Infinity and Continuity The mutual dependence and distinctness of multiplicity and wholeness

Daniel Strauss

Talk given at the TENTH BIENNIEL MEETING OF THE NORTH AMERICAN FICHTE SOCIETY. Title of conference: "Interpreting Fichte's Bestimmung des Menschen", in Lisbon, Portugal, 27 April, 2010, 4.30-5.15pm.

Infinity, which is by nature a highly elusive concept, and in a way, the most elusive concept of all, has been a matter for merely mythological and cosmological consideration. It is due to Anaximander, the Pythagoreans and Zeno of Elea that the notion was gradually put into a more rational and “mathematical” perspective, and ultimately gained its first systematic analysis with Aristotle. Here, by “mathematical”, I mean the Aristotelian account of infinity which was implemented in the work of the Greek mathematicians. The Aristotelian analysis of the notion predominated the Western thought until the set-theoretic approach emerged. Although, infinity is now purified from its once-accepted cosmological connotations, it still seems that the discovery of the true nature of the notion can never be attained from a purely mechanical and materialistic viewpoint. In this paper, I investigate the views on infinity from the Greek period up until today, highlighting certain mathematicians and thinkers who made the most remarkable and intriguing contributions to the matter.

The concept of infinity had, in ancient times, an indistinguishable development between mathematics and philosophy. We could also say that his real birth and development was in Magna Graecia, the ancient South of Italy, and it is surprising that we find, in that time, a notable convergence not only of the mathematical and philosophical point of view, but also of what resembles the first “computational approach” to “infinitely” or very large numbers by Archimedes. On the other hand, since the birth of philosophy in ancient Greece, the concept of infinite has been closely linked with that of contradiction and, more precisely, with the intellectual effort to overcome contradictions present in an account of Totality as fully grounded. The present work illustrates the ontological and epistemological nature of the paradoxes of the infinite, focusing on the theoretical framework of Aristotle, Kant and Hegel, and connecting the epistemological issues about the infinite to concepts such as the continuum in mathematics.

Infinity and Continuity The mutual dependence and distinctness of multiplicity and wholeness D.F.M. Strauss Dean's Office University of the Free State P.O. Box 339 Bloemfontein 9300 South Africa dfms@cknet.co.za Paper presented at the Free University of Brussels October 15, 2006 Contents Infinity and Continuity Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Probing some foundational designs . . . . . . . . . . . . . . . . . . . . 2 The impasse of logicism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Contradiction and the meaning of analysis . . . . . . . . . . . . . . . . . . 4 Tacit assumptions of axiomatic set theory . . . . . . . . . . . . . . . . . . . 6 The primacy of natural numbers and their succession (induction) . . . . . 7 Ordinality versus cardinality. . . . . . . . . . . . . . . . . . . . . . . . . . 8 Wholeness and Totality – the irreducibility of the Whole-Parts Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Some crucial structural features. . . . . . . . . . . . . . . . . . . . . . 13 The inevitability of employing analogical elementary basic concepts . . 14 The theory of modal aspects . . . . . . . . . . . . . . . . . . . . . . . 16 Numerical and spatial terms . . . . . . . . . . . . . . . . . . . . . . . . . 18 Deepening our understanding of infinity . . . . . . . . . . . . . . . . . 21 Mathematics and Logic . . . . . . . . . . . . . . . . . . . . . . . . . . 24 The circularity entailed in set theoretical attempts to arithmetize continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Aristotle and modern mathematics – infinity and continuity . . . . . . . 27 Aristotle and Cantor: an ‘intermodal’ solution . . . . . . . . . . . . . . 30 Concluding remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Sketches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Index of Subjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Index of Persons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Infinity and Continuity The mutual dependence and distinctness of multiplicity and wholeness D.F.M. Strauss Practicing mathematicians, consciously or not, subscribe to some philosophy of mathematics (if unstudied, it is usually inconsistent) (Monk, 1970:707) 1 Introduction Although every single academic discipline employs concepts an explicit account of the nature of concept-formation is almost never encountered – in general the concept of a concept is absent. Normally key (or: supposedly basic) concepts are defined straightaway. Yet various disciplines did realize that definitions employ terms and that one cannot simply continue to define these terms without ending in a regressus in infinitum. Concept-formation and definition therefore ultimately rest upon terms which are not defined and cannot be defined. How does one know these indefinable (primitive) terms? This is an epistemological issue which is rooted in philosophical assumptions about the world in which we live – and therefore involve ontological commitments. Within the history of philosophy the underlying problem concerns the relation between unity and diversity and it centers in the question regarding the coherence of irreducibiles. Greek mathematics commenced with an approach evidenced in the Pythagorean statement: “everything is number.” The discovery of incommensurability by Hippasos of Metapont (about 450 B.C. – cf. Von Fritz, 1945) generated an emphasis on a new approach which attempted to consider our intuition of space as being more basic than that of number. Because intuition can grasp continuity all at once “Greek mathematics and philosophy were inclined to consider continuity to be the simpler concept” (Fraenkel et al., 1973:213). During the later part of the 19th century mathematics once again reverted to an arithmeticistic perspective – a process initiated by Bolzano and carried through by Weierstrass, Dedekind, and Cantor. What is amazing about this course of events is that what seemed to have been solved twice still turned out to form the center of the problems involved, namely the discovery of irrationality (incommensurability) and the difficul1 Infinity and Continuity ties surrounding the theory of functions in Germany and France. Fraenkel et al. remark: Though the arguments have changed, the gap between discrete and continuous is again the weak spot – an eternal point of least resistance and at the same time of overwhelming scientific importance in mathematics, philosophy, and even physics (1973:212-213). It is indeed amazing that the entire history of mathematics only explored the following two options: either reduce space to number or reduce number to space.1 The obvious third option was never examined: accept both the uniqueness (i.e., irreducibility) of number and space and their mutual interconnectedness! In order to investigate some of the implications entailed in such a third alternative we start by looking at the implicit assumptions of modern axiomatic set theory. 2 Probing some foundational designs Although Euclid already employed the axiomatic method it received its modern rigour from Hilbert's work on the foundations of geometry (Grundlagen der Geometrie, 1899). In this work Hilbert abstracts from the contents of his axioms, based upon three undefined terms: “point,” “lies on,” and “line.” 2.1 The impasse of logicism Frege, Dedekind and Russell advanced a logicistic approach to the foundations of mathematics. Dedekind started from an actual infinity of “objects” within my “Gedankenwelt” (translated by Rucker with the descriptive term “mindscape” – Rucker, 1982:47). Russell defines number with the aid of his supposedly purely logical concept of class. The logical concept, he claims, enables the reduction of mathematics to logic. For example, the number “2” is “defined” in the following way: “1 + 1 is the number of a class w which is the logical sum of two classes u and v which have no common terms and have each only one term. The chief point to be observed is, that logical addition of numbers is the fundamental notion, while arithmetical addition of numbers is wholly subsequent” (Russell, 1956:119). The irony, however, is that Russell already had to use the meaning of number in order to distinguish between different (“logical”) classes. After all, he speaks about the sum of “two” classes where each of them contains “one” element. This presupposes an insight into the numerical meaning of the numbers “1” and “2”! Consequently, the number “2,” which had to appear as the result of “logical addition,” is presupposed by it! In his discussion of number and 1 It should be kept in mind that Frege eventually showed an affinity with the latter position. Dummett refers to Frege's own “eventual expedient of reducing arithmetic to geometry” (1995:319). With regard to a preference for geometrical intuitions Wang writes: “An alternative course would be to consider our geometrical intuitions, as Plato and Bernays (and, I understand, also Frege in his later years) apparently preferred” (1988:203). 2 D F M Strauss the concept of class Cassirer displays a clear understanding of this circularity (1953:44 ff.).1 Although Dedekind asserts that the idea of infinity should form part of the logical foundation of mathematics, it soon turned out that the meaning of infinity precedes logic. Hilbert points out that in contrast to the early attempts of Frege and Dedekind he is convinced that as a precondition for the possibility of scientific knowledge certain intuitive representations and insights are indispensible and that logic alone is not sufficient.2 Fraenkel et al. also affirm: “It seems, then, that the only really serious drawback in the Frege-Russell thesis is the doubtful status of InfAx,3 according to the interpretation intended by them” (1973:186). Myhill mentions the fact that the axioms of Principia do not determine how many individuals there are: “the axiom of infinity, which is needed as a hypothesis for the development of mathematics in that system, is neither provable nor refutable therein, i.e., is undecidable” (Myhill, 1952:182). In anticipation of our subsequent analysis of the inter-modal coherence between aspects we have to point here at the foundational role of the meaning of number with respect to that of the meaning of analysis. Every attempt to deduce the meaning of number from the meaning of analysis (or: logic) is faced with a vicious circle. Cassirer is also quite explicit in this regard. He claims that a critical analysis of knowledge, in order to side-step a regressus in infinitum, has to accept certain basic functions which are not capable of being “deduced” and which are not in need of a deduction.4 David Hilbert also points at this “catch 22” entailed in the logicist attempt to deduce the meaning of number from that of the logical-analytical mode. In his Gesammelte Abhandlungen Hilbert writes: “Only when we analyze attentively do we realize that in presenting the laws of logic we already had to employ certain arithmetical basic concepts, for example the concept of a set and partially also the concept of number, particularly as cardinal number [Anzahl]. Here we end up in a vicious circle and in order to avoid paradoxes it is necessary to come to a partially simultaneous development of the laws of logic and arithmetic” (1970:199). 1 Singh also points out that Russell's attempt makes him a victim of the “vicious circle principle” (1985:76). 2 “Im Gegensatz zu den früheren Bestrebungen von Frege und Dedekind erlangen wir die Überzeugung, daß als Vorbedingung für die Möglichkeit wissenschaftlicher Erkenntnis gewisse anshauliche Vorstellungen und Einsichten unentbehrlich sind und die Logik allein nicht ausreicht” – Hilbert, 1925:190. 3 InfAx = Axiom of Infinity. 4 “Denn die kritische Analyse der Erkenntnis wird, wenn man nicht einen regressus in infinitum annehmen will, immer bei gewissen Urfunktionen Halt machen müssen, die einer eigentlichen 'Ableitung' weder fähig noch bedurftig sind” (1957:73). 3 Infinity and Continuity 2.2 Contradiction and the meaning of analysis The initial stage of mathematical set theory, as developed by Georg Cantor (between 1874 and 1899), got stuck in contradictions. By 1895 Cantor himself discovered that his set theory contains anomalies. Cantor proved e.g. the proposition that for every set A of ordinal numbers an ordinal number exists which is greater than every ordinal number contained in the set. Consider however the set W of all ordinal numbers. Since this set is a set of all ordinal numbers, the foregoing proposition implies that an ordinal number exists which is greater than every ordinal number contained in W – but this is contradictory, since the set W supposedly already contains all ordinal numbers. A similar contradiction holds with regard to Cantor's cardinal numbers (cf. Meschkowski, 1967:144-145 and Singh, 1985:73). The ensuing axiomatization of set theory, for example that of ZermeloFraenkel, proceeds on the basis of (i) first order predicate calculus1 and (ii) it introduces as undefined term the specific set-theoretical primitive binary predicate Î which is called the membership relation (Fraenkel et al., 1973:23).2 This approach follows a general pattern: an axiomatic theory (axiomatic theories of logic excluded) “is constructed by adding to a certain basic discipline – usually some system of logic (with or without a set theory) but sometimes also a system of arithmetic – new terms and axioms, the specific undefined terms and axioms under consideration” (Fraenkel et al., 1973:18). The first order predicate calculus assumed by ZF contains a set of connectives enabling the expression of negation (Ø), conjunction (Ù), disjunction (Ú), conditional (®), biconditional («) and the two quantifiers: the universal quantifier (" / for all) and the existential quantifier ($ / there exist). This means that the underlying logic provides ZF Set Theory with the following primitive symbols: connectives, quantifiers and, in addition, also variables. These connectives testify to the fact that the analytical mode is normed in the sense that it makes possible concept-formation and argumentation which may or may not conform to normative logical principles [such as the principle of identity, (non-)contradiction, the excluded middle, and so on]. Negation makes it possible to assess illogical thinking and contradictions – a statement and its negation cannot both be true in the same context. Negation as a connective presupposes the logical principles of identity and (non-) contradiction – in what is analyzable A is A and A is not non-A. Thus analysis presupposes unity and multiplicity. First of all this relates to arithmetical phenomena. 