraction, he had notfaith the abstraction process illustrated by Dedekind could provide such items nei-ther. Here again Dummett is not bringing anything new to the table.

46See section 6.3 for Russell’s “charitable” reading of Dedekind that avoids the conclusionof number as structural-only objects.

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There is, however, an original argument he proposes in the same occasion. Thepoint is that, according to the introduction of numbers as elements of a simpleinfinity, Dedekind has no way to structurally distinguish a succession starting with0 with another starting with 1, as the only thing characterizing the first number,whether it is 0 or 1, is its position as the base-element of the simple infinity; if wetake numbers as primarily elements of the φ-order, there is no structural way todistinguish a succession starting with 0 from another one, starting with 1. Likewise,0 and 1 cannot be properly differentiated.At first glance, it looks like a good argument. The theory presented does indeedfall in its scope; Dedekind decides to start the succession of natural numbers with1 instead of 0,47 but there seems to be no way to structurally distinguish the twosuccessions; therefore, there is no way to differentiate 1 and 0 neither. However, 1and 0 are clearly two different numbers, with different properties.Parsons (2008), while defending is own brand of structuralism, suggests the exis-tence of other ways to distinguish a simple infinity whose base-element is 1 fromanother whose base-element is 0, once arithmetic is fully reconstructed; for instance,the base-element which is neutral with respect to addition is the number nought.The mention of Parsons’ book is quite important at this point. In fact, althoughneither Dummett or the neo-logicists such as Crispin Wright cared for Dedekindvery much, there is a scenario in modern day philosophy of mathematics where he ispositively remembered, and, indeed, hailed as a forefather: structuralism. Roughlyput, structuralism in philosophy of mathematics is the theory according to whichwhat mathematicals theories describe are not numbers as objects, but structures(usually understood in a set-theoretical fashion). Horsten (2012) thus describesstructuralism: “Structures consists of places that stand in structural relations toeach other. Thus, derivatively, mathematical theories describe places or positionsin structures. But they do not describe objects. The number three, for instance, willon this view not be an object but a place in the structure of the natural numbers.”Steward Shapiro, an important exponent of structuraism, thus describes the the-ory, while already suggesting his favourite version of it: “[P]ure mathematics is thestudy of structures, independently of whether they are exemplified in the physicalrealm, or in any realm for that matter”.48 This is the case for instance in modernalgebra, where systems of objects are studied precisely as satisfying relational con-ditions which make them elements of groups, fields, modules and so forth.Many structuralist are quite happy to think of Dedekind’s foundational writings,especially Was sind und was sollen die Zahlen?, as one of the first expressions ofstructuralism. For instance Hellman: “The idea that mathematics is concerned prin-cipally with the investigation of structures of various types in complete abstraction

47Both Frege and Peano start with the number 0; it could be argued that Dedekind’sneglect of the number nought has something to do with the fact that his set theory allowsfor the empty set only with some degree of reluctance. Yet, according to Dedekind, numbersare not sets.

48Shapiro (1997, pp. 75).

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from the nature of individual objects making up those structures is not a novelone, and can be traced at least as far back as Dedekind’s classic essay, Was sindund was sollen die Zahlen? ” 49 , but also Shapiro himself: “a direct forerunner of[...] structuralism is [...] Dedekind. [...] His presentation of the natural numbers viathe notion of Dedekind-infinity, in Was sind und was sollen die Zahlen?, and someof his correspondence constitute a structuralist manifesto.”50

In a very general sense, Dedekind was a structuralist. The notion of simple infinityrequired as the ultimate source of all arithmetic describes a set-theoretical con-struct, and the numbers introduced are similarly characterized as elements of thatconstruct: what matters, is the the φ-order, that is to say, the structure. Let’s seeonce again this passage from Definition 73: “The relations or laws which are derivedentirely from the conditions α, β, γ, δ, in [71] and therefore are always the same inall ordered simply infinite systems, whatever names may happen to be given to theindividual elements form the first object of the science of numbers or arithmetic.”Additionally, Dedekind’s interest in the operation of abstraction when introducingnatural numbers is very similar to what structuralists usually claim (see the earlierquote from Geoffrey Hellman): the structure is focused by the mathematician inabstraction from the particular nature of the elements so structured.

To sum it all up, it seems that Dedekind has been subject to two main inter-pretations: one, as a psychologist, and the other, as a structuralist. These labelsattached to the mathematician from Braunschweig are both partial and tentative,yet they both point to extremely compelling problems in the interpretation of hisview on numbers. In particular:

• The notion of abstraction is central for Dedekind, but how is it to be inter-preted? Is it a mere act of subjective neglect? Or is it something more, an actof psychological creation, as sometimes Dedekind states? Or is it somethingnot psychological altogether?

• The numbers so obtained through “abstraction”: what are they? Do theyonly have structural (arithmetical) properties? Are they objects? What is anobject? Does the operation of abstraction, whatever it is, characterize themwith sufficient precision, as the Frege of the Grundgesetze denies?

• what is the ultimate object of study of arithmetic (and so, ultimately, ofall mathematics)? Is it number-objects? Or the structure itself of a simpleinfinity? And, if that is the case, what is the nature of this structure?

All of these unresolved dynamics undermine Dedekind’s claim that Was sind undwas sollen die Zahlen? is able to offer a final clarification of numerical notions, thenature of numbers as objects, and, ultimately, a rigorous foundations for the whole

49Hellman (1989, p. vii).50Shapiro (1997, p. 14).

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of mathematics. But how to answer them on the slim evidences Dedekind offers?51

I suggest that the best way to make sense of Dedekind’s position through this wholeset of problems is by focusing on the claim that Dedekind was a structuralist ; byunderstanding what “structuralism” is taken to mean I will clarify what “abstrac-tion” is supposed to mean as well, and what number-objects are taken to be. Inthis way: it will also come out what the ultimate ground of disagreement betweenDedekind and Frege truly is.This is what I’m going to do in the next, final section.

