The Significance of a Non-Reductionist Ontology for the Disciplines of Mathematics and Physics–an Historical and Systematic Analysis
The attempt to reduce what is truly unique to something else leads to the deification of something or some aspect within creation, normally accompanied by imperialistic “all”-claims such as, “everything is number,” “everything is matter,” “everything is feeling,” “everything is historical” or “everything is is interpretation.” The distortions thus created inevitably result in insoluble antinomies.
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 anything within creation.
1. The contemporary intellectual climate
In spite of the decline of positivism within the domain of the philosophy of science many scholars in the various academic disciplines (special sciences) still advocate its neutrality postulate. As examples of “exact sciences” mathematics and physics are normally lifted out. These two disciplines, according to the positivistic view, are objective and neutral – they rule out the possibility of any and all presuppositions exceeding the boundaries of these “exact” disciplines. Alternatively, insofar as historicism and its relativistic consequences gave rise to what is known as postmodernism, grand stories (“meta-narratives”) are questioned and truth uprooted – every person has her own “story” to tell.
Amidst all of this the leading philosophers of science during the 20th century increasingly acknowledged the inevitability of an ultimate commitment in scholarship as well as the presence of a (philosophical) theoretical view of reality underlying all academic endeavours. Some of them explicitly reject reductionism. Popper straightforwardly states: “As a philosophy, reductionism is a failure” (Popper, 1974:269). In order to capture problematic situations within the disciplines (and their logic) the term reductionism emerged by the middle of the 20th century. In 1953 Quine used it in his discussion of “The Verification Theory and Reductionism” (see Quine, 1953:37 ff.) and in the early seventies the work “Beyond Reductionism” appeared (see Koestler & Smythies, 1972). Smith (1994) considers the scientist-philosopher Michael Polanyi to be “perhaps the severest and most comprehensive critic of reductionism” because he “was a major scientist of this century and was drawn into philosophical debate primarily because of the threat to scientific freedom, political democracy, and to humane values that he saw in reductionism”. To this he adds the remark:
His works The Contempt of Freedom, The Logic of Liberty, Science Faith and Society, Personal Knowledge, and The Tacit Dimension have as a common theme the criticism of reductionism in all its scientific, cultural and moral forms.
The best way to challenge both positivism (objectivity and neutrality) and postmodernism (historicism and relativism) is to confront the supposedly “exact” sciences, mathematics and physics, with the implications of a non-reductionist Christian philosophy.
Practicing mathematicians, consciously or not, subscribe to some philosophy of mathematics (if unstudied, it is usually inconsistent) (Monk, 1970:707)
2. Are there different standpoints in mathematics?
Before we investigate relevant historical perspectives and systematic distinctions it is worth challenging this claim of Fern by quoting a number of statements:
The mathematician Kline writes:
The developments in the foundations of mathematics since 1900 are bewildering, and the present state of mathematics is anomalous and deployrable. The light of truth no longer illuminates the road to follow. In place of the unique, universally admired and universally accepted body of mathematics whose proofs, though sometimes requiring emendation, were regarded as the acme of sound reasoning, we now have conflicting approaches to mathematics (Kline, 1980:275-276)
In respect of formalization in intuitionistic mathematics the Dutch logician Beth remarks:
Meanwhile, for the intuitionists this formalization has in no way the meaning of a foundation as it does for the logicists. On the contrary, formalistic expression is in a position to produce no more than an inadequate picture of intuitionism (Beth, 1965:90).
The intuitionistic mathematician, Heyting, explains what is basic to intuitionism
every logical theorem … is but a mathematical theorem of extreme generality; that is to say, logic is a part of mathematics, and can by no means serve as a foundation for it (Heyting, 1971:6).
Of course intuitionism represents an authentic mathematical stance:
The intuitionists have created a whole new mathematics, including a theory of the continuum and a set theory. This mathematics employs concepts and makes distinctions not found in the classical mathematics (Kleene, 1952:52).
In fact intuitionism created an entirely new mathematics. Beth explains:
It is clear that intuitionistic mathematics is not merely that part of classical mathematics which would remain if one removed certain methods not acceptable to the intuitionists. On the contrary, intuitionistic mathematics replaces those methods by other ones that lead to results which find no counterpart in classical mathematics (Beth, 1965:89).
Perhaps the most perplexing observation comes from Stegmüller:
The special character of intuitionistic mathematics is expressed in a series of theorems that contradict the classical results. For instance, while in classical mathematics only a small part of the real functions are uniformly continuous, in intuitionistic mathematics the principle holds that any function that is definable at all is uniformly continuous” (Stegmüller, 1970:331).
3. Two apparently simple questions with
(i) Is 2 + 2 = 4? and (ii) is a straight line the shortest distance between two points?
We may relate question (i) to the idea that mathematics is objective and neutral – as asserted by Fern:
Mathematical calculations are paradigmatic instances of universally accessible, rationally compelling argument. Anyone who fails to see “two plus two equals four” denies the Pythagorean Theorem, or dismisses as nonsense the esoterics of infinitesimal calculus forfeits the crown of rationality (Fern, 2002:96-97).
The statement that “a straight is line the shortest distance between two points” indeed seems to be as self-evident as the statement that “2 + 2 = 4”. In an earlier phase of his development Bertrand Russell ‘corrected’ this definition: “A straight line, then, is not the shortest distance, but is simply the distance between two points” (Russell, 1897:18). The three key terms in this statement concern spatial configurations, namely the terms ‘line’, ‘point’ and ‘shortest’. Yet the crucial element maintained in Russell’s improved definition echoes something of our awareness of numerical relations: distance.1 If this is indeed the case it may turn out that an analysis of this statement will at once get entangled in the consideration of arithmetical and spatial issues, which means that it cannot be analyzed purely in spatial (or geometrical) terms.
4. Historical detour
Early Greek mathematics followed the arithmeticistic approach of the Pythagorean school with its claim that “everything is number.” Although the Pythagoreans believed that numerical relationships ordered the cosmos, they discovered that geometrical figures and lines can be construed that cannot be expressed by the relation between integers. The discovery of incommensurability by Hippasus of Metapont (450 B.C.) therefore caused a crisis since within the assumed form-giving function of number the formless (infinite) was revealed. Laugwitz remarks: “Every numerical relationship allows for a geometric representation, but not every line-relationship can be expressed numerically. This established the primacy of geometry over arithmetic and as a result the Books of Euclid treat the theory of numbers as a part of geometry” (Laugwitz, 1986:9).2 This geometrization of mathematics inspired a space metaphysics lasting at least until Descartes and Kant. During the nineteenth century, however, Cauchy, Weierstrass, Dedekind and Cantor once again pursued the path of an arithmeticistic approach. Of particular significance in this regard is set theory as it was developed by Cantor (including his theory of transfinite arithmetic).
When Russell and Zermelo independently discovered the fundamental inconsistency of Cantor’s set theory in 1900 and 1901, mathematics gave birth to three schools of thought, namely the logicist school (Russell, Gödel), the intuitionist school (Poincaré, Brouwer, Heyting, Weyl and Dummett), as well as the axiomatic formalist school (guided by the foremost mathematician of the twentieth century, David Hilbert and still largely dominating the scene of contemporary mathematics).
In 1781, the first edition of his influential work, “Critique of Pure Reason” (CPR) appeared. What is remarkable is that the main systematic subdivisions of this work provide the springboard for the three mentioned diverging trends in 20th century mathematics – intuitionism (exploring the “transcendental aesthetic” of the CPR), logicism (oriented to the “transcendental analytic”) and axiomatic formalism (affirming the thrust of the “transcendental dialectic”). We already noted that the fundamental differences within the discipline of mathematics caused a situation where what is true within intuitionistic mathematics may be false within formalism, while what is mathematically accepted by formalism, such as Cantor’s theory of transfinite numbers, is rejected as a phantasm by intuitionism (see Heyting, 1949:4) and as non-existent.3
In 1900 the French mathematician, Poincaré, made the proud claim that mathematics has reached absolute rigour. In a standard work on the foundations of set theory, however, we read: “ironically enough, at the very same time that Poincaré made his proud claim, it has already turned out that the theory of the infinite systems of integers – nothing else but part of set theory – was very far from having obtained absolute security of foundations. More than the mere appearance of antinomies in the basis of set theory, and thereby of analysis, it is the fact that the various attempts to overcome these antinomies, …, revealed a far-going and surprising divergence of opinions and conceptions on the most fundamental mathematical notions, such as set and number themselves, which induces us to speak of the third foundational crisis that mathematics is still undergoing” (Fraenkel, A. et al, 1973:14).
In this context the history of Gotllob Frege is perhaps the most striking. In 1884 he published a work on the foundations of arithmetic. After his first Volume on the basic laws of arithmetic appeared in 1893 Russell’s discovery (in 1900) of the antinomous character of Cantor’s set theory for some time delayed the publication of the second Volume in 1903 – where he had to concede in the first sentence of the appendix that one of the corner stones of his approach had been shaken. Russell considered the set C with sets as elements, namely all those sets A that do not contain themselves as an element. It turned out that if C is an element of itself it must conform to the condition for being an element, which stipulates that it cannot be an element of itself. Conversely, if C is not an element of itself, it obeys the condition for being an element of itself.
