# The ontological status of mathematical entities: the necessity for modern physics of an evaluation of mathematical systems.

FAR FROM BEING A PURELY ESOTERIC CONCERN of theoretical
mathematicians, the examination of the ontological status of
mathematical entities, I submit, has far-reaching implications for a
very practical area of knowledge, namely, the method of science in
general, and of physics in particular. Although physics and mathematics
have since Newton's second derivative been inextricably wedded,
modem physics has a particularly mathematical dependence. Physics has
moved and continues to move further away from the possibility of direct
empirical verification, primarily because of the increasingly complex
logistical problems of experimentation within the parameters of the very
large and of the very small. As certain areas become more and more
theoretical, with developments of this century in astrophysics,
cosmology, and quantum mechanics, and more specifically, with the
postulation of new hypothetical elementary particles based almost
exclusively upon mathematical data, physics is forced to depend
increasingly upon mathematics as a method for verifying physical
possibility. Typically, a mathematical formulation descriptive of an
empirically established phenomenon x is manipulated and made subject to
derivation on the assumption that the new formulation will continue to
correspond with physical reality, and may even yield new information
about the phenomenon's behavior. Why, however, should a coherence
between the empirically-defined world and mathematical processes be
assumed? This coherence is, above all, dependent upon a hidden
metaphysically strong presupposition about the ontological status of
mathematical entities and their systems.

That there is a metaphysically strong presupposition of the sort to which I refer is not immediately obvious, and I would like here to address three common refutations of this position initially given. Perhaps the most immediate is the insistence that mathematics serves a purely descriptive function in the sciences, that it acts only as a kind of language. Although this characterization is certainly applicable in some cases, it cannot possibly justify the present use of mathematics to make hypotheses and predictions in physics. It cannot explain the prescriptive use of mathematics to verify and suggest physical possibility.

Assuming the prescriptive use of mathematics, another argument can be made that mathematics is simply logic, in its most absolute, noncontroversial tautological sense. Thus, the use of mathematics in physics simply ensures the same consistency, although in a much more easily manipulatable form, that would occur by our following out the implications of theories using what amounts to common sense reason, for it is obvious that our knowledge of physical reality (physics) must be limited by, or at least not be inconsistent with, our own mental principles of logic. Unfortunately, this tautological view of mathematics too is untenable; for besides its rather narrow view of the role of mathematics, it makes the mistaken assumption that mathematics as used in physics is in fact logical, never mind tautological. One need only think of the prominent use in physics of complex numbers and common surds such as the exponential function and pi to realize how many mathematical inconsistencies have been wholeheartedly embraced without question and with success. The intuitionist school of mathematics, very much concerned with consistency and solid grounding, deems the use of infinity as unacceptably anti-intuitive, yet where would its absence leave calculus, a veritable cornerstone of the foundations of physics? A merely tautological system would severely limit the present scope of the physical and even social sciences.

The final and most common argument against inherent assumptions of mathematical Platonism in physics is simply that of cold pragmatism, which claims that we use the mathematical systems that we use not because we endow them with any real ontological status, but because they are effective. They make possible certain coherent explanations of the world around us; if they did not, we would not hesitate to adopt the next expedient mathematical system that did. This view is problematic on many different levels. First, the adoption of more than one mathematical system within one body of knowledge threatens to make any kind of coherence impossible. The movement in modem physics is currently toward simplification, the search for the grand unification theory that will tie all of the area-specific theories together, yielding consistency within the large mass of knowledge known as physics. While quantum mechanics uses four-dimensional Riemann-space, the theory of relativity operates in infinitely dimensional Hilbert space. Most of classical mechanics functions with Euclidean geometry. For the grand unification theory to be successful, it will be necessary to reconcile the three geometries so they fit within one large coherent and structurally sound system. The basis for the physicist's optimism in this project lies in his or her usually unexamined assumption of a Platonistic ideal of mathematics, that is, that there is only one true system, and that it corresponds to reality as he or she knows it. Differing competing mathematical processes that describe one area of physics or set of physical phenomena, even more so than differing processes describing areas on opposite ends of the size spectrum (as in the case of quantum mechanics and relativity theory), would wreak havoc on any attempt to synthesize the whole of this vast field of knowledge.

Second, "effective" does not mean "true." This in itself would not be such a large problem (we can, after all, be more modest in our expectations) except for the presence of still a third problem, namely, that there exists the possibility that one is simply limiting one's array of physical hypotheses by the kind of mathematical questions one chooses to pose, for these questions are a product of the particular mathematical system being used. If one poses a physical question in strictly mathematical terms, one should see the assumptions inherent within that particular mathematical description reflected in the final physical outcome (an outcome, in fact, which is supposed to validate its mathematical vehicle). Since there is the potential for one's mathematically defined hypothesis to determine the result as much as the scientific reality does, one finds that the term "effective" is not particularly reassuring in this context. The issue is being approached backward. As the mathematician Morris Kline has stated:

The disagreements concerning what correct mathematics is and the variety

of differing foundations affect seriously not only mathematics

proper but most vitally physical science. As we shall see, the most well-developed

physical theories are entirely mathematical.... Hence scientists,

who do not personally work on foundational problems, must

nevertheless be concerned about what mathematics can be confidently

employed if they are not to waste years on unsound mathematics.(1) Physicists must first judge which kinds of mathematical questions are acceptable and which are not before they can even begin to evaluate the answers to these questions.