6 Dedekind as a structuralistAs we have seen, Dedekind has been sometimes listed amongst the ranks of struc-turalists as one of their forefathers. As modern structuralism in philosophy of math-ematics is not an omogeneous position, it would be best to describe the spectrumof possibilities and check whether or not Dedekind fits the description. As far as Iknow, there are three main variants of structuralism in philosophy of mathematics:set-theoretical structuralism, eliminativist structuralism, and ante rem structural-ism. Let’s see them in turn, followed by a more original alternative (from Reck(2003)) more sensitive to Dedekind’s intellectual background.

6.1 Set-theoretical structuralismAccording to Benacerraf (1965) there is no point in arguing, from a mathematicalpoint of view, whether numbers, if taken as sets, are identified with Zermelo’s orVon Neumann’s ordinals; this is because those successions are isomorphic and theyare therefore elementarily equivalent; the same rules of arithmetic are satisfied bythem. Although this fact could be taken to mean (as Benacerraf himself did) thatnumbers should not be understood as objects in the first place, perhaps the point isthat what the arithmetician is interested in, is the ordering structure, whatever theelements exemplifying it are; where this structure is understood set-theoretically, asin the case of Zermelo’s and Von Neumann’s ordinals, this is sometimes called set-theoretical structuralism. According to set-theoretic structuralism, all that mathe-matics talks about are sets, used for model-theoretical purposes, and abstractionis a mere act of ignoring the non-set-theoretical characteristics of their elementssince, from a mathematical point of view, as long as those models are equivalent,they do not really matter.This theory, applied to Was sind und was sollen die Zahlen?, would entail thatthe structure the arithmetician is interested in, is an arbitrary simple infinity. And

51The only place where Dedekind carefully reconstructs for the reader the train ofthoughts leading to Was sind und was sollen die Zahlen? is the (already mentioned)1890 letter to Keferstein. See Dedekind (1932).

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the abstraction Dedekind talks about in Definition 73 is nothing but the subjec-tive act of neglecting the peculiarities of its elements, as they are not interestingfrom a mathematical point of view. There is slim evidence that, when Dedekindwas thinking about natural numbers, he was thinking about elements of a sim-ple infinity whatsoever. Nonetheless, someone could indeed think of Dedekind asa set-theoretical structuralist: it is true, first of all, that the way he introducesabstraction in Definition 73 suggests that he is speaking about the elements ofsome simple infinity, and that such operation only consists in deliberately ignoringsome of their properties;52 furthermore, when he introduces the number-series inDefinition 73, he employs the same notation he used to denote an arbitrary simpleinfinity in Definition 71; that is to say, he used the simple capital N, instead ofintroducing a different notation. Remark 134 also suggests that every rule of arith-metic so obtained has great generality and is true of all simple infinity, since theyare all similar (and therefore form an equivalence class).However, there is, I believe, a simple argument against this structuralist interpre-tation of Was sind und was sollen die Zahlen?. Keep in mind that:

• according to set-theoretical structuralism, abstraction is a subjective act thatdoes not alter the non-arithmetical properties that the elements of a structurehave per se. Neither does it create new objects.

• because of Theorem 132, the choice between simple infinities is arbitrary forthe goals of arithmetic.

• Theorem 66 and Theorem 72 do not state that there is a single simple infinite.There could be more than one.

These three points, taken together, generate an absurd consequence for Dedekind’stheory of naturals, when interpreted as a form of set-theoretical structuralism. Thechoice between multiple simple infinities, with elements with different properties,is arbitrary; but, as that choice dictates what numbers ultimately are, as objects,this means that numbers have non-arithmetical properties, yet it is arbitrary whichones they have. And this is clearly unacceptable. If numbers are elements of oneof the simple infinities in the equivalence class established in passages from 132to 134, they have to be the elements of one of them in particular: yet there is nogrounds for deciding.Dedekind explicitly introduces number as objects. Therefore, if they are elementsof some simple infinity, it must be an additional “new” simple infinity, obtained byabstraction, and not one of the initial, arbitrary ones (the ones constituted by theworldy progressions we encounter, such as a progression of points in space, or ofinstants in time). In the light of this argument, I believe that the constancy in the

52“If in the consideration of a simply infinite system N set in order by a transformationφ we entirely neglect the special character of the elements; simply retaining their distin-guishability and taking into account only the relations to one another in which they areplaced by the order-setting transformation φ, then...”. Emphasis mine.

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use of “N” from Dedekind constitutes a minor mistake.53

Besides, Dedekind’s emphasis on abstraction as an act of creation tell us thatDedekind’s abstraction must be something more than the simple process of “ignor-ing some non-structural properties of the elements of a simple infinity”.Therefore the question now is whether Dedekind should be taken as claiming thatnumbers are not objects, or, if they are, that they are “special” objects, with onlyarithmetical (structural) properties; this is what I’m going to do, in turn, in thenext two subsections, introducing two other versions of structuralism.

6.2 Eliminativist structuralismOne, following Benacerraf, could follow structuralism as stating that mathematicsdoes not commit to us to number-objects to begin with; all mathematical state-ments are to be taken as universally quantified over all models; so, for instance, anarithmetical statement concerning number 5 could be restated as: in all models forarithmetic, the 5th element... In this way, the apparent commitment to numbers inarithmetic is dissolved.54 As the name suggests, eliminativist structuralism strivesfor parsimony in the postulation of new mathematical items, and this is accom-plished through universal quantification: this is what abstraction amounts to, inthis version of the theory.

It is not worthless to point out that eliminativist structuralism basically coincideswith Russell’s charitable reconstruction of Dedekind’s view after the criticisms of-fered in the Principles of Mathematics; and, following Russell, Dummett statedthat this brand of structuralism is followed by Dedekind in complete harmony witheveryday mathematical practice. Russell must have thought something along theselines: Dedekind could not have believed that numbers are objects only character-ized by being in a new, mysteriously privileged progression. Therefore, they are notobjects to begin with, and all of the relative statements have to be reformulatedthrough quantification to all progressions (all models of arithmetic). In his words:“what Dedekind presents to us is not the numbers, but any progression alike, and hisdemonstration nowhere -not even where he comes to cardinals- involve any prop-erty distinguishing numbers from other progressions”.55 And similarly Dummett,talking about mathematical practice:

“One may speak, for example of ’the’ five-element non-modular lattice. There are,53A relevant change in notation was employed by Dedekind in an earlier draft of the

booklet, but mysteriously disappeared in the 1888 published version. See Reck (2003, pp.405).