1 Van Heijenoort remarks that “an axiomatization of set theory is usually embedded in a logical calculus, and it is Weyl's and Skolem's approach to the formulation of the axiom of separation that is generally adopted” (1967:285). 2 Note that within Zermelo-Fraenkel Set Theory (ZF) the terms “set” and “element” are synonymous (1973:24), implying that this theory will avoid the phrase “x is an element of y” (1973:23, note 2). Their terminology, in terms of the membership relation, is such that “x Îy” is read as “x is a member of y” or as “x belongs to y” (which is synonymous with “x is contained in y” / “y contains x (as a member)” (1973:23). 4 D F M Strauss Therefore the meaning of the logical principle of identity and that of non-contradiction analogically reflect this basic arithmetical meaning of unity and multiplicity. Whatever is given as a discrete unity (as being distinct) is identical to itself (the basis of the principle of identity) and is different from whatever it is not (the basis of the principle of non-contradiction) (cf. Strauss, 1991). In themselves these two principles therefore (in a positive and negative way) reveal the coherence between two irreducible modal aspects of reality, namely the logical and the numerical modes. In passing we may mention that arithmetical addition differs from a logical synthesis – something understood by Immanuel Kant. Where he argues for the synthetic nature of mathematical judgments in his Critique of Pure Reason (CPR), he clearly realizes that pure logical addition (a merely logical synthesis) cannot give rise to a new number (cf. CPR, 1787:15 where he considers the proposition that 7+5=12). In a different way Frege made the same point: entitary directed abstraction can only proceed to more abstract entities, but never to any number as such. The logical addition of ‘ones’ or ‘twos’ cannot but end with the repeated identification of another number of the same kind: having identified a ‘two’ and another ‘two’ still provides us only with the ‘abstract’ notion of ‘twoness’. The presence of variables within the first order predicate calculus in yet another way also highlights the original meaning of number – but this time in coherence with the kinematical and the physical aspects. Physical changes presuppose constancy. Constancy as a term actually finds its “modal seat” within the kinematical aspect of pure movement – evidenced in Einstein's special theory of relativity where the velocity of light is postulated as being constant in a vacuum. The original meaning (modal seat) of change (variability) is found in the physical aspect where energy-operation causes certain effects (changes) to occur. Remark: The conditional as connective analogically reflects the physical cause-effect relation – if this then one can conclude to that. Weyl (1966:32) touches upon the relation between the (physical) relationship between cause and effect on the one hand and ground and conclusion on the other without realizing that a basic concept of logic is at stake which analogically reflects the coherence between the logical and the physical aspects of reality – the relation between logical grounds and conclusions reflects the original physical relation of cause and effect. He does realize that the logical relation may have its foundation in an applicable essential law, a “causal connection” or “an empirical regularity,”but nontheless claims that the “sign ®“expresses a purely logical conclusion (‘Folge’)” without realizing that the inevitability to employ this term ‘Folge’ demonstrates the undeniable analogical link between the logical and the physical aspect of reality. 5 Infinity and Continuity In the footsteps of Plato and Galileo it was clear to modern physics and to Einstein that changes could only be detected on the basis of constancy, explaining the close connection of constants and variables also in logic.1 Furthermore, the mere fact that both these terms appear in the plural makes it plain that the practice in logic to employ the terms constants and variables at once also presupposes the original quantitative meaning of number. 2.3 Tacit assumptions of axiomatic set theory The first four axioms of ZF do not guarantee the existence of a set (or an object) at all (cf. Fraenkel et al., 1973:39, note 2).2 In addition to these four Axioms the Axiom of Subsets (v) and the Axiom of Infinity (vi) are introduced.3 The Axiom of Infinity asserts explicitly that some set exists (cf. Fraenkel et al., 1973:39, note 2).4 Let us now look at some of the tacit assumptions of axiomatic set theory. In the absence of a foundational ontological consideration of the interrelationships between the numerical, the spatial, the kinematical, the physical and the analytical aspects of reality an extremely fundamental circulus vitiosus is concealed. Set theory is appreciated as a purely arithmetical theory (as intended by Cantor and most of his successors). Yet, in order to construct an axiomatic foundation for set theory, as we have seen, the aid of an underlying logic is required. Does this mean that logic in itself can provide a sufficient foundation for set theory (or: mathematics)? Whereas Russell claims that logic and mathematics are identical [1956:(v)] and made an attempt to derive the number concept from the logical class concept, we have seen that this entire procedure begs the question. The claims of logicism are untenable because both the logical class concept as well as the Axiom of Infinity make an appeal to the basic meaning of number. In other words, the very meaning of analysis (i.e, the logical mode of reality) presupposes the meaning of number and therefore cannot serve as a foundation for it. 1 We shall see that Weierstrass tried to eliminate the notion of variables through the postulation of a static infinite domain. 2 These Axioms are (Fraenkel et al., 1973:27-35): (i) the Axiom of Extensionality (if x Í y and y Í x, then x = y); (ii) the Axiom of Pairing [for any two elements a and b there exists the set y which contains just a and b (i.e., a and b and no different member)]; (iii) the Axiom of Union / Sumset (for any set a there exists the set whose members are just the members of the members of a); (iv) the Axiom of Powerset (for any set a there exists the set whose members are just all the subsets of a). 3 The former (also known as the Axiom of Separation) states that for any set a and any condition B (x) on x there exists the set that contains just those members x of a which fulfil the condition B (x), while the latter (the Axiom of Infinity) secures the existence of infinite sets by postulating them (cf. Fraenkel et al., 1973:46). We shall presently return to the form of the InfAx. 4 For the moment we leave aside the Axiom Schema of Replacement/Substitution, the Axiom of Choice and the Axiom Schema of Foundation. 6 D F M Strauss The pretended foundation of set theory in logic therefore side-steps the crucial issue: in order to provide an axiomatic foundation for an analysis of the meaning of number an underlying logic is required which in itself presupposes this very meaning of number! We have pointed out that this already follows from the inevitable presence of quantifiers and a multiplicity of (constants and) variables assumed in first order predicate calculus. The intuition of multiplicity is made possible by the unique quantitative meaning of the numerical aspect – first of all accounted for in the mastery of the natural numbers and in the fact that succession is also inherent to our understanding of natural numbers (the best known mathematical “application”of this order of succession is found in [mathematical/complete] induction).1 Already in 1922 Skolem had a firm grip on these issues: Those engaged in doing set theory are normally convinced that the concept of an integer ought to be defined and that complete induction must be proved. Yet it is clear that one cannot define or provide an endless foundation; sooner or later one encounters what is indefinable or unprovable. Then the only option is to ensure that the first starting points are immediately clear, natural and beyond doubt. The concept of an integer and the inferences by induction meet this condition, but it is definitely not met by the set theoretic axioms such as those of Zermelo or similer ones. If one wishes to derive the former concepts from the latter, then the set theoretic concepts ought to be simpler and employing them then ought to be more certain than working with complete induction – but this contradict the real state of affairs totally (1979:70). 2 3 The primacy of natural numbers and their succession (induction) The intuition of one, another one, and so on generates the most basic meaning of infinity – literally without an end, endlessly, infinitely. The awareness of a multiplicity is at least accompanied by an awareness of succession. Kant realized that succession differs from causation: the day succeeds the night and the 1 According to Freudenthal, Dedekind was perhaps the first one (cf. Dedekind, 1887, par. 59, 80) to call the conclusion from n to n + 1 complete induction (“vollständige Induktion”). Neither Bernoulli nor Pascal is the founder of this principle. Its discovery must be credited to Francesco Maurolico (1494-1575) (cf. Freundenthal, 1940:17). In a mathematical context, where “bad induction” is supposed to be excluded (as Freundenthal remarked – 1940:37), no adjective is necessary to qualify the term induction. 2 “Die Mengentheoretiker sind gewöhnlich der Ansicht, dass der Begriff der ganzen Zahl definiert werden soll, und die vollständige Induktion bewiesen werden soll. Es ist aber klar, dass man nicht ins Unendliche definieren oder begründen kann; früher oder später kommt man zu dem nicht weiter Definierbaren bzw. Beweisbaren. Es ist dann nur darum zu tun, dass die ersten Anfangsgründe etwas unmittelbar Klares, Natürliches und Unzweifelhaftes sind. Diese Bedingung ist für den Begriff der ganzen Zahl und die Induktionsschlüsse erfüllt, aber entschieden nicht erfüllt für mengentheoretische Axiome der Zermelo'schen Art oder ähnliches; sollte man die Zurückführung der ersteren Begriffe auf die letzteren anerkennen, so müssten die mengentheoretischen Begriffe einfacher sein und das Denken mit ihnen unzweifelhafter als die vollständige Induktion, aber das läuft dem wirklichen Sachverhalt gänzlich zuwider.” 7 Infinity and Continuity night succeeds the day, but neither the day nor the night causes night or day. We shall argue below that whenever a multiplicity is grasped collectively (i.e., at once), an appeal is made to our intuition of space. Weyl says that the starting-point of mathematics is the series of natural numbers, the law according to which the number 1 is brought forth from nothing and where every number in turn gives rise to its successor (1921:57). On the next page Weyls calls the “always another one” the original mathematical intuition (die mathematische Urintuition) – it is not possible and it is not required to provide a further foundation for this Urintution. Upon this basis Weyl categorically holds that from the intuitionistic stand-point complete induction secures mathematics from being an enormous tautology and impregnates its statements as synthetic (non-analytic).1 With a reference to Weyl also Skolem mentions the fact that the concept of an integer and of induction constitutes the logical content of Hilbert's metamathematics.2 Number displays an order of succession which is embodied in the application of induction. It is therefore significant that already in 1838 De Morgan used the equivalent expression “successive induction” (Freundenthal, 1940:36-37). Weyl does not stop to emphasize that mathematics in its entirety, even regarding the logical form in which it operates, is dependent upon the essence of the natural numbers.3 3.1 Ordinality versus cardinality Those who are convinced that (axiomatic) set theory provides a sufficient basis for mathematics give priority to the concept of cardinal number. This concept seems to be more abstract and general than that of ordinal number, because in the latter case the relation between the members of a set is taken into account whereas it is absent in the case of cardinal numbers. According to Smart the main purpose of Cassirer's “critical study of the history of mathematics is to illustrate and confirm the special thesis that oridinal number is logically prior to cardinal number, and, more generally, that mathematics may be defined, in Leibnizian fashion, as the science of order” (1958:245). The notion of a cardinal number faces two problems on the way to its foundational claim. First of all, it is dependent upon the concept of a whole, and 1 “Unabhängig aber davon, welchen Wert man dieser letzten Reduktion des mathematischen Denkens auf die Zweieinigkeit beimißt, erscheint vom intuitionistischen Standpunkt die vollständige Induktion als dasjenige, was die Mathematik davon bewahrt, eine ungeheure Tautologie zu sein, und prägt ihren Behauptungen einen synthetischen, nicht-analytischen Charakter auf” (1966:86). 2 “In der Tat basiert sich ja Hilbert sehr wesentlich auf dem Begriff der ganzen Zahl und der vollständigen Induktion in der Metamathematik, und diese stellt ja den logischen Inhalt seiner Theorie dar” (Skolem, 1929:89). What Skolem calls the “logical content” actually refers to the primtive meaning of number which is presupposed in logic – an insight also emphasized by Weyl (see the main text below and the next footnote)! 