My argument is that the mathematical assumptions upon which much of modem theoretical physics is grounded are characteristically Platonic, in the sense that the language of mathematics is thought to correspond to the actual physical order of the universe. By positing an actual connection between the external world of physics and the seemingly internal mental world of mathematics, we can deduce certain truths about that external world by the internal method. Thus, both physical and mathematical truths are implicitly understood as "discovered," and are meant to reflect more than merely the discoverer's own private creative process.

The problem is that while physics implicitly makes certain assumptions regarding the objectivity of its mathematical vehicle, allowing for only one true description, mathematicians are not at all unanimous in their support of the Platonic school. In fact, there are currently four general schools of thought. of which the Platonic school is but one. The scenario of equally valid yet conflicting mathematical systems (leading to conflicting results or predictions) has the potential to create a conflict which physics is at this time ill-equipped to face.(2) The Nobel prize-winning physicist Eugene Wigner worried that eventually,

The argument could be of such abstract nature that it might not be possible

to resolve the conflict, in flavor of one or of the other [mathematically

derived] theory, by an experiment. Such a situation would put a

heavy strain on our faith in our theories and on our belief in the reality

of the concepts which we form. It would give us a deep sense of frustration

in our search for what I called the "ultimate truth." The reason

that such a situation is conceiving is that, fundamentally, we do not

know why our theories work so well.(3) Physicists must subject their choice of mathematics to foundational scrutiny in order to be secure in their own foundations of method.

This kind of scrutiny requires an evaluation of the different mathematical schools of thought, and the implications of each for the method of physics. The most fundamental ideological division between them is whether they understand their project as the creation of mathematical systems or their discovery. Is the order that we see, the symmetry, a product solely of our minds, or is it rather a real reflection of the fabric of the universe? For purposes of clarity, I propose first that we define three basic standards for judging the validity of mathematical, and by extension, physical systems. The first is internal coherence. This litmus test for validity merely prohibits self-contradiction: one part of the theory may not contradict another part, nor may different theories within one physical system be mutually exclusive. Initially, we may use as examples created mathematical systems, such as Hilbert's nonstandard analysis (which as of yet has no physical applications), or any other formalistic system; its only requirement is that it be internally consistent. The second candidate is external coherence (or "correspondence"), by which I mean that the theory must be consistent with observed (empirical) data, as well as be able to ground predictions with regard to that physical data. Absolute internal consistency assumes a secondary role, and is not as rigorous. Thus, a particular mathematical system, such as non-Euclidean geometry, must have practical applications to be valid by external coherence. The third candidate for criterion of mathematical validity, which I am describing loosely as "Platonic," requires both internal and external coherence, bridging the gap between human intuition and experience.

In addition, there are two different senses of which we may speak of the "closure" of mathematical systems. The first is the more open of the two (as well as the more useful), and states: Given x finite assumptions, every intelligible question has an intelligible answer with respect to x. The second, more restrictive closure entails that given x finite assumptions, there can be drawn y finite intelligible questions, as well as the corresponding intelligible answers with respect to x. The only difference between the two is the restriction on the number of questions; the second closure requires not only a finite number of assumptions, but also that there will be a limited number of questions arising from these assumptions. The second type of closure, being axiomatic, seems to be restricted by Godel's proof; the first type does not. Likewise, the first two general candidates of internal and external coherence allow for the possibility of competing systems, while the third requires that the existence of only one particular system be necessary, thereby excluding all others. Obviously, one can see that a method which yields necessary results as opposed to merely possible ones would be deemed more useful by anyone seeking knowledge, if for no other reason than that it provides more clues. We must, however, question to what extent our security of necessary results may actually be a false security. By virtue of what, other than pure pragmatism at best, or arbitrariness at worst, do we decide that one method is superior to another?