54This version of structuralism also has a modal element, in the sense that, other thanbeing quantified, mathematical sentences are also modalized; this is of no interest rightnow. Hellman (1989, 2001) offers a defence of this modal variant of eliminativist struc-turalism. Also see (Reck and Price 2000).

55Russell (1903, pp. 249). Emphasis mine.

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of course, many non-modular lattices with five elements, all isomorphic to one an-other; if you ask him [the mathematician] which of these he means, he will reply,’I was speaking of the abstract five-element non-modular lattice’. But then, evenif he retains a lingering belief in the operation of abstraction, his way of speakingis harmless: he is merely saying what holds good of any five-element non-modularlattice. This is how neo-Dedekindians such as Paul Benacerraf, who have under-standably jettisoned the doctrine of abstraction, would have us suppose it to bewith the natural numbers”.56

This neutralization of Dedekind’s view on numbers, although “charitable”, is notvery honest. Although some passage in Remark 134 about the equivalence of allsimple infinities as models of arithmetic could be taken as suggesting some kind ofuniversal quantification over them, Dedekind makes it very clear, in Definition 73,that numbers, as elements of some simple infinity or another, are objects and inno point in Was sind und was sollen die Zahlen? he seemed to express any kindof eliminativist attitude. In fact, he had no problems with abstract entities of anykind, and he seemed quite hapy to take set-theory at face-value as referring tospecial objects, known as “sets” or “systems”.57

6.3 Ante rem structuralismIf we refuse to take Dedekind as a set-theoretical structuralist, that could be takento mean that, according to him, numbers could not have non-mathematical prop-erties such as those of points, or instants; and relatedly, abstraction cannot simplybe the choice of “not taking into consideration” such additional features.There is a reason for this position. Dedekind was very clear about the necessity ofexpunging foreign notions from arithmetic and mathematics, even set-theoreticalnotions. He didn’t agree with Heinrich Weber’s Fregean-like suggestion of takingnumbers as extensions of sets (and refused Lipschitz’s suggestion about Stetigkeitund irrationale Zahlen about identifying real numbers with cuts on rationals, with-out the introduction of any new element) exactly because, in his words: “one willsay many things about a class (e.g., that it is a system of infinitely many elements,namely all similar ones) that one would surely attach only very unwillingly to thenumbers (as a weight)”. 58 Numerical notions should be rigorous, and number-objects should be “pure” elements; in the words of Reck (2003, pp. 387): “Set-theoretic notions, while more closely related to arithmetic ones than geometric or

56Dummett (1991a, pp. 52). Whether or not eliminativist structuralists suchs as Benac-erraf are really “neo-Dedekindians” (whatever that means) is of no importance here. AlsoParsons (2008) introduces eliminativist structuralism mentioning Dedekind, although hedoes not explicitly attribute it to him.

57This again has to to with his idea of objects being objects of thought, as he believedthe notion of set to correspond with a“grouping” faculty of the human mind.

58(Dedekind 1932, p. 490).

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empirical notions, still bring in aspects that are “foreign”, “disturbing”, and shouldbe avoided”.This suggests another way of taking the structuralist route about mathematics;from a model-theoretic point of view, if arithmetic has no preference when it comesto equivalent set-theoretical constructs, this is because what arithmetic is really in-terested about (its real object of study) is not any of them in particular, but ratherwhat they all have in common and makes them equivalent. According to StewardShapiro59, the most prominent supporter of ante rem structuralism, the actual ob-ject of mathematics is not any set-theoretical construct out of objects, but rather astructure-type (perhaps of a Platonic kind) that is related to the former by a rela-tion very similar to, if not identical to, instantiation. This structure-type does notdisplay any object in it, but “places” only characterized by the relational propertiesthat determine their position in the overall structure. The ante rem structuralistmay or may not want to say that places in the structure-type are full-blown objects;sometimes they talk about those places as sui generis objects, with an indepen-dent existence from the usual ones that fill the places in the structure, and generatemathematical structures as we know them.60 According to Shapiro (1997, pp. 74-83), “a structure is the abstract form of a system”, and a system (a mathematicalstructure) is “a collection of objects with certain relations”.As the reader can easily guess, ante rem structuralism is non-eliminativist positionwhich is not scared of introducing new mathematical items: not only set-theoreticalconstructs (as set-theoretical structuralism) but also peculiar structure-types, andother shadowy entities such as independently existing places in that structure.In the discussion of Russell’s criticism in the Principle of Mathematics (see theprevious section) the author argued that the series of natural numbers could notbe what all simple infinities have in common, since all that they have in commonare the α-δ conditions. But perhaps ante rem structuralism is a way to make senseof this claim while preserving its literal meaning: according to an ante rem inter-pretation of Dedekind’s structuralism, if the series N of naturals, as characterizedby those structural conditions, is the structure-type all different simple infinitieshave in common, and numbers are merely identified with places in that structure.The number 2, for instance, is different from any 2th element in a progression, butonly has the mathematical properties conferred by the simple infinity.61 The suigeneris objects employed by the ante rem structuralism are very much alike thepure elements Dedekind identified natural numbers with. There seems to be someground for attributing this position to Dedekind: numbers seem indeed to only havearithmetical properties, as I argued that we cannot identify them with objects of a

59Shapiro (1997). Also see Parsons (2008) for a discussion on this subject.60From a purely philosophical point of view, this means that relations exist prior to

their relata: this is why this position is called ante rem structuralism.61It is the image of the base-element of N. More precisely, since in N there are no

independent objects exemplifying the structure, it is the place in the first iteration of theφ-function.