3 “die Mathematik ist ganz und gar, sogar den logischen Formen nach, in denen sie sich bewegt, abhängig vom Wesen der natürlichen Zahl” (1921:70). 8 D F M Strauss secondly cardinality can only be established by implicitly applying ordinality. Surely the notion of order does not exhaust the meaning of number – as realized by Cassirer: “As soon as we proceed from the mere succession of numbers to a specific multiplicity (Vielheit) we encounter the transition from ordinal numbers to cardinal numbers as it was developed by Dedekind, Helmholtz and Kroneker” (1910:53). The intimate link between the notion of a cardinal number and the supposition of a totality is also seen from the perspective of the concept of a one-to-one correspondence – Cassirer uses the term “Gleichzahligkeit” (1910:58) – which clearly presupposes the intuition of at once (simultaniety; Gleichzeitigkeit). The concept of equivalence implied in a one-one correlation underlies the difference between a straightforward multiplicity (how many/Wieviel) and the concept “just as many” / “equalling” (Gleich-viel) (Cassirer, 1910:62). The crucial question is whether the intuition of a whole (totality), which is given at once, stems from our numerical intuition of succession or whether it has an irreducible meaning transcending the confines of relations of number? Is it possible to bring succession and simultaneity under the same denominator or are they irreducibly different? Skolem mentions the fact that Zermelo pointed out that the expressions ordinal numbers and cardinal numbers are employed when sets are compared (Skolem, 1929:80). But what is required to compare sets? How is it possible to establish the equality of two cardinal numbers? The only way to achieve this is to order the members of both sets one by one, because otherwise no on-to-one correspondence could be assumed. In other words, comparing cardinals presupposes some or other order of succession! On the basis of a similar argument Weyl (also employing the term “Gleichzahligkeit”) is therefore fully justified in his claim that “ordinal number is primary”. It has been contested many times whether cardinal number is the primary concept and ordinal number the secondary. ... This definition is not only limited to finite sets; latching on to it Cantor developed his theory of infinite cardinal numbers within the context of his general set theory. But the possibility of mapping, present in the criterion of equi-numerosity, can only be assessed if the act of mapping takes the one after the other, in an ordered temporal succession and when the elements of both sets are thus themselves ordered. If the comparison of two sets is in an abstract way separated in determining the number of each set and subsequently comparing the numbers, then it is not allowed to order the individual sets themselves by designating one element after the other in time (as it is any way necessary for a set to be given individually; and numbers concern only sets such as these, that we use in everyday life). Therefore it seems to me incontestable that the ordinal number is primary. Modern 9 Infinity and Continuity mathematical foundaitonal research, once again disrupted by dogmatic set theory, confirms this throughout. 1 Dummett considers it to be a shortcoming that Frege did not pay enough attention to Cantor's work. Had he done that he “would have understood what it revealed, that the notion of an ordinal number is more fundamental than that of a cardinal number” (1995:293). Dummett continues by pointing out that this is even true of the finite case: “after all, when we count the strokes of a clock, we are assigning an ordinal number rather than a cardinal. If Frege had understood this, he would therefore have characterised the natural numbers as finite ordinals rather than as finite cardnals” (Dummett, 1995:293). Cassirer is correct when he emphasizes that the “determination of number by the equivalence of classes presupposes that these classes themselves are given as a plurality” (1923:52). Although he holds that the transition from ordinal number to cardinal number does not entail any new mathematical content, he does discern a “new logical function” in the “formation of the cardinal number” (1953:42): “As in the theory of ordinal number the individual steps are established and developed in definite sequence, so here the necessity is felt of comprehending the series, not only in its successive elements, but as an ideal whole” (Cassirer, 1953:42). Cassirer does not realize that the supposed “new logical function” actually unmasks a new key element transcending the mere awareness of a succession, namely the idea of a whole. 4 Wholeness and Totality – the irreducibility of the Whole-Parts Relation Although the terms “whole” and totality” are closely related to the term “continuity” it seems difficult to define the meaning of continuous extension – also realized by Dantzig: “From time immemorial the term continuous has been applied to space, ..., something that is of the same nature in its smallest parts as it is in its entirety, something singly connected, in short something continuous! don’t you know .... any attempt to formulate it in a precise definition invariably ends in an impatient: ‘Well, you know what I mean!’” (Dantzig, 1 Es ist viel darüber gestritten worden, ob nicht umgekehrt die Kardinalzahl das Erste und die Ordinalzahl der sekundäre Begriff sei. … Es ist diese Definition nicht einmal auf endliche Mengen beschränkt; die an sie sich knüpfende Theorie der unendlichen Kardinalzahlen hat G. Cantor im Rahmen seiner allgemeinen Mengenlehre entwickelt. Aber die Möglichkeit der Paarung, von der im Kriterium der Gleichzahligkeit die Rede ist, läßt sich nur prüfen, wenn die Zuordnungsakte einer nach dem andern, in geordneter zeitlicher Folge, vorgenommen und damit die Elemente beider Mengen selber geordnet werden. Reißt man die Vergleichung zweier Mengen abstraktiv auseinander in Zahlbestimmung jeder Menge und nachfolgenden Vergleich der Zahlen, so ist es also unerläßlich, die einzelne Menge selber zu ordnen durch Aufweisung eines Elementes nach dem andern in der Zeit (wie es ohnehin nötig ist, damit ein Inbegriff individuell gegeben sei; und nur von solchen Inbegriffen handeln die Zahlen, deren wir uns im täglichen Leben bedienen). Daher scheint es mir unbestreitbar, daß die Ordinalzahl das Primäre ist. Die moderne mathematische Grundlagenforschung, welche die dogmatische Mengenlehre wieder zerstört hat, bestätigt dies durchaus (Weyl, 1966:52-53). 10 D F M Strauss 1947:167).). Synonyms like “uninterrupted,” “connected,” “coherent,” and so on, simply repeat what is meant by continuity, in stead of defining it! Yet, that a “continuous” whole allows for an infinite number of divisions was already discovered by the Greeks. Zeno's B Fr. 3 reads: If things are a multiplicity, then it is necessary that their number must be identical to their actual multiplicity, neither more nor less. But if there are just as many as there are, then their number must be limited (finite). If things are a multiplicity, then necessarily they are infinite in number; for in that case between any two individual things there will always be other things and so on. Therefore, then, their number is infinite. The assumption that things are many serves two opposite conclusions! Apparently the two sides of the (spatial) whole-parts relation lies at the foundation of this argument. If the multiplicity of the first section refers to the many parts of the world as a whole, it stands to reason that taken together they constitute the unity of the world as a whole (and that their number would be limited). If, on the other hand, one starts with the whole and then tries to account for its parts, one must keep in mind that between any two of the many parts there will always be other, indicating an infinity of them. H. Fränkel explicitly uses the whole-parts relation to explain the meaning of this fragment (1968:425 ff., 430). If this interpretation is sound, then Zeno not only understood something of the whole-parts relation, but also, for the first time, realized that spatial continuity is characterized by being infinitely divisible. This perspective can also be used in support of the interpretation given to Zeno's B Fr.1 by Hasse and Scholz (1928:10-13). This first fragment (which we inherited from Simplicius) states that if there exists a multiplicity, then simultaneously it must be large and small; large up to infinity and small up to nothingness. Hasse and Scholz clarify this fragment by interpreting it as follows: If it is permissible to conceptualize a line-stretch as an aggregate of infinitely many small line stretches, then there are two and only two possibilities. Every basic line segment either has a finite size (larger than zero), in which case the aggregate of line-stretches transcends every finite line-stretch; or the supposed line-stretches are zero-stretches in the strict sense of the word, in which case the composed line is also a zero-stretch, because the combination of zerostretches can always only produce a zero-stretch, however large the number of zero-stretches used may be (1928:11). 1 Besides the fact that we can render the two mentioned fragments of Zeno perfectly intelligible by using the whole-parts relation, further support for this understanding may also be drawn from the account which Aristotle gave of Zeno's arguments (cf. Metaph. 233 a 13 ff and 239 b 5 ff.). One of the standard expositions of Zeno's argumentation against the reality of motion is completely dependent on the employment of the whole-parts relation with its im1 We shall later on see that on the basis of measure theory and non-denumerable sets Grünbaum tried to side-step the last remark in order to accomplish an arithmetization of the continuum. 11 Infinity and Continuity plied trait of infinite divisibility. Guthrie explains this argument by saying that according to Zeno Motion is impossible because an object moving between any two points A and B must always cover half the distance before it gets to the end. But before covering half the distance it must cover half of the half, and so ad infinitum. Thus to traverse any distance at all it must cover an infinite number of points, which is impossible in any finite time (1980:91-92). For the larger part of 2000 years continuity (with its implied whole-parts relation), dominated the scene – both within the domains of philosophy and mathematics. Early modernity does witness atomistic theories of nature, but only in the course of the nineteenth century it penetrated the mathematical treatment of continuity. Bolzano explored an atomistic approach and it was brought to fruition by Weierstrass, Dedekind and Cantor. The new assumption in the approach of Weierstrass holds that we have to define limits in terms of a static domain encompassing all real numbers. Boyer (1959:286) refers to this in the following explanation: In making the basis of the calculus more rigorously formal, Weierstrass also attacked the appeal to the intuition of continuous motion which is implied in Cauchy's expression – that a variable approaches a limit. Previous writers generally had defined a variable as a quantity or magnitude which is not constant; but since the time of Weierstrass it has been recognized that the ideas of variable and limit are not essentially phoronomic, but involve purely static considerations. Weierstrass interpreted a variable x simply as a letter designating any one of a collection of numerical values. A continuous variable was likewise defined in terms of static considerations: If for any value xo of the set and for any sequence of positive numbers d1, d2, ......., dn, however small, there are in the intervals xo – di, xo + di others of the set, this is called continuous. Weyl characterizes the new apparent success of the aim to arithmatize mathematics as atomistic (1921:56, 72). According to him within a continuum it is certainly possible, through divisions, to generate partial continua, but it is not clever (“unvernünftig”) to assert that the total continuum is composed out of the limits and these partial continua. Two years later, in 1923, he remarks that certain derived concepts come closer to an indication of the domain of study of geometry, such as a striaght line, and the occurrence of point and line (see Lorenzen, 1986:151). In general Weyl holds that “[A] true continuum after all coheres within itself and cannot be divided into separate pieces; it contradicts its essence.”1 Weyl mentions that he changed his own view by accepting the position taken by Brouwer in this regard. Nonetheless one does not have to 1 “Erscheint dies dem heutigen Mathematiker mit seiner atomistischen Denkgewöhnung anstößig, so war es in früheren Zeiten eine allen selbstverständliche Ansicht: innerhalb eines Kontinuums lassen sich wohl durch Grenzsetzung Teilkontinuen erzeugen; es ist aber unvernünftig, zu behaupten, daß das totale Kontinuum aus der Grenze und jenen Teilkontinuen zusammengesetzt sei. Ein wahrhaftes Kontinuum ist eben ein in sich Zusammenhängendes und kann nicht in getrennte Bruchstücke aufgeteilt werden; das widerstreitet seinem Wesen” (Weyl, 1921:73). 12 D F M Strauss adhere to the intuitionistic understanding of (infinity and) continuity to realize that the whole-parts relation and the totality-character of continuity stands in the way of a complete (atomistic) arithmetization of continuity. Paul Bernays, the co-worker of David Hilbert, senses the irreducibility of the spatial whole-parts relation (i.e., the totality feature of spatial continuity) with an astonishing lucidity: The property of being a totality “undeniably belongs to the geometric idea of the continuum. And it is this characteristic which obstructs a complete arithmetization of the continuum.”1 Compare his remark in a different context where he even states that the classical foundation of the real numbers given by Cantor and Dedekind does not “manifest a complete arithmetization” (1976: 187-188). To this he adds the remark: It is in any case doubtful whether a complete arithmetization of the idea of the continuum could be justified. The idea of the continuum is any way originally a geometrical idea (1976:188). Something remarkable emerges from this situation. If the nature of totality (wholeness / the whole-parts relation) cannot be arithmetized how does one explain the entailed property of being infinitely divisible? Does this property imply that the (numerical) intuition of succession (literally without an end / endless / infinite) is an indispensible building block of continuity? Does it therefore mean that “number” is “built into” the nature of (spatial) “continuity”? 