Introduction

Some two and a half millennia ago, so the story goes, Thales found himself at the foot of the great pyramid of Cheops. It had already been there some 2000 years when he paid it a visit. Thales was immensely impressed, we can imagine, and immediately tried to measure the hight of this colossal volume in the desert. The pyramid had been built by the Pharaoh Cheops to remind people how small they are in the vastness of things. His architects seem to have wanted everyone to admit that between this great monument and the humble man there was no common measure. Yet not so Thales! What appeared to be strikingly incommensurable with any form of thinking applied at the time for asking reasonable questions, to him posed an interesting challenge. That‘s why he is considered the first real ‘thinker’ because unlike the wise man before him, scribes and priests, accountants and storytellers, Thales did what they did as well, reciting formulas of worship, counting money, memorizing the important tales, but also something very different from that. He was asking questions of which he did not know how to answer them. What is thinking? or: What is the connection between what I think and what really exists?

In many ways, so I would like to argue, we find ourselves in a similar situation today. Instead of the pyramids we have all bundles of effects that scare us and that are due to the networked dynamics of very large numbers, which we can so easily operate with in our information technology based digital logistics. The logistics of our times reveals what we call ‘population effects’ of a global order, like those which provoke the idea that there were limits to the growth of humanity. The Club of Rome has famously used the first large-scale simulation platform to compute ‘reasonable’ scenarios. The means for computation, in purely quantitative terms, has grown in capacity so much since these early days of digital computation that today, we are awaiting the emergence of a supernatural intelligence to greet us (if we choose a friendly vision of it). We might well name it Singularity (and not Cheops), but we really don‘t know how to put ourselves in relation with its indicators and announcements. Populations are not accumulations of individuals-as-particles, as we begin to realise. Thus the basic idea of analysis, that when making statements about the whole nothing should be taken into account which is not already inherent within its elementary particles, is meeting its limits. Neither elementary geometry nor the fundamental theorem of arithmetics, which holds that all integers can be factorized unambiguously into powers of primes, can provide us with applications that would allow us to measure the effects of population-based phenomena. Algebraic means, on the other hand, do allow us to measure statistically; yet unlike geometry and arithmetics, it does not offer any clear sight to what we must hold as basic or prime. Hence we call algebraic procedures and their outcomes contingent, or model-theoretic, because the form of polynomial equations constitute and determine the solution space, irreducibly so [1]. Algebraic solution spaces are engendered by performing the so-called Dedekind Cuts, a set-theoretical procedure of partition which allows for logical definition of various numerical domains [2] – yet at the cost of no longer assuming continuity to be a necessary property of the order of magnitudes, as the Leibnizian dictum that change, in nature, be continuous (i.e. nature makes no jumps), still holds. Instead what opens up, with Dedekind‘s mechanism of partition, is a peculiar kind of rendering-coherent which affects not only proportions but also ratios, an intellectual capacity which affects both, the order of real world magnitudes and that of numerical valency, the order of multitudes.

1 Seeking to approach abstract reasoning in a most general fashion

For the predominant number theoretic interest throughout the nineteenth century, we can perhaps consider the solvability of various algebraic equations as the main motive, especially their solvability by extensions of the rational number space into the real and the complex number space. What soon became clear along such lines is that in some of these extensions, the Fundamental Theorem of Arithmetic—asserting the unique factorisation of all integers into powers of primes— fails. The question became then whether a suitable alternative to the Fundamental Theorem could be found. Of main relevance for our concern here is the introduction of a notion of Ideal Numbers [3] in such terms that the theorem of unique factorisation could be recovered also for algebraic integers [4]. This move led almost immediately to striking progress, yet the precise nature of these new mathematical objects was left unclear, as were the basis for their introduction and the range of applicability of the technique. This formed the larger background to the interest of seeking a logical approach to abstract reasoning in a most general fashion. For, if logical forms are held to be metaphysically “factual”, it implies not only that they have a distinct type of content, but furthermore that their application may also arbitrary and contingent. This conflicts severely with what sets mathematical reasoning apart from any other, empirically based reasoning, namely that its outcomes are meant to be necessary. The challenge, thus we can summarise, was to find a way to account for logical inference on the grounds of purely formal order, equally independent of any spatio-temporal intuition or psychological, incommunicable interiority [5].