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simple infinity whatsoever. They have to be something new : and a structure-typeseems to fit perfectly in the description. This structure-type, as characterized bythe four conditions in Definition 71 is the true subject of arithmetic and is theresponsible of the equivalence of all simple infinities as stated in Remark 134.This is indeed a very tempting interpretation of Dedekind’s theory of naturals inWas sind und was sollen die Zahlen?, and indeed, the idea of introducing a newelement different from the set-theoretical constructs which correspond to it is notnew to Dedekind: it was already present in the introduction of real numbers inStetigkeit und irrationale Zahlen. The are problems, however:

• Dedekind thinks of numbers as “normal” objects rather than something new.In the same way, he sometimes identifies the number series with a simpleinfinity, that is to say a determinate object of thought (Ding). This, as thereader knows, is the evidence in favour of a set-theoretical reading of Def-inition 73. There is a single instance, the already mentioned 1890 letter toKeferstein, where he refers to the number series as the “abstract types” ofall simple infinities, yet both type and token are, for Dedekind, objects: theyare not on different levels, as the exemplification relation would suggest: theyare merely isomorphic.A closely related issue is the following: if there are no independently existingobjects in the number series, but only places as sui generis objects, how couldthe number-series be isomorphic with other simple infinities, or how couldthere be any mapping whatsoever between them? How to redefine the notionof function in a context where a mathematical structure lacks a domain (astraditionally intended) is a problem of modern ante rem structuralism, yetthere is no trace of it in Dedekind.

• the second problem, a very important one, is the role of abstraction. Dedekind’semphasis on the act of abstraction as an act of creation must be alwaysconfronted with the possibility that what he had in mind was ultimately apsychological act, say, an act of the human mind. In the case of ante remstructuralism, this means that Dedekind could have thought to the numberseries N as a type created by the mind; this is very close to what is knownin philosophy as a conceptualist view about universals.

There is, however, a non psychological reading of Dedekind’s abstraction in thisante rem structuralism reading of Was sind und was sollen die Zahlen?. It is a boldsuggestion: the isomorphism between simple infinities established in Theorem 132could provide a principle of logical abstraction sufficient to introduce the structure-type as new mathematical objects as envisioned by the neo-Fregeans. This proposalis quite interesting as it closes a crucial gap between Frege’s introduction of num-bers by definition and Dedekind’s introduction of numbers by abstraction.62 More

62At the time Frege did not think that naturals could be introduced through Hume’sPrinciple and other abstraction principles without an explicit definition, but now neo-Fregeans seem to think it.

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precisely, we have identity conditions for structure-types, as relative to the set-theoretical constructs that exemplify them. For instance:

(ab) The structure of N = the structure of M iff N and M are isomorphic

where N and M are simple infinities, defined in Definition 71. Since all simpleinfinities are isomorphic (Theorem 132), that points to the existence of a singlenew entity that is so introduced, that is to say, the structure-type that is commonof all of them. Tait’s description of “Dedekind’s abstraction” looks reminiscent ofthis proposal:

“In set theory we construct the system <ω, φ, σ> of finite von Neuman ordi-nals, where σx = xUx. We may now abstract from the particular nature of theseordinals to obtain the system N of natural numbers. In other words, we introduceN together with an isomorphism between the two systems. [...] In this way, thearbitrariness of this or that particular ’construction’ of the numbers [...] noted inconnection with the numbers in Benacerraf (1965), is eliminated”.63

Keep in mind, however, that the structure-type that we are so introducing is notthe equivalence class of all simple infinities (we can however think of the structure-type and the equivalence class as the intensional and the extensional side of thesame notion).

6.4 Reck’s logical structuralism“Abstraction”, as we have seen, can mean many things; yet if we take Dedekind’sabstraction as something more than a mere act of subjective neglect, we are morelikely to give a psychologistic interpretation to it; in fact, as we have seen, Dedekindcalled numbers “a free creation of the human mind” (Definition 73), more likelycoming from a “creative powers [...] concerning mental things” (letter to Weber).Yet Dedekind does not offer any enlightening remark about how to answer Frege’sdevastating criticisms towards psychologism; therefore, a psychologistic reading ofDedekind is highly problematic.Reck (2003) offers a new historically-sensitive reconstruction of Dedekind struc-turalism, that differs from any other version in the modern debate, which ulti-mately leads to a consistent reconstruction of Dedekind’s brand of structuralism:Reck calls this view “logical structuralism”. According to this approach, Dedekind’sabstraction/creation is not a psychological properties, and furthermore, numbersonly have arithmetical properties; this, as in the case of ante rem structuralism,will ultimately allow to rethink to the general notion of “object”.

63Tait (1986, pp. 369, footnote 12) A similar hypothesis is brought forward by FrancescoGana in his 1963 commentary to the Italian translation of Was sind und was sollen dieZahlen?.

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This thesis is sensitive of Dedekind’s cultural background; and Dedekind’s culturalbackground was, most of all, in Göttingen. Independence from geometrical and aposteriori intuitions in the clarification of the basic conceptual elements of arith-metic is not the only legacy the Gottingen school left to Dedekind: there was also apoint in mathematical methodology. For instance, the driving thrust of Riemann’sresearch on non-Euclidean geometries was to explore new conceptual possibilitiesin a fully consistent way, even if, as some claimed, there could be no intuition in ourminds as source for this geometrical knowledge. More generally, the mathematicsof the Göttingen school was methodologically without boundary, from neither anepistemological or philosophical source. Also, Georg Cantor writes in (Grundlageneiner allgemeinen Mannigfaltigkeitslehre), that:

“Mathematics is in its development entirely free and is only bound in the self-evident respect that its concepts must both be consistent with each other and alsostand in exact relationships, ordered by definitions, to those concepts which havepreviously been introduced and are already at hand and established”.

Therefore, apparently, the only boundary in the determination of conceptual schemesin mathematics should be (logical) consistency. Dedekind’s insensibility to con-structivistic and finitistic concerns such as the ones from his colleague and rivalLeopold Kroneker is explicit in Was sind und was sollen die Zahlen?, where itseems that the mathematical methodology of the Göttingen school, and of Cantoras well, is proudly embraced.