5 Some crucial structural features At the beginning of our reflections we have mentioned the opposing one-sidedness present in the history of mathematics and philosophy regarding the relationship of discreteness and continuity. We have highlighted the key role of terms such as order, succession, induction, ordinal number, cardinal number, infinity (endlessness), at once, wholeness (totality), the whole-parts relation and (infinite) divisibility. In addition we have mentioned the basic role of multiplicity which, within logic (and mathematics), is reflected in the concepts of (constants and) variables. On top of that we had to refer to the connectives of first order predicate calculus [negation, conjunction, disjunction and (bi-)conditional]. Sometimes disciplines do pay attention to the supposedly basic concepts they employ. We have already suggested that these basic concepts are closely related to the primitive terms encountered in a discipline and the implied relation between unity and multiplicity is centered in the way in which one accounts for the coherence of irreducibiles. Rucker, for example, elsewhere remarks: “The discrete and continuous represent fundamentally different aspects of the mathematical universe” (Rucker, 1982:243). Moore proceeds a step further by discerning two ‘clusters’ of concepts which dominate the his1 “Und es ist auch dieser Charakter, der einer vollkommenen Arithmetisierung des Kontinuums entgegensteht” (1976:74). 13 Infinity and Continuity tory of the notion of infinity. In the first cluster the following terms are categorized: “boundlessness; endlessness; unlimitedness; immeasurability; eternity; that which is such that, given any determinate part of it, there is always more to come; that which is greater than any assignable quantity” (1990:1). Within the second cluster he mentions: “completeness; wholeness; unity; universality; absoluteness; perfection; self-sufficiency; autonomy” (Moore, 1990:1-2). These characterizations are congruent with the quoted weak spot alluded to by Fraenkel et. al – regarding the , “the gap between discrete and continuous” (1973:212-213). On the previous page they present a stronger formulation: “Bridging the gap between the domains of discreteness and of continuity, or between arithmetic and geometry, is a central, presumably even the central problem of the foundation of mathematics” (Fraenkel, A., et al., 1973:211). Though this formulation may be consistent with the history of the problem (which explored the one-sided options of reducing number to space or reducing space to number), it may be fruitful alternatively to acknowledge the difference between discreteness and continuity while holding on to the mutual connectedness of both these perspectives. However, exploring this alternative does require distinctions not familiar within the special sciences. The guiding perspective required is an account of the analogical basic concepts employed by the disciplines. Though not universally acknowledged by the various disciplines the reality of using such concepts is ubiquitous. 6 The inevitability of employing analogical elementary basic concepts Contemporary postmodernity emphasizes language as horizon, but in doing that it does not sufficiently distinguish between (i) entitary analogies and (ii) modal analogies. Ordinary and scientific language designate analogies between entities with metaphors (such as the foot of the mountain or the wave theory of light). Whereas such metaphors (entitary analogies) could be replaced by entirely different ones, the substitution of modal analogies is only meaningful if synonomous terms are employed. A spatial term such as domain, for example, could be replaced by others, such as range, scope or sphere. Although, for technical purposes, mathematicians may decide to attach slightly different connotations to these terms, the fact that they all share a generic spatial meaning cannot be denied. In order to understand what is at stake in the case of truly analogical basic concepts we look at the classical legacy about the infinite divisibility of continuity. In his article on the infinite commemorating Karl Weierstrass (published in 1925), Hilbert looks at this issue from the perspective of the infinitely small and the infinitely large (1925:163 ff.). The discovery of energy quanta on the one hand and Einstein's 14 D F M Strauss theory of relativity, on the other hand, eliminate both possibilities.1 The following distinction is therefore necessary: that between mathematical space and physical space. Whereas the former – in a purely abstract and functional perspective – is both continuous and infinitely divisible, physical space is neither continuous nor infinitely divisible. Since it is bound to the quantum structure of energy physical space cannot be subdivided ad infinitum. Energy quanta indeed represent the limit of the divisibility of energy.2 An analogy is present whenever differences are shown in what is similar. In this case: both mathematical space and physical space are extended (their similarity), but in being discontinuous and not infinitely divisible (their differences) the latter differs from the former. The concept physical space is an instance of an analogical basic concept. It is based upon the fact that within the structure of the physical aspect the meaning of space is analogically reflected. The multiplicity of properties combined in the unity of a (logical) concept, in turn, analogically reflects the meaning of number (one and many) within the analytical aspect of reality. In general it is the case that each aspect of reality expresses its coherence with the other aspects through analogical moments. The Dutch philosopher, Herman Dooyeweerd, created a unique theory of “modal law-spheres” to explain this complex state of affairs. The classical distinction between properties and entities runs parallel to that between function and entity.3 Entities function in the various aspects of reality.That is to say, the various aspects have an ontic status and are not mere products of thought. Since Descartes an implicit nominalist assumption accompanies the understanding of functional (modal) aspects. Remark: The classical realist position in philosophy accepts a threefold existence of the so-called “universalia”: ante rem in God's mind (the legacy of Plato's transcendent ideal forms), in re within things (a la Aristotle – as their universal substantial forms), and post rem (as subjective universal concepts – with as criterion of truth the correspondence between thought and reality (”adequatio intellectus et rei”). Nominalism rejects the first two and only acknowledges universal concepts (or: words) within the human mind – outside the human mind only 1 A more extensive explanation is found in his oration on Naturerkennen und Logik (Hilbert, 1930:380-381). In order to account for the discrete nature of the omission or absorbtion of energy, Planck postulated that radiant energy is quantized, proportional to the frequency v in the formula E = hnv – where n is an integer, v the frequency, and h the quantum of action (Wirkungsquantum) with the value 6.624 ´ 10-34. 2 Owing to gravitation (“curved space”) the universe is considered to be finite though unbounded. 3 The quoted work from Cassirer, Substanzbegriff und Funktionsbegriff, made an extremely valuable contribution to an understanding of this issue. Often function concepts are also designated as concepts of relation (“Relationsbegriffe”). 15 Infinity and Continuity uniquely individual entities exist. (Within nominalism a distinction is made between those nominalists who accept universal concepts: conceptualism and the more extreme position which only accepts words.) In their ordinary understanding sets are universals and therefore they partake, in the words of Fraenkel et al., in “the well-known and amply discussed classical problem of the ontological status of the universals” (1973:332). The three main traditional answers given to this problem, namely realism, nominalism and conceptualism, are related with their modern counter-parts known as platonism, neo-nominalism, and neo-conceptualism (Fraenkel et al., 1973:332). Stegmüller explains the mathematical relevance of these different positions as follows: the three mentioned ontological positions, namely nominalism, conceptualism, and platonism, are mapped in terms of the quantitative categories “finite totality (Gesamtheit) – denumerable infinite totality – non-denumerable infinite totality” (1965:117-118). In his Principles of Philosophy Descartes says “that number and all universals are only modes of thought” (Part I, LVIII). David Hilbert (implicitly) still continues this orientation when he argues that after we have established the finiteness of reality in two directions (the infinitely small and the infinitely large), it may still be the case that the infinite does have a justified place within our thinking!1 The implication is clear: “reality” is exhausted by entities, modal (functional) properties are located in the realm of thought. 7 The theory of modal aspects The theory of modal law-spheres first of all acknowledges the ontic givenness of the modal aspects. Hao Wang remarks that Gödel is very “fond of an observation that he attributes to Bernays”: “That the flower has five petals is as much part of objective reality as that its color is red” (Wang, 1982:202). The quantitative side (aspect) of things (entities) is not a product of thought – at most human reflection can explore this given (functional) trait of reality by analyzing what is entailed in the meaning of multiplicity. Yet, in doing this (theoretical and non-theoretical) thought explores the given meaning of this quatitative aspect in various ways, normally first of all by forming (normally called: creating) numerals (i.e., number symbols). The simplest act of counting already explores the ordinal meaning of the quantitative aspect of reality. Frege correctly remarks “that counting itself rests on a one-one correlation, namely between the number-words from 1 to n and the objects of the set” (quoted by Dummett, 1995:144). However, in the absence of a sound and thought-through distinction between the dimension of concretely existing entities (normally largely identi1 “Die Endlichkeit des Wirklichen haben wir nun in zwei Richtungen festgestellt: nach dem Unendlichkleinen und dem Unendlichgroßen. Dennoch könnte es sehr wohl zutreffen, daß das Unendliche in unserem Denken einen wohlberechtigten Platz hat und die Rolle eines unentbehrlichen Begriffes einnimmt” (1925:165). 16 D F M Strauss fied with “physical” or “space-time existence”) and the dimension of functional modes (aspects) of ontical reality, which cannot be observed through sensory perception, mathematicians oftentimes struggle to account for the epistemic status of their “subject matter.” Perhaps the awareness for the need of acknowledging this distinct dimension of reality is best articulated in Wang's discussion of Gödel's thought. Wang discusses Gödel's ideas regarding “mathematical objects” and mentions his rejection of Kant's conception that they are “subjective.” Gödel holds: “Rather they, too, may represent an aspect of objective reality, but, as opposed to the sensations, their presence in us may be due to another kind of relationship between ourselves and reality” (quoted by Wang, 1988:304, cf. p.205). To this Wang adds his support: “I am inclined to agree with Gödel, but do not know how to elaborate his assertions. I used to have trouble by the association of objective existence with having a fixed ‘residence’ in spacetime. But I now feel that ‘an aspect of objective reality’ can exist (and be ‘perceived by semiperceptions’) without its occupying a location in spacetime in the way physical objects do” (Wang, 1988:304). Of course Wang could have referred to the important insights of Cassirer in this regard. Already in his article on Kant and modern mathematics (1907), and particularly in his influential work: Substance and Fucntion (1910), Cassirer distinguishes between entities and functions. He crearly realizes that quantitative properties are not exhausted by any individual entity: “”number is to be called universal not because it is contained as a fixed property in every individual, but because it represents a constant condition of judgment concerning every individual as an individual” (1953:34). If we set aside the (neo-)Kantian undertones of this statement, Cassirer already saw something of the modal universality of the arithmetical aspect of reality. Every aspect has an undefinable core (or: nuclear) meaning (also designated as the meaning-nucleus) which qualifies all the analogical meaning-moments within a specific aspect. These analogical moments may refer backwards to ontically earlier aspects (known as retrocipations) or forwards to ontically later aspects (known as anticipations). Earlier and later are taken in the sense of cosmic time as it is called by Dooyeweerd. The aspects of reality are fitted in an inter-modal coherence of earlier and later: the most basic aspect is that of number (meaning-nucleus: discrete quantity), which is followed by the aspect of space (continuous extension), the kinematical aspect (core: constancy), the physical (change/energy-operation), the biotic (life), the sensitive (feeling), the logical (analysis), the cultural-historical (formative control/power), the sign-mode (symbolical signification), and so on. Cosmic time has an order-side and a duration-side – we are acquainted with the numerical time-order of succession, the spatial time-order of simultaneity, the kinematical time-order of uniform flow, the irreversible physical time-order, the biotical time-order of birth, growth, maturation, aging and dying (cf. Dooyeweerd, 1996-I, pp.30-32). 