We may very well ask whether a Platonistic understanding of mathematics, even if merely a myth, is a necessary myth for physics. Put another way: even if a Platonistic understanding is not correct, can the method of physics function without at least postulating the Platonistic understanding hypothetically, and then correcting for the hypothetical element by later empirical verification of the results? What are the limitations imposed by the scientific method on physics and physical knowledge? For mathematical derivations to be of use in predicting empirical phenomena, we must make certain assumptions not only about the validity of mathematical coherence, but about the coherence, or order, of the physical world. The latter assumption is evident in the physicists' project itself; if physical properties changed constantly and randomly, then there would be no laws of nature to discover.(4) A pragmatic definition of "order" requires a reference to an ability to predict. If mathematics (intuitively rational) represents order, and the physical world is in fact ordered, then mathematical order should be able to be used to predict occurrences and properties in the physical world. The essence of Platonism (with a metaphorical rather than the literal interpretation of his theory of Forms) is the necessary joining of the intuitive and the empirical.

The Platonic understanding of mathematics can be characterized as having mathematical entities linked to perfect templates of their imperfect representations; it is therefore desirable first to discuss what is meant by Plato's perfect templates. the Forms. They first appear as well-defined entities in the Phaedo and the Republic, although even in his earlier dialogues, where the theory of Forms is not yet full-blown, Plato refers to nonsensible "things," such as the better (Gorgias 489d) and holiness (Euthyphro 6d). Plato makes the strong claim that we do not simply require the Forms in a kind of metaphorical sense in order to understand particulars, but that these particulars actually do participate in the Forms. White explains that "objects in the world somehow fall out into "natural kinds," and that certain ways of collecting things together yield genuine groups and others do not, quite independently of the way in which we ourselves happen to classify things."(5)

The Platonic dialectic, by which we are able to get closer to understanding the Forms, requires that we seek definitions. This process involves moving from the particular to the more general. Admittedly, there is a circular aspect to this attempt: the way to understand the Forms is to come closer to their true definitions. This requires one to engage in a continual process of clarification, yet the only way that one knows if a particular definition is correct is through a comparison of it with the Forms. We can resolve the issue at least partially by moving from an intuitive (remembering) knowledge to intellectual (creative) knowledge. Even after we have moved past the problem of finding a correct definition, however, the role of the definition remains ambiguous, for it is unclear whether, once a definition is found, one can logically deduce truths about the Forms. Or, once one comes close enough to the Forms through definitions, can one then intuit information independently of that definition?(6)

Perhaps the characteristic of the Forms most crucial to mathematical Platonism is the theory of recollection presented in the Meno. Socrates tries to explain to Meno that knowledge, particularly knowledge about virtue, cannot be taught, but rather is remembered. In Meno 80-86 Socrates uses a slave boy as an example of how a person without instruction "knows" certain mathematical truths; in this case, the fact that the "square on the diagonal of the original square is double its area." This assumption is also that made by modern Platonic mathematicians, who believe that the purpose of mathematics is to discover an originally existing mathematical structure rather than to create it, and that any person possessing reason, regardless of time, place, or culture, may find it. Recollection provides an answer to the paradox of Meno's question,

But how will you look for something when you don't in the least know

what it is? How on earth are you going to set up something if you don't

know what is the object of your search? To put it another way, even if

you come right up against it, how will you know that what you have

found is the thing you didn't know?(7) Plato presents his theory of mathematics within the context of the Forms. One of the most explicit presentations of the Forms is given in Plato's discussion of the divided line (Republic 509d-511c). Ideas are described as eternal and one, while things are characterized as temporal and multiple. Mathematical entities are somewhere in the middle, being both eternal and multiple.

According to Plato, mathematicians seek to understand at the abstract level of the Forms: not the individual imperfect manifestations of a man-made square here or a diagonal appearing in nature there, but rather the "square and the diagonal itself."(8) The relationship of mathematics to the Forms stands in contrast with that of the dialectic, which also attempts to take one to a final Form or Idea. In comparison to the dialectic, the mathematical method is somehow less pure, for although both begin with hypotheses, the dialectic aims at establishing irrefutable premises, while mathematical premises always remain at the level of hypotheses (Republic 510c). The second major difference between dialectic and the mathematical method is that, unlike dialectic, the mathematical method is dependent on sensibles, although mathematics itself is not about sensibles but rather about the Forms. In other words, although "square itself' is not actually a square, we begin to conceive the former through visual contact with the latter's imperfect exhibition of the Forms. This connection between geometry and vision is later expanded on by Kant in the Critique of Pure Reason,(9) on which I will elaborate later. In both cases, the fact that we are able to abstract and idealize geometric forms (which should not be confused with generalization from observed objects), shows us that we must already have some underlying understanding of extension, limit, and symmetry so as to be able to categorize and synthesize our sensory perceptions. It is this "underlying understanding" that is remembered by the slave boy in the Meno. Robinson makes the claim that this process is "analogous to grasping a visible object by direct vision [as compared to] grasping it through its shadow or reflection."(10) Within the line's division of the intelligible and the visible, mathematics therefore straddles the boundary between the two divisions. It is dependent on both the intelligible (since it deals with Ideas) and the visible (since it requires vision at least to initiate the inquiry). Unlike geometry, however, other areas of mathematics, such as number theory and set theory, do not seem to be dependent on vision or any other sense. It could be argued that number theory is grounded in the same division of one and two that the dialectic depends upon for its binary disjunctions, while the dialectic's dependence on categorization is also the basis of set theory. To my knowledge, however, Plato does not discuss the problem except in the context of geometry.