But what has this to do with the interpretation of Dedekind’s structuralism?The question we had to answer was: how to take Dedekind’s notion of “creation”seriously enough, but in a non-psychologistic way? According to Reck, “new light isshed on Dedekind’s remarks about ’free creation’. They now reveal themselves alsoas the endorsement of a certain mathematical methodology [...] opposed to bothempiricist and materialist and to Kroneckerian strictures, and that, more positively,encourages the ’free’ exploration of conceptual possibilities as exemplified in theworks of Riemann and Cantor”.64

Reck notices what he calls Dedekind’s principle of determinatedness for objects,to be found in the very first passages of the booklet: “a thing [a Ding, an ob-ject of thought] is completely determined by all that can be affirmed or thoughtabout it.” In the case of system, this amounts to the principle of extensionalityregulating their identification. But what about numbers? Number have to be de-terminated in a way that what has been said about them in Was sind und wassollen die Zahlen? is both complete relativey to arithmetic and consistent. Wherecan we find completeness and consistency for numerical notions in Was sind undwas sollen die Zahlen?. The semantic completeness of Dedekind’s theory of num-bers can be proved by looking at the proof of categoricity (of the notion of single

64Reck (2003, pp. 393).

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infinity) in Theorem 132. Numerical notions need not to lead to contradictions,as well; however it seems that no proof of consistency is given when it comes tosimple infinities or their elements. The closest thing we have, that perhaps worksfor Dedekind as a proof of consistency, is Theorem 66 (which is about infinite, andnot simply infinite sets). But Theorem 66, as controversial as it is, is a proof ofexistence, not consistency. How can it proves the consistency of numerical notions?According to Reck, we must take the notion of determinatedness (of objects) veryseriously, in the light of the methodological issues pointed out earlier: mathemati-cal items are not found in space-time, they are not intuited, they are not sociallyor psychologically built: they are determined as complete and consistent possibil-ities.65 Their creation amounts to their determination; and their existence, theirdeterminatedness, is only decided by the aforementioned conditions of consistencyand completeness. This is why, in the notorious 1890 Keferstein letter, Dedekindrefers to the “proof” of existence in Theorem 66 as a claim about consistency: inshort, the only way to prove that there is an infinite set is by proving that thereis nothing wrong in the corresponding concept; and this is done by considering analleged instance of it. It therefore appears that there is nothing more to the exis-tence of an infinite set than the consistency of its concept.66 Dedekind wondered,after the definition of an infinite set in Definition 64:

“Does such a system exist at all in the realm of ideas? Without a logical proofof existence it would always remain doubtful whether the notion of such a systemmight not perhaps contain internal contradictions. Hence the need for such proofs”.

Reck’s logical structuralism ultimately consists in the following 4 theses:

• objects are logical determinates. Natural numbers, in particular, are merelydetermined by structural properties (e.g., 5 is only determined by the prop-erty “being the 5th element in a simple infinity”).

• Formal theory of existence; that is to say, existence amounts to a state offull logical determinatedness. There is no “thick existence” beyond that.67

65Does that mean that they are possible, rather than actual objects? After all, as in thecase of Riemann’s study of non-Euclidean geometries, the determination of such geometriesas “conceptual possibilities”, does not state whether our world is Euclidean or not.

66It is what we would call a semantic proof of consistency, rather than a syntactic one.Dedekind always moved in a model-theoretic environment which supports a semantic-first approach; also Theorem 132 as a proof of completeness is a proof of semanticalcompleteness; it ultimately depends on the notion of simple infinity, rather than a set ofaxioms.

67Göttingen was not the only place where similar approaches were employed; the Aus-trian philosopher Alexius Meinong developed, in the same period, a peculiar theory ofintentional objects; in his Gegenstandstheorie [theory of objects] (1904) being, or exis-tence (Dasein) was to be separated from mere subsistence (Sosein), which only consistedin objects having properties and being determined by the mind, as an intentional object

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• Creation is not a psychological, but a logical act of determination: to create orto “bring into existence” a mathematical item means to explore a conceptualpossibility about mathematics.

• There is no psychology in abstraction. “Abstraction” rather means that whenwe create (in the sense intended) mathematical items, we take into considera-tion the elements of a succession whatsoever, and we discard all the propertiesthat are of no interest from a mathematical point of view.

According to this interpretation, what Dedekind is ultimately doing in Was sindund was sollen die Zahlen? is exploring the possibility of natural numbers beingnothing but a succession of ordinals. The number-series N is “identified as a newsystem of mathematical objects, one that is neither located in the physical, spatio-temporal world, nor coincides with any of the previously constructed set-theoreticsimple infinities. [...] It is that simple infinity whose objects only have arithmeticproperties, not any of the additional, ’foreign’ properties objects in other simpleinfinities have”.68

Furthermore, this appeal to the methodological freedom proper of mathematicsmay dispel the need for a further (perhaps philosophical) justification for Dedekind’sordinal approach to natural numbers. Let’s remember why Russell in the Princi-ples of Mathematics said that ordinals were “more complicated” than cardinals: weuse natural numbers to put things in a succession; but what is it that makes usthink that numbers form a succession themselves? Yet, as long as the theory willbe consistent, there will be no need of a further justification for investigating itfurther.

7 ConclusionsAll the important problematics have been, I believe, introduced. As a conclusion,there are two questions that need to be answered about Dedekind’s value as alogicist:

• what is the best interpretation of Dedekind’s philosophical theory in Wassind und was sollen die Zahlen?

• how is it different from Frege’s logicism as expressed in the Grundlagen andGrundgesetze?

(an object of the mind) is. From 1915, Meinong additionally spoke about some objects(without being) as incompletely determinated, lacking some property, e.g., a triangle atthe highest level of generality. For an object to exist it is required from it to be com-pletely determined. Meinong’s most famous work, the 1904 paper on the theory of objectis reprinted and translated in Chisholm (1960).

68Reck (2003, pp. 400).

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Unfortunately, success in philosophy is rarely achieved by knock-out of the oppo-nent; it is more a matter of taking points and weighting different theories in thelight of their internal and reciprocal advantages and disadvantages. Therefore, I willweight the answers to all the questions given about the interpretation of Dedekind,and try to proclaim a winner. Secondly, I will measure the difference between this(allegedly) conclusive philosophical interpretation of Dedekind’s theory of naturals,and Frege’s.