Close to the end of his life, in 1924/25, Frege not only reverted to a geometrical source of knowledge, but also explicitly rejected his initial logicist position. In a sense he completed the circle – analogous to what happened in Greek mathematics after the discovery of irrational numbers. In the case of Greek mathematics this discovery prompted the geometrization of their mathematics, and in the case of Frege the discovery of the untenability of his “Grundlagen” also inspired him to hold that mathematics as a whole actually is geometry:
So an a priori mode of cognition must be involved here. But this cognition does not have to flow from purely logical principles, as I originally assumed. There is the further possibility that it has a geometrical source. … The more I have thought the matter over, the more convinced I have become that arithmetic and geometry have developed on the same basis – a geometrical one in fact – so that mathematics in its entirety is really geometry (Frege, 1979: 277).
What is therefore the upshot of the history of mathematics? It emerged under the spell of Pythagorean arithmeticism (“everything is number”), then, owing to the discovery of irrational numbers (incommensurability) it experienced a fundamental geomatrization and during the 19th century it once again explored the avenue of arithmeticism, thus closing the circle of arithmeticism. Finally, in the thought of Frege also the circle of space was closed, because he once more thought that all of mathematics fundamentally is geometry.
5. Starting-points for a third alternative?
From the perspective of a non-reductionist ontology there is an obvious alternative never pursued throughout the history of mathematics:
Accept the uniqueness and irreducibility of number and space as well as their mutual interconnectedness and coherence.
In order to highlight this alternative way we may use the above-mentioned argument regarding 2 + 2 = 4 as our angle of approach. This will introduce considerations stemming both from the domains of number and space, particularly through the introduction of another ‘sum’ – one in which we may suggest that “2+2” is not equal to 4 but equals√8.
6. Numerical and spatial addition in the context
of the law-subject distinction
The conviction that mathematics is objective and neutral may here be defended by reinforcing the original claim (namely that 2+2=4), i.e. by referring to the addition of 2 fingers and another two 2 fingers, which indeed adds up to 4 fingers. Apparently this specified addition conclusively confirms the soundness of the initial statement that 2+2 is equal to 4. Unfortunately the issue is more complicated than it may seem at first sight, for the alternative assertion, namely that 2+2=√8 implicitly changed the context of addition. When a person walks 2 miles north and afterwards 2 miles east, then that person will be √8 miles away from the initial point of departure. This context concerns an instance of spatial addition that is mathematically treated in vector analysis, where a vector possesses both distance (magnitude) and direction.4 One can capture this altered context by underscoring the numerals involved in order to specify the fact that we are dealing with vectors: 2+2=√8. The upshot is that we now clearly have two different kinds of facts related to addition at hand: a numerical fact (designated as 2+2=4) and a geometrical fact (in the “right-angle-case” designated as 2+2=√8). In order to capture the specifications of this example one may construct the following figure:
These facts are not unqualified – that is to say, they are distinct because they are differently qualified or structured, respectively as numerical and as spatial facts. They are therefore not simply ‘facts’ in themselves. In their factuality they are delimited by alternative order-determinations. The operation of numerical addition displays an order-determination different from the operation of spatial addition, as is clearly manifested in the alternative sums: 4 and √8. In our example the underlying “order diversity” therefore makes possible the indicated distinction between numerical and spatial facts.
But there is something else present in this distinction between these two kinds of facts, namely the reference to the operation of addition. Modern mathematical set theory normally first of all approaches this domain in terms of the algebraic structure of fields – where the (binary) operations called addition (+) and multiplication (.) meet the field axioms (specified as laws).5
Let us give one step back and initially extract from this mathematical practice merely the operations (laws) of addition and multiplication. The fact that addition and multiplication within a specific system of numbers (such as the system of natural numbers) yield numbers still belonging to the initial set is also mathematically articulated by saying that the system of numbers under consideration is closed under the operations (laws) of addition and multiplication. Since ancient Greek philosophy it was understood that conditions (laws) and whatever meets these conditions are both distinct and strictly correlated. The most basic instance of such a strict correlation between (arithmetical) laws and arithmetical subjects (numbers) conditioned by these laws is found in the system of natural numbers. It is immediately evident that the addition and multiplication of any two natural numbers once more yield natural numbers (s = system; t = set):
system of na- operations / laws: (+,×)
tural numbers N s numerical subjects: Nt = (1, 2, 3, …)
The designation ‘system’ therefore comprises both arithmetical laws and arithmetical subjects – in the sense that the laws (operations) not only determine the behavior of the subjects but also delimit them. What has been explained above therefore means that the system of natural numbers finds its determination and delimitation in the operations of addition and multiplication that are closed over the set of natural numbers – in the sense that adding or multiplying any two natural numbers will always yield another natural number. The ultimate presupposition of these operations is found in the numerical order of succession. The Peano axioms (for the positive integers) yield a mathematical articulation of this primitive arithmetical order of succession. The correlation of the operations of addition and multiplication and their delimiting and determining role in respect of numerical subjects are consistent with Peano’s axioms because they are entailed in the complete ordered field of real numbers (see Berberian, 1994:230).
Introducing further arithmetical laws or operations will invariably call for additional (correlated) numbers that are factually subjected to the determining and delimiting arithmetical laws. For example, if the operation of subtraction is added to those of addition and multiplication, the correlating set of integers (Zt) is constituted – and considered in their correlation this yields the system of integers (see Ebbinghaus, et.al, 1995:19).
system of – operations / laws: (+,×, –)
integers Is numerical subjects: It = (0, +1, -1, +2, -2, …)
Likewise, extending the arithmetical operations by introducing division the correlating set of fractions is needed within the system of rational numbers.
system of ratio- operations / laws: (+,×, –, : )
nal numbers Qs numerical subjects: Qt = (a/b; a,b є Zt / b≠0)
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, 1932:23 ff. & 29 ff.). 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, 1995:22).
Against this background it is clear that the systematic arithmetical statement 2+2=4 does not designate a “brute fact” (a fact “in itself,” “an sich”), since the factual relation specified for numerical subjects (selected from the set of natural numbers) that are involved in it, exhibits the measure of the numerical law of addition. One can also say that this statement conforms to the determining and delimiting effect of the arithmetical law of addition. Consequently, the statement that 2+2 is equal to 4 concerns a law-conformative (arithmetical) state of affairs – it displays a specific lawfulness or orderliness for it meets the conditions set by the presupposed arithmetical order.
If there are multiple laws known to be arithmetical laws then one may speak of a unique sphere of arithmetical laws strictly correlated with diverse arithmetical subjects (sets of numbers) subjected to these laws. Another way to capture this situation is to speak of a numerical sphere in which arithmetical laws are strictly correlated with arithmetical subjects (numbers); in other words within this numerical domain a distinction is made between its law-side (order side) and its factual side. Myhill, who appreciates Brouwer as the originator of “constructive mathematics,” introduces the notion of a ‘rule’ (the equivalent of what we have designnated as “law-side”) as “a primitive one in constructive mathematics”; “We therefore take the notion of a rule as an undefined one” (Myhill, 1972:748). (Myhill received his Harvard Ph.D. under W.V. Quine). In his encompassing introduction to set theory (the third impression), Adolf Fraenkel refers to the peculiar constructive definition of a set which accepts as a foundation the concept of law and the concept of natural number as intuitively given (Fraenkel, 1928:237).6
The geometrical sum – 2+2=√8 belongs to a different domain, to a different sphere of laws, one where it is also possible to distinguish between a law-side (order side) and a factual side. The sphere of spatial laws differs from the sphere of numerical laws – in an exemplary way expressed in the difference between 2+2=4 and 2+2=√8.
Remark: At this stage it should be mentioned that the aim of our analyses is not in the first place directed at interconnections between different mathematical sub-disciplines. The goal is to show that number and space are not only unique and irreducible aspects of (ontic) reality, but also to argue that they mutually cohere in many ways (eventually highlighted with reference to what will be designated as analogies on the law-side and on the factual side of these aspects). Whenever interconnections between mathematical sub-disciplines are highlighted the aim is to demonstrate the underlying ontic interconnections between the aspects of number and space.
7. Distance: highlighting the mutual coherence
between number and space
We may now return to the mentioned key element in the modified definition given by Russell, distance: a line “is simply the distance between two points.” The after-effect of the Greek geometrization of mathematics is seen in the long-standing and persistent use of the term ‘Größe’ (‘magnitude’ – for numbers) up to 19th century mathematicians – such as Bolzano and Cantor (in spite of their ‘arithmetizing’ intentions their designation of numbers still used the gateway of the spatial aspect). Greek mathematics already indirectly wrestled with spatial magnitudes – such as lengths, surfaces and volumes – although the ratios contemplated by them were treated in non-numerical contexts. By comparing spatial figures (such as line-segments, surfaces and solids) Euclidean geometry used ratios of magnitudes within a non-numerical context (sometimes a physical one) in their measurements. Naturally the Greeks were fully aware of specific numerical properties of spatial figures, because otherwise they would not have had a concept of a triangle, i.e. of a figure with three sides and three angles.
