# The Significance of Unity and Diversity for the Disciplines of Mathematics and Physics

**I. Introduction**

Closely related to a longstanding over-estimation of human rationality, the scholarly disciplines of mathematics and physics were always appreciated as the best examples of rationality at work and as pure instances of â€œexactâ€ scientific thinking. Of course the embarrassing fact is that in spite of the reign of â€œobjectiveâ€ and â€œneutralâ€ reason, not even these two â€œexactâ€ sciences managed to side-step serious differences of opinion, manifested in opposing schools of thought.

*Spiral staircase in the now mostly abandoned Histadrut building in Ramat Gan, Israel.*

The history of mathematics reflects a continued attempt either to reduce space to number or number to space. Greek mathematics started with the belief that everything is number but the discovery of irrational numbers generated a crisis that caused the geometrization of mathematics. Since early modernity (Descartes) mathematics slowly but certainly found its way back to number, accompanied by its second foundational crisis, given in the nature of the infinite combined with the mathematical idea of a limit. During the 19th century the rise and development of set theory appeared to have transcended these problems, but unfortunately Russell and Zermelo discovered that the plain everyday understanding of sets harbours intrinsic antinomies. It is strange that mathematics never opted for a third alternative, one generated by the biblical idea that we acknowledge the diversity in creation without attempting to reduce it to one or another privileged mode of explanation. Such a non-reductionist ontology suggests the sought after third option: accept both number and space in their uniqueness as distinct modal aspects of reality while at the same time investigate their inter-connections and mutual coherence.

A similarly problematic situation is found in the discipline of physics. Although the positivistic philosophy of science is still quite alive amongst practising natural scientists, the positivist criterion of truthâ€”sense perceptionâ€”cannot withstand the test of critical reflection. If it is only the senses that can provide us with knowledge, then the history of physics cannot be accounted for. This is the case because the concept of matter, for example, was theoretically described by employing alternative modes of explanationâ€”starting with number, then moving to space and motion and finally resulting in 20^{th} century physics which had to acknowledge the qualifying role of physical energy-operation. These aspectual terms themselves, used in (successively alternative) characterizations of matter, are not open to sensory perception. The diversity in creation cannot be reduced to *a theory of everything*. Only when a non-reductionist ontology is employed is it possible to be freed from all forms of monistic reductionism.

**II. Mathematics**** **

*A Few Perspectives on the History of Mathematics *

During the development of mathematics this discipline appears twice to have completed the same circle. In ancient Greece the Pythagoreans were convinced that *everything is number*, i.e. they believed that fractions (rational numbers) are capable of expressing the â€œessenceâ€ of reality. However, in his dealing with a regular pentagram Hippasus of Metapont (450 B.C.) discovered that the ratio of certain line segments is *incommensurable* (cf. K. von Fritz, 1965:271 ff., especially pp.295-297), marking the discovery of irrational numbers. What the Pythagoreans supposed to be within what they considered the *form-giving *(delimiting and ordering) function of number, thus confronted them with an unbounded and infinite series, indicating something formless and unlimited. To escape the â€œfateâ€ of irrational numbers, they translated all their arithmetical problems into *spatial* terms (any spatial figure has a *definite* and limited form). This possibility of handling irrational numbers in a geometrical way caused a fundamental geometrization of Greek mathematics. This is not only an outcome directed by the basic tension between *matter* and *form*, since it also means a shift in the focus of explanation. Instead of sticking to the arithmetical mode of explanationâ€”everything is numberâ€”the spatial point of view entered the scene.

This new orientation lasted until the modern era where we slowly see mathematics increasingly moving back to an arithmetizing orientation. Sometimes the developments in Greek mathematics are depicted as the first foundational crisis of this discipline. The second crisis was generated by the combination of the idea of infinity and the technical mathematical concept of a limit (17th-18th centuries). Soon the new arithmeticism of the 19th century found a powerful alley in the development of modern *set theory*. In the line from Bolzano, Cauchy, Weierstrass, and Dedekind up to Georg Cantor, this theory offered the hope of supplying mathematics with a firm and secure foundation. Between 1874 and 1899 Cantor developed mathematical set theory in such a way that mathematicians soon considered it to be the ultimate foundation of mathematics.

*The Crisis of Set Theory*

Yet once more this new foundation experienced a severe blow. In 1900, Bertrand Russell and Ernst Zermelo independently discovered that the everyday concept of a setâ€”as being constituted by clearly distinct elements bound together into a wholeâ€”is inconsistent. Just consider the set C, which has as its elements those sets that do not contain themselves as elements. Now contemplate two options; the one supposing that C is an element of C, and the other that C is not an element of C. It should be kept in mind that the *condition* for a set to be an element of C is that it cannot contain itself as an element. The upshot is perplexing.