What set theory means, in somewhat schematic terms, is the possibility for infinitesimal reasoning within a combinatorics of details, i.e. a combinatorics of items, partial components. It gives us the means to truly elaborate ideas, to articulate them by building up prospects and concepts from simple parts into an overall, objective, representation. The cost that goes along with this is what Reck refers to as the cutting of „the umbilical cord“: what we regard thereby as parts, while modelling and calculating within such a combinatorics of details, is an unproblematic notion of simplicity, or elementarily. This cost builds the background, Reck argues, to understanding what lies behind the eminent interest which logicians and mathematicians held throughout the nineteenth century for what we will summarise here as the program of arithmetisation [8]. In arithmetic, so Gottlob Frege, who is along with Richard Dedekind one of the two key figures concerned with these problems at the time, famously stated in his The Foundations of Arithmetic that „[i]n arithmetic, we are not concerned with objects which we come to know as something alien from without through the medium of the senses, but with objects given directly to our reason and, in its nearest kin, utterly transparent to it“ [9]. This citation makes explicit what was at stake – namely a frame of reference freed from any empirical complexity, a givenness of mathematical objectivity ‘directly to our reason’, by making the idea of factorisation into powers of primes the foundation of all inferential reasoning. The argument I would like to sketch here does not criticise the philosophical value ascribed to unique factorisation as a principle for inferential reasoning, but it does criticise the striving to establish the philosophical conditions for postulating an immediate accessibility of the objects of mathematics through understanding them in terms of natural kinship with our reason. Or perhaps, to be more precise, what is to be reviewed is such postulated relation of kinship as spiritual kinship, as one which seeks to nullify this very relation. Frege‘s perspective tried to establish this quite clearly: „A third realm must be recognised. Anything belonging to this realm has it in common with ideas that it cannot be perceived by the senses, but has it in common with things that it does not need an owner so as to belong to the contents of consciousness“ [10]. The further development of attempting to decouple knowledge from any kind of psychological maintenance or individual ownership, ethical responsibility and politically liberal subjectivity that could be tied to it, can be well illustrated by referring to the stance taken by Karl Popper in his theory of the three worlds: „We may first distinguish the following three worlds or universes: first the world of physical objects or physical states; secondly, the world of states of consciousness, or of mental states, or perhaps of behavioral dispositions to act; and thirdly, the world of objective contents of thought, especially the scientific and poetic thoughts and of works of art“ [11] From this, we can extrapolate easily the radical twist Frege‘s dangerous idea has obviously taken over time (even by not attending to the straight-forward political ‘success’ of the concept in Europe [12]: where Frege named the example of the Pythagorean Theorem as an example of such objective knowledge, Popper referred to Beethoven‘s 9th Symphony as an example. What happens here, through such annihilation of abstraction in reasoning (by making it the nearest kin, and as such transparent to reason in general), is the total annihilation of an individual persona‘s intellectuality and competency in developing abilities. We will refer to this interest in generalising intellectuality, by annihilating abstraction, as ‘intramundanistic’.

Any discussion of the philosophical challenges that go along with an account for the applicability of arithmetic to the world participates in the wider puzzle of explaining the link between experience, language, thought, and the world. Obviously, the two paths of Dedekind and Frege express two very different world views. They both also involve, inseparable form their accounting of the applicability of abstract reasoning, an accounting of how to deal with issues of modality that accompany such applications. In the case of Frege, any modality issue concerning this is sought to be annihilated completely (transparency of those objects to our individual minds, as ‘Reason‘s Nearest Kins’). He wants to establish the domain of natural numbers as the very foundation of philosophy at large. To this domain, we cannot itself refer. He wants it to be immediate to our reasoning. Frege‘s interest was to restrain the rôle of modality for philosophy by delegating it altogether to the two realms which in his word view of ‘intramundane finitism’ are subjected to his Third Empire of Logical Objectivity, accompanying a First Empire of Spatio-Temporal Being in the physical world, and a Second Empire of Psychological Being on the other. Quite different in the case of Dedekind. For him, modality issues related to the applicability of logical reasoning turn so fundamental that they indeed affect all established distinctions: any collection of objects, for Dedekind, counts as a set that can subsequently be considered a subset of another. His main concern, with his program of arithmetisation, was that such fundamental arbitrariness need not be conceived as absolute relativism, but can be dealt with in terms of conceptual compatibility. For whatever can be supported by conditions of provability, logically rigorous foundations can be provided. Hence his interest in an infinitary treatment of categoricity, completeness, independence, etc. The key assumption he has to make, thereby, is to consider abstraction as an act of engendering. To view abstraction in such a literal sense as creative sounds, of course, at first oddly and outright mythical. Yet consider this. The consideration of an intellectuality in general, as the Frege legacy is suggesting, appears by no means less mythical – for it assumes an intramundane realm where, also very literally, only one Genus rules. Now, does this not sound slightly Olympian?

My intention is neither ironical nor proclamatory in any sense. The interest in thinking about genera and generality in such ‘animistic’ terms, as being the products of creative intellectuality, follows Dedekind in his interest in providing the conditions for non-psychologistic predication, his interest in putting predicability on the basis of an algebraic notion of modality. Dedekind‘s objectivity-within-ideality, I would like to suggest, can be viewed as such. After all, the difficulty with abstraction and its psychological connotations would be, foremost, that what is abstracted is mental, that what I abstract is mine and what you abstract is yours. The main argument against psychologism in the context of abstraction is – at first sight – not that the source of judgement about the abstracted objects is not in some sense to be found in the common human psyche, but rather that the objects abstracted should not be found in the individual psyche [18]. However, the Fregean idea that intellectuality ought to be well-founded only if it absolutely general, this does suggest at least a certain impatience with humanistic values. Frege himself was, by the way, well aware and concerned with these problems of explaining scientific progress. One major challenge for him was, after all, to overcome the Kantian implications of the a priori / a posteriori distinction, which grants arithmetics – along with geometry – a synthetical status based on spatio-temporal intuition. The Fregean notion of logical objectivity, to put it short, had to be backed up somehow with his claim that analytic reasoning is not merely explanatory, as Kant held, but can also be ampliative [19]. Analytical reasoning must be able to explain the evolution of knowledge. This plays, undoubtedly, an important role for his invention of the three world theory.

