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The Significance of a Non-Reductionist Ontology for the Discipline of Mathematics: A Historical and Systematic Analysis

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Abstract

A Christian approach to scholarship, directed by the central biblical motive of creation, fall and redemption and guided by the theoretical idea that God subjected all of creation to His Law-Word, delimiting and determining the cohering diversity we experience within reality, in principle safe-guards those in the grip of this ultimate commitment and theoretical orientation from absolutizing or deifying anything within creation. In this article my over-all approach is focused on the one-sided legacy of mathematics, starting with Pythagorean arithmeticism (“everything is number”), continuing with the geometrization of mathematics after the discovery of irrational numbers and once again, during the nineteenth century returning to an arithmeticistic position. The third option, never explored during the history of mathematics, guides our analysis: instead of reducing space to number or number to space it is argued that both the uniqueness of these two aspects and their mutual coherence ought to direct mathematics. The presence of different schools of thought is highlighted and then the argument proceeds by distinguishing numerical and spatial facts, while accounting for the strict correlation of operations on the law side of the numerical aspect and their correlated numerical subjects (numbers). Discussing the examples of 2 + 2 = 4 and the definition of a straight line as the shortest distance between two points provide the background for a brief sketch of the third alternative proposed (inter alia against the background of an assessment of infinity and continuity and the vicious circles present in contemporary mathematical arithmeticistic claims).

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Notes

  1. The focus of our considerations will be on the interconnections between space and number. We shall argue that there are structural features that are inherent in these two facets of reality prior to the actual definition of metrical spaces (in 1906 by Fréchet). Mac Lane accepts space as “something extended” and on the basis of the notion of “distance” defines a metric space (see Mac Lane 1986, pp. 16–17). It is clear that the notions of extension and distance precede the definition of a metrical space. An explanation of the mutual relation between discreteness and continuity within a topological context requires a different argument. A starting-point for such a discussion is found in White (1988, pp. 1–12).

  2. In connection with the history of the concept of matter we shall return to Greek philosophy.

  3. Already Kronecker, a prominent early intuitionist (and contemporary of Cantor) defended a Kantian view on the a priori nature of arithmetic while denying it in respect of geometry and mechanics (see Kronecker 1887, p. 265). Al great master of twentieth century intuitionist mathematics, Brouwer also holds: “However weak the position of intuitionism seemed to be … it has recovered by abandoning Kant's apriority of space but adhering the more resolutely to the apriority of time” (Brouwer 1964, p. 69). Gödel's remarks that there is a close relationship between the concept of a set explained in footnote 14 and the categories of pure understanding in Kant’s sense” (see Gödel 1964, p. 272). Finally, Hilbert refers to Kant’s understanding of reason ideas in order to justify his own justification of the mathematical usefulness of the actual infinite (see Hilbert 1925, p. 190.

  4. Just compare the remarks quoted above concerning different standpoints in mathematics.

  5. A field is defined as a set F such that for every pair of elements a, b the sum a + b and the product ab are still elements of F subject to the associative and commutative laws for addition and multiplication, and combined to the presence of a zero element and a unit (or identity) element (see Bartle 1964, p. 28; Berberian 1994, 1ff). This definition of a field is then expanded to that of an ordered field and it is finally connected to the idea of completeness. The existence of least upper bounds differentiate the real numbers from all other ordered fields” (Berberian 1994, pp. 11–12).

  6. This explanation, in terms of the strict correlation between operations at the law-side and numerical subjects at the factual side, is formally similar to the way in which Klein introduces negative numbers and fractions (by means of the reverse operations of addition and multiplication—see Klein 1932, 23ff & 29ff). Ebbinghaus et al. points out that in a paper on “Pure Number Theory” (Reine Zahlenlehre) Bolzano already developed a theory of rational numbers, “and in fact a theory of those sets of numbers that are closed with respect to the four elementary arithmetic operations” (Ebbinghaus et al. 1995, p. 22).

  7. In the course of our analysis it will become clear that the word “meaning” fulfils a specific systematic role. It brings to expression that the meaning of an aspect only reveals itself in its coherence with other aspects. Therefore a reference to the meaning of an aspect intends to capture this idea of uniqueness in coherence.

  8. “[Das] Problem von der Geraden als kürzester Verbindung zweier Punkte” (see Hilbert 1970, p. 302).

  9. “… der Unterschied zwischen Diskretem (algebraische Strukturen) und Kontinuierlichem (topologische Strukturen)” (Laugwitz 1986, p. 12).