- If C is an element of C, it must conform to this condition, i.e. that it does not contain itself as an elementâ€”if C is an element of C, then C is not an element of C
- If C is not an element of C, then it meets the condition for being an element of C – If C is not an element of C, then C is an element of C
- Therefore, C is an element of C
*if*and*only if*it is not an element of C!

The apparently innocent combination of *multiplicity* and *wholeness* therefore caused havoc within the discipline of mathematics, giving rise to conflicting schools of thought within this special science.

*Diverging Schools of Thought in Mathematics*

While the logicism of Russell held that mathematics is actually logic, the intuitionist school and that of axiomatic formalism accepted a pre-logical or extra-logical subject matter. Intuitionism reacted both to logicism and formalism and in particular to what Russell and Zermelo discoveredâ€”and in doing that it generated a whole new mathematics:

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).

This explicit and unqualified trust in â€œmathematical reasonâ€ apparently did not take notice of Morris Kline’s assessment, almost three decades ago:

The developments in the foundations of mathematics since 1900 are bewildering, and the present state of mathematics is anomalous and deplorable. 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. Beyond the logicist, intuitionist, and formalist bases, the approach through set theory alone gives many options. Some divergent and even conflicting positions are possible even within the other schools. Thus the constructivist movement within the intuitionist philosophy has many splinter groups. Within formalism there are choices to be made about what principles of metamathematics may be employed. Non-standard analysis, though not a doctrine of any one school, permits an alternative approach to analysis which may also lead to conflicting views. At the very least what was considered to be illogical and to be banished is now accepted by some schools as logically sound (Kline, 1980:275-276).

Perhaps the most astounding observation comes from StegmÂ¸ller:

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 (StegmÂ¸ller, 1970:331).

*The Third Foundational Crisis of Mathematics *

In 1900, the French mathematician, PoincarÃˆ, 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 PoincarÃˆ made his proud claim, it has already turned out that the theory of the infinite systems of integersâ€”nothing else but part of set theoryâ€”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 undergoingâ€ (Fraenkel et al., 1973:14).

Zermelo introduced his axiomatization of set theory in order to avoid the derivation of â€œproblematicâ€ sets and Hilbert dedicated the greater part of his later mathematical life to developing a proof of the consistency of mathematics. But when GË†del demonstrated that in principle it is not possible to achieve this goal, Hilbert had to revert to intuitionistic methods in his proof theory (â€œmeta-mathematicsâ€). After Hilbert died in 1943, his student, Hermann Weyl, who switched to an intuitionistic orientation, wrote: â€œIt must have been hard on Hilbert, the axiomatist, to acknowledge that the insight of consistency is rather to be attained by intuitive reasoning which is based on evidence and not on axiomsâ€ (Weyl, 1970:269).

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 1903â€”where he had to concede in the first sentence of the appendix that one of the corner stones of his approach had been shaken.

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 circleâ€”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. In the case of Frege, the discovery of the untenability of his â€œGrundlagenâ€ also inspired him to hold that mathematics as a whole is actually geometry:

So an

a priorimode 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 basisâ€”a geometrical one in factâ€”so that mathematics in its entirety is really geometry (Frege, 1979: 277).

*The Three Options of the History of Mathematics and a**n Alternative Approach: Accept the Uniqueness of Number and Space and Explore Their Mutual Coherence*

The history of mathematics clearly opted for at least *three* different possibilities:

- attempt exclusively to use the quantitative aspect of reality as mode of explaining the whole of mathematicsâ€”Pythagoreanism, modern set theory (Cantor, Weierstrass), and axiomatic set theory (axiomatic formalismâ€”Zermelo, Fraenkel, Von Neumann and Ackermann);
- explore the logical mode as point of entryâ€”the logicism of Frege, Dedekind and Russell; and,
- the intermediate period during which the geometrical nature of mathematics was asserted, once again taken up by Frege close to the end of his life.

In opposition to all forms of reductionism, evinced in the multiplicity of â€œismicâ€ positions found within philosophy and the various scholarly disciplines, the positive contribution of the philsophical heritage handed to us in the thought of Dooyeweerd and Vollenhoven is given in their emphasis on a non-reductionist ontology. Looking at the history of mathematics and the dominance of an arithmeticistic axiomatic formalism within contemporary mathematics, the obvious observation to be made in terms of a non-reductionist ontology is the following one: Acknowledge the uniqueness and irreducibility of every aspect inevitably involved in practicing mathematics without attempting to reduce anyone of the aspects involved to any other aspect.

Dooyeweerd shows that whenever this anti-reductionist approach is not followed, theoretical thought inescapably gets entangled in theoretical antinomies. His claim is that the logical principle of non-contradiction finds its foundation in the more-than-logical (cosmological) principle of the excluded antinomy (*principium exclusae antinomiae*) (see Dooyeweerd, 1997-II:37 ff.). A Christian attitude within the domain of scholarship, while observing the *principium exclusae antinomiae*, will attempt to avoid every instance of a one-sided deification of anything within creation. The biblical perspective that God is Creator and that everything within creation is dependent upon the sustaining power of God, opens the way to the life-encompassing consequences of the redemptive work of Christ, for in Him we are in principle liberated from the sinful inclination to search within creation for a substitute for God. We are in principle liberated from this inclination in order to be ableâ€”albeit within this dispensation always in a provisional and fallible wayâ€”to respect the creational diversity with the required intellectual honesty for what it is: creaturely reality in its dependence upon God.