3    Dedekind‘s legacy: compute-ability.

Dedekind‘s theory of engendering abstraction surely ought to be read in relation to these problems too. Instead of postulating a kind of intramundane and immediate idealism, as Frege did, he wanted to provide the means to understand and reflect the applicability of abstract reasoning. In that sense, you could say, Dedekind was much more indebted to Kant‘s critical philosophy than Frege was, who chose the path of an outspoken cognitive fundamentalism. Both have postulated a sphere of ideality for the objects of abstract reasoning. Yet unlike that of Frege, Dedekind‘s ideality is structural. It is neither positive nor negative. It is created as an objective ideality by his infinitary procedure of partition, the cut. This allows him to understand his ideality as being objective, capable of being organised into fields according to structural definitions. The Ideal for Dedekind is not, as I will argue in the following, the element of knowledge but that of an infinite ‘learning’. He replaced Kummer‘s ‘ideal numbers’ by his own theory of Ideals, and introduced the concepts of  fields, rings, lattice, module into algebraic number theory [20]. His Ideals should play the same role as Kummer‘s ideal numbers with respect to providing the conditions for unique factorisation in fields of algebraic integers. The crucial difference, however, consists in his nonfinitist, non-constructive approach – even though this meant assuming, in principle, semantic arbitrariness. “It is not what numbers ‘are’ intrinsically that concerns Dedekind“, writes Howard Stein and continues, „[H]e is not concerned, like Frege, to identify numbers as particular ‘objects’ or ‘entities’; he is quite free of the preoccupation with ‘ontology’ that so dominated Frege” [21]. Yet this liberation of being preoccupied with issues of ontology comes at the cost of having to develop, as we have already mentioned, a notion of abstraction that involves a genuine kind of ‘creation’ in the sense that there really is something new engendered by abstraction. Dedekind himself is cautious and doesn‘t speak of ‘objects’. If at all, they would have to be described as structurally-determinable, algebraic-numerical and hence symbolic objects. Such symbolic objectivity does not represent the structures of the world, but constitutes the structures that help us to develop our abilities in relating experience, language, thought, and the world, in learning-to-think.

The language of ‘creation’ indeed infuses Dedekind’s writings. It is not merely an empty phrase to be interpreted in a purely figurative manner, for in his view, it does indeed generate entirely new objects. In discussing the real numbers in Stetigkeit und irrationale Zahlen, Dedekind writes: “Whenever, then, we have to do with a cut (A1, A2) produced by no rational number, we create a new, an irrational number α” [22]. His procedure of logical definition of numbers, and indeed of any set of items, the cut, engenders entirely new objects. In one of Dedekind’s letters to his friend and collaborator Heinrich Weber he provides with great care further commentary on this point, together with the related discussion concerning the integers. Weber had evidently suggested that the natural numbers be regarded primarily as cardinal rather than ordinal numbers – the view Frege held, for example, and the view which was later, in a different way than Frege‘s own account, defined by Russell [23]. Dedekind replies to this suggestions:

„If one wishes to pursue your way – and I would strongly recommend that this be carried out in detail – I should still advise that by number … there be understood not the class (the system of all mutually similar finite systems), but rather something new (corresponding to this class), which the mind creates. We are of divine species [Wir sind göttlichen Geschlechts] and without doubt possess creative power not merely in material things (railroads, telegraphs), but quite specially in intellectual things. This is the same question of which you speak at the end of your letter concerning my theory of irrationals, where you say that the irrational number is nothing else than the cut itself, whereas I prefer to create something new (different from the cut), which corresponds to the cut … We have the right to claim such a creative power, and besides it is much more suitable, for the sake of the homogeneity of all numbers, to proceed in this manner.“ [24]

Now, if Dedekind is not concerned with making ontological claims, why then does he talk repeatedly about the creation of something new? And if his assertions do not at least entail certain ontological commitments, how then are we to construe this talk of creating something new? In his Preface to the first edition of the 1888 essay Was sind und was sollen die Zahlen, Dedekind writes directly and simply: “My answer to the problems propounded in the title of this paper is, then, briefly this: numbers are free creations of the human mind” [25]. They are ‘free’ not in the sense that they are unrestricted or lacking in constraints – they are well defined precisely because of their structural restrictions and constraints, which can be logically deduced and established by applying the procedure of the cut. Rather, then, numbers are ‘free’ in the sense that they are ‘free’ from any other ‘content’ – i.e., from any accidental and arbitrary properties that particular elements may happen to possess. For Dedekind, numbers are the manifest products of abstraction-as-learning-to- think, not as representation of abstract knowledge. And the question of the applicability of arithmetics, that so preoccupied Frege, is raised, by Dedekind, to another level: in order to be clear and reflected about the applicability of arithmetics, he wants to be clear and reflected about the applicability of his logical procedure of partition, the cut, which, within his theory of ideals, is capable of providing the conditions of possibility for applying arithmetics for solving complex equations. His algebraic procedure can provide a fully determinable objectivity within ideality. This objectivity is, we have already pointed this out, not in any sense representational. We have tentatively described its character as that of auxiliary constructions which can help us to think more relaxed and clearly within situations, where decisions are to be made, and where abstract reasoning is to be applied.