In itself this already shows that spatial figures (such as triangles) reveal an unbreakable coherence with the meaning of number. Of course it should be remembered that the overemphasis of number as a mode of explanation caused the Pythagoreans to see spatial figures as numbers. Kurt von Fritz remarks: “Likewise, so they said, ‘are’ the geometrical figures in reality the numbers or bundles of numbers that constitute the length relationships of their sides; through them their form is determined and through them they can therefore be expressed” (Von Fritz, 1965:287). It was only through the analytical geometry of Descartes and Fermat that numerical magnitudes were eventually contemplated – assigned to line segments, surfaces and solids. Savage & Ehrlich remarks: “Euclidean geometry compares lengths, areas, and regions by comparing physical, non-numerical ratios of these magnitudes and in effect uses such ratios in the place of our numbers” (see Savage & Ehrlich, 1992:1 ff.). However, the lacking understanding of the interconnections between number and space caused the mistaken identification of a line with its length (distance).
The first observation to be made in this connection is to establish that the notion of a ‘line’ as the ‘distance’ between two ‘points’ concerns spatial realities. A line is a spatial subject (configuration), not an arithmetical one. Yet the crucial question is: how can one designate the ‘distance’ between two points? The answer is: by specifying a number (for example by saying it is 3 inches long). The problem with this answer is that something spatial, namely a ‘line’, is now apparently equated with something numerical, namely ‘distance’! In passing we note that the term ‘distance’ in yet a different way evinces an intrinsic connection with the meaning of number because a line is supposed to be the ‘distance’ between two points. Multiplicity (‘two’) is numerical; yet a multiplicity of points is spatial. Furthermore, the term ‘inch’ here has the function of the unit of measurement, i.e. the unit length. Therefore this unit is on a par with the notion of distance, because the number 1 and the number 3 respectively represent these two lengths. Does this mean that the domains of space and number are coinciding? If it is the case, then a question of priority arises (which embraces which one): is space numerical (then a ‘line’ is identical to ‘distance’, i.e. to number), or is number after all spatial in nature (then number, i.e. ‘distance’ is identical to space, i.e. to a ‘line’)? As we noted this concise dilemma reflects the basic contours of the history of mathematics as a discipline. After the initial Pythagorean claim that everything is number the discovery of irrational numbers turned mathematics into geometry. Then, during the 19th century arithmeticism once more gained the upper hand, although Frege close to the end of his life, reverted once more to the view that mathematics essentially is geometry. The situation is further complicated by the fact that the number specified (such as ‘3’) does not stand on its own, i.e., it appears within a non-numerical context – one in which the general issue of magnitude prevails, with length as a one-dimensional magnitude. And to add insult to injury, we now suddenly have to account for another spatial notion: dimensionality! New problematic questions are now generated, for in our example of “3 inches” – related to the extension of a line – the reference to length brought with it the (spatial) perspective of one dimension (length specifies magnitude in the sense of one dimensional extension). On the one hand this points at extension which, presumably, essentially belongs to our awareness of space, while at the same time, just as in the case of the term ‘distance’, it reveals a connection with number as well, for one can speak about 1-dimensional extension (magnitude; i.e. of length), 2-dimensional extension (magnitude; i.e. of area), 3-dimensional extension (magnitude; i.e. of volume), and so on. Even if priority is given to the spatial context by admitting that the distinction between different dimensions is indeed something spatial, no one can deny that in some or other way number here plays a foundational role, for without number the given specification (regarding 1, 2, or 3 dimensions) is unthinkable.
Clearly, the term ‘distance’ is embedded within the domain of space and it also evinces a strict correlation between an order of extension (the law-side of this domain – i.e. dimensionality) and factually extended spatial subjects – spatial figures (such as 1-dimensional ones, i.e. lines), 2-dimensional ones, i.e. areas) and 3-dimensional ones (i.e. volumes).
The complexities generated by considering an order of extension correlated with factually extended spatial subjects (spatial figures) add weight to the suggestion that although something like a line has a spatial nature, its extension reveals that its true spatial meaning depends upon the meaning of number. The reason for this acknowledgement is found in the intrinsic role of numerical terms that are ‘coloured’ by space, such as distance and dimension. Within a numerical context, such as what is mathematically known as “real analysis,” one can easily dispense with the concept of distance. But textbooks on real analysis sometimes still acknowledge that the geometric meaning of the term ‘distance’ may be useful, for “instead of saying that |a – b| is ‘small’ we have the option of saying that a is ‘near’ b; instead of saying that ‘|a – b| becomes arbitrarily small’ we can say that ‘a approaches b’, etc.” (Berberian, 1994:31).
8. Back to space
Two years after Russell gave his mentioned modified definition of a line as the distance between two points, the German mathematician, David Hilbert, published his axiomatic foundation 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.” Suddenly the term ‘distance’ disappeared. The next year, when Hilbert attended the second international mathematical conference in Paris, he presented his famous 23 mathematical problems that co-directed the development of mathematics during the 20th century in a significant way – and in problem 4 he provides a formulation that opens up a new perspective on this issue, for instead of speaking of the distance between two points he talks of a straight line as the (shortest) connection of two points.7 This choice of words completely avoids the traditional view, even found in the work of a contemporary mathematician like Mac Lane who still believes that the “straight line is the shortest distance between two points” (Mac Lane, 1986:17).
Hilbert’s German term ‘Verbindung’ (‘connection’) does not define a line since it presupposes the meaning of continuous extension. Every part of a continuous line coheres with every adjacent part in the sense of being connected to it. Although it is tautological to say that the parts of a continuous line are fitted into a gapless coherence, it says nothing more than to affirm that the parts are connected. In this sense the connection of two distinct spatial points also highlights the presence of (continuous) spatial extension between the points that are connected to each other. In other words, Hilbert’s formulation suggests that two points cannot be connected by a third point, but only by means of a line, i.e. through continuous spatial extension.
Combined with the primitive terms employed in his axiomatic foundation of geometry (‘line’, ‘lies on’ and ‘point’) the term ‘connection’ no longer equates a line with its distance. Once ‘liberated’ from this problematic bondage, alternative options emerge in order to account for the meaning of the term ‘distance’. If distance is the 1-dimensional measure of factual (continuous spatial) extension, then one can do two things at once:
(i) acknowledge the spatial context of this measure (1-dimensional magnitude) and
(ii) account for the reference to number that is evident both in the ‘1’ of 1-dimensional extension and in the (numerically specified) length evident in ‘distance’ as a specified (factual) spatial magnitude.
The core meaning of space, related to the awareness of extension and dimensionality, now acquires a new appreciation, further supported by the undefined nature of the term ‘line’ in Hilbert’s 1899 work. The message is clear: if the core meaning of space (extension) is indefinable and primitive, then it is impossible to attempt to define a line by using a term revealing a reference to what is not original within space, namely the number (!) employed in the specification of the ‘distance’ between two points. On the one hand, distance as the measure of extension of a (straight) line depends upon and presupposes the existence of the line in its primitive 1-dimensional extension and can therefore never serve as a definition of it, and on the other hand it reveals a connection with the meaning of number. Therefore the ‘definition’ of a (straight) line as “the distance between two points” (Russell, Mac Lane) presupposes what it wants to define and consequently begs the question.
9. What is presupposed in space?
In our discussion of the question whether or not the domains of space and number are coinciding we have started by analyzing some consequences of the option that they do coincide. Aristotle already explored this possibility, but without success, because he employed the biological method of concept formation (of a genus proximum and differentia specifica) in a context where it does not fit. As genus his category of “quantity” is then differentiatedinto a discrete quantity and a continuous quantity: “Quantity is either discrete, or continuous” (Categoriae, 4 b 20). “Number, … is a discrete quantity” (Categoriae, 4 b 31). The parts of a discrete quantity have no common limit, while it is possible in the case of a line (as a continuous quantity) to find a common limit to its parts time and again (Categoriae, 4b 25ff., 5 a 1ff.). In this account the aspects of number and space are brought under one umbrella and this approach precludes an insight into the uniqueness and irreducibility of number and space.
Rejecting Aristotle’s approach calls for an acknowledgement of the fact that every specification of spatial configurations is unavoidably connected with terms reflecting in some or other way the coherence of space with the meaning of number (magnitudes and the number of dimensions). This outcome opens the way to the alternative option: of investigating the consequences of the assumption that although space and number are unique and distinct they still unbreakably cohere. The new question to be analyzed is then.