Therefore, while respecting the uniqueness and diversity of various aspects within created reality, it should be realized that no single aspect could ever be understood in its isolation from all the other aspects. Clearly, if a serious attempt is made to side-step the conflicting *ismic* trends operative throughout the history of mathematics, the most obvious hypothesis is contained in conjecturing the following thesis: *Accept the uniqueness and irreducibility of the various aspects of created reality, including the aspects of quantity, space, movement, the physical, the logical-analytical, and the lingual (or: sign) mode, while at the same time embarking upon a penetrating, non-reductionist analysis of the inter-modal connections between all these aspects.*

Of course this proposal is crucially dependent upon a more articulated account of the theory of modal aspects as such. At the same time the incredibly rich legacy of special scientific knowledge within the domain of mathematics ought to be integrated into such an alternative approach. Yet this does not entail that such a Christian approach will have to deal with a *different reality*â€”simply because such an understanding already fundamentally misunderstands the biblical perspective. The latter actually is the only life-orientation operative within world history emphasizing the *unity* and the *goodness* of creation in its entiretyâ€”by realizing that the directional antithesis between what is good and bad (redemption and sin) may never be identified with the structure of God’s creation. Christians and non-Christians are not living in different worlds and they are not doing different thingsâ€”but they indeed do the same things differently! They share with all human beings the ability to think, to discern and to argue. But given the supra-theoretical biblical starting-point of Christian scholarly reflection within the disciplines, Christians are called to take serious the demand not to absolutize anything within creation.

Although axiomatic set theory proceeds under the flag of being fully arithmeticistic, it does not realize that its entire analysis of the â€œcontinuumâ€â€”interpreted as an analysis of the real numbersâ€”inherently depends on â€œborrowingâ€ something crucial from our spatial intuition, namely the awareness of â€œat onceâ€ (simultaneity) and the feature of being a *totality* (a whole with its parts). Zermelo-Fraenkel set theory accepts the primitive binary predicate designated as the membership relation. This move only apparently conceals any connection with our spatial intuition, for the moment in which we set out to investigate what is at stake, it is clear that the undefined status of the term â€˜setâ€™ (or, alternatively, the â€˜membership relationâ€™) borrows the two above-mentioned key features from the spatial mode, namely *simultaneity* and the *whole-parts relation*.

Therefore mathematical set theory in fact ought to be seen as a spatially deepened theory of number. In this context it is noteworthy that Hao Wang informs us that Kurt GË†del speaks of sets as being â€œquasi-spatialâ€ and then adds the remark that he is not sure whether GË†del would have said the â€œsame thing of numbersâ€ (Wang, 1988:202)!

Particularly in confrontation with the dominant claim that mathematics has been arithmetized completely, these insights should be embedded within the context of an inter-modal understanding of the meaning of number and space. Without explaining this point in a technical way, the inherent circularity entailed in this whole position could be highlighted on the basis of the distinction between what should be designated as the *successive infinite* (traditionally known as the *potential infinite*) and the *at once infinite* (traditionally: *the actual infinite*). The introduction of these phrases took into account the rich legacy of philosophical and mathematical reflections on the nature of infinity. By â€œlocatingâ€ the interconnections between these two kinds of infinity relative to the respective meanings of number and space, the aforementioned attempted arithmetization of mathematics stands and falls with the acceptance of non-denumerable setsâ€”and it can be shown that the only basis upon which the latter could be introduced is by employing the idea of the *at once infinite*. But in order to employ the at once infinite one has to account for the theoretical deepening and opening up (disclosure) of the primitive numerical intuition of *succession* in its anticipation to the spatial meaning of *simultaneity* (at once) underlying the (regulative) hypothesis of viewing successively infinite sequences as if all their elements are present at once. Therefore, implicitly or explicitly, the use of the at once infinite has to make an appeal to the meaning of spaceâ€”i.e., to the spatial (time-order) of at once (*simultaneity*), entailing the feature of *totality* which is irreducible to *succession*. Consequently, spatial continuity could be reduced to number if and only if its irreducibility is assumed (in the inevitable acceptance of the at once infinite).

Although Paul Bernays, the co-worker of the foremost mathematician of the 20^{th} century, David Hilbert, and the author of a distinct variant of modern axiomatic set theory, did not develop the necessary theoretical distinctions advanced in this account of the (inter-modal) meaning of the at once infinite (actual infinity), he does have a clear understanding of the futility of arithmeticistic claims. He writes:

It should be conceded that the classical foundation of the theory of real numbers by Cantor and Dedekind does not constitute a complete arithmetization. … 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).