Dedekind‘s interest in the rôle of abstraction in learning, rather than in representing knowledge, expresses itself in a remarkable distinction between the notion of generality and that of abstraction. Where Frege wants to establish a transparency of abstraction itself, in favour of a total regime of generality in the realm of abstract objects, Dedekind is interested in making abstraction itself accessible to reason, by understanding how it proceeds. The cut is for Dedekind, I would like to suggest, a symbolic means or medium, a tool for what could be called ‘structural inception’ [26]. One aim of any structural theory is to determine what operations are necessary and sufficient, and what mathematical objects must be invented, to provide a solution for a problem of a prescribed kind [27]. What we suggest here to call the rôle of abstraction in terms of ‘structural inception’ for learning to think, seeks to establish the conditions for logical definability of concepts under a twofold criterium:

1) these conditions need to be wrought such that the structures they constrain be inceptive, rather than definitive,

2) they need to allow a characterisation of the elements by purely arithmetic (computational, mechanical, grounded in algorithms) procedure.

Each such field of structural definability is at one and the same time rigorous yet relative, relative to a well-defined field of numbers, real or complex, as a simple infinite set for which unique factorisation (arithmetics) holds. What characterises such ‘structural inception’ is, quite contrary to finitist approaches, an infinitary treatment of the finite. We suggest to call such conditions for rigorous definability that allow for structural inception ‘constructured’, in order to distinguish this approach from constructivist, finitist approaches. Infinitary treatment of the finite brings along, however, one severe problem: such constructured conditions for rigorous definability-in-potential bear, as an inceptive structural ground, the capacity to engender a variety of cases as solution for a problem of a prescribed kind: of such varieties, all possible cases are, in principle, equivalent.

Dedekind‘s approach to abstraction rests, as we have argued, on his preference in relating abstraction to learning-to-think rather than to an objectivity of knowledge. As such, what we have called his ‘universal set’ or ‘totient’ – “the totality of all things that can be objects of my thought” – is not meant, itself, to ever be an object of knowledge in a well-defined sense. An infinitist treatment of the finite means precisely this: semantically, the finite in focus can never conclusively and coherently be defined by purely arithmetic means. It accounts that the necessary abstraction involved in applying arithmetic means – that aspect which Frege wants to annihilate – yields fruit, so to speak. Dedekind‘s ‘totient’ ought to be understood, I would like to suggest, as a purely operative quantifier that complements his procedure of partition, the cut, within his overall algebraic approach. His universal set is meant to serve as the most general auxiliary construction necessary within Dedekind‘s method of quantifying logics in a non-representational, non-definitive, but rather structurally inceptive – or as he calls it, creative – way. If we consider this, Dedekind‘s way of logical quantification was indeed very different from that introduced by Frege. Dedekind engages with logicist methods in terms of what they have to offer for his interest in categoricity. This is very different from Frege, who saw in it a syntactical means to define concepts extensionally.

For Dedekind, in short, the structural criterium for well-definedness, namely completeness, is just as relevant as it is for model-theoretic, syntactic approaches. However, for him his approach to it „is to be understood in a semantic sense, as based on categoricity; similarly, consistency is to be understood semantically, as satisfiability by a system of objects“ [28]. It has been argued that Dedekind‘s notion of creative abstraction is an irrelevant assumption of his, due to personal world views, as Tait points out: „An objection often leveled at the abstractions of Cantor and Dedekind is that the abstractions do no work – they play no role in proofs.“ Tait is perfectly right in pointing out further „[B]ut it would seem that logical abstraction, as it is described here, does play a role, not in proofs, but in that it fixes grammar, the domain of meaningful propositions, concerning the objects in question, and so determines the appropriate subject matter of proofs“ [29]. This kind of preparing the grounds, by abstraction, for further computations, is precisely what we mean by ‘structural inception’. Like in farming, you cannot grow any fruit on just any ground. Here lies the great promise of what we call the Dedekind-Legacy: It allows, enhanced by machinic computations in computers, enormous flexibility and agility in terms of finding concepts and their definitions that are right not primarily for generalised cases, but for specified tasks. And this not despite of but precisely because such ‘constructured’ conditions for rigorous definability bear, as an inceptive structural ground, the capacity to engender a variety of cases as possible solutions for a problem of a prescribed kind, all of which are, in principle, equivalent. This means, the question of right definitions is not restricted to some sort of general adequacy, but can also involve desiderata of nearly any kind. Reck gives the following examples of the desiderata Dedekind himself applied in his working out of foundational definitions for mathematics: fruitfulness, generality, simplicity, and ‘purity’, i.e., the elimination of aspects ‘foreign’ to the case at hand [30]. As a concrete example Reck refers to the definition of prime numbers which must be at just the right level of generality [31].