17 Infinity and Continuity The meaning of an aspect finds expression in its coherence with other aspects (retrocipations and anticipations). Retrocipatory analogies are captured in the elementary basic concepts of a discipline. The first challenge in an analysis of the elementary (analogical) basic concepts of the various academic disciplines is to identify the modal “home” or “seat” of particular terms. Within an aspect their is a difference between the order-side (also known as the law-side) and its correlate, the factual side (that which is subjected to the law-side and delimited and determined by the latter). The numerical time-order of succession belongs to the law-side of the arithmetical aspect, and any ordered sequence of numbers appears at its factual side (think of the natural numbers in their normal succession). With the exception of the numerical aspect (which only have subject-subject relations), all the other aspects in addition also have subject-object relations at their factual side.1 (Cf. Sketch 1on page 37). 7.1 Numerical and spatial terms It is intuitively clear that our awareness of succession and multiplicity (underlying the concept of an ordinal number and induction) makes an appeal to the quantitative aspect of reality. These terms therefore have their modal “seat” (“home”) in the arithmetical aspect. Of course it is natural that special scientists will attempt to reduce apparently primitive terms to familiar and more basic ones. But if such an attempt becomes circular, or even worse, contradictory, then it may be the case that the primitive terms involved are truly irreducible! Phrased differently: an attempt to define what is undefinable may end up in antinomic reduction.2 Sometimes the challenge is not to get out of the circle, but to get into it(s irreducible meaning)! In the course of our preceding discussion the following cluster of terms probably transcend the confines of the numerical aspect: simultaneity (at once), completedness, wholeness (totality), and the whole-parts relation. The most prominent recognition of the spatial “home” of wholeness and totality is found in the thought of Bernays. He writes that it is recommendable not to distinguish the arithmetical and geometrical intuition according to the moments of the spatial and the temporal, but rather by focusing on the difference between the discrete and the continuous.3 Being fully aware of the 1 The identifiability and distinguishability of something represents its latent logical object-function. When it is identified and distinguished by a thinking subject, this analytical object-function is made patent. 2 The classical example is Zeno's attempt to define movement in static spatial terms – and just doing that eliminated the meaning of motion. 3 “Es empfiehlt sich, die Unterscheidung von "arithmetischer" und "geometrischer" Anschauung nicht nach den Momenten des Räumlichen und Zeitlichen, sondern im Hinblick auf den Unterschied des Diskreten und Kontinuierlichen vorzunehmen” (1976:81). 18 D F M Strauss arithmeticistic claims of modern analysis it is all the more significant that Bernays questions the attainability of this ideal of a complete arithmetization of mathematics. He unambiguously writes: We have to concede that the classical foundation of the theory of real numbers by Cantor and Dedekind does not constitute a complete arithmetization of mathematics. It is anyway very doubtful whether a complete arithmetization of the idea of the continuum could be fully justified. The idea of the continuum is after all originally a geometric idea (Bernays, 1976:187-188).1 Particularly in explaining the difference between the potential and the actual infinite the difference between succession and at once and the irreducibility of the notion of a totality surfaces. Hilbert introduces the difference between the potential and the actual (or: genuinely) infinite by using the example of the “totality of the numbers 1, 2, 3, 4, ...” which is viewed as a unity which is given at once (completed): If one wants to provide a brief characterization of the new conception of infinity introduced by Cantor. one can indeed say: in analysis where the infinitely small and the infinitely large feature as limit concept, as something becoming, originating and generated, that is, as it is stated, with the potential infinite. But this is not the true infinite. We have the latter when, for example, we view the totality of the numbers 1, 2, 3, 4, ... as a completed unity or when we observe the points of a line as a totality of things, given to us as completed. This kind of infinity is designated as the actual infinite. 2 According to Lorenzen the understanding of real numbers with the aid of the actual infinite cannot camouflage its ties with space (geometry): “The overwhelming appearance of the actual infinite in modern mathematics is therefore only understandable if one includes geometry in one's treatment. ... The actual infinite contained in the modern concept of real numbers still reveals its descent (Herkunft) from geometry” (1968:97). Lorenzen highlights the same assumption when he explains how real numbers are accounted for in terms of the actual infinite: One imagines much rather the real numbers as all at once actually present – even every real number is thus represented as an infinite decimal fraction, as if 1 “Zuzugeben ist, daß die klassische Begründung der Theorie der reellen Zahlen durch Cantor und Dedekind keine restlose Arithmetisierung bildet. Jedoch, es ist sehr zweifelhaft, ob eine restlose Arithmetisierung der Idee des Kontinuums voll gerecht werden kann. Die Idee des Kontinuums ist, jedenfalls ursprünglich, eine geometrische Idee.” 2 “Will man in Kürze die neue Auffassung des Unendlichen, der Cantor Eingang verschafft hat, charakterisieren, so könnte man wohl sagen: in der Analysis haben wir es nur mit dem Unendlichkleinen und dem Unendlichengroßen als Limesbegriff, als etwas Werdendem, Entstehendem, Erzeugtem, d.h., wie man sagt, mit dem potentiellen Unendlichen zu tun. Aber das eigentlich Unendliche selbst ist dies nicht. Dieses haben wir z. B., wenn wir die Gesamtheit der Zahlen 1, 2, 3, 4, ... selbst als eine fertige Einheit betrachten oder die Punkte einer Strecke als eine Gesamtheit von Dingen ansehen, die fertig vorliegt. Diese Art des Unendlichen wird als aktual unendlich bezeichnet” (Hilbert, 1925:167). 19 Infinity and Continuity the infinitely many figures (Ziffern) existed all at once (alle auf einmal existierten) (1972:163). These modes of speech highlight the inevitability of employing terms with a spatial descent even when the pretention is to proceed purely in numerical terms. Lorenzen correctly points out that arithmetic by itself does not provide any motive for the introduction of the actual infinite (1972:159). The fundamental difference between arithmetic and analysis in its classical form, according to Körner, rests on the fact that the central concept of analysis, namely that of a real number, is defined with the aid of actual infinite totalities (“aktual unendlicher Gesamtheiten” – 1972:134). Without this supposition Cantor's proof on the non-denumerability of the real numbers collapses into denumerability. While rejecting the actual infinite, intuitionism interprets Cantor's diagonal proof of the non-denumerability of the real numbers in a constructive sense – cf. Heyting (1971:40), Fraenkel et al. (1973:256,272) and Fraenkel (1928:239 note 1). However, in order to reach the conclusion of non-denumerability, every constructive interpretation falls short – simply because there does not exist a constructive transition from the potential to the actual infinite (cf. Wolff, 1971). Remark: Husserl's Philosophie der Aritmetik1 In 1887 Edmund Husserl completed his habilitation with a study concerned with a psychological analysis of the concept of number – followed in 1891 by his extensive “Philosophie der Arithmetik”. In this work he connected the infinite both with the psychological nature of a collective synthesis (1970:64 ff.), and with the principle of succession (p.220). He questioned the actual infinite in the light of the finiteness of arithmetic. He even planned a second volume of his “Philosophie der Arithmetik,” but eventually realized that he could not achieve his aim without using the actual infinite. L. Eley (in the Preface to Husserliana, Vol. XII) saw in this failure the reason why this second volume was never published. From the unpublished manuscripts it is nevertheless clear that Husserl did not succeed in giving a foundation to general arithmetic without the acceptance of the actual infinite (which shows that he confronted himself with classical analysis). It seems to be impossible to develop set theory without “borrowing” key-elements from our basic intuition of space, in particular the (order of) at once and its factual correlate: wholesness / totality. Since spatial subjects are extended their multiple parts exist all at once. This multiplicity is at the factual side of the spatial aspect a retrocipation to the meaning of number – i.e., multiple parts analogically reflect the meaning of number (multiplicity) within space. Bernays did not have a theory of modal aspects at his disposal and therefore lacks the possibility of articulating explicitly the intermodal connections between number and space. Yet, in stead of saying that the mathematical analy1 Edmund Husserl completed his habilitation in 1887 and published in 1891 his study: Philosophie der Aritmetik. 20 D F M Strauss sis of the meaning of number reveals an anticipation to the meaning of space, he states that the idea of the continuum is a geometrical idea which analysis expresses with an arithmetical language.1 8 Deepening our understanding of infinity We have already pointed out that the most primitive meaning of infinity relates to the arithmetical time-order of succession. It has its foundation in the order-side or the law-side of the numerical aspect. This order of succession determines every infinitely proceeding sequence of numbers as well as the different operations at the law-side of the numerical aspect [operations such as addition, multiplication, subtraction, division, as well as the principle of induction – which can be disclosed by (non-theoretical as well as theoretical mathematical) thinking]. Because the expression potential infinite lacks an immediate intuitive clarity, the most appropriate phrase suitable to capture the meaning of this most basic kind of infinity is to call it the successive infinite. We have noted that the Greeks already turned the infinite “inwards” – with their discovery of the infinite divisibility of continuity. In terms of the intermodal coherence between the aspects of number and space this infinite divisibility analogically (i.e., as a retrocipatory analogy) reflects the successive infinite at the law-side of the numerical aspect. The system of rational numbers (fractions), in turn, analogically (i.e., as an anticipatory analogy) reflects this infinite divisibility evinced at the factual side of the spatial aspect. Consequently, the rational numbers represent an anticipation to a retrocipation (see Sketch 2 on page 37).2 When, under the guidance of our theoretical (i.e., modally abstracting) insight into the meaning of the spatial order of simultaneity, the original modal meaning of the numerical time-order is disclosed (deepened), we encounter the regulatively deepened anticipatory idea of actual or completed infinity. Any sequence of numbers may then, directed in an anticipatory way by the spatial order of simultaneity, be considered as if its infinite number of elements are present as a whole (totality) all at once. In this context it is noteworthy that Hao Wang informs us that Kurt Gödel speaks of sets as being “quasi-spatial” and then adds that he is not sure whether Gödel would have said the “same thing of numbers” (1988:202). This mode of speech is in line with our suggestion that the undefined term “element of” employd in ZF set theory actually harbours the totality feature of continu1 “Die Idee des Kontinuums ist, jedenfalls ursprünglich, eine geometrische Idee, welche durch die Analysis in arithmetischer Sprache ausgedrükt wird” (1976:74). 2 In passing we note that the numerical anticipation to a spatial retrocipation – revealed in the denseness of the rational numbers – is not sufficient to characterize any real number in a mathematical satisfactory way – at least not in a deepened and disclosed way. Yet intuitionism uses this semi-disclosed meaning of number in its characterization of the “continuum.” 