7.1 The best possible Dedekind• About structuralism There is conclusive evidence against the identifica-

tion of Dedekind as a set-theoretical structuralist; the succession of naturals,as a simple infinity, is probably a set-theoretical construct, but a special one.In particular abstraction needs to be something more than a act of ignoringsome properties, as it is usually claimed by some structuralists; furthermore,numbers cannot have non-arithmetical properties.As for eliminativist structuralism, although its quantificational aspect res-onates with some passages of Dedekind’s second foundational booklet, thereseems to be no trace of eliminativism about new mathematical objects init.69 Dedekind’s structuralist, then, could either be identified with ante remstructuralism (an existing brand of structuralism in the debate) or Reck’slogical structuralism (an ad hoc proposal).No position such as ante rem structuralism can be found in Was sind undwas sollen die Zahlen? ; no type/token distinction is introduced, and nei-ther is discussed the possibility of numbers, as result of abstraction, beingspecial sui generis objects such as places in an abstract structure-type. Fur-thermore, ante rem structuralism has to deal with Dedekind’s claim thatnumbers, and the series they generate, is a free creation of the human mind.He must either have thought of the series of naturals as a universal-like struc-ture generated by the mind as conceptualists now think, and therefore openhis position to Frege’s anti-psychologist charge, or something along the lineof a Fregean-like logical abstraction as I suggested. In neither of the twoDedekind’s foundational writing there is an explicitly philosophical discus-sion on psychologism, so I think that at this point we should at least considerthe possibility of stopping being charitable with him, and stopping assumingthings about his position he didnt’ say: he explicitly talked, in multiple oc-casions, about numbers being created by the mind and the creative powersof the spirit, so we might has well take that literally. Perhaps the Kantianattitude so widespread in German universities of his time had some influ-ence on Dedekind as well. Of course Kant cannot be so simply charged ofpsychologism; but it was the same attitude in the philosophy of arithmeticto which Frege reacted when he was criticizing psychologism.

69On the contrary, Dedekind is at peace with sets, and sometimes, even something more.

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Reck’s logical structuralism is on the other hand an interesting and appar-ently complete option; yet his theory of objects as logical determinates needsmore elucidation. And a formal theory of existences clashes with Dedekind’sanalogy of the creation of numbers with the creation of railroads and tele-graphs; the existence of these objects is not formal, in any significant senseof the word.What can we say, then, about the best possible Dedekind? First of all, thathe was perhaps a psychologist, or alternately, that he believed in some kind oflogical abstraction. And secondly, because numbers as objects are a creationof the human mind, he must have implicitly had a theory of objects (and/orexistence), either, as Reck claims, as logical constructs, or as mere places inthe special structure all simple infinities share.

• Abstraction vs definition Frege and Russell accused Dedekind of postulat-ing the existence of natural numbers instead of doing the real job and provingit. Additionally, Frege stressed out in the second volume of the Grundgesetzethat Dedekind di not provide any definition for natural numbers.This is bothan epistemological point (how do we know those things really exist?) and apoint about methodology in the introduction of new mathematical objects.Frege stated:

“If there are logical objects at all —and the objects of arithmetic are suchobjects— then there must be a means of apprehending, or recognizing, them.This service is performed for us by the fundamental law of logic that permitsthe transformation of an equality holding generally into an equation [i.e.,Basic Law V]. Without such a means a scientific foundation for arithmeticwould be impossible”.70

According to our reconstruction of Dedekind’s theory, this is (partially) un-fair. Although it is true that no explicit definition is ever provided, a naturalnumber can be properly characterized: it is what only has arithmetical /structural properties, those put forward in the definition of a simple infinityin Definition 71. The number one, for instance, is whatever object can beexhaustively described as “being a base-element”. The Julius Caesar problemdoes not threaten Dedekind, as obviously the famous Gaul conqueror hadnon-mathematical properties.Now, for the epistemological point: did Dedekind merely postulate new en-tities? Not quite. At least, that was not his intention. Dedekind did notpretend the reader to believe in systems and simple infinities: this is whyhe introduced Theorem 66 and Theorem 72. It might be (rightly) arguedthat Theorem 66 doesn’t prove much: in this respect, Was sind und wassollen die Zahlen? has proven to be failure. But in his eyes, they were

70Frege (1903, pp. 278-279).

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meant to show that there was nothing inconsisent or ill-conceived in the no-tion of simple infinity, so that the succession of numbers could be extractedthrough abstraction from other progressions of elements. Furthermore, theproof of isomorphism provided by Dedekind was to ensure the uniqueness ofwhat was abstracted from all the different progressions. Therefore, althoughDedekind’s proof were (partially) unsatisfactory, he was clearly not “magi-cally”postulating natural numbers into existence: what he was working onwas to be both conceptually consistent and unique.71 With this understand-ing in mind, the advantages of “defintion” over “abstraction” are harder tosee.As for abstraction itself, Frege is right to claim that the operation of abstrac-tion, so crucial for Dedekind, is not explicitly investigated (at least if we ex-clude the aforementioned Keferstein letter) nor there ar clear boundaries toit; Dedekind, in this respect, did not provide any help for his commentatorscoming from a philosophical point of view. But I have earlier claimed thatDedekind’s abstraction is either logical or psychological abstraction, and, inboth cases, the objects so abstracted are no ordinary objects like rocks, ortrees. According to Reck, they are logical determinates. According to an anterem structuralist interpretation, they are mere “places” of a structure thatis either obtained by abstraction principles such as (ab), or a psychologi-cal generalization out of worldly progression of elements. In the first case,Dedekind’s introduction of natural numbers would not be, after all, thatmuch different from Frege’s.

So neither of those arguments has much force. Dedekind did not merelypostulate the existence of numbers: they are either psychologically or log-ically constructed out of abstraction, and Theorem 66 and 72 (allegedly)provided the consistency of the corresponding numerical notions. And nei-ther did he fail to characterize them properly; this is the task of Definition 71.

7.2 Final comparison with FregeWe now have a good grasp of both Frege’s and Dedekind’s introduction of naturalnumbers, and of their importance in the debate; it is thus time to summarize theircore similarities and differences. First of all, I will list what they agreed on.