^{1}

**III. Physics**

An understanding of modern physics is crucially dependent upon a clear distinction between the four most basic aspects of reality, namely: number, space, movement, and the physical aspect. The uniqueness and coherence of these fundamental aspects of reality are indispensible in an assessment of the implications of a non-reductionist ontology for the foundation of the discipline of physics.

*Faith in Reason*

Interestingly, the dominant philosophical orientation amongst the special sciences during the first half of the 20^{th} century wanted to restrict science to the â€œpositive factsâ€ that were assumed to be the sole guide to â€œobjective scientific truth.â€ â€œSense dataâ€ were supposed to be the only source of reliable knowledge, and this position supported the postulate of the *neutrality* of human rational endeavors. The latter conviction (!) erroneously labeled 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 one or another *diverging direction-giving orientation*. Consequently, despite the fact that positivism acknowledged that there are universal structural conditions for theory making, it never allowed that deep, extra-scientific convictions could be among them. This overlooks the point made by Stegmuller to the effect that there is no single domain in which a self-guarantee of human thinking existsâ€”one already has to *believe* in something in order to *justify* something else (StegmÂ¸ller, 1970:314).

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 advance by implicitly proceeding from a particular life-orientation. There are not simply â€˜scientificâ€™ people *liberated* from any and all supra-rational convictions, and â€œnon-scientificâ€ people blurred by the â€˜evilâ€™ 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*.^{2}

*The Concept of Matter*

We have seen that the Pythagoreans held the view that everything is number, but that after the discovery of irrational numbers they reverted to a spatial approach. In respect of the nature of material things the most important consequence of this switch is that the Greek-Medieval legacy only acknowledges *concrete material extension*. Extension characterizes the nature of material things.

In line with the Aristotelian tradition 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 and even Immanuel Kant. 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: â€œThat the nature of body consists not in weight, hardness, colour, and the like, but in extension aloneâ€ (Descartes, 1965:200â€”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 separates that which belongs to sensation, such as impenetrability, hardness, color, etc., then from this empirical intuition something else is left, namely extension and shape.^{3}

It should not surprise us therefore that Descartes straightaway applied the feature of (mathematical) continuity to material things and even to atoms which 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 such 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.”

*Galileo: Motion as Principle of Explanation*

The truly modern era in physics begins with Galileo, who formulated his law of inertia. 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 that motion is something *given* and that therefore, instead 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*)â€”and this does need a *physical force*. The well-known German physicist Von Weizsâ€°cker 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 (â€˜BewegungsgrË†ï¬‚eâ€™) (Von Weizsâ€°cker, 2002:172).

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 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)â€”i.e. without an enduring subject there is nothing to â€œhold on to,â€ 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.^{4}

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 the constancy of the velocity of light in a vacuum. Although he often merely speaks of â€œthe principle of the constancy of the speed of lightâ€,^{5} he naturally intends â€œthe principle of the vacuum-velocityâ€ (â€œdas Prinzip der Vakuumlichtgeschwindigkeitâ€â€”see Einstein, 1982:30-31; and also Einstein, 1959:54). From this it follows that Einstein primarily aimed at a theory of *constancy*â€”whatever moves, moves relative to this element of constancy. It was merely a concession to the historicistic *Zeitgeist* at the beginning of the 20^{th} century that he gave prominence to the term â€˜relativityâ€™â€”all movement is relative to the constant *c*. It also implies, by taking into consideration the interconnection between the kinematic and physical aspects, that a more precise formulation of the first main law of thermodynamics (the law of energy-conservation) ought to be designated as the law of *energy-constancy* (an analogy of the foundational kinematic aspect on the law-side of the physical aspect of energy-operation).

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).^{6} 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 â€œmoving body.â€ 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).^{7} Leibniz continues this legacy in his belief that physical events can be explained mechanisticlly in terms of magnitude, figure, and motion.^{8}

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 â€˜restâ€™ is a state of movement (cf. Stafleu, 1987:58). Unique and irreducible modes of explanation are not *opposites*â€”for they are mutually cohering and irreducible.^{9}

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: â€œThe Principles of Mechanics developed in a New Context.â€ This caused him (and Russell) to view the concept of *force* as something intrinsically antinomous.