10. What is the interrelation between space and number?
If the measure of the factual (one dimensional) extension of a straight line could be specified by its distance, then the distance of a line not only presupposes its spatial extension since it also presupposes the intrinsic interconnection between the meaning of space and the meaning of number. Various mathematicians had an appreciation of this state of affairs.
But let us consider further options. The mere possibility to juxtapose two distinct ‘facts’, such as the statements that 2+2=4 and 2+2=√8, points in the direction of acknowledging two unique domains – each with its own sphere of laws and correlated subjects. Laugwitz refers to the approach of Bourbaki according to which there is a difference between what is discrete (algebraic structures) and what is continuous (toplogical structures).8
Of course modern mathematicians are inclined to give preference to the meaning of number and infinity. Tait remarks: “Surely the most important philosophical problem of Frege’s time and ours, and one certainly connected with the investigation of the concept of number, is the clarification of the infinite, initiated by Bolzano and Cantor and seriously misunderstood by Frege” (Tait, 2005:213). What therefore needs to be clarified is summarized in the following two issues:
(i) which one of these two domains (‘poles’) is more fundamental, in the sense of foundational, to the other?and
(ii) how should one account for the interconnections (interrelations) between these two domains (‘poles’)?
10.1 Which region is more basic?
Let us start with the approach of Bernays where he considers the way in which one can distinguish between our arithmetical and geometrical intuition. He rejects the widespread view that this distinction concerns time and space, for according to him the proper distinction needed is that between the discrete and the continuous.9 Rucker also states: “The discrete and continuous represent fundamentally different aspects of the mathematical universe” (Rucker, 1982:243). Fraenkel et.al even consider the relation between discreteness and continuity to be the central problem of the foundation of mathematics: “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). But then the question recurs: what is the relationship between the ‘discrete’ and ‘continuous’?10 In terms of the distinction between the domain of number and that of space the term “pattern” in the first place derives its meaning from spatial configurations or patterns. Only afterwards can one stretch this term – metaphorically or otherwise – in order to account for quantitative relations as well.
Whatever the case may be, speaking of a “discrete patterns” just as little bridges the gap between discreteness and continuity than referring to the “domain of number” does it (where the term “domain” is also derived from the meaning of space). The issue at stake in this connection is one falling outside the scope of this article for it concerns what should be treated in an analysis of the elementary and compound basic concepts of a scholarly discipline (such as mathematics).
Fraenkel et.al. even speak of a ‘gap’ in this regard and add that it has remained an “eternal spot of resistance and at the same time of overwhelming scientific importance in mathematics, philosophy, and even physics” (Fraenkel et.al., 1973:213). These authors furthermore point out that it is not obvious which one of these two regions – “so heterogeneous in their structures and in the appropriate methods of exploring” – should be taken as starting-point. Whereas the “discrete admits an easier access to logical analysis” (explaining according to them why “the tendency of arithmetization, already underlying Zenon’s paradoxes may be perceived in [the] axiomatics of set theory”), the converse direction is also conceivable, “for intuition seems to comprehend the continuum at once,” and “mainly for this reason Greek mathematics and philosophy were inclined to consider continuity to be the simpler concept” (Fraenkel et.al., 1973: 213).
Of course the modern tendency towards an arithmetized approach (particularly since the beginning of the 19th century) chose the alternative option by contemplating the primary role of number. Although Frege – as mentioned above – by the end of his life equated mathematics with geometry (consistent with the just mentioned position of Greek mathematics), his initial inclination certainly was to opt for the foundational position of number. Already in 1884 he asked if it is not the case that the basis of arithmetic is deeper than all our experiential knowledge and even deeper than that of geometry?11
From our discussion of the difference between an arithmetical and a spatial sum and in particular from our remarks about the term ‘distance’ it is possible to derive an alternative view on the order relation between the regions of discreteness and of continuity. Suppose we consider the idea that discreteness constitutes the core meaning of the domain of number and that continuous extension highlights the core meaning of space. Then these core meanings guarantee the distinctness or uniqueness of each domain. The domain of number, with its sphere of arithmetical laws and numerical subjects, is then seen as being stamped, characterized or qualified by this core meaning of discreteness. Likewise the domain of space, with its sphere of spatial laws and spatial subjects, is then viewed as being qualified by the core meaning of continuous extension.
But we have seen that a basic spatial subject, such as a (straight) line, cannot be understood without some or other reference to the meaning of number, for observing the measure of the line’s extension requires the notion of ‘distance’ that involves number. Furthermore, since a line a spatial figure is extended in 1-dimension, it clearly only has a determinate meaning in subjection to the first order of spatial extension (namely one dimension). We have argued that in both domains (number and space) there is a strict correlation between the law-side and the factual side. In the case of space it is therefore possible to discern a reference to number both at the law-side and the factual side. Speaking of one or more than one dimensions presupposes the meaning of number on the law-side and this mode of speech at once specifies the meaning of the one dimensional extension, i.e. magnitude, of something like a line where the meaning of the number employed in the designation of the length of the line presupposes the original (primitive) meaning of number. The domain of number therefore appears to be more basic because an analysis of the meaning of space invariably calls upon foundational arithmetical features.
This conclusion is further supported by the approach of Maddy where she argues that most recent textbooks “view of set theory as a foundation of mathematics” (Maddy, 1997:22; see also Felgner, 1979:3) and that a set theoretic foundation can “isolate the mathematiccally relevant features of a mathematical object” in order to find a “set theoretic surrogate” for those features (Maddy, 1997:27, 34).12 Bernays categorically asserts that “the representtation of number is more elementary than geometrical representations” (Bernays, 1976:69).13 In general one may view the arithmeticism of Weierstrass, Dedekind and Cantor as an (over-estimated) acknowledgement of the foundational position of the domain of number.
We may summarize the thrust of our preceding argument in favour of the foundational position of number in respect of space as follows:
The core meaning of space – namely continuous extension – entails factual extension in one or more dimensions; and specifying “one or more” dimensions presupposes the natural numbers 1, 2, 3, … At the same time the 1-dimensional extension of a straight line comes to expression in the measure of this extension, designated as its length – and the latter (its length) is specified by using a number – showing that the meaning of spatial extension intrinsically presupposes (“builds upon”) the meaning of number.
10.2 Interconnections between functional domains
A metaphorical way to capture this state of affairs is to use an image from human memory by saying that within the meaning of space (both at the law-side and the factual side), we discover configurations reminding us of the core meaning of number. A key element in all metaphorical descriptions is found in the connection between similarities and differences. Whenever what is different is shown in what is similar one may speak of analogies. Yet we want to broaden the scope of an analogy in order to include more than what is normally accounted for in a theory of metaphor. Our first designation already achieves this goal, for whenever differences between entities and properties bring to expression what is similar between those entities or properties, we meet instances of an analogy.14 Implicit in the nature of an analogy is the distinction between something original and something else which ‘reminds’ one of what is originally given but that is now encountered in a non-original context, i.e. within an analogical setting. This is exactly what we have noticed in the terms ‘distance’ and ‘dimension’ – for in both cases the use of numerical terms in a spatial context remind us of their original (non-spatial) quantitative meaning. In terms of the idea of an analogy one can say that there is an analogy of number on the law-side of the spatial aspect (one, two, three or more dimensions) and that there is an analogy of number at the factual side of the spatial aspect (magnitude – as the correlate of different orders of extension: in one dimension magnitude appears as length, in two dimensions it appears as area, in three as volume). An account of the basic position of number can now be articulated in terms of the idea of analogies, for since basic numerical analogies are presupposed within the domain of space, the original meaning of number is indeed foundational to the meaning of space.
In the previous paragraph we introduced a new word in order to refer to the domains of number and space, namely the term ‘aspect’. The underlying hypothesis of this usage is found in the theory that the various aspects of reality belong to a distinct dimension which is fundamentally different from the concrete what-ness of (natural and social) entities (such as things, plants, animals, artifacts, societal collectivities and human beings).
These concrete entities (and the processes in which they are involved) all function within the different aspects of reality. Questions about the way in which entities exist concern their how-ness, their mode of being. Aspects in this sense are therefore (ontic) modes of being. That my chair is one and has four legs reveal its function within the quantitative mode of reality; that it has a certain shape and size highlights its spatial function; that one can identify and distinguish it highlights its logical-analytical function, that it has a certain economic value demonstrates its function within the economic mode of reality, that it is beautiful or ugly brings to expression its aesthetic function, and so on.
This dimension of functions or aspects can also be designated as that of modalities or modal functions. What has already been said about the domains of number and space concern properties that may serve to define the nature of an aspect. Of course any description of modal aspects inevitably employs metaphors (involving entitary analogies). Fore example, one may say that aspects are ‘points of entry’ to reality, that they provide an ‘angle of approach’ to reality, and so on. Conversely, the modal aspects provide access to the dimension of entities – they may serve as modes of explanation of concrete reality.