• Numbers are objectsAlthough this claim means two different things for Frege and Dedekind (as Iwill explain in the disagreements list) there still is a point of contact. Theyboth refuse, in analysing numerical concepts, to take statements where num-bers figure in an adjectival position as prior to the others.

71Perhaps, as I earlier suggested, we can even think that Dedekind gave identity condi-tions to structures such as the one of natural numbers.

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Frege’s “sleight of hand” as Dummett calls it, in §56 of the Grundlagen con-sisted exactly in the presupposition that numbers should be taken as object,and statements such as “3 is the number of the concept F” as more funda-mental than “there are 3 Fs”.Dedekind was not interested in non-arithmetical statements such as “thereare 3 cats”; he was more concerned with inner-arithmetical expressions, suchas “7+5=12”. Nonetheless, after the introduction of simple infinities in Defi-nition 71, he felt the need almost immediately to introduce natural numbersas elements of an additional simple infinity in Definition 73, so, qua objects.He could have used the notion of simple infinity to analyse numerical state-ments of cardinality of adjectival form first; but keep in mind that cardinalnumbers are introduced later on in Was sind und was sollen die Zahlen?,and they do not have much importance.He was perhaps guided by inner-mathematical motives. As Gillies (1982, pp.42) puts it: “if we take our reductionist’s prescription seriously [to eliminatecommitment to numbers as objects by paraphrasing all arithmetical state-ments in adjectival form like “there are n Fs”], we would be forced to abandonvirtually all higher number theory, and most of Peano arithmetic. We wouldnot be able to get much beyond simple numerical equations”.

• Numerical succession (and mathematical induction).In the preface of the second edition of Was sind und was sollen die Zahlen?(1893), Richard Dedekind wrote:

“However different the view of the essence of number adopted in [Frege’sGrundlagen] is from my own, it contains, particularly from §79 on, points ofvery close contact with my paper, especially with my definition (44) [chain].The agreement, to be sure, is not easy to discover on account of the differentform of expression; but the positiveness with which the author speaks of thelogical inference from n to n+1 [. . . ] shows plainly that here he stands uponthe same ground with me.”

In the Grundlagen der Arithmetik Frege never specifically introduces or de-fines the series, or succession, of natural numbers. On the contrary, it some-time seems that such a definition is implied when he speaks, discussing an-cestral and weak ancestral, about “following from... in a series”. I don’t thinkthat is the case: the idea of succession or series of naturals is not prior to thenotions introduced in section IV of the Grundlagen, but it is shaped by them.It is easy to achieve this result with the machinery he provides: for instance,the series of naturals can be considered the set of numbers (definition §72)bearing the weak ancestral relation of R with 0 (where R is defined alongthe lines of §76 as “immediate predecessor”) as ordered by the R-relation ofimmediate predecessor itself. The notion of chain as in Definition 71 of Wassind und was sollen die Zahlen? is perfectly mirrored by the combined def-

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initions of “immediate predecessor” in §76 and that of weak ancestral of aR-series in §81 of the Grundlagen of Arithmetik.Furthermore, both of these constructs serve the purpose of showing thatmathematical or complete induction, the passage from n to n+1 can besubsumed under a particular kind of logical inference.

Lastly, I will list the disagreements between Frege and Dedekind when it comes tonatural numbers.

• What an object isFor both Frege and Dedekind natural numbers are object. Yet this claim leadthem to parallel conclusions only regarding the logical analysis of numericalstatements, as they disagree on what an “object” ultimately is.According to Frege, numbers are objective non-physical objects (§26, §58); heconsiders them to be objective as they are independent, at least in principle,from any idea or intuition that we may have of them; but of course we cannotexpect to find natural numbers in space-time, therefore “I distinguish what Icall objective from what is handleable or spatial or actual”.72 Object, on theother hand, is one horn of his basic ontological repartition between objectsand concepts, that is to say, between saturated and unsaturated entities.For Dedekind as well, it is quite difficult to understand what objects are;in Definition 1 of Was sind und was sollen die Zahlen?, he introduces them(Ding) as objects of thought, and he makes it immediately clear that systemsare objects. In the light of the previous discussion on the nature of Dedekind’sstructuralism and abstraction, there are two candidate answers:

– mathematical objects are logical determinates, whose existence is whollyformal (Reck’s logical structuralism)

– numbers are sui generis objects, say, places in the structure-type (anterem structuralism)

Surely the ante rem interpretation is right to claim that numbers are objectsof a particular kind, as it is settled at this point that Dedekind’s naturals,unlike other objects, only have structural and arithmetical properties. Yethe does not explicitly speak about the existence of this kind of objects. Onthe other hand he does speak about all objects as standing in the need ofdetermination (Definition 1), thus supporting the idea that objects studiedby mathematics are ultimately logical constructs, disciplinary possibilitiesexplored for the sake of research itself.When it comes to objects, it is quite difficult to understand what Dedekindhad in mind. Nonetheless, I think it is safe to assume that although Fregeand Dedekind agreed on the claim that numbers are objects, they disagreedon what “object” amounts to; that makes it quite hard to draw any parallelbetween their ontologies.

72Grundlagen der Arithmetik, §26.

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• Cardinals vs ordinalsLet us say one final time that for Frege natural numbers are primarily car-dinals, while for Dedekind, they are first of all ordinals. This is a crucial dif-ference between them, one that highlights their backgrounds: Frege followedin the more philosophical tradition of treating cardinality applications asfundamental to the concept of number, though non-mathematical in nature.Dedekind’s conception of number, then, is more minimal and closely relatedto mathematical practice. Dummett (1991a, pp. 48) thus says: “Dedekind’scentral concern was to characterise the abstract structure of the system ofnatural numbers; what those numbers are used for was for him a secondarymatter. In this respect Frege, pioneer as he was, was old-fashioned”.The impact of this divergence is surely of great value; and, furthermore, onemust consider the fact that both of them need to take into account the otheruse of natural numbers: Frege must take into account ordinals and Dedekindcardinals. Specifically, when defining ordinals, Frege ultimately agrees withDedekind, as, through the notion of weak ancestral of a relation R (Grundla-gen §81) natural numbers can be considered all of those numbers that followin the series starting with 0. But “number” here is already defined as thenumber of a concept (as of §72), so as a cardinal number.Dedekind, on the other hand, refuses to take cardinal numbers as sets orextensions in a Fregean fashion, as explicitly suggested to him by HeinrichWeber, because of the foreign notions such an analysis would bring into arith-metic; he introduces cardinal numbers starting by Definition 161 of Was sindund was sollen die Zahlen?, by saying that a n is the number of a set, whereasn is presumably an element of the simple infinity N introduced in Definition73.Their differences, rather than bring forth incompatibilities between their ac-counts, may shed some light on different uses of the same numbers, both asreferenced in inner-arithmetical statements, and in extra-arithmetical claimsof cardinality.