*Acknowledging the Physical as Distinct Mode of Explanation*

As soon as the physical aspect of reality surfaced it opened up the way for 20^{th} century physics to explore it as a distinct mode of explanation and to arrive at an even more nuanced understanding of reality. For example, in his *protophysics* Paul Lorenzen distinguishes *four* units of measurement reflecting the first four modes of explanation: *mass*,* length*, *duration* and *charge *(Lorenzen, 1976:1 ff.). 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).^{11} Einstein is equally explicit in his negative attitude towards â€œthe mechanistic framework of classical physicsâ€ (see Einstein, 1985:146). And a decade after Max Planck discovered his â€œWirkungsquantumâ€ 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 mineâ€”DFMS) of unchangeable, similar mass-points or mass-elements.^{10}

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.^{12}

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 20^{th} 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 = mc^{2}

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 from a purely kinematic perspective all processes are *reversible*. Einstein refers to Boltzmann who realized that thermodynamic processes are irreversible.^{13} Already in 1824 Carnot discovered irreversible processes. Since 1850 Clausius and Thompson independently developed the second main law of thermodynamics, known as 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 strives towards uniform dissipation (see Apolin, 1964:440 and Steffens, 1979:140 ff.). Planck remarks that â€œthe irreversibility of natural processesâ€ confronted â€œthe mechanical conception of natureâ€ with â€œinsurmountable problemsâ€ (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 â€˜conservationâ€™â€”as if energy is â€œheld on to.â€ 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.

*The Impasse of Positivism*

We may now consider the claim of positivism, namely that sensory perception is the ultimate source of scientific knowledge. Let us explore this issue in some more detail. In order to highlight the limitations of the senses in the acquisition of knowledge, we only have to consider the aforementioned sketch of the history of the concept of matter. We have referred to the fact that the Pythagoreans adhered to one statement above all else: everything is number. After the discovery of irrational numbers we saw that Greek mathematics as a whole was transformed into a spatial mode (the geometrization after the initial arithmetization). As a consequence, material entities were no longer described purely in arithmetical terms. The aspect of space now provided the necessary terms required to characterize material entities. This spatial angle of approach remained in force until the rise of modern philosophy, since philosophers like Descartes and Kant still saw the â€˜essenceâ€™ of material things in their *extension*. Particularly through the work of Galileo and Newton, the main tendency of classical physics eventually underwent a shift in perspective by attempting to describe all physical phenomena exclusively in terms of (kinematic) *motion*. 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.

From this brief historical analysis it is clear that different aspects served to characterize matterâ€”starting with the perspective of number, and then proceeding to the aspect of space, the kinematic aspect, and eventually the physical aspect of reality. The implication of this is that the positivistic appeal to *sense data* is problematic, because the theoretical â€˜toolsâ€™ employed in the *description* of what is observed always utilize *terms* that are not susceptible to â€œempirical observationâ€ themselves. The moment we proceed from what has been *observed *to a *description* of what has been observed the positivist criterion collapses, because the terms employed in such a description derive from aspects that are not open to sensory perception. Can these modal aspects be observed in a sensory way. Can they be *weighed*, *touched*, *measured* or *smelled*? The answer must be negative, for they are not *things* but aspects of things (or rather aspects within which concretely existing things *function*). The first step positivism had to take in order to digest â€œsense dataâ€ theoretically has already eliminated the restriction of reliable knowledge to sense data!

The renowned physicist, Max Planck, who eventually became sharply critical of Mach’s positivism, distinguished between the real outside world, the world of the senses and the (theoretical) world of the science of physics, which he equates with the â€œphysikalisches Weltbildâ€ (the physical world picture) (Planck, 1973:208). The abstractions that belong to the â€˜Weltbildâ€™ are not sensorily perceptibleâ€”they embrace, according to him, the known law-conformities and concepts such as space, time and causality (see Vogel, 1961:149).

*Matter and the First Four Modes of Explanation*

When StegmÂ¸ller explains the problems attached to an understanding of the nature of matter the first four aspects of reality suddenly acquire a new significance. In the first place he distinguishes two global basic conceptions regarding the nature of matter, and he points out that currently these conceptions once again, as previously, occupy a prominent place in the discussions. He calls these two basic conceptions the *atomistic conception* and the *continuity conception*.^{14} Laugwitz also points out that insofar as physics subjects itself to auxiliary means from mathematics it cannot escape from the polarity between continuity and discreteness.^{15}

Suddenly the question concerning the infinite divisibility of matter once again occupies a central position, thus highlighting anew the important distinction between *physical space* and *mathematical space*. It is clear that this distinction between â€˜atomismâ€™ and â€˜continuityâ€™ is based upon number and space as the two most basic modes of explanation of reality. But this is not yet the end of the dependence upon unique modes of explanation. For according to StegmÂ¸ller these two conceptions were designed in order to bring to a solution the following two problems (StegmÂ¸ller, 1987:91):

- The apparent
*indestructibility*of matter; and, - The apparent or real limitless transformability of matter.