How, we might ask, did Dedekind come up with such ideas? It is difficult for us to remember, today, that his time was a time under the freshly emerging, and all-affecting paradigm of thinking in terms of balances, and the indirect control of balanced systems following the insights of thermodynamics [32]. To denote the newly recognised kind of latency involved in balanced systems, scientists had started to speak about what they called ‘invariances’, systemic properties which do not change throughout processes of permutation and change. To paraphrases Bell in his History of Mathematics: „Invariance is changelessness in the midst of change, permanence in a world of flux, the persistence of configurations that remain the same despite the swirl and stress of countless hosts of curious transformations“ [33]. Digesting these newly emerged fundamental concepts, which allowed to think in terms of controlling balances, in the fields of energy and electricity, algebraists had, along the same lines as physicists had, started to consider the laws of conservation – they had started to consider laws of thought. The promise of algebraic structural methods, very generally, was to identify such invariants. Dedekind‘s relevancy in this tradition is well documented with the important rôle he played for Emmy Noether, one of the main protagonists of the emerging kind of science [34].

The problem with this approach is, undoubtedly, that it severely shakes the idea of axiomatic foundations resting on the assumption that mathematical reasoning involves judgements which are purely deductive, absolutely and unambiguously necessary, and conclusive. Quite to the contrary of what we might associate with ‘laws’ today [35], speaking of laws of thought then seemed at least to hold the promise of introducing some kind of empiricism into abstract reasoning itself [36]. However, these difficulties are due to reasons immanent within the evolution of mathematical thought throughout the nineteenth century, and are shared by Dedekind with other algebraic structuralists as well [37]. The crucial point about invariance theory concerns the method of factorisation. We have already briefly touched upon this, lets now be a bit more precise. While up until polynomials of the fourth degree, factorisation of the exponents into roots within the space of the real numbers is possible, factorisation of polynomials of higher order involves the complex number space. The Theorem of Algebra, as formulated by Gauss, holds that polynomials of n degree can be solved, within the complex number space, in such a way that they resolve into n solutions. Their treatment relies on auxiliary constructions and procedures, the provision of which is the main task of a.o. Kummers‘, Dedekinds‘, Kroneckers‘ theories Ideal Numbers. The assumption of algebraic, symbolic ideality allows to define domains of relatively, yet determinately and uniquely factorisable integers. Hence they allow for arithmetic treatment of n-th order polynomials. Sophus Lie‘s observation of 1874, in a letter to a friend, gives a clear idea of the situation: „In the theory of algebraic equations before Galois only these questions were proposed: Is an equation solvable by radicals, and how is it to be solved? Since Galois, among other questions proposed is this: How is an equation to be solved by radicals in the simplest way possible?“ Lie not only observes these complications for higher order algebraic equations, but continues by extending their range: „I believe the time is come to make a similar progress in differential equations“ [38]. What was lacking, in short, are clear criteria for deciding upon one solution rather than another [39].

Dedekind is treating numbers in radically algebraic ways. Taking the analogy between the laws of conservation and those of thought a step further, we must of course ask what it might be, that is being conserved by the laws of thought. But let us first consider whether there is an argument to support such speculation. On the one hand, this is simply a path pursued also by many other pioneers in the methods of symbolic algebra of the time [40]. The technical inventions that accompanied this new paradigm in thinking simply was the main field of application for variation calculus, partial differential equations, invariant theory. In support from a different angle, it doesn‘t seem all too far reached to understand Dedekind‘s infinitist quantification operator – ‘all that can be object to his thought’ – along these lines. It would support our argumentation that the objection which even Cantor held, namely that his notion of ‘all objects of thought’ was an ‘inconsistent totality’ [41], is unjustified. As we have already argued, this would only be accurate if Dedekind were 1) interested in coming up with a definition of his universal set, and 2) if he would hold the principle claim that logical definitions, in order to count as such, ought not be in conflict among each other. I have already argued that both of these are not likely the case. In fact, what marks the Dedekind Legacy vis-à-vis the Frege Legacy is precisely the radical arbitrariness concerning the range of variety between sets, objects, their mapping functions, and the possibly well-definable correlations among them. But let us return to the analogy between the laws of conservation and those of thought. In the case of Dedekind there is a further hint pointing in this direction. Namely Dedekind‘s radical interest not directly in thinking, but in learning to think. This requires to understand the logical conditions of knowledge as being object to evolution. It requires to view those conditions as capable of refinement and development by human intellect, and its capacity and capability of learning. Pursuing such a perspective further would involve an architectonic theory of reason, its faculties, and their interplay. To my awareness, Dedekind is not explicitely elaborate in this direction, he doesn‘t seem to go far here. However, he clearly writes in his introduction to Was sind und was sollen die Zahlen, that arithmetics, algebra, analysis, for him, are only components or parts of what he considers to be logics. „In speaking of arithmetic (algebra, analysis) as part of logic I mean to imply that I consider the number concept entirely independent of the notions or intuitions of space and time, that I consider it an immediate result from the laws of thought“ [42]. Rather than arithmetics, and a well-defined concept of the natural numbers, as for Frege, for Dedekind it is the ability of the mind to think, which is „the absolutely indispensable foundation“ on which the whole science of numbers is to be established. Let us cite the full passage:

„In speaking of arithmetic (algebra, analysis) as part of logic I mean to imply that I consider the number concept entirely independent of the notions or intuitions of space and time, that I consider it an immediate result from the laws of thought. My answer to the problems propounded in the title of this paper is, then, briefly this: numbers are free creations of the human mind; they serve as a means of apprehending more easily and more sharply the difference of things. It is only through the purely logical process of building up the science of numbers and by thus acquiring the continuous number domain that we are prepared accurately to investigate our notions of space and time by bringing them into relation with this number- domain created in our mind. If we scrutinize closely what is done in counting an aggregate or number of things, we are led to consider the ability of the mind to relate things to things, to let a thing correspond to a thing, or to represent a thing by a thing, an ability without which no thinking is possible. Upon this unique and therefore absolutely indispensable foundation […] must, in my judgement, the whole science of numbers be established.“ [43]

Even if Dedekind does not elaborate on a corresponding architectonics of reason, his words make one thing very clear: he considers learning as the sole possible foundation for his logicist mathematics. Just like the phenomena of the physical objects are not given by the laws of conservation, but need to be distinguished and defined on their basis, the laws of thinking do not give us objective knowledge, but can provide a stable basis for distinguishing and defining what we consider as such in consistent ways. Let us be clear here and summarise. Dedekind‘s relying on “the totality of all things that can be objects of my thought” grants that subsequently, arbitrary subsets of that totality are being considered and can be defined by recursion. For Dedekind, abstraction is creative because he views it as capable of literally engendering pre-specific, particular cases that can only subsequently be generalised as new objects of thought [44]. Hence, was sets his abstractionism apart from any psychologist account, is that there are clear criteria for a thought to qualify as objective. The objectivity within thought has to be produced.

conclusion

„In this sense there is a mathesis universalis corresponding to the universality of the dialectic. If Ideas are the differentials of thought, there is a differential calculus corresponding to each Idea, an alphabet of what it means to think. Differential calculus is not the unimaginative calculus of the utilitarian, the crude arithmetic calculus which subordinates thought to other things or to other ends, but the algebra of pure thought, the superior irony of problems themselves – the only calculus ‘beyond good and evil’. This entire adventurous character of Ideas remains to be described.“ [45]

The next big question, hence, concerns this capacity of our reason to train itself and develop its faculties. Our main argument, here, has been to read Dedekind‘s notion of abstraction as providing auxiliary constructions for thought to create objects as an active encoding. The numbers that can be logically founded and defined by Dedekind‘s procedure of the cut are all, in principle, computable [46]. Our suggestion is to integrate computability philosophically, as a faculty within human reason‘s architectonic. Given this fundamental emphasis of the rôle of abstraction in learning to think, we assume that the Dedekind ideality is an ideality that can be cultivated by learning to think. Ideas and thoughts are grounds, they can be cultivated in order to produce intersubjective objects of thought – ideal objects, in the Dedekindian sense – objects for which it is determinate how to reason about. If the problems are grounded in the conditions that can be constructured to solve them, there is a direct link between their gravity and our capacities to think. In the face of our global IT based institutional, economical, political, technological – in short: urban infrastructures and the ubiquitously distributed capacities for machinic computation, the question of arithmetics‘ applicability can no longer be separated from a contractual politics. The interest in cultivating these conditions into ever more capacious conditions should need no further legitimisation – despite the challenging degree of abstraction contained in it. This stands, of course, in stark contrast to the notions of computability which is common sense today in cognitive science, at least where what we have stigmatised as the Fregean legacy, is being continued. Here, the condition that a procedure „demands no insight or ingenuity on the part of the human being carrying it out“ in order to count as a well-defined, logically founded, non-arbitrary computational method, is often explicitly stated [47].

1) focussing on entire systems of objects and on general laws, in such a way that results can be transferred from one case to another.

2) the move away from particular formulas, or from particular symbolic representations, and towards more general characterisations of the underlying systems of objects, specifically in terms of relational and functional properties.

3) more particularly, the consideration of homomorphisms, automorphisms, isomorphisms, and features invariant under such mappings.

Very generally, the methodical cultivation of these points, which I have largely citing here from E.Recks text on “Dedekind’s Contributions to the Foundations of Mathematics” [48] will owe much to the investigation of novel, abstract concepts – fantastic notions! [cf my post on this] – introduced by learning to specify particular cases, cases which are genuinely engendered by abstraction, towards their generalizability.

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[1] „Die Aussage, dass jede algebraische Gleichung n-ten Grades im Bereich der komplexen Zahlen stets genau n nicht notwendig voneinander verschiedene Lösungen hat, wird allgemein als Fundamentalsatz der Algebra bezeichnet.“ [That each algebraic equation of n-th power has n solutions within the complex number space, solutions which are not necessarily distinguishable, this is generally referred to as the Fundamental Theorem of Algebra“ (transl. VB)]. Alten, H.-W. et al. (2005) 4000 Jahre Algebra: Geschichte, Kulturen, Menschen, Springer, Berlin p. 283.