8 BibliographyBolzano, Bernard (1851). Paradoxien des Unendlichen, C.H. Reclam (Paradoxes ofthe Infinite; Ewald 1996: 249–92 (excerpt)).Cantor, Georg (1883). Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Math-ematische Annalen 21:545–586.Cantor, Georg (1932). Ernst Zermelo, ed., Gesammelte Abhandlungen mathema-tischen und philosophischen inhalts, Berlin: Springer. Carnap, Rudolf (1931). Dielogizistische grundlegung der mathematik. Erkenntnis 2 (1): 91–105.Chisholm, Roderick (ed.) (1960). Realism and the Background of Phenomenology.Free Press. 76–117.Dedekind, Richard. 1872. Stetigkeit und irrationale Zahlen, Braunschweig:Vieweg

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(reprinted in Dedekind (1930– 32), Vol. 3, pp. 315–334; English trans., Continuityand Irrational Numbers, in Dedekind (1963), pp. 1–22).Dedekind, Richard (1888). Was sind und was sollen die Zahlen?, Braunschweig:Vieweg (reprinted in Dedekind (1930–1932), Vol. 3, pp. 335–391; English trans.,The Nature and Meaning of Numbers, in Dedekind (1963), pp. 29–115).Dedekind, Richard (1930–1932). Gesammelte Mathematische Werke,Vols 1–3, R.Fricke et al., eds., Braunschweig: Vieweg.Dedekind, Richard. (1963). Essays on the Theory of Numbers, W.W. Beman, trans.and ed., New York: Dover (originally published by Open Court: Chicago, 1901).Dedekind, Richard (1982). Scritti sui fondamenti della matematica, a cura diFrancesco Gana. Napoli : Bibliopolis.Dummett, Michael (1991a). Frege: Philosophy of Mathematics. Harvard UniversityPress.Dummett, Michael (1991b). Frege and Other Philosophers. Clarendon Press.Ferreirós, José (1999). Labyrinth of Thought: A History of Set Theory and its Rolein Modern Mathematics, 2nd ed., Boston: Birkhäuser, 2007.Ferreirós, José (2007). The rise of pure mathematics as arithmetic in Gauss, in C.Goldstein et al., eds., The Shaping of Arithmetic after C.F. Gauss’s DisquisitionesArithmeticae, Berlin: Springer: 235–268. Ferreirós, José (2009). ‘Hilbert, logicism,and mathematical existence’, Synthese, 170, 33–70.Frege, Gottlob (1884). Die Grundlagen der Arithmetik, Breslau: Marcus (Englishtrans., The Foundations of Arithmetic, J.L. Austin, trans., Northwestern Univer-sity Press, 1980).Frege, Gottlob (1893/1903). Grundgesetze der Arithmetik, Vols 1–2, Jena: Pohle(English trans. (selections), The Basic Laws of Arithmetic, M. Furth, trans. anded., University of California Press, 1964; cf. also Frege (1997) for excerpts).Gillies, Donald (1982). Frege, Dedekind and Peano on the foundations of arith-metic. Van Gorcum and Comp.Grassmann, Hermann (1861).Lehrbuch der Mathematik für höhere Lehranstalten,Band 1. Berlin: Enslin.Hellman, Geoffrey (1989). Mathematics Without Numbers: Towards a Modal-StructuralInterpretation. Oxford University Press.Hellman, Geoffrey (2001). Three varieties of mathematical structuralism. PhilosophiaMathematica 9 (2):184–211.Kreiser, Lothar (2001). Gottlob Frege: Leben–Werk–Zeit, Frankfurt: Meiner.Peano, Giuseppe (1889). Arithmetices principia nova methodo exposita. Englishtranslation in H. C. Kennedy (ed.) Selected works of Giuseppe Peano. Allen andUnwin. 1973:101–134.Parsons, Charles (2008). Mathematical Thought and its Objects. Cambridge Uni-versity Press.Reck, Erich and Price, Michael (2000). Structures and structuralism in contempo-rary philosophy of mathematics. Synthese 125 (3):341–383.Reck, Erich (2003). Dedekind’s structuralism: An interpretation and partial de-

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fense. Synthese 137 (3): 369–419.Reck, Erich (2013). Frege, Dedekind, and the Origins of Logicism. History andPhilosophy of Logic 34 (3): 242–265.Russell, Bertrand (1903). Principles of Mathematics. Routledge.Shapiro, Steward (ed.) (2005). The Oxford Handbook of Philosophy of Mathemat-ics and Logic. Oxford University Press.Shapiro, Steward (1997). Philosophy of Mathematics: Structure and Ontology. Ox-ford University Press.Stein, Howard (1988). Logos, logic, and logistiké: Some philosophical remarks ontransformations in nineteenth century mathematics’, inW. Aspray and P. Kitcher,eds., History and Philosophy of Modern Mathematics, Minneapolis: University ofMinnesota Press: 238–259.Sundholm, Göran (2001). Gottlob Frege, august bebel, and the return of Alsace-Lorraine: The dating of the distinction between “Sinn” and “Bedeutung” ’, Historyand Philosophy of Logic, 22, 57–73.Tait, William (1986). Truth and proof: The platonism of mathematics. Synthese69 (3):341–370.Zermelo, Ernst (1908). Investigations in the foundations of set theory I. Englishtranslation in J. van Heijenoort (ed.) From Frege to Gödel. Harvard UniversityPress. 1967: 199–215.

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