When these two problems are assessed in relation to one another it is immediately clear that they depend upon the third and fourth ontic modes of explanation given in reality, namely on the meaning of kinematic persistence (â€˜immutabilityâ€™) and physical changefulness (â€˜transformabilityâ€™).^{16}

As soon as we do this, the key points of our historical survey of physics are again brought into play a decisive conditioning role in our theoretical reflections. The â€œthing-nessâ€ of material entities once and for all transcends the limited nature of the unique angles of approach (modes of existence and modes of explanation) that served our understanding of matter. Things function at once within all these modes and yet, in spite of this aspectual many-sidedness, things are never exhausted by any one of these modal aspects. And it seems that the *mystery* surrounding material entities derives from this multi-aspectual *but-at-once more-than-merely* aspectual nature of such entities.

It is precisely this more-than-merely-aspectual-nature of material things that sheds a negative light on any *monistic* attempt to develop a â€œtheory of everything.â€ Greene, for example, wants a framework that will combine all insights into a seamless whole, into a ‘single theory that, in principle’ is capable of describing all phenomena (Greene, 2003: viii). He indeed presents â€œsuper string theoryâ€ as the â€œUnified Theory of Everythingâ€ (Greene, 2003:15; cf. also pp.364-370, 385-386). However, he does not realize that although he has a *purely physical theory* in mind, the *meaning* of the physical aspect of reality inherently points beyond itself to its inter-modal coherence with other aspects, first of all with those aspects that are foundational to the physical aspect (namely the aspects of number, space, and movement). Even the way in which he phrases his goal cannot escape from terms that have their original seat within some of these aspects. Just consider his reference to a â€œseamless wholeâ€ and his use of the quantitative meaning of number while referring to a â€œsingle theory.â€ The idea of â€œeverythingâ€ also makes an appeal to the quantitative meaning of the *one *and the *many*. Likewise the idea of a â€œseamless wholeâ€ reflects the core meaning of space (i.e. *continuous extension*) which underlies our awareness of *wholeness* and of *coherence* (â€œseamlessâ€).

The unity of physical (material) entities can never be found in one privileged or elevated mode of explanation. Only a non-reductionistic ontology can liberate us from this untenable â€œunity-ideal.â€ Material entities exceed the confines of every modal aspect in which they functionâ€”thus underscoring the crucial role of acknowledging *unity and diversity* also within the domain of the discipline of physics.

Accepting the unity and diversity within creation presupposes both the acknowledgement of God as Creator and the insight that nothing within creation can be elevated to serve as a substitute for God. Reifying or divinizing any aspect always lead to theoretical antinomies. Only when a non-reductionist ontology is employed is it in principle possible to do justice to the unity and diversity within creation.

**Endnotes**

^{1} â€œDie hier gewonnenen Ergebnisse wird man auch dann wÂ¸rdigen, wenn man nicht der Meinung ist, daï¬‚ die Â¸blichen Methoden der klassischen Analysis durch andere ersetzt werden sollen. Zuzugeben ist, daï¬‚ die klassische BegrÂ¸ndung der Theorie der reellen Zahlen durch Cantor und Dedekind keine restlose Arithmetisierung bildet. Jedoch, es ist sehr zweifelhaft, ob eine restlose Arithmetisierung der Idee des Kontinuums voll gerecht werden kann. Die Idee des Kontinuums ist, jedenfalls ursprÂ¸nglich, eine geometrische Idee. Der arithmetisierende Monismus in der Mathematik ist eine willkÂ¸rliche These. Daï¬‚ die mathematische Gegenstâ€°ndlichkeit lediglich aus der Zahlenvorstellung erwâ€°chst, ist keineswegs erwiesen. Vielmehr Iassen sich vermutlich Begriffe wie diejenigen der stetigen Kurve und der Flâ€°che, die ja insbesondere in der Topologie zur Entfaltung kommen, nicht auf die Zahlvorstellungen zurÂ¸ckfÂ¸hrenâ€ (Bernays, 1976:187-188).

^{2} A penetrating analysis is given by Clouser in terms of the hidden role of religious belief in theories (see Clouser, 2005).

^{3} â€œSo, wenn ich von der Vorstellung eines KË†rpers das, was der Verstand davon denkt, als Substanz, Kraft, Teil-barkeit usw., imgleichen, was davon zur Empfindung gehË†rt, als Undurchdringlichkeit, Hâ€°rte, Farbe usw. absondere, so bleibt mir aus dieser empirischen Anschauung noch etwas Â¸brig, nâ€°mlich Ausdehnung und Gestaltâ€ (Kant, 1781/1787-B:35).

^{4} â€œWenn in dem bestâ€°ndigen Flusse aller Dinge nichts Festes, Ewiges beharrte, wÂ¸rde die Erkennbarkeit der Welt aufhË†ren und alles in Verwirrung stÂ¸rtzenâ€ (Frege, 1884:VIIâ€”Einleitung).

^{5} â€das Prinzip der Konstanz der Lichgeschwindigkeit”â€”cf. Einstein, 1982:32.