21 Infinity and Continuity ity, because then it does entail that set theory (in an anticipating way) is dependent on “something spatial”! This also amounts to a confirmation of the unbreakable coherence between the law-side and the factual side of the numerical and the spatial aspects. The modal anticipation from the numerical time-order to the spatial time-order must therefore have its correlate at the factual side. At the factual side of the numerical aspect we encounter the sequence of natural numbers and integers (expressing the primitive meaning of numerical discreteness). Introducing the dense set of rational numbers manifests (as an anticipation to a retrocipation) the semi-disclosed meaning of number. When we employ the anticipation at the law-side of the numerical aspect to the law-side of the spatial aspect we encounter the intermodal foundation of the notion of actual infinity – although the basic intuitions at play suggest a more suitable designation for the actual infinite. But before we discuss a more appropriate naming of the actual infinite a brief remark is required to explain the “as if” character of this deepened notion of infinity. The anticipation from number to space at the law-side determines the correlated multiplicity of natural numbers, integers and rational numbers which are, under the employment of the actual infinite, considered as if they are present as completed (though infinite) wholes or totalities. Remark: “As if”: the actual infinite as a regulative hypothesis Vaihinger developed a whole philosophy of the “as if” (Die Philosophie des Als Ob), in which he tries to demonstrate that various special sciences may use, with a positive effect, certain fictions which in themselves are considered to be internally antinomic. The infinite, both in the sense of being infinitely large and infinitely small, is evaluated by Vaihinger as examples of necessary and fruitful fictions (cf. 1922:87 ff., p.530). Ludwig Fischer presents a more elaborate mathematical explanation of this notion of a fiction. In general he argues: “The definition of an irrational number by means of a formation rule always involves an 'endless', i.e. unfinished process. Supposing that the number is thus given, then one has to think of it as the completion (Vollendung) of this unfinished process. Only in this ... the internally antinomic (in sich widerspruchsvolle) and fictitious character of those numbers are already founded” (1933:113-114). Without the aid of a preceding analysis of the modal meaning of number and space, this conclusion is almost inevitable. Vaihinger and especially Fischer simply use the number concept of uncompleted infinity (the successive infinite) as a standard to judge the (onto-)logical status of the actual infinite. Surely, within the closed (not yet deepened) meaning of the numerical aspect, merely determined by the arithmetical time-order of unfinished succession, the notion of an actual infinite multiplicity indeed is self-contradictory. However, the meaning intended by us for the actual infinite transcends the limits of this concept of number since, in a regulative way, it refers to the core meaning of the spatial aspect which (in an anticipatory sense) underlies the hy22 D F M Strauss pothetical use of the time-order of simultaneity (the “all” viewed as being present at once). Paul Lorenzen echoes something of this approach in his remark that the meaning of actual infinity as attached to the “all” shows the employment of a fiction – “the fiction, as if infinitely many numbers are given” (1952:593). In this case too, we see that the “as if” is ruled out, or at least disqualified as something fictitious, with an implicit appeal to the primitive meaning of number. As long as one sticks to the notion of a process, one is implicitly applying the yardstick of the successive infinite to judge the actual infinite. Paul Bernays did see the essentially hypothetical character of the opened up meaning of number, without (due to the absence of an articulated analysis of the modal meaning coherence between number and space) being able to exploit it fully: “The position at which we have arrived in connection with the theory of the infinite may be seen as a kind of the philosophy of the 'as if'. Nevertheless, it distinguishes itself from the thus named philosophy of Vaihinger fundamentally by emphasizing the consistency and trustworthiness of this formation of ideas, where Vaihinger considered the demand for consistency as a prejudice ...” (1976:60). Although the deepened meaning of infinity is sometimes designated by the phrase completed infinity, this habit may be misleading. If succession and simultaneity are irreducible, then the idea of an infinite totality cannot simply be seen as the completion of an infinite succession. When Dummett refers to the classical treatment of infinite structures “as if they could be completed and then surveyed in their totality” he equates this “infinite totality” with “the entire output of an infinite process” (1978:56). The idea of an infinite totality transcends the concept of the successive infinite. A remarkable ambivalence in this regard is found in the thought of Abaraham Robinson. His exploration of infinitesimals is based upon the meaning of the at once infinite. A number a is called infinitesimal (or infinitely small) if its absolute value is less than m for all positive numbers m in Â (Â being the set of real numbers). According to this definition 0 is infinitesimal. The fact that the infinitesimal is merely the correlate of Cantor's transfinite numbers is apparent in that r (not equal to 0) is infinitesimal if and only if r to the power minus 1 (r-1) is infinite (cf. Robinson, 1966:55ff). In 1964 he holds that “infinite totalities do not exist in any sense of the word (i.e., either really or ideally). More precisely, any mention, or purported mention, of infinite totalities is, literally, meaningless.” Yet he believes that mathematics should proceed as usual, “i.e., we should act as if infinite totalities really existed” (Robinson, 1979:507). Cantor explicitly describes the actual infinite as a constant quantity, firm and determined in all its parts (1962:401). Throughout the history of Western philosophy and mathematics, all supporters of the idea of actual infinity implicitly or explicitly employed some form of the spatial order of simultaneity. What should have been used as an anticipatory regulative hypothesis (the idea of actual infinity), was often (since Augustine) reserved for God or an eternal 23 Infinity and Continuity being, accredited with the ability to oversee any infinite multiplicity all at once. This anticipatory regulative hypothesis of actual infinity does not cancel the original modal meaning of number, but only deepens it under the guidance of theoretical thought. Perhaps also here it would be helpful to introduce new terms for this well-known expression. In stead of speaking about the actual infinite we should rather talk about the at once infinite. Alongside the successive infinite the at once infinite already surfaced in the disputes of the early 14th century about the infinity of God.1 These new expressions relate directly to our basic numerical and spatial intuitions, viz., our awareness of succession and simultaneity – and their mutual irreducibility is based upon the irreducibility of the aspects of number and space.2 A truly deepened and disclosed account of the real numbers cannot be given without the aid of the at once infinite. That this anticipatory coherence between number and space always functioned prominently in a deepened account of the real numbers, may be shown from many sources. It will suffice to mention only one in this context. But before we do that we have to return briefly to the relationship between mathematics and logic. 9 Mathematics and Logic If the meaning of analysis presupposes the original quantitative meaning of unity and multiplicity, turning the logicistic attempt to deduce the meaning of number from the logical mode into a vicious circle, then it appears to be strange that we still have to concede that the discipline of mathematics is in need of a logical foundation?! Cassirer first of all approaches this problem in terms of the numerical analogy within the logical-analytical mode. His question is that it is not understandable why one only accepts logical identity and difference, which enter the set concept as necessary elements, as such basic functions, but that one does not do the same with regard to numerical unity and difference? He claims that a truly satisfactory deduction of the one from the other is also not 1 Compare the expressions infinitum successivum and infinitum simultaneum (Maier, 1964: 77-79). 2 Dooyeweerd did not accept the idea of the at once infinite (actual infinity) owing to the fact that he was strongly influenced by the intuitionistic mathematicians Brouwer and Weyl in this regard. Cf. Dooyeweerd, 1996-I:98-99 (footnote 1) and 1996-II:340 (footnote 1). 24 D F M Strauss achieved by set theory, which entails a persistent suspicion that all similar attempts will continue to harbor a concealed epistemological circle.1 Yet, although it is true that the meaning of analysis presupposes the meaning of number, the (theoretical) analysis of number remains a thought act which is qualified by the logical aspect. This explains why the discipline of logic occupies such a founding role with respect to all disciplines, including mathematics. 10 The circularity entailed in set theoretical attempts to arithmetize continuity The nuclear meaning of space is indefinable. If one tries to define the indefinable two equally objectionable options are open: (i) either one ends up with a tautology – coherence, being connected, and so on, are all synonymous terms for continuity – or, even worse, (ii) one becomes a victim of (antinomic) reduction, i.e. one tries to reduce what is indefinable to something familiar but distinct. While the idea is ancient, modern Cantorian set theory again came up with the conviction that a spatial subject such as a particular line must simply be seen as an infinite (technically, a non-denumerable infinite) set of points. If the points which constitute the one dimensional continuity of the line were themselves to possess any extension whatsoever, it would have had the absurd implication that the continuity of every point is again constituted by smaller points than the first type, although necessarily they also would have had some extension. This argument could be continued ad infinitum, implying that we would have to talk of points with an ever-diminishing “size.” In reality such “diminishing” points do not at all refer to real points, since they simply reflect the nature of continuous extension, which as we have seen, is infinitely divisible. Such points build up space out of space. Anything which has factual extension has a subject-function in the spatial aspect (such as a chair) or is a modal subject in space (such as a line, a surface, and so forth). A point in space, however, is always dependent on a spatial subject since it does not itself possess any extension. Remark: We are now in a better position to explain the structural meaning of the three undefined terms introduced by Hilbert in his mentioned axiomatization of geometry in 1899. These terms instantiate the spatial subject-object relation at the factual side of the spatial aspect. The term “line” represents the factual subject-side (factual, one-di1 “In der Tat ist nicht einzusehen, warum man lediglich logische Identität und Verschiedenheit, die als notwendige Momente in den Mengenbegriff eingehen, als solche Urfunktionen gelten lassen und nicht auch die numerische einheit und den numersichen Unterschied von Anfang an in diesen Kreis aufnehmen will. Eine wirklich befriedigende Herleitung des einen aus dem anderen ist auch der mengentheoretischen Auffassung nicht gelungen, und der Verdacht eines verstckten erkenntnistheoretischen Zirkels blieb gegenüber allen Versuchen, die in dieser Richtung gemacht werden, immer bestehen” (Cassirer, 1957:73-74). 25 Infinity and Continuity mensional spatial extension); the term “point” represents the factual object-side; while the relation of dependence of the latter upon the former is represented by the “relational term”: “lies on”. The length, surface or volume of a point is always zero – it has none of these. If the measure of one point is zero, then any number of points would still have a zero-measure. Even a(n denumerable) infinite set of points would never constitute any positive distance, since distance presupposes an extended subject.1 Remark: The following classical “definition” of a line is well-known: A straight line is the shortest distance between two points. A straight line, however, is a factual spatial figure extended in one dimension. The measure of this extension is indicated by the numerical analogy of distance (magnitude). We can say in a particular instance that the length (i.e. the numerical analogy) of a line is so much. The so much of a line, however, is not the line. In other words, the extension of the line cannot be defined by the indication of its length. The length of a line presupposes the factual extension of the line – from which it remains distinct. For this reason Hilbert imported the term line as an undefined term in his mentioned axiomatic foundation of geometry (cf. 1899). The fourth problem mentioned by him in his presentation to the International Mathematical Conference at Paris in 1900 correctly refers to the problem of the straight line as the shortest connection of (and not: distance between!) two points (Hilbert, 1970:302). A. Grünbaum has combined insights from the theory of point-sets (founded by Cantor) with general topological notions and with basic elements in modern dimension theory in order to arrive at an apparently consistent conception of the extended linear continuum as an aggregate of unextended elements (1952:288 ff.). From his analysis it is clear that he actually had “unextended unit point-sets” in mind and not simply a set of “unextended points” (1952: 295). Initially he starts with a non-metrical topological description and then, later on, introduces a suitable metric normally used for Euclidean spaces (point-sets). The all-important presupposition of this analysis is the acceptance of the linear Cantorean continuum (arranged in an order of magnitude, i.e. the class of all real numbers) (cf. 1952:296). 1 The following classical “definition” of a line is well-known: A straight line is the shortest distance between two points. A straight line, however, is a factual spatial figure extended in one dimension. The measure of this extension is indicated by the numerical analogy of distance (magnitude). We can say in a particular instance that the length (i.e. the numerical analogy) of a line is so much. The so much of a line, however, is not the line. In other words, the extension of the line cannot be defined by the indication of its length. The length of a line presupposes the factual extension of the line – from which it remains distinct. For this reason Hilbert imported the term line as an undefined term in his mentioned axiomatic foundation of geometry (cf. 1899). The fourth problem mentioned by him in his presentation to the International Mathematical Conference at Paris in 1900 correctly refers to the problem of the straight line as the shortest connection of (and not: distance between!) two points (Hilbert, 1970:302). 