[2] In particular, Dedekind cuts among the real numbers may be considered defined as cuts among the rationals. They hence allow for a clear definition of rational numbers and integers, despite their infinity in terms. It turns out that every cut of reals is identical to the cut produced by a specific real number. In other words, the number line where every real number is defined as a Dedekind cut of rationals is a complete continuum without any further gaps: „Whenever, then, we have to do with a cut produced by no rational number, we create a new irrational number, which we regard as completely defined by this cut […]. From now on, therefore, to every definite cut there corresponds a definite rational or irrational number […].“ Richard Dedekind, Continuity and Irrational Numbers, Section IV. Cf. the introductory essay by Erich Reck, “Dedekind’s Contributions to the Foundations of Mathematics”, in The Stanford Encyclopedia of Philosophy (Fall 2011 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/fall2011/entries/ dedekind-foundations/> (14.2 2012).

[3] origninally introduced by E. Kummer (1844) De numeris complexis, qui radicibus unitatis et numeris realibus constant, Gratulationsschrift der Universität Breslau zur Jubelfeier der Universität Königsberg, available in Kummerʻs Collected Papers Vol. 1, p. 165-192. (cf footnote 4), but also treated by Gauss, Dirichlet, Dedekind and Kronecker a.o.

[4] the idea has also been pursued a.o. by Leopold Kronecker, although in a way that leads more to a constructivist philosophical attitude than to a structuralist one, which would more aptly characterize the attitude of Dedekind. Cf. L. Kronecker, Grundzüge einer Theorie der algebraischen Grössen, Journal für reine und angewandte Mathematik Vol. 92, 1882, p. 1-122; and as an engl. collection and translation of Dedekinds respective writings in R.Dedekind, Theory of Algebraic Integers, Cambridge University Press 1996 [1877].

[5] This is where Frege failed to meet his own goals, as we will see in a moment, when he fell back from his early, purely context and syntax dependent account, to introducing an extensional dimension of definitions. Cf. Potter, M. (2000) Reasonʻs Nearest Kin, Philosophies of Arithmetic from Kant to Carnap, Oxford University Press, Oxford, ch. 1, pp. 20-54.

[6] Reck, E.H. (2003) „Frege, Natural Numbers, and Arithmeticʻs Umbilical Cord“ in: Manuscrito 26:2, 2003, Special Issue: Logic, Truth and Arithmetic: Essays on Gottlob Frege, M. Ruffino, guest editor, pp. 427-470.

[14] It is generally held that Dedekind did not proceed towards to formulating an axiomatic theory. This is often interpreted as a failure on the side of Dedekind, an interpretation which does not seem likely to us, as we will discuss further on. In this case, Dedekind had published the same four conditions for constructing simply infinite sets in the context of his proof, which Peano has published a little bit later, and in a somewhat different notation than Dedekind. Peano had dedicatedly called them axioms. Hence it became the convention to refer to those conditions as the Dedekind-Peano Axioms. Cf. for a brief summary discussion cf. Reck, E. “Dedekind’s Contributions to the Foundations of Mathematics”, The Stanford Encyclopedia of Philosophy (Fall 2011 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/fall2011/ entries/dedekind-foundations/>; for an elaborate discussion see Ferreirós (1999) Labyrinth of Thought: A History of Set Theory and its Role in Modern Mathematics, Birkhäuser, Boston, ch. 7; and Ferreirós (2005) “Richard Dedekind (1888) and Giuseppe Peano (1889), Booklets on the Foundations of Arithmetic, Chapter 47 of Landmark Writings in Western Mathematics, 1640–1940, I. Grattan-Guinness (ed.), Elsevier, Amsterdam, pp. 613–626.

[15] Dedekind, R. (1888) Was sind und was sollen die Zahlen? Vieweg, Braunschweig (originally published as a separate booklet); reprinted in Dedekind, R. (1930–32). Gesammelte Mathematische Werke, Vols. 1–3, R. Fricke, E. Noether & Ö. Ore, eds., Vieweg, Braunschweig Vol. 3, pp. 335–91.

[16] Dedekind, R. (1888) Was sind und was sollen die Zahlen? Vieweg, Braunschweig (originally published as a separate booklet); reprinted in Dedekind, R. (1930–32). Gesammelte Mathematische Werke, Vols. 1–3, R. Fricke, E. Noether & Ö. Ore, eds., Vieweg, Braunschweig Vol. 3, pp. 335–91.

[17] cf. Temple Bell, E. (1915) An Arithmetic Theory of Certain Numerical Functions, University of Washington, Seattle, p. 9ff.

[18] cf Tait, W.W. (1997) “Cantor versus Frege and Dedekind: On the Concept of Number”, in W. W. Tait (ed.) Early Analytic Philosophy, Open Court, Chicago.

[19] cf for a discussion on this: Potter, M. (2000) Reason’s Nearest Kin, Philosophies of Arithmetic from Kant to Carnap, Oxford University Press, Oxford, pp. 55ff.

[20] cf Corry L. (2004) Modern Algebra and the Rise of Mathematical Structures, second edition (revised), Birkhäuser, Boston; especially Chapter 2: “Richard Dedekind: Numbers and Ideals”, pp. 66–136., McLarty, C. (2006) “Emmy Noether’s ʻSet Theoreticʼ Topology: From Dedekind to the Rise of Functors”, in The Architecture of Modern Mathematics, J. Ferreirós & J. Grey (eds.), Oxford, Oxford University Press, pp. 187– 208.