^{6} Maier remarks that â€œDescartes und seine Schule” indeed pursued a â€œrein phoronomisches Idealâ€ and attempted to explain â€œalles Sein und Geschehen in der Welt lediglich aus Bewegungen.â€

^{7} â€œG. Galilei zâ€°hlt als primâ€°re Qualitâ€°ten der Materie arithmetische (Zâ€°hlbarkeit), geometrische (Gestalt, GrË†ï¬‚e, Lage, BerÂ¸hrung) und kinematische Eigenschaften (Beweglichkeit) aufâ€ (Hucklenbroich, 1980:291).

^{8} On October 9, 1687 Leibniz wrote in a letter that we â€œmust always explain nature mathematically and mechanicallyâ€ (Leibniz, 1976:38). In a footnote the Editor of Leibniz’s work writes that Leibniz’s approval of the corpuscular philosophy of Boyle ought to be understood as â€œany philosophy which explains physical events mechanistically or in terms of magnitude, figure, and motionâ€ (Leibniz, 1976:349, note 14).

^{9} For that reason we also number and space ought not to be seen as opposites as it was asserted by Lakoff en NË™Ã’ez (2000:324) owing to their inability to appreciate the unique and mutually cohering nature of these aspects.

^{10} â€œDiejenige Naturanschauung, die bisher der Physik die wichtigsten Dienste geleistet hat, ist unstreitig die mechanische. Bedenken wir, daï¬‚ dieselbe darauf ausgeht, alle qualitativen Unterschiede in letzter Linie zu erklâ€°ren durch Bewegungen, so dÂ¸rfen wir die mechanische Naturanschauung wohl definieren als die Ansicht, daï¬‚ alle physikalischen Vorgâ€°nge sich vollstâ€°ndig auf Bewegungen von unverâ€°nderlichen, gleichartigen Massenpunkten oder Massenelementen zurÂ¸ckfÂ¸hren lassenâ€ (Planck, 1973:53).

^{11} It is therefore strange that the contemporary physical scientist from Cambridge, Stephen Hawking, still writes: â€œThe eventual goal of science is to provide a single theory that describes the whole universeâ€ (Hawking, 1988:10).

^{12} â€œDie Tragweite einer strengen Unterscheidung phoronomischer (im folgenden kinematisch genannt) und dynamischer Argumente mË†chte ich an einem Beispiel erlautern, das … aus der Protophysik stammt. Die Aussage â€œein KË†rper kann seine Geschwindigkeit nur stetig â€°ndernâ€ kann von der modernen Physik nur dynamisch verstanden werden. Geschwindigkeitâ€°nderungen sind Beschleunigung, d.h. als Zweite Ableitung des Weges nach der Zeit definiert. Zeit wird von der Physik als ein Parameter behandelt, an dessen Erzeugung durch eine Parametermaschine (â€œUhrâ€) de facto bestimmte Homogenitâ€°tserwartungen geknÂ¸pft sind … Bezogen auf den Gang einer angeblich so ausgewâ€°hlten Parametermaschine kann eine KË†rper seine Geschwindigkeit deshalb nicht unstetig, d.h. mit unendlich groï¬‚e Beschleunigung â€°nderen, weil dazu eine unendlich groï¬‚e Kraft erforderlich wâ€°reâ€ (Janich, 1975:68-69).

^{13} â€œEr hat damit das Wesen der im Sinne der Thermodynamik â€œnicht umkehrbarenâ€ Vorgâ€°nge erkannt. Vom molekular-mechanischen Gesichtspunkte aus gesehen sind dagegen alle Vorgâ€°nge umkehrbarâ€ (Einstein, 1959:42).

^{14} â€œSelbst die beiden groï¬‚en Grundkonzepte Â¸ber die Natur der Materie stehen heute nach wie vor zur Diskussion, wenn auch mannigfaltig verschleiert hinter Bergen von Formeln. Diese beiden Grundkonzepte kann man als die *atomistische Auffassung *und als die *Kontinuumsauffassung *der Materie bezeichnenâ€ (StegmÂ¸ller, 1987:91).

^{15} â€œDie Physik, insofern sie sich mathematischer Hilfsmittel bedient oder sich gar der Mathematik unterwirft, kann an der Polaritâ€°t von Kontinuierlichem und Diskretem nicht vorbeiâ€ (Laugwitz, 1986:9).

^{16} The physicist Rollwagen holds the view that the â€œdualismâ€ of wave and particle introduced a new dimension, namely the â€œpossibility of the … mutual transformation of elementary energy-structuresâ€ (Rollwagen, 1962:10).

**Literature**

Bernays, P. 1976. Abhandlungen zur Philosophie der Mathematik. Darmstadt: Wissenschaftliche Buchgesellschaft.

Clouser, R.A. 2005. The Myth of Religious Neutrality: An Essay on the Hidden Role of Religious Belief in Theories. Notre Dame: University of Notre Dame Press (new revised edition, first edition 1991).

Descartes, R. 1965. A Discourse on Method, Meditations and Principles, translated by John Veitch, Introduced by A.D. Lindsay. London: Everymanâ€™s Library.