  10. “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” (Bernays 1976, p. 81).

  11. Below we shall ‘liberate’ the idea of ‘discreteness’ from the arithmeticistic habit to distinguish between discrete, dense and continuous sets—in order to allow discreteness to play its role as meaning-kernel of the numerical aspect that qualifies all kinds of number, even when such kinds of number may imitate spatial features (such as wholeness, divisibility and continuity).

  12. We noted above that the problem concerning which one is more basic—number or space—cannot be solved by specifying a genus proximum—albeit that of Aristotle with his distinction between a discrete quantity and a continuous quantity or that of the structuralist Resnik with his distinction between discrete patterns and continuous patterns (cf. Aristotle: “Quantity is either discrete, or continuous”—Categ. 4 b 20; and Resnik 1997, 201ff, 224ff).

  13. “Liegt nicht der Grund der Arithmetik tiefer als der alles Erfahrungswissens, tiefer selbst als der der Geometrie?” (Frege 1884, p. 44).

  14. The arithmeticistic practice to distinguish between discrete, dense and continuous actually merely highlights disclosed structural moments through which different types of numbers, in anticipatory way, point at structural features of the spatial aspect (such as wholeness, infinite divisibility and continuity) without ever “escaping” from the qualifying role of discrete quantity. Even every real number remains distinct and unique. As Laugwitz states it: “From the outset the set concept is constructed in such a way that what is continuous escapes from its grip, for according to Cantor a set concerns the ‘bringing-together’ of clearly distinct things … the discrete rules” (Laugwitz 1986, p. 10). We shall return to this issue below.

  15. Already in 1910 Grelling recognized set theory as the foundation of mathematics as a whole: “Zuerst ausgebildet als Hilfsmittel der Untersuchung bei gewissen Fragen der Analysis, hat sich die under den Händen inhres Schöpfers Georg Cantor und sein Schüler zu einer selbständigen metahmatischen Disziplin entwickelt, die heute die Grundlage der gesamten Mathematik bildet.” [“In the first place developed as an auxiliary tool of the investigation of certain questions of analysis (set theory) in the hands of Cantor and his pupils (it was) developed into an independent mathematical discipline. Currently it constitutes the foundation of mathematics in its entirety” (Grelling 1910, p. 6).

  16. He also states: “For our human understanding the concept of number is more immediate than the representation of space” (Bernays 1976, p. 75).

  17. Whenever entities are involved in the figurative mode of speech such designations are considered to be metaphorical. But as soon as similarities and differences between modal functions (as they will be explained below) are captured, these purely aspectual interrelations represent a domain of analogies distinct from metaphors. When purely intermodal connections (analogies) are metaphorically explored, an element of the entitary dimension of reality will always be present (such as it is found in the metaphor of a person being “reminded” of an original domain).

  18. The term ‘natural’ intends to capture the realms (‘kingdoms’) found in reality—things, plants and animals and they are distinct from societal realities. The latter are lived out through the social activities of human beings and the latter are guided by the normative considerations of the logical and post-logical aspects. Although such normatively guided actions may take the lead of any normative aspect, human beings are not qualified by any one of them. For that reason humankind cannot be subsumed in any ‘kingdom’—the normative structure qualifies the temporal Gestalt of being human, but in itself it is not qualified by any aspect—the human being has an eternal destination.

  19. See also Brouwer 1924, p. 554. When a “species” π does not contain a continuum as part it is of dimension 0 in the Menger-Urysohn sense.

  20. 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.

  21. The classical example is Zeno’s attempt to define movement in static spatial terms.

  22. “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” (Bernays 1976, p. 81).

  23. “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.”.

  24. “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, p .167).

  25. “Die Idee des Kontinuums ist, jedenfalls ursprünglich, eine geometrische Idee, welche durch die Analysis in arithmetischer Sprache ausgedrükt wird” (Bernays 1976, p. 74).

  26. Compare the expressions infinitum successivum and infinitum simultaneum (Maier 1964, pp. 77–79).

  27. 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, 1997-I, pp. 98–99 (footnote 1) and 1997–II, p. 340 (footnote 1).

  28. “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, p. 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.”.

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    D. F. M. Strauss

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Strauss, D.F.M. The Significance of a Non-Reductionist Ontology for the Discipline of Mathematics: A Historical and Systematic Analysis. Axiomathes 20, 19–52 (2010). https://doi.org/10.1007/s10516-009-9080-5

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