Every aspect contains a sphere of modal (functional) laws (at its law-side); a factual side (subjected to modal laws); and a core meaning qualifying, characterizing or stamping all the structural moments discernable within an aspect (in particular also the analogical elements pointing to the meaning of other modal functions of reality). This core meaning or meaning-nucleus guarantees the uniqueness and irreducibility of every aspect and it underlies the inevitable use of primitive (= indefinable) terms by those disciplines that explore a specific modal aspect as angle of approach to reality. Some of these structural features of an aspect are captured in the sketch on the next page.
11 The irreducible meaning of space underlying
Hilbert’s primitive terms
Within the arithmetical aspect the factual relation between numbers is constituted as subject-subject relations – as it is present in the addition of numbers, the multiplication of numbers or establishing the numerical difference between numbers (subtraction). However, at the factual side of the spatial aspect there are not only subject-subject relations (such as intersecting lines), for there are also subject-object relations present, mainly expressed in the idea of a boundary.
Already in his abstraction theory Aristotle employed the notion of a boundary (or limit) – which is intuitively immediately associated with spatial notions (Aristotle used the term eschaton). By the 13th century AD Thomas Aquinas accounts for a 1-dimensional line by means of a descending series of abstractions. In contradistinction to natural bodies, all mathematical figures are infinitely divisible. The Aristotelian legacy is clearly seen in his definition of a point as the principium of a line (cf. Summa Theologica, I,II,2), which indicates the fact that a determinate line-stretch has points at its extremities (“cuius extremitates sunt duo puncta” – Summa Theologica, I,85,8). This legacy returns in a somewhat more general form in the 18th century (the era of the Enlightenment). Kant remarks:
Area is the boundary of material space, although it is itself a space, a line is a space which is the boundary of an area, a point is the boundary of a line, although still a position in space (Kant, 1783, A:170).
In 1912 Poincaré discussed similar problems. Concerning the way in which geometers introduce the notion of three dimensions he says: “Usually they begin by defining surfaces as the boundaries of solids or pieces of space, lines as the boundaries of surfaces, points as the boundaries of lines” (cf. Hurewicz & Wallman, 1959:3). Although only related to three dimensions, Poincaré here provides us with an intuitive approach to dimension, implicitly stressing the unbreakable correlation between the law-side and the factual side in the spatial aspect:
… if to divide a continuum it suffices to consider as cuts a certain number of elements all distinguishable from one another, we say that this continuum is of one dimension; if, on the contrary, to divide a continuum it is necessary to consider as cuts a system of elements themselves forming one or several continua, we shall say that this continuum is of several dimensions (Hurewicz & Wallman, 1959:3).
Before 1911 the problem of dimension was confronted with two astonishing discoveries. Cantor showed that the points of a line can be correlated one-to-one with the points of a plane, and Peano mapped an interval continuously on the whole of a square. The crucial question was whether, for example, the points of a plane could be mapped onto the points of an interval in both a continuous and one-to-one way. Such a mapping is called homeomorphic. The impossibility to establish a homeomorphic mapping between a “m-dimensional set and a (m+1)-dimensional set (h > 0)” was solved by Lüroth for the case where m = 3 (Brouwer, 1911:161). Brouwer provided the first general proof of the invariance of the number of a dimension (see Brouwer, 1911:161-165). Exploring suggestions of Poincaré, Brouwer introduced a precise (topologically invariant) definition of dimension in 1913, which was independently recreated and improved by Menger and Urysohn in 1922 (cf. Hurewicz & Wallman, 1959:4). Menger’s formulation (still adopted by Hurewicz and Wallman) simply reads:
- the empty set has dimension -1,
- the dimension of a space is the least integer n for which every point has arbitrarily small neighborhoods whose boundaries have dimension less than n (Hurewicz & Wallman, 1959:4, cf. p.24).15
Whereas a spatial subject is always factually extended in some dimension (such as a 1-dimensional line, a 2-dimensional area, and so on), a spatial object merely serves as a boundary (in a delimiting way). The boundaries of a determined line-stretch are the two points delimiting it (with the line as a one-dimensional spatial subject). But these boundary points themselves are not extended in one dimension. Within one dimension points are therefore not spatial subjects but merely spatial objects, dependent upon the factual extension of the line. Yet a line may assume a similar delimiting role within two dimensions – for the lines delimiting an area are not themselves extended in a two dimensional sense. Likewise a surface can fulfil the role of a spatial object, namely when it delimits three dimensional spatial figures (such as a cube).
In general it can therefore be stated that whatever is a spatial subject in n dimensions is a spatial object in n+1 dimensions. A point is a spatial object in one dimension (an objective numerical analogy on the factual side of the spatial aspect), and therefore a spatial subject in no dimension (i.e. in zero dimensions). In terms of the fundamental difference between a spatial subject and a spatial object, it is impossible to deduce spatial extension from spatial objects (points). It is therefore unjustifiable to see a line as a set of points. But it falls outside the scope of this presentation to highlight the circularity present in Grünbaum’s attempt to argue for a consistent conception of the extended linear continuum as an aggregate of unextended elements (see Grünbaum, 1952). Grünbaum did not realize that the actual infinite – or, as we prefer to call it: the at once infinite – depends upon a crucial spatial feature, namely the spatial order of simultaneity. In the idea of the at once infinite the meaning of number points towards the meaning of space in an analogical way. Every known attempt to reduce space to number employs the at once infinite – and since the latter pre-supposes the irreducibility of the spatial order of at once these attempts all turn out to be circular (in the sense that one can reduce space to number if and only if one assumes the irreducibility of space). This remark also applies to the ideas advanced by Carl Posy in connection with building a “continuous manifold” out of “real numbers” (see Posy, 2005:321 ff.).
We can now account for the three primitive terms in Hilbert’s axiomatization of geometry in the context of the spatial subject-object relation. The term ‘line’ reflects the primary existence of a (one dimensional) spatial subject, the term ‘point’ highlights the primary existence of a (one dimensional) spatial object and the phrase ‘lies on’ accounts for the relation between a spatial subject and a spatial object – in other words, it highlights the spatial subject-object relation.
From our discussion thus far it is clear that the theory of modal aspects constitutes a key element of a non-reductionist understanding of reality.
12. 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 identified with ‘physical’ or “space-time existence”) and the dimension of functional modes (aspects) of ontic 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 clearly 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” (Cassirer, 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 the cosmic time-order 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. At the factual side of each aspect there are subject-ob ject relations (except for the numerical aspect where within which there are only subject-subject relations).
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 we discerned 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.16
13. The impasse of artimeticism
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.17 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.18 Being fully aware of the 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 categorically 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).19
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.20
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 (Lorenzen, 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 the infinitely many figures (Ziffern) existed all at once (alle auf einmal existierten) (Lorenzen, 1972:163).
Our discussion regarding 2+2=√8 argued that within the quantitative aspect the order of succession (on its law-side) provides a basis for arithmetical operations such as addition and multiplication and their inverses and it also makes possible our basic numerical awareness of greater and lesser. The arithmetical order of succession therefore determines our most basic intuition of infinity, in the literal sense of one, another one, and so on, without an end, endlessly, indefinitely, infinitely. The traditional designation of this kind of infinity, known as the potential infinite, lacks an intuitive appeal. But when we alternatively refer to the ‘successive infinite’ this shortcoming is left behind. The other kind of infinity, traditionally known as the actual infinite, also calls for an “intuitively transparent” designation – such as the at once infinite. The successive infinite, presupposed in the infinite divisibility of continuity, makes possible induction, which, according to Weyl, guarantees that mathematics does not collapse into an enormous tautology (Weyl, 1966:86). According to Gödel non-“tautological” relations between mathematical concepts “appears above all in the circumstance that for the primitive terms of mathematics, axioms must be assumed” (Gödel, 1995:320-321). In the case of finitism where the “general concept of a set is not admitted in mathematics proper … induction must be assumed as an axiom” (Gödel, 1995:321).
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 (Lorenzen, 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).
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: wholeness / 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. For example, in stead of saying that the mathematical analysis 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.21
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 continuity. The implication is that in an anticipatory way set theory 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 first of all encounter the sequence of natural numbers (expressing the primitive meaning of numerical discreteness). Then there are the integers (keeping in mind that the term ‘integer’ derives from wholeness and therefore points forward to what is non-integral, namely fractions). Introducing the dense set of rational numbers imitates the infinite divisibility of spatial continuity. Since this divisibility embodies the successive infinite it represents, within space, a retrocipation to the numerical time-order of succession. Therefore, as an anticipation to a retrocipation,) the rational numbers represent 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 here are better served by the phrases suggested above, namely the successive and the at once ininfinite. The fact that the at once infinite deepens the meaning of number requires a brief remark explaining the “as if” character of this disclosed notion of infinity. The anticipation from number to space on the law-side determines the multiplicity of natural numbers, integers and rational numbers which are correlated with it. Uor suggestion is that under the guidance of the actual infinite these sequences of numbers are considered as if they are present as completed (though infinite) wholes or totalities given at once.
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 an example of a necessary and fruitful fiction (cf. Vaihinger, 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” (Fischer, 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 hypothetical 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” (Lorenzen, 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 …” (Bernays, 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 simply 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 (Cantor, 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 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.