26 D F M Strauss On the basis of certain distance axioms, the real function d(x,y) (called the distance of the points x,y which have the coordinates xi,yi) is used to define the length of a point-set constituting a finite interval on a straight line between two fixed points (the number of this distance is its length). For example, the length of a finite interval (a,b) is defined as the non-negative quantity b – a (disregarding the question about the interval's being closed, open, or half-open). In the limiting case of a = b, the interval is called “degenerate” with length zero (in this case we have a set containing a single point) (cf. Grünbaum, 1952:296). Furthermore, division as an operation is only defined on sets and not on their elements, implying that the divisibility of finite sets consists in the formation of proper non-empty subsets of these (surely, the degenerate interval is indivisible by virtue of its lack of a subset of the required kind) (1952:301). Finally, the following two propositions are asserted and are considered to the perfectly consistent: “1. The line and intervals in it are infinitely divisible” and “2. The line and intervals in it are each a union of indivisible degenerate intervals” (1952:301). If we confront this analysis of Grünbaum with our characterization of the nature of the actual infinite (the at once infinite), we soon realize that his whole approach is circular. We have seen that, on the basis of the regulative hypothesis of the at once infinite, not only the set of real numbers but also the number of line segments having a common end point could be considered as non-denumerable infinite totalities. In the latter case (i.e., in the case of a group of line segments), we may identify, within the modal structure of space, a retrocipation to an anticipation (a mirror-image of the structure of the system of rational numbers). This retrocipation to an anticipation ultimately underlies Grünbaum's statement: “the Cantorean line can be said to be already actually infinitely divided” (1952:300). Seemingly, the objection that any denumerable sum of degenerate intervals (with zero-length) must have a length of zero, does not invalidate Grünbaum's claim that a positive interval is the union of a continuum of degenerate intervals, because in the latter case we are confronted with a non-denumerable number of degenerate intervals – obviously, if we cannot enumerate them, we cannot add them to form their sum (for this reason, measure-theory also side-steps the mentioned objection, valid for the denumerable case). (Any attempted “addition” would leave out at least one of them.) 11 Aristotle and modern mathematics – infinity and continuity In order to overcome the problems posed by Zeno's paradoxes, Aristotle introduced his notion of the potential infinite (cf. also De gen. et corr. 316 a 14 ff.). The infinite divisibility of a line is granted by him, with the qualification that this divisibility is only potential and can therefore never be carried through actually. Furthermore, Aristotle upholds that no moving object is “counting” 27 Infinity and Continuity while it moves, since then the objection of Zeno would have been valid: in order to traverse a finite spatial distance it is required to actually count an infinite sequence of numbers. Aristotle's notion of division specified an important trait of continuity: when we divide a continuous distance into two halves, we have to take the same point twice – in the first instance as a starting-point and in the second instance as an end-point. To this he adds that the infinite number of halves thus obtained is never actual but only potential halves (Phys. 263 a 23 ff.). Aristotle firmly opposes the acceptance of the actual infinite, mainly in terms of the following objections (cf. Phys. 204 a 20 ff., Metaf. 1066 b 11 ff., and Metaf. 1084 a 1 ff.): (i) if the actual infinite is composed out of parts, everyone of these parts should taken by itself would be actual infinite, implying the absurdity that the whole is no longer prior to (cf. Pol. 1253 a 19-20) (or greater as) a part; (ii) if the actual infinite is composed out of finite parts we would be able to perform the impossible by counting the infinite, or else there must exist transfinite (cardinal) numbers being neither even or odd (it turned out that both objections actually saw something essential of the actual infinite!). [Since Bolzano an infnite set is defined as one in which it is possible to establish a one-to-one correspondence between the set and a proper subset of it. Owing to the fact that communtativity does not hold for Cantor's transfinite ordinal numbers, one can characterize omega (w) as being both even and odd, and as neither being even nor odd (cf. Cantor, 1962:178-179).] Aristotle considers it impossible to explain the continuity of a straight line in terms of the (infinite) number of its points. If “that which is infinite is constituted by points, these points must be either continuous or continuously in contact with one another” (Physica, 231a29-31). Points, however, are indivisible (a point has no parts), while “that which exists between two points is always a line” (Physica, 231b8). According to Aristotle it is clear that “everything which is continuous is divisible into divisible parts which can be divided infinitely: since if it was divisible into indivisible parts, we would have the divisible and indivisible in contact since the limits of continuous things are one (i.e. the same – DFMS)” (Physica, 231b15ff.). 1 In the 19th century the new arithmeticistic tendency aimed at an arithmetical definition of spatial continuity. Bernard Bolzano already illuminates this tendency in par.38 of his work on the paradoxes of the infinite (cf. Bolzano, 1921). He mentions the objection that the attempt to construct extension out of 1 Von Weizsäcker says that the domain in which figures are defined displays, when compared with natural numbers, the property of continuity. In this context he calls upon Aristotle's view of continuity: “Continuity is defined by Aristotle as that which could be divided endlessly in similar parts. The parts of a continuum cannot be counted; but one can measure continua” (1993:115). 28 D F M Strauss parts which themselves are not extended will contain a hidden circle, but he believes that the problem disappears when it is realized that “each whole” exactly “has numerous properties absent in the parts” (1920:72). This is the first instance of claiming an “emergent” property flowing from the whole. M.J. White explicitly employs the term “emegrent” when he explains that when topology considers large classes of points collectively “continuity becomes an ‘emergent’ property” (1988:8, cf. pp.9, 12). The criterion which Bolzano sets for a continuum is that “a continuum is present there, but also only there where a set (Inbegriff) of simple objects (of points ...) finds itself, which is situated in such a way that every single object has at least an environment (Nachbar) (of points – DFMS) in this set for every distance (Entfernung) however small” (1920:73). Cantor criticizes the criterion used by Bolzano as insufficient since a set constituted e.g. by distinct continua (and therefore being as a whole discontinuous) would still be continuous in terms of Bolzano's definition (the end points of each distinct continuum would after all still contain further points of the particular set in an arbitrarily small environment – cf. Cantor, 1962:194). Cantor declares that he has no other choice but to posit a “general purely arithmetical concept of a point-continuum” supported by the way in which he defines real numbers (1962:192). He defines a point-continuum as a perfectly coherent set. A set is perfect when every point of the set is a limit-point and when all limit-points of the set belong to the set. He calls “T a coherent point set, when for every two points t and t’ of this set, at a given arbitrarily small number e there are always a finite number of points t1, t2, t3, ..., tn of T’ present in multiple ways, so that the distances tt1, t1t2, ...tnt’ are all together smaller than e” (1962:194). Comment: Cantor's definition of coherence concerns a metrical characteristic of the continuum. In modern topology, however, continuity is described in terms of open sets (abstracted from the characteristics of a metrical space – cf. Willard, 1970:16-19). P.S. Alexandroff defines the continuum as a non-empty compactly coherent set (1956:163ff., 201ff.). A set is compact if every infinite subset has at least one limit-point. (A point x is a limit-point of a set A when every environment of x contains at least one point of A different from x). This implies that a set in an Euclidean space is only compact if it is delimited. In terms of Alexandroff's definition an infinite straight line is therefore not continuous, while for Cantor it is (cf. also Meschkowski, 1967:55). Dedekind follows the continuity of a straight line arithmetically by successively introducing new numbers: If now, as is our desire, we try to follow up arithmetically all phenomena in the straight line, the domain of rational numbers is insufficient and it becomes necessary that the instrument R constructed by the creation of rational numbers be essentially improved by the creation of new numbers (namely irrational numbers – DFMS) such that the domain of numbers shall gain the same 29 Infinity and Continuity completeness, or as we may say at once, the same continuity, as the straight line (Dedekind, 1901:9). On this foundation Dedekind describes his well-known concept of a cut which characterizes continuity: when “all real numbers break up into two classes U1, U2, such that every number a1 of the class U1 is less than every number a2 of the class U2 then there exists one and only one number by which this separation is produced” (1901:20). Dedekind's notion of a cut is dealt with in analysis textbooks in such a manner that the real number which brings about the division is greater than or equal to every element in the one set and smaller than or equal to all the elements of the other set (cf. e.g. Bartle, 1964:51). Cantor himself refers to the relation which exists between his view of a perfect set and Dedekind's cut theorem (1962:194). G. Böhme strikingly shows how Cantor's definition of the continuum contains two stipulations which both meet the twofold Aristotelian definition of a continuum, namely coherence and a characteristic which ensures the existence of dividing points for infinite division (1966:309). Only allowing a Dedekind-cut at divisions, Böhme justifies his statement as follows: when a Cantorian continuum as such is divided into two by means of the indication of a point so that the one set contains those points which are in numerical value greater than or equal to the indicated point, while the other set contains those points of which the numerical value are smaller than or equal to the numerical value of the indicated point, both parts are again continuous. Such divisions are possible into infinity (due to the perfection of the continuum), and the parts are still coherent in the Aristotelian sense (i.e. their limit-points are the same) (1966:309). This is a remarkable situation: the Cantor-Dedekind description of the continuum presupposes the use of the actual infinite (in particular by using the actually infinite set of real numbers), but nonetheless meets Aristotle's two requirements for a continuum – and this while Aristotle explicitly rejects the actual infinite and only recognizes potential infinity! Did Aristotle actually use the actual infinite implicitly, or is the Cantor-Dedekind definition in the last instance not purely arithmetically founded? 12 Aristotle and Cantor: an ‘intermodal’ solution We have concluded the previous paragraph by raising the question: how is it possible that both employ the same criteria (infinite divisibility and taking any point of division twice [as a starting-point and as an end-point]), but fundamentally oppose each other in connection with the infinite? This almost perplexing situation is immediately clarified in terms of our analysis of the intermodal coherence between number and space. We saw that the original spatial whole-parts relation is indeed founded on the numerical time-order of succession (displaying the primitive meaning of infinity as endlessness), explaining the infinite (endless) divisibility of spatial continuity (this feature manifests the retrocipation at the factual side of the spatial aspect 30 D F M Strauss to the law-side of the numerical aspect). If one restricts oneself to this original meaning of the spatial aspect (without, in addition, considering the opened up meaning of the foundational aspect of number), as Aristotle implicitly did (understandable in the light of the geometrization of Greek mathematics), then no meaning could be given to the notion of actual infinity – explaining the relative correctness in the claim (put forward by Aristotle, Kant and many others in the history of philosophy and mathematics) that a line is not divided in the sense of the actual infinite (at once infinite), but only divisible in the sense of the potential infinite (successive infinite). It is relatively correct, because the foundational meaning of number (analogically reflected in the multiplicity of successive spatial divisions) is not itself as yet opened up by the anticipatory hypothesis of the at once infinite (referring the meaning of number, under the guidance of our theoretical thinking, to the modal meaning of space). But this is exactly what Cantor did – he effectively used this anticipatory meaning of number in his description of a perfectly coherent set of real numbers (taken in their linear order), without realizing, however, that implicitly his use of the at once infinite still presupposes the primitive and irreducible spatial order of simultaneity. But given this disclosed approach, it is no longer possible to exclude the actually infinite dividedness of a line in terms of its endless divisibility. The notion of endless divisibility, employing the number-concept of the successive infinite, may be deepened with the foundational aid of the opened up numerical use of the number-idea of actual infinity.1 Consequently, the diverging approaches to continuity present in the thought of Aristotle and Cantor (in spite of their formal agreement concerning the two mentioned criteria for continuity) may be summarized as follows: Both characterize continuity, but each one chooses his own view-point – Aristotle's angle of approach is the spatial aspect (which does not necessarily need the at once infinite), whereas Cantor's perspective uses the numerical anticipation to space (an approach which does need the at once infinite)! The irreducibility of the spatial time-order of simultaneity to the numerical time-order of succession is ultimately dependent on the irreducibility of the modal meaning of space to that of number. From this it directly follows that the spatial whole-parts relation, determined by the spatial order of simultaneity, is also irreducible – explaining why the typical totality character of the continuum reveals an unavoidable circularity in the attempted purely arithmetical ‘definition’ of continuity. In other words, the modal meaning of space, qualified by the primitive meaning-nucleus of continuous extension (expressing itself at the law-side as a simultaneous order for extension and at the factual side as dimensionally determined extension – with or without a de1 Perhaps Hegel could have abused this perspective in the following way: a continuous whole is the “absolute” an sich; in its infinite divisibility, it reaches out to its opposite, the discrete nature of number; and finally, in the at once infinite as an anticipation to space the absolute idea returned to itself – the infinite whole, in itself divided in the sense of actual infinity. 