[21] H. Stein, “Logos, Logic, and Logistiké: Some Philosophical Remarks on Nineteenth Century Transformations in Mathematics”, in W. Aspray and P. Kitcher (eds.) History and Philosophy of Mathematics (Minneapolis: University of Minnesota Press, 1988), p 247.

[22] Dedekind, R. (1872) Stetigkeit und irrationale Zahlen, Vieweg, Braunschweig (originally published as a separate booklet); reprinted in Dedekind R. (1930–32), Vol. 3, pp. 315–334, In: Gesammelte Mathematische Werke, Vols. 1–3, R. Fricke, E. Noether & Ö. Ore, eds., Vieweg, Braunschweig.

[26] from L. inceptionem (nom. inceptio) for „a beginning, undertaking,“ a compound of in– and –capere for „take, seize, comprehend“, and sharing common roots with capable, L. for “able to hold much, broad, wide, roomy”, here used in the sense of increasing the capacities proper to the auxiliary constructions which abstraction creates within Dedekindʻs objective ideality.

[27] Reck comments on Dedekindʻs structuralist methods as follows: „Not only does he study systems of objects or whole classes of such systems; and not only does he attempt to identify basic concepts. Dedekind also tends to do both, often in conjunction, by considering mappings on the systems studied, especially structure-preserving mappings (homomorphisms etc.), and what is invariant under them. This implies, among others, that what is crucial about a mathematical phenomenon may not lie on the surface (concrete features of examples, particular symbolisms, etc.), but go deeper. And while the deeper features are often captured set-theoretically by him (Dedekind cuts, ideals, quotient structures, etc.), this points beyond set theory in the end, towards category theory“. Reck, E. “Dedekind’s Contributions to the Foundations of Mathematics”, The Stanford Encyclopedia of Philosophy (Fall 2011 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/fall2011/entries/dedekind-foundations/>. cf. also Corry, L. (2004) Modern Algebra and the Rise of Mathematical Structures, second edition (revised), Birkhäuser, Boston; especially Chapter 2: “Richard Dedekind: Numbers and Ideals”, pp. 66–136.

[37] cf for example the chapters „Emergence of Structural Analysis“ and „Invariances“ in: Bell, E.T. (1992) [1940] The Development of mathematics. 2nd edition, Dover, New York, pp. 245-269, and pp. 420-468.

[38] Lie, S. letter to A. Meyer, cited in Bell, E.T. (1992) [1940] The Development of mathematics. 2nd edition, Dover, New York, p. 436.

[39] Of all algebraic solutions possible, Bell writes in relation to a list of determinants by Cayley, „some are useful, the vast majority merely curious.“ Bell, E.T. (1992) [1940] The Development of mathematics. 2nd edition, Dover, New York, p. 426.

[40] cf footnote 36, and also: Bell, E.T. (1992) [1940] The Development of mathematics. 2nd edition, Dover, New York.

[41] Reck, E. “Dedekind’s Contributions to the Foundations of Mathematics”, The Stanford Encyclopedia of Philosophy (Fall 2011 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/fall2011/ entries/dedekind-foundations/>.

[42] Dedekind, R. (1901) Essays on the Theory of Numbers, W.W. Beman, ed. and trans., Open Court Publishing Company, Chicago, pp. 31-32; reprinted by Dover, New York, 1963.

[43] Dedekind, R. (1901) Essays on the Theory of Numbers, W.W. Beman, ed. and trans., Open Court Publishing Company, Chicago, pp. 31-32; reprinted by Dover, New York, 1963.

[44] Dedekind himself did not use the expression of ʻpre-specificityʼ for such structurally ideal objects. It is a term I have coined for describing the kind of ʻcomputable idealityʻ in the context of a theory of design: Bühlmann, V., Wiedmer M. (eds.) (2008) Pre-Specifics. Some Comparatistic Investigations on Research in Art and Design, JRP| Ringier Press, Zurich, especially: Bühlmann, V. „Pseudopodia. Prolegomena to a Theory of Design“ pp.21-80; also: Buehlmann, V., (2010), ‘Pre-specifics, Considering the Design of Mediality’, in Freek Lomme, Michael Capio (eds.), Pre-specifics: Access X! Onomatopee 52, Eindhoven, pp. 15-33.

[45] Deleuze, G. (1994) [1968] Difference and Repetition, Columbia University Press, New York, pp. 181/82

[46] they are computable within their structural definitions, which supports their ennumerability and hence, their service for the mechanization of arithmetic operations within the orderability they embody.

[47] Copeland, B. Jack, “The Church-Turing Thesis”, The Stanford Encyclopedia of Philosophy (Fall 2008 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/fall2008/entries/church-turing/>.

[48] As Reck suggests in: Reck, E. “Dedekind’s Contributions to the Foundations of Mathematics”, The Stanford Encyclopedia of Philosophy (Fall 2011 Edition), Edward N. Zalta (ed.), URL = <http:// plato.stanford.edu/archives/fall2011/entries/dedekind-foundations/>.

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* This post draws a lot from a paper given at the Turing 2012 Conference, University of Manila, Philippines in March 2012.

1   computation

2   abstraction

3   architectonic computing

4   computability

6   materiability

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