The new phrases for speaking of the potential and actual infinite, namely the successive and at once infinite, already had surfaced in the disputes of the early 14th century concerning the infinity of God.22
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.23
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.
14. 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-denumerably 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 (see our earlier discussion, pp.).
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.
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 (Grünbaum, 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” (Grünbaum, 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. Grünbaum, 1952:296).
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) (Grünbaum, 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 <%-2>to an anticipation ultimately underlies Grünbaum’s statement: “the Cantorean line can be said to be already actually infinitely divided” (Grünbaum, 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.)
In this argumentatyion the irreducibility of the spatial time-order of simultaneity to the numerical time-order of succession is presupposed – 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 defined 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. Therefore the attempt to reduce space to number is circular, for it has to employ the idea of the at once infinite which presupposes (in an anticipatory way) the irreducible meaning of space.
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), 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.24 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) (Bernays, 1976:188).
The aritmeticistic claims of set theory are circular for proving the non-denumarability of the real numbers requires (as anticipatory hypothesis) the at once infinite which in turn presupposes the irreducibility of space. Therefore, space can be reduced to number if and only if it cannot be reduced to number.
It is sufficient to consider, for example, the discussions concerning the nature of space and time, determinism, indeterminism and causality, the continuous or discontinuous characteristics of matter and energy, the infinity or finiteness of the universe that have been produced by the developments of contemporary physics: even when they have been led by professional and outstanding physicists, they were of a genuine philosopical nature, as it can be easily seen if we consider that they also occupied the mind of several outstanding professional philosophers of our time. Moreover, they count among the most typical and classical questions of the philosophy of nature of all times, and the fact that they are still the object of lively discussions also after the acceptance, say, of relativity and quantum physics clearly indicates that they have not been solved by such theories, but simply further problematized as a consequence of them (Agazzi, 2001:10).
In addition to the aspects of number and space that played a central role in our discussion of the implications of a non-reductionist ontology an assessment of the foundation of the discipline of physics will have to analyze the meaning of the kinematic and physical aspects as well.
Interestingly the dominant philosophical orientation amongst the special sciences during the first half of the 20th century wanted to restrict science to the “positive facts” that were assumed to be the sole guide to “objective scientific truth.” “Sense data,” were supposed to be the only source of reliable knowledge, and it supported the postulate of the neutrality of human rational endeavors. The latter conviction (!) erroneously labelled any ultimate commitment (conviction) operative within the domain of rationality as a disturbing factor that should be eliminated from science.
However, without an implicit trust or faith in reason this postulate itself cannot be maintained. All human beings are endowed with the capacity to think and to argue rationally, but they do this from diverging direction-giving orientations. Consequently, in spite of acknowledging the universality of the structural conditions making possible human thinking in the first place, no single human being can escape from some or other deepest conviction. Stegmüller holds that there is no single domain in which a self-guarantee of human thinking exists – one already has to believe in something in order to justify something else (Stegmüller, 1969:314).
An analysis of the structure of scientific activities therefore does not aim at securing a domain of the good by protecting it from the evil influence of direction-giving ultimate commitments, for any such analysis can only proceed by implicitly proceeding from a particular life-orientation.
There are not simply ‘scientific’ people liberated from any and all supra-rational convictions, and “non-scientific” people blurred by the ‘evil’ of adhering to some or other conviction. Whatever the life-orientation of thinkers may be, they all equally share in the dimension of rationality (or: logicality) and all of them are inevitably in the grip of a more-than-rational ultimate commitment.25
The positivistic appeal to sense data is problematic, because the theoretical ‘tools’ employed in the description of what is observed always utilize terms that are not susceptible to “empirical observation” themselves. In order to demonstrate this point it will be instructive to consider the history of the concept of matter.
15. Historical perspective on the concept of matter
The early Greek philosophers have chosen some or other fluid element as principle of origin, such as water (Thales), fire (Heraclitus) and air (Anaximenes). Of course the subsequent development should take into account the significant role of the school of Pythagoras. The contribution of this school is that it articulated the insight that rational knowledge cannot be divorced from numerical relationships. Naturally this school went too far in its above-mentioned one-sided claim that everything is number. This thesis rests on the conviction that with the aid of the relation between integers, i.e. by merely using normal fractions, it is possible to describe the ‘essence’ of whatever there is in numerical terms.
However, soon the developments within Greek culture became sensitive to spatial configurations – such as the shape of the calyx leafs found in nature, for this shape appeared as an instantiation of a regular pentagram. An investigation of the geometrical properties of a regular pentagram led to the discovery that it is not possible to express the ratio between any side and any diagonal of the regular pentagram with the aid of normal fractions, i.e. in terms of the ratio of two whole numbers / integers: a/b. This limitation at once embodied the discovery of ‘incommensurable’ quantities – something completely unacceptable for the Pythagoreans because suddenly within the limiting and form-giving nature of number the apeiron (the unbounded-infinite) appeared, i.e. irrational numbers were discovered.
Flowing from the inherent tension in Greek thought between what is limited and what is unlimited (the peras and the apeiron) the discovery of irrational numbers (or in modern mathematical terms: real numbers) inspired the search for an alternative principle of explanation – one that can escape from the unbounded (infinite) present in number.
The alternative mode of explanation that entered the scene was found in space. The spatial aspect allowed for the acceptance of static forms and it also opened the possibility to observe any spatial figure at once, without any before and after. The implication was that the acquisition of concepts is enclosed within the now and in the school of Parmenides this resulted, as we have seen, in the equation of thought and being.
It is known that on the basis of Babylonian observations Thales accurately predicted an eclipse of the sun in the year 585 B.C. He also had the remarkable geometrical skill to calculate the height of a pyramid from a sun shadow of 450 (keeping in mind that a pyramid differs from something like a tree where it is possible to establish its height perpendicular to its base). Thales also knew that the diagonals of a rectangular triangle are equidistant and according to Lorenzen he provides the starting-point for geometry as a coherent theoretical system (Lorenzen, 1960:45-46).
The important feature of this development is that the spatial figures of Greek geometry were idealized. It meant that a straight line, circle and square are not perceivable in a sensory way – they can merely be contemplated intellectually. Plato’s account of human knowledge reflects this conviction because he explicitly states that the conclusions reached do not use “sensory objects”:
Then by the second section of the intelligible world you may understand me to mean all that unaided reasoning apprehends by the power of dialectic, when it treats its assumptions, not as first principles, but as hypotheses in the literal sense, things ‘laid down’ like a flight of steps up which it may mount all the way to something that is not hypothetical, the first principle of all; and having grasped this, may turn back and, holding on to the consequences which depend upon it, descend at last to a conclusion, never making use of any sensible object, but only of Forms, moving through Forms from one to another, and ending with Forms (Politeia, 510D).
Plato’s dialogue Meno, where the leader of the conversation used leading questions in order to allow the conversation partner to produce a geometrical proof, caused Oskar Becker to remark that this gave birth to the appreciation of the a priori nature of mathematics (Becker, 1965:X).
The effect of the discovery of irrational numbers was not only that mathematics was geometrized for it also paved the way for a speculative theory of reality attempting to explain the entire universe in terms of a spatial perspective – as a substitute for the outdated arithmetical orientation of the Pythagoreans. The implication was that Greek thought now understood matter in terms of spatial extension. An entity is identified with the place it occupies. Something is its place.
It should be noted, however, that Parmenides hardly disposed over an independent space concept. He also did not contemplate the idea of an empty space. When something is its place, then the absence of something implies that the subject to which the predicate ‘place’ applies is not present. Herman Fränkel writes: “With the assertion of a complete filling of space … the existence of a mere empty space is rather denied than acknowledged.” (Fränkel, 1968:181, note 4).
In its denial of movement the school of Parmenides, in particular the arguments of Zeno, merely formulated the consequences of over-emphasizing the spatial aspect as mode of explanation. If something indeed is its place then it can never move, for passing from one place to another place will entail a change of essence!
The metaphysical overextension of the static nature of space even motivated a remarkable denial of the spatial whole-parts relation.
In order to understand this properly we have to keep in mind what we have explained earlier, namely that whatever is continuously extended in a spatial sense allows for an infinite divisibility (see above). The spatial whole-parts relation turns the original numerical meaning of succession – the successive infinite – ‘inwards’, embodied in the successive infinite divisibility of a continuum. In terms of the inter-modal coherence between number and space and in the light of the foundational role of number it indeed belongs to the meaning of the spatial whole-parts relation that it contains the possibility of endless divisions.
The spatial metaphysics of Parmenides, for that matter, inspired Zeno to defend a view of unitary wholeness that excludes plurality. In other words, Zeno wants to deny the ‘part’-element of the spatial whole-parts relationship while at the same time holding on to the ‘wholeness’ which entails it.