31 Infinity and Continuity fined metric), not only implies that this meaning-nucleus of space is irreducible to number, but also that the spatial order of simultaneity at the law-side and the whole-parts relation at the factual side of the spatial aspect are ultimately irreducible. Although he did not pay attention to the law-side of the spatial aspect (obviously because he did not dispose of an articulated meaning-analysis of the structure of number and space), we have seen that Paul Bernays does appreciate the irreducibility of the spatial whole-parts relation (the totality feature of spatial continuity) (Bernays, 1976:74). The property of being a totality “undeniably belongs to the geometric idea of the continuum. And it is this characteristic which resists a complete arithmetization of the continuum” (“Und es ist auch dieser Charakter, der einer vollkommenen Arithmetisierung des Kontinuums entgegensteht – 1976:74). Laugwitz realized that the real numbers, in terms of Cantor's definition of a set, are still individually distinct and in this sense 'discrete'. According to him the set concept was designed in such a way that what is continuous withdraws itself from its grip, for according to Cantor a set concerns the uniting of well-distinguished entities, implying that the discrete still rules.1 Although this objection actually shows that Laugwitz did not understand the difference between the successive and the at once infinite properly, in its own way it could be seen as an objection to the arithmeticistic claims of modern mathematics. In this regard Laugwitz implicitly supports Bernays's deeply felt reaction against the mistaken and one-sided nature of modern arithmeticism, expressed in his words: The arithmetizing monism in mathematics is an arbitrary thesis. The claim that the field of investigation of mathematics purely emerges from the representation of number is not at all shown. Much rather, it is presumably the case that concepts such as a continuous curve and an area, and in particular the concepts used in topology, are not reducible to notions of number (Zahlvorstellungen) (1976:188). 13 Concluding remark Although continuity belongs to the core meaning of the spatial aspect, and in this role qualifies the retrocipatory analogies to number within the spatial aspect, the meaning of spatial continuity expresses itself in its coherence with the (foundational) meaning of number. The apparently contradicting positions of Aristotle and Cantor-Dedekind underscores this perspective, because 1 “Der Mengenbegriff ist von vornherein so angelegt worden, daß sich das Kontinuierliche seinem Zugriff entzieht, denn es soll sich nach Cantor bei einer Menge ja handeln um eine “Zusammenfassung wohlunterschiedener Dinge ... – das Diskrete herrscht” (Laugwitz, 1986:10). And on the next page we read: “So kommt man dazu, die Frage nach der Mächtigkeit der Menge der reellen Zahlen als ‘Kontinuumproblem’ zu bezeichnen. In dieser Auffassung wird der Unterschied zwischen Diskretem und Kontinuierlichem verwischt: Je zwei Teilpunkte sind wohl voneinander unterschieden, aber ihre Gesamtheit soll das Kontinuum repräsentieren; dieses würde also durch Diskretes dargestellt.” 32 D F M Strauss it highlights that the former explores a spatial point of view (merely requiring the successive infinite) whereas the latter approaches continuity from the perspective of the spatially disclosed and deepened meaning of number (which has to employ the at once infinite). At the same time this analysis enabled us to argue that modern mathematical set theory actually is a spatially deepened theory of number. Bernays correctly states (as quoted on page 21 above): The idea of the continuum is a geometrical idea which analysis expresses in terms of arithmetic (1976:74).1 Not realizing this spatially deepened character of set theory causes the vicious circle entailed in all attempts to argue for a complete reduction of space to number on the basis of a theory employing the numerical anticipation to space (by accepting the at once infinite). In opposition to the attempt of Greek mathematics to reduce number to space and the modern pursuit to arithmatize space, the third alternative presented here aims at acknowledging both the uniqueness and mutual coherence of number and space. 14 Literature Alexandroff, P.S. (1956): Einführung in die Mengenlehre und die Theorie der reellen Funktionen, Berlin. Bartle, G. (1964): The Elements of Real Analysis, New York. Bernays, P. (1976): Abhandlungen zur Philosophie der Mathematik, Darmstadt 1976. Beth, E.W. (1965): Mathematical Thought, Amsterdam. Böhme, G. (1966). Unendlichkeit und Kontinuität. In: Philosophia Naturalis, Volume 11, (pp.304-317). Bolzano, B. (1920): Paradoxien des Unendlichen (1851), Leipzig. Boyer, C.B.(1959): The History of the Calculus and its Conceptual Development, New York. Boyer, C.B.(1968): A history of Mathematics, New York: John Wiley & Sons. Cantor, G. (1962): Gesammelte Abhandlungen Mathematischen und Philosophischen Inhalts, Hildesheim: Oldenburg Verlag (1932). Cassirer, E (1910): Substanzbegriff und Funktionsbegriff, Berlin 1910, reprinted: Darmstadt 1969. Cassirer, E. (1953): Substance and Function, New York 1953 (first edition of the English translation of Substanzbegriff und Funktionsbegriff: 1923). Cassirer, E. (1957): Das Erkenntnisproblem in der Philosophie und Wissenschaft der neueren Zeit, Stuttgart: Kohlhammer Verlag. Dantzig, T. (1947): Number the Language of science. Dedekind, R. (1901): Essays on the Theory of Numbers, Chicago. Dedekind, R. (1887): Was sind und was sollen die Zahlen, Braunscweig (10th ed.) 1969. 1 We have mentioned that a topological description of continuity may abstract from any given metric. 33 Infinity and Continuity Dooyeweerd, H. (1996): A New Critique of Theoretical Thought, Series A (Volumes I-IV) of The Collected Works of Herman Dooyeweerd, General Editor D.F.M. Strauss, Lewiston: Mellen Press.. Dummett, M.A.E. (1978): Elements of Intuitionism, Oxford: Clarendon Press. Dummett, M.A.E. (1995): FREGE, Philosophy of Mathematics, Harvard University Press, Cambridge, Second Printing. Eley, L. (1970): Preface, in: Husserliana, Vol. XII. Felgner, U. (editor) (1979): Mengenlehre, Wissenschaftiche Buchgesellschaft, Darmstadt. Fischer, L. (1933): Die Grundlagen der Philosophie und der Mathematik, Leipzig. Fränkel, H. (1968): Zeno von Elea im Kampf gegen die Idee der Vielheit, in: Um die Begriffswelt der Vorsokratiker, Wege der Forschung, Band IX, ed. H-G Gadamer, Darmstadt. Fraenkel, A.A. 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(1930): Naturekennen und Logik, in: David Hilbert, Gesammelte Abhandlungen, Vol.3, Second Edition, Berlin: Springer. Husserl, E. (1970): Philosophie der Arithmetik (1891), Husserliana, The Hague: Martinus Nijhof, Vol. XII. Kant, I. (1787): Kritik der reinen Vernunft, (Hamburg: Felix Meiner Edition 1967), 1st edition 1781. Körner, S. (1972): Mathematik als die Wissenschaft formaler Systeme: Exposition, in: Meschkowski, 1972. Laugwitz, D.D. (1986). Zahlen und Kontinuum. Eine Einführung in die Infinitesimalmathematik. Mannheim: B.I.-Wissenschaftsverlag. Lorenzen, P. (1952): Ueber die Widerspruchfreiheit des Unendlichkeitsbegriffes, in: Studium Generale, Zeitschrift für die Einheit der Wissenschaft en im Zusammenhang ihrer Begriffsbildungen und Forschungsmethoden, Vol.10, Berlin. Lorenzen, P. (1972): Das Aktual-Unendliche in der Mathematik, in: Grundlagen der Mathematik, ed. H. Meschkowski, Darmstadt. 34 D F M Strauss Lorenzen, P. (1986). Die Theoriefähigkeit des Kontinuums. 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(1991): The Ontological Status of the principle of the excluded middle, in: Philosophia Mathematica II Vol.6, n.1 (pp.73-90). Vaihinger, H. (1922): Die Philosophie des Als Ob, 8th ed. Leibzig. Van Heijenoort, J. (1967): From Frege to Gödel – a source book in mathematical logic 1879-1931, Cambridge, Mass 1967. Von Fritz, K. (1945): The Discovery of Incommensurability by Hippasus of Metapontum, Annals of Mathematics, 46. Von Weizsäcker, C.F. (1993): Der Mensch in seiner Geschichte, München : DTV. Wang, Hao (1988): Reflections on Gödel, MIT Press, Cambridge, Massachusetts. Weyl, H. (1921): Ueber die neue Grundlagenkrise der Mathematik, Mathematische Zeitschrift, Band 10. 35 Infinity and Continuity Weyl, H (1926): Die heutige Erkenntnislage in der Mathematik, Erlangen. Weyl, H. (1931): Die Stufen des Unendlichen, Jena. Weyl, H. (1946): Mathematics and Logic, A brief survey serving as preface to a review of The Philosophy of Bertrand Russell, American Mathematical Monthly, Vol. 53. 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(1971): Zur Problematik der absoluten Überabzählbarkeit, in: Philosophia Naturalis, Band 13. 36 D F M Strauss Sketches Sketch 1 The structure of a modal aspect Time-order Law-side / Norm-side Meaning-nucleus qualifying all analogies ante- and/or retrocipations [analogies] Guarantees a) uniqueness b) irreducibility and c) indefinability of the nuclear meaning of each modal aspect subject-subject and subject-object relations Factual side Time-duration Sketch 2 The mutual coherence and irreducibility of discreteness and continuity Law-side Numerical time-order of succession The Successive Infinite (SI) Anticipation n atio cip tro Re An Nuclear meaning: discrete quantity Re tic ipa tio Spatial time-order of at once The At Once Infinite (AI) Nuclear meaning: n continuous extension tro N (atural numbers) cipa The at once infinite dividedness tio n Z (integers) of a line Spatial figures Anticipation Q (fractions) (the whole-parts relation) R (real numbers) successive infinite divisibility Factual side 37 38 Index of Subjects L A actual infinite 19-20, 22-24, 27-28, 30-31 actually infinite dividedness 31 analogical basic concepts 14 anticipation to a retrocipation 21-22 anticipations 17-18 arithmetical aspect 17-18 arithmeticism 32 arithmetization of continuity 13 at once infinite 24, 31-33 axiom of infinity 3 axiom of separation 4 axiomatization 4, 25 law-side 18, 21-22, 31-32 linear Cantorean continuum 26 logical addition 2, 5 logical synthesis 5 logicism 2, 6 M measure theory 11 membership relation 4 metaphors 14 modal analogies 14 modal aspects 5, 16, 21 multiplicity 1, 4-5, 7-9 D degenerate intervals 27 E elementary basic concepts 14, 18 endlessness 13-14, 30 energy quanta 15 entitary analogies 14 equi-numerosity 9 eternity 14 extended linear continuum 26 N natural numbers 7-8, 10, 28 Nominalism 16 non-contradiction 5 non-denumerability 20 numerical aspect 7, 18, 21-22, 31 O objective reality 16-17 ontological commitments 1 ordinal number 4, 8-9 P F factual extension 26 G geometrization 31 given at once 9, 19 gravitation 15 I incommensurability 1 individuals 3 induction 7-8, 13, 18, 21 infinite divisibility 31 infinite succession 23 infinite totality 16, 23 infinitely divisible 11, 13, 15, 25, 27 infinitesimals 23 infinity 2-3, 24, 31 intuition 1-2, 8, 21, 24 irreducibility 2, 10, 13, 19, 24, 31-32 platonism 16 potential infinite 19, 21, 27, 31 primitive symbols 4 primitive terms 14, 18 principle of identity 4-5 Q quantifiers 4, 7 quasi-spatial 22 R regressus in infinitum 1, 3 retrocipations 17-18 retrocipatory analogy 21 S semi-disclosed 21-22 semiperceptions 17 set theory 2, 4, 6-10, 20, 22, 25, 33 simultaneity 9, 18, 21, 23-24, 31 39 spatial subject 20, 25 spatial terms 18 static domain 12 V variables 6-7, 13 vicious circle 3, 24, 33 T time-order of simultaneity 18, 23, 31 time-order of succession 18, 21, 30-31 totality 9 W wholeness 1, 13-14, 18 whole-parts relation 11-13, 18, 30-32 Wirkungsquantum 15 40 Index of Persons A Alexandroff 29, 33 Aristotle 28, 30-32, 36 B Bartle 30, 33 Bernays 2, 13, 16, 18-19, 21, 23, 32-33 Beth 33 Böhme 30, 33 Bolzano 1, 12, 28-29, 33 Boyer 12, 33 C Cantor 1, 4, 6, 9-10, 19, 32-33, 35 Cassirer 15, 25, 33, 35 D Dantzig 10, 33 Dedekind 1-3, 7, 19, 29-30, 32-33 Descartes 15-16 Dooyeweerd 24, 33 Dummett 2, 10, 17, 23, 34 E Eley 20, 34 F Felgner 34-35 Fischer 22, 34 Fraenkel 1, 3-4, 6, 14, 16, 20, 34 Fränkel 11, 34 Frege 2-3, 5, 10, 17, 35 Freudenthal 7, 34 K Kant 5, 7, 17, 31, 34 Körner 20, 34 Kroneker 9 L Laugwitz 32, 34 Lorenzen 12, 19-20, 23, 34 M Maier 24, 34 Meschkowski 4, 29, 34-35 Monk 1, 35 Moore 14, 35 Myhill 3, 35 R Robinson 23, 35 Rucker 2, 14, 35 Russell 2-3, 6, 35 S Schilpp 35 Simplicius 11 Singh 3-4, 35 Skolem 4, 8-9, 35 Smart 8, 35 Stegmüller 16, 35 Strauss 5, 33 V Vaihinger 22-23, 35 Van Heijenoort 4, 34-35 Von Fritz 1, 35 Von Weizsäcker 28, 35 G Guthrie 12, 34 H Hasse 11, 34 Hegel 31 Helmholtz 9 Heyting 20, 34 Hilbert 2-3, 8, 15, 19, 26, 34 Hippasos of Metapont 1 Husserl 20, 34 W Wang 2, 16-17, 22, 35 Weierstrass 1, 6, 12, 15 Weyl 4, 8-10, 12-13, 24, 35 White 29, 36 Willard 29, 36 Wolff 20, 36 Z Zeno 18, 27-28, 34 Zermelo 4, 7, 9 41

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