Dooyeweerd, H. 1997. A New Critique of Theoretical Thought, Collected Works of Herman Dooyeweerd, A Series Vols. I-IV, General Editor D.F.M. Strauss. Lewiston: Edwin Mellen.

Einstein, A. 1959. Autobiographical Notes. In: Albert Einstein, Philosopher-Scientist. Edited by P.A. Schilpp. New York: Harper Torchbooks.

Einstein, A. 1982. GrundzÂ¸ge der Relativitâ€°tstheorie. Reprint of the 1969 Braunschweig edition (original edition 1922). Wiesbaden: Friedrich Fieweg & Sohn.

Einstein, A. 1985. Relativity, the Special and General Theory. Bristol: Arrowsmith (reprint of the first 1920 translation).

Fraenkel, A., Bar-Hillel, Y., Levy, A. & Van Dalen, D. 1973. Foundations of Set Theory, 2nd revised edition. Amsterdam: North Holland.

Frege, G. 1979. Posthumous Writings. Oxford: Basil Blackwell.

Greene, B. 2003. The Elegant Universe. New York: W.W. Norton & Company.

Hawking, S.W. 1988. A Brief History of Time, London: Transworld Publishers.

Hilbert, D. 1970. Gesammelte Abhandlungen, Vol.3, Second Edition, Berlin: Verlag Springer.

Hucklenbroich, P. 1980. Der physikalische Begriff der Materie. In: Ritter, Historisches WË†rterbuch der Philosophie. Volume 5, Stuttgart: Schwabe & Co Verlag (pp.921-924).

Janich, P. 1975. Tragheitsgesetz und Inertialsysteem. In: Frege und die moderne Grundlagenforschung, red. Chr. Thiel, Meisenheim am Glan: Hain.

Kant, I. 1787. Kritik der reinen Vernunft, 1st Edition (references to CPR A). Hamburg: Felix Meiner edition (1956).

Kant, I. 1787. Kritik der reinen Vernunft, 2nd Edition (references to CPR B). Hamburg: Felix Meiner edition (1956).

Kleene, S.C. 1952. Introduction to Metamathematics. Amsterdam: North-Holland Publishing Company.

Kline, M. 1980. Mathematics, The Loss of Certainty, New York: Oxford University Press.

Lakoff, G. & NË™Ã’ez, R.E. 2000. Where Mathematics Comes From, How the Embodied Mind Brings Mathematics into Being, New York: Basic Books.

Laugwitz D. 1986. Zahlen und Kontinuum. Eine EinfÂ¸hrung in die Infinitesimalmathematik. Mannheim: B.I.-Wissenschaftsverlag.

Leibniz, G.W.L. 1976. Philosophical Papers. Edited by Leroy E. Loemker. Synthese Historical Library, Volume 2. Dordrecht-Holland: D. Reidel (first print 1969).

Lorenzen, P. 1976. Zur Definition der vier fundamentalen Meï¬‚grË†ï¬‚en. In: Philosophia Naturalis, Volume 16 (pp.1-9).

Maier, A. 1949. Die Vorlâ€°ufer Galileis im 14. Jahrhundert, Roma: Edizioni di Storia e letteratura.

Planck, M. 1973. Vortrâ€°ge und Erinnerungen, 9th reprint of the 5th edition. Darmstadt: Wissenschaftliche Buchgesellschaft.

Reid, C. 1970. Hilbert, With an appreciation of Hilbertâ€™s mathematical work by Hermann Weyl. New York: George Allen & Unwin.

Rollwagen, W. 1962. Das Elektron der Physiker. Munich: Max HÂ¸ber.

Stafleu, M.D. 1987. Theories at Work: On the Structure and Functioning of Theories in Science, in Particular during the Copernican Revolution, Lanham: University Press of America.

StegmÂ¸ller, W. 1970. Main Currents in Contemporary German, British and American Philosophy. Dordrecht: D. Reidel Publishing Company, Holland.

StegmÂ¸ller, W. 1987. HauptstrË†mungen der Gegenwartsphilosophie. Volume III, Stuttgart: Alfred KrË†ner Verlag.

Vogel, H. 1961. Zum Philosophischen Wirken Max Plancks. Seine Kritik am Positivismus. Berlin: Akademie-Verlag.

Von Fritz, K. 1965. Die Entdeckung der Inkommensurabilitâ€°t durch Hippasos von Metapont. In: Becker, 1965:271-307.

Von Weizsâ€°cker, C.F. 2002. Groï¬‚e Physiker, Von Aristoteles bis Werner Heisenberg. MÂ¸nchen: Deutscher Taschenbuch Verlag.

Wang, H. 1988: Reflections on GË†del. Cambridge Massachusetts: MIT Press.

Weyl, H. 1970. David Hilbert and His Mathematical Work. In: Reid, 1970 (pp. 243-285).