His position is that reality is both one and indivisible. Yet, in order to argue for his position, he explored the whole-parts relation in his argument that is aimed at the denial of plurality! The reason why Zeno considers plurality to be self-contradictory is that plurality requires a number of (indivisible) units and because it also implies that reality is divisible (see Guthrie, 1980:88). But divisibility threatens the wholeness of a unit, since anything divisible has to be a magnitude which must be infinitely divisible. The supposed indivisibility of a unit clashes with its infinite divisibility. “Hence, since plurality is a plurality of units, there can be no plurality either” (Guthrie, 1980:89).
The antinomies of Zeno (including those of Achilles and the tortoise and the flying arrow) represent the strating-point of a long speculative tradition in which the meaning of space was metaphysically explored within the context of a speculative theory of being that finds in God – as the Highest Being (ipsum esse) – its conclusion.
By exchanging two modes of explanation and attempt to strip them from their intrinsic connections caused multiple distortions. The school of Parmenides did realize that space provides an original mode of explanation but in the attempt to ‘purify’ space from number it challenged a foundational condition of space, given in the nature and meaning of a multiplicity. By ignoring the foundational role of a numerical multiplicity Zeno distorted the meaning of number and at once also skewed the meaning of space by questioning the divisibility of a spatial continuum. The infinite divisibility of a continuous whole within space is a reminder of the original successive meaning of number lying at the basis of space. Just as little as it is possible to separate space from other aspects of reality can it be separated from the numerical aspect. Even in the most extreme examples of arithmeticism in modern mathematics, aiming at reducing space to number, key features of space were needed. In the case of axiomatic set theory it turned out to be unavoidable to use undefined (‘primitive’) terms derived from the spatial whole-parts relation, such as set or element of (see Fraenkel et.al., 1973:21 ff).
The original numerical meaning of the number one as an integer analogically appears within the spatial aspect. The unity of a spatial subject is found in its wholeness. In other words, a spatial unity is constituted as a genuine whole or totality, a unitary whole allowing an infinite divisibility. The speculative (metaphysical) idea of a unitary whole precluding multiplicity robs both number and space from their unique meaning as well as from their mutual coherence.
In respect of the nature of material things the most important consequence is that the Greek-Medieval legacy only acknowledges concrete material extension. Extension characterizes the nature of material things.
Within the Aristotelian legacy it was believed that celestial bodies obey laws that are different from those that hold for entities on earth. In addition it was believed that the movement of anything required a cause. The problem of motion increasingly acquired a more prominent position, although it did not mean that the powerful influence of the classical space metaphysics immediately lost its hold. The power of this spatial orientation is indeed still evident in the thought of Descartes (1596-1650) and even Immanuel Kant (1724-1804). In their understanding of nature both these philosophers continued to assign a decisive role to spatial extension. For Descartes extension serves as the essential characteristic of material bodies – res extensa, for he writes: “That the nature of body consists not in weight, hardness, colour, and the like, but in extension alone” (Descartes, 1965:200 – Part I, IV). Kant’s characterization of material bodies is also oriented to space. When our understanding leaves aside everything accompanying their representation, such as substance, force, divisibility, etc., and likewise also separate that which belong to sensation, such as impenetrability, hardness, color, etc., then from this empirical intuition something else is left, namely extension and shape.26
It should not surprise us therefore that Descartes straightaway applied the feature of (mathematical) continuity to material things and even to atoms that since Greek antiquity were supposed to be the last indivisible material particles. He holds that there cannot be atoms or material particles that are inherently non-divisible.
We likewise discover that there cannot exist any atoms or parts of matter that are of their own nature indivisible (Descartes, 1965:209; Part I, XX).
In this context (XX) he even introduces the idea of God in order to make acceptable the infinite divisibility of matter. He argues that although God can make a particle small enough that no creature can divide it this does not set any limits to the Divine capacity to divide. Therefore it should be assumed that matter is indeed infinitely divisible:
Wherefore, absolutely speaking, the smallest extended particle is always divisible, since it is such of its very nature.
That Descartes continues to hold on to extension as the essential trait of matter embodies his connection with the long-standing Greek-Medieval tradition. However, what he has to say regarding the infinite divisibility of matter breathes the spirit of the early modern functionalistic orientation.27 This new functionalistic attitude soon attempted to explain concrete things completely in functional terms. Yet Descartes at the same time did pay attention to motion, which he defined as the “action by which a body passes from one place to another” (Descartes, 1965:210; Part I, XXIV). This new point of view finds itself on the cross-road of the transition from the Medieval to the modern era.
Although Buridan (early 14th century) did contribute to the uprooting of the dominant position of spatial extension and the transition to the modern era, it should be kept in mind that the impetus idea itself cannot be equated with the nature of inertia. There is a conception that the mechanics of Buridan and classical physics are fundamentally similar, entailing that the impetus theory practically already brought to expression the law of inertia.
This convergence is first of all sought in the supposed correlation between the scholastic view of impetus and the dynamic element of inertial motion. In the second place it was believed that the assumption of permanence present in Buridan’s view of impetus already discovered the law according to which motion not disturbed in any way will be everlasting (cf. Maier, 1949:142).
It is indeed striking that an impetus that was transferred in a celestial sphere (supposed to follow a circular path) was supposed to be free from any resistance. Therefore it seemed proper to compare it with the underlying idea of inertia. But inertia concerns uniform (rectilinear) motion, not circular motion. In the case of the impetus theory there was a difference between what happened in the heavens and happened on earth. According to the Medieval Scholastic understanding a force without resistance cannot produce motion. But since impetus is artificially and forcefully superimposed upon some or other obstacle (i.e. an interfering force of motion), such an obstacle can only be overcome when the impetus itself is altered in the process of overcoming the resistance. Yet the decisive difference between these two views is given in that element of inertial movement from which one cannot abstract, namely the inertia of a mass-point.28 It is possible to abstract from external obstacles and forces, but it is impossible to abstract from that which is crucial for inertia, namely the mass of whatever moves. According the classical mechanics the latter is the real factor in the continuation of movement (inertia). In the case of the impetus theory inertial mass serves as resistance (obstacle) for the movement of the impetus that caused it. Consequently there is an unbridgeable gap between the impetus theory and the basic idea if inertia, namely the possibility of an everlasting rectilinear motion.29
Galileo formulated his law of inertia with the aid of a thought experiment. Suppose a body moves on a friction-free path extended into infinity, then this movement will simply continue endlessly. Opposed to the traditional Aristotelian-Scholastic conception according to which the movement of a body is dependent upon a causing force the law of inertia implies the motion is something given and that therefore in stead of trying to deduce or explain it it should be accepted as a mode of explanation in its own right. Motion is original and unique and indeed embodies a distinct mode of explanation different from those used by the Pythagoreans (number) and the Eleatic school of Parmenides (space). If motion does not need a causing force, then at most it is possible to speak of a change of motion (acceleration or deceleration) – and this does need a physical force. The well-known German physicist remarks:
Since the law of inertia has shown that no force is required for a change of place the most natural thing to do is to accept that force causes a change of speed, or, as Newton says, the magnitude of motion (‘Bewegungsgröße’) (Von Weizsäcker, 2002:172).30
The idea of a uniform (rectilinear) motion on the one hand expands the inherent limitations attached to number and space as modes of explanation, and on the other it at once opens the way to consider another problem that already captured Greek thought. This problem concerns the relation between persistence (think about the nature of inertia) and dynamics (consider the change of motion requiring a physical force).
The important insight of Plato is that change can only be established on the basis of constancy (persistence) – i.e. without an enduring subject there is nothing to “hold on to,” nothing to which the alleged changes can be attributed. Of course this insight does not force us to join the specualative account which Plato gave for it in his metaphysical theory of static, super-sensory ideal forms – although it is true that his solution did form a lasting attraction for many scholars. Even Frege said that amidst the on-going flow of events something lasting, something with eternal durability must exist for otherwise the knowability of the world would be canceled and everything would collapse in confusion.&31
The proper elaboration of Plato’s insight, namely that change presupposes constancy, is found in Galileo’s formulation of the law of inertia and in Einstien’s theory of relativity. The core idea of Einstein’s theory is after all die constancy of the velocity of light in a vacuum. Although he often merely speaks of “the principle of the constancy of the speed of light”,32 he naturally intends “the principle of the vacuum-velocity” (“das Prinzip der Vakuumlichtgeschwindigkeit” – see Einstein, 1982:30-31; and also Einstein, 1959:54). From this it follows that Einstein primarily aimed at a theory of constancy – whatever moves move relative to this element of constancy. It was merely a concession to the historicistic Zeitgeist at the beginning of the 20th century that he gave prominence to the term ‘relativity’ – all movement is relative to the constant c.
However, a certain ambiguity is still found in the thought of Descartes and his followers for in spite of the fact that they viewed extension as the essential property of matter, they also simultaneously pursued the kinematical ideal to explain everything that exists and happens exclusively in terms of movement (cf. Maier, 1949:143).33 It is generally known that Thomas Hobbes took the full step to the exploration of movement as principle of explanation in his intended rational reconstruction of reality. According to the newly established natural science ideal he first demolished reality to a heap of chaos in order afterwards to build up, step by step, a new rationally ordered cosmos, guided by the key concept “moving body.” His acquaintance with the mechanics of Galileo enabled him to exceed the limits of space as mode of explanation. Galileo himself embodies the long history of our understanding of matter up to this phase of its development because he explicitly explores the three modes of explanation thus far highlighted in our discussion. He accounts for arithmetical properties (countability), geometrical properties (form, size, position and contact) and kinematic features (motion).34 Leibniz continues this legacy in his belief that physical events can be explained mechanisticlly in terms of magnitude, figure, and motion.35
Writing on the foundations of physics, David Hilbert refers to the mechanistic ideal of unity in physics but immediately adds the remark that we now finally have to free ourselves from this untenable ideal (cf. Hilbert, 1970: 258).36
As soon as the kinematic mode of explanation is acknowledged in its own right the necessity to find a cause for motion disappears. The classical opposition between being at rest and moving is therefore untenable, because from a kinematic perspective ‘rest’ is a state of movement (cf. Stafleu, 1987:58). Unique and irreducible modes of explanation are not opposites – for they are mutually cohering and irreducible.37
Although Descartes and Newton did employ the concept force, it may in general be said that modern physics since Newton is characterized by its mechanistic main tendency. The mechanistic view consistently attempts to reduce all physical phenomena to a kinematic perspective. However, already in the course of the 19th century modern physics started to explore the nature of energy. The founder of physical chemistry, Wilhelm Ostwald, developed his so-called Energetik (enegetics) that even influenced the later views of Heisenberg. Vogel refers to Heisenberg’s work “Wandlungen in den Grundlagen der Naturwissenschaft” (Stuttgart 1949) where the latter explicitly speaks of energy as the basic stuff that constitutes matter in its threefold stable forms: electrons, protons and neutrons (Vogel, 1961:37). Yet Ostwald‘s Energetik did not exert a lasting influence upon the physics of the 20th century, probably because it was attached to a specific view of continuity opposed to an atomistic approach. Niels Bohr particularly mentions the excessive skepsis found in the thought of Mach regarding the existence of atoms.38
The last prominent physicist who consistently adhered to the mechanistic approach was Heinrich Herz. Soon after Hertz’s death in 1894 the work in which he attempted to restrict the discipline of physics to the concepts mass, space and time, reflecting the three most basic modes of explanation of reality, namely the modes of number, space and movement, appeared: “The Principles of Mechanics developed in a New Context.” This caused him (and Russell) to view the concept of force as something intrinsically antinomous.
The Latin designation of mass during the medieval period was “quantitas materiae” (see Maier, 1949:144). From this it appears that number (quantitas) plays a key role in the concept mass. Mass concerns a physical quantity, but it is also possible to observe the quantity of energy from the perspective of the kinematical modality. In this case the technical expression is kinetic energy that indicates the action capacity inherent to a moving body (see Maier, 1949:142).
As soon as the physical aspect of reality surfaced it opened up the way for 20th century physics to explore it as an independent mode of explanation and to arrive at an even more nuanced understanding of reality. For example, in his protohpysics Paul Lorenzen distinguishes four units of measurement reflecting the first four modes of explanation: mass, length, duration and charge (Lorenzen, 1976:1 ff.).
A decade after Max Planck discovered his “Wirkungsquantum” he explicitly addressed the intrinsic untenability of the mechanical understanding of reality.
The conception of nature that rendered the most significant service to physics up till the present is undoubtedly the mechanical. If we consider that this standpoint proceeds from the assumption that all qualitative differences are ultimately explicable by motions, then we may well define the mechanistic conception as the conviction that all physical processes could be reduced completely to the motions (the italics are mine – DFMS) of unchangeable, similar mass-points or mass-elements.39
Einstein is equally explicit in his negative attitude towards “the mechanistic framework of classical physics” (see Einstein, 1985:146).
Eventually the distinction between the kinematic and physical aspects of reality thus became common knowledge. According to Janich the scope of an exact distinction between phoronomic (subsequently called kinematic) and dynamic arguments could be explained in terms of an example. Modern physics has to employ a dynamic interpretation of the statement that a body can alter its speed only continuously. Given certain conditions a body can never accelerate in a discontinuous way, that is to say, it cannot change its speed through an infinitely large acceleration, because that will require an infinite force.40
The idea of an attracting force, initially conceived of in connection with magnetism, eventually brought Newton to the insight that magnetism is a force that cannot be explained through motion, although in its own right, foundational to the physical aspect, motion is a mode of explanation. Stafleu points out that the rejection of the Aristotelian distinction between the physics of celestial bodies and the physics of things on earth paved the way, in the footsteps of Galileo and Descartes, to realize that the same physical laws apply to both domains, i.e. that physical laws display modal universality (i.e. they hold universally) (Stafleu, 1987:73). He also remarks that Newton (just as Kepler) indeed already appreciated force positively as a principle of explanation that is distinct from motion as an original principle of explanation (see Stafleu, 1987:76). Stafleu summarizes this process through which the physical aspect emerged as an equally original mode of explanation as follows:
In Newtonian mechanics, a force is considered a relation between two bodies, irreducible to other relations like quantity of matter, spatial distance, or relative motion. Though an actual force may partly depend on mass or spatial distance, as is the case with gravitational force, or on relative motion, as is the case with friction, a force is conceptually different from numerical, spatial or kinematic relations (Stafleu, 1987:79).
Since the introduction of the atom theory of Niels Bohr in 1913, and actually already since the discovery of radio-activity in 1896 and the discovery of the energy quantum h, modern physics realized that matter is indeed characterized by physical energy operation. It is therefore understandable that 20th century physics eventually had to come to a general acknowledgement of the decisive significance of energy operation for the nature and understanding of the physical world, as it is strikingly captured in Einstein’s famous formula:
E = mc2
It was also realized that physical processes are irreversible. In itself this observation also justifies the distinction between the kinematic and the physical aspects of reality. Both Planck and Einstein knew that in terms of a purely kinematic perspective all processes are reversible. Einstein refers to Boltzmann who realized that thermodynamic processes are irreversible.41 Already in 1824 Carnot discovered irreversible processes – a discovery that was elaborated independently from each other in 1850 to the second main law of thermodynamics (the law of non-decreasing entropy). This law accounts for the fundamental irreversibility of natural processes within any closed system. The term entropy itself was introduced by Clausius only in 1865. In 1852 Thomson explains that according to this law all available energy strive towards uniform dissipation (see Apolin, 1964:440 and Steffens, 1979:140 ff.). Planck remarks that “the irreversibility of natural processes” confronted “the mechanical conception of nature” with “insurmountable problems” (Planck, 1973:55).
It is only on the basis of an insight into the foundational position of the kinematic aspect in respect of the physical aspect that an appropriate designation of the first law of thermodynamics is made possible. Although we are used to employ the familiar designation of it as the law of energy conservation there is an element of ambiguity attached to the term ‘conservation’ – as if energy is “held on to.” When, on the law-side, the retrocipation from the physical aspect to the kinematic aspect is captured by the phrase energy constancy this ambiguity disappears and then we have at hand a concise and precise formulation of this law.
16. The mystery of matter
The preceding historical sketch made it clear that although each one of the four modes of explanation did open up a legitimate angle of approach none of them can claim to be the exclusive and/or exhaustive source of our knowledge of material things. Whatever their worth, they merely provide us with a partial perspective, one that will always be co-determined by a totality view exceeding the scope of any specific mode of explanation. Such a totality perspective actually exceeds the scope of any special science since it inevitably rests upon some or other philosophical view of reality. It entails the necessity to employ modal terms in concept-transcending ways.
These considerations intimately cohere with the modal universality inherent in each modal aspect. Acknowledging the modal universality of the different modal aspects is constitutive for an account of typicality, individuality and concept-transcending knowledge (idea-knowledge).
The impressive power of theoretical thinking fist of all derives from exploring the modal universality of specific modal aspects. The philosophically informed physicist Von Weizsäcker implicitly draws upon this insight when he appreciates quantum theory as the central theory of contemporary physics. His explanation highlights the modal universality of the physical aspect, for this modal universality is not restricted by the typical nature of any (type of) entity – it cuts across all typical differences. We have noted that Von Weizsäcker says:
Quantum theory, formulated sufficiently abstract, is a universal theory for all classes of entities (Von Weizsäcker, 1993:128).42
In addition to this appeal to modal universality Von Weizsäcker also explicitly articulates the fundamental philosophical insight that everything coheres with everything: (“Alles hängt mit allem zusammen” – Von Weizsäcker, 1993:134).
The modal universality of each one of the four modes of explanation that we have discerned in their successive decisive roles during the history of our understanding of physical nature entails that the scope of each one of them is unspecified. This means that whatever concretely exist functions within every one of these modes of existence. We have referred to the law of gravity as an example of the unspecified modal universality of the physical aspect.