In the metaphor of the Cave, mathematics floats in an intermediate stage. Robinson states:

In the figure of the cave everything from the first moment of the prisoners'

release to the last moment of looking at shadows and reflections

outside the cave is mathematics, and everything from the first look at

the real things outside the cave to seeing the sun is dialectic. It follows

that the state of the unreleased prisoners is everything below mathematics.(11) Because of the two criticisms given of geometry--of its uncertified hypotheses and its dependence on sensation--Plato makes the implicit claim at the end of his discussion of the line that mathematics is preferable to science (because science depends even more on the empirical, which provides not only its method, but also its subject matter), and also that mathematics itself would have a higher status as knowledge if it were to give a logos to its hypotheses (Republic 511c). How it is to do so is unclear, but perhaps Plato would for this reason put the less visually intuitive arithmetic and set theory on a higher rank than geometry.

It is important to note that Plato's understanding of mathematics was highly influenced by Pythagoras.(12) Unlike Aristotle, both Plato and Pythagorus believed that mathematics and its laws can act as a basis for other kinds of knowledge, such as morality or aesthetics. In his dialogues, Plato examines the question of what constitutes mathematics; he also uses it as a tool, providing illustrations, analogies, and even direct correlations with his treatment of other subjects. In the Republic, he compares temperance to harmony (books 2-4), uses the divided line's mathematical proportions to explain different kinds of knowledge (book 6), and enumerates the desirability of lives of each type in the tyrant's number (books 8-9). As in the Meno, Plato uses geometrical diagrams frequently. This conflation of mathematics with other kinds of knowledge is crucial because it is a precursor to the interrelatedness of mathematical perfection and science that would later be taken almost for granted, from the Ptolemaic system's pure circular orbits to the elegant simplicity of Maxwell's equations for the electromagnetic field. The thinkers of the Enlightenment also had a peculiar fascination with mathematics that went beyond the realm of science, and used mathematical models to explain both philosophy and society.

Today, Platonism in a modified form occupies a place among the major competing theories about the nature of mathematics, although absolute Platonism has proved to be untenable under the Russell-Zermelo paradox.(13) Unlike the formalists, such as Whitehead and Russell, who found the need to dismantle all of Platonism, some realized that they could keep most Platonic underpinnings simply by imposing a few restrictions. The chief restriction was to reject the so-called principle of totality, the assumption that there exists a totality of mathematical entities (the second type of closure earlier defined). Subsequent modifications along these lines include those of Kronecker and Brouwer, who proposed the further elimination of the assumption of a presupposed totality of integers. The rejection of the principle of totality would seem to have widespread physical implications, for it in fact rejects the notion on which infinite series problems are based, as well as the foundations of calculus. Without it, one cannot look at the "end" of an infinite process, but only continue with a potentially infinite number of finite repetitions (of the n + 1 variety). Bernays explains,

A general theorem about numbers is to be regarded as a sort of prediction

that a property will present itself for each construction of a number;

and the affirmation of the existence of a number with a certain property

is interpreted as an incomplete communication of a more precise proposition

indicating a [particular] number having the property in question

or a method for obtaining such a number; Hilbert calls it a "partial

judgement."(14) In Bernays' statement we can see two different interpretations of infinity: as a closed system (Platonic) and as an open, repetitive system (formalistic). Hilbert also describes them respectively as "actual infinity" versus the "potential infinite."(15) Put in the context of Plato's theory of recollection of the Forms, we can say that while Euclid's system of axioms is described by Euclid himself as constructed (that is, we can construct a line from point A to point B), Hilbert's axiom system is set in the language of discovery (that is, between any two points there will be found one straight line).

So far we can see that there are evidently at least three distinct strains in contemporary mathematics, each assuming a greater degree of necessity, and each claiming a stronger independent ontological position. The first is that of the formalist, who believes that we can produce many mathematical systems, none more correct than the other, as long as they all remain logically consistent. The second postulates that, given our existence, mind structure, and so forth, we can conceive of mathematical entities in only one way. This Kantian(16) approach I will classify as "intuitionist." although it contains elements from both the conceptualist and intuitionist schools. The third approach is Platonic, holding that mathematical entitles exist in only one way, before and independently of our existence. perhaps within the very pattern of the universe. This. as mentioned before, can be said to be the most important assumption made by theoretical physicists in using mathematical representations to discover nonempirically tested physical phenomena. Plato would undoubtedly be pleased to see the increasing dependence on mathematics in contemporary theoretical physics.

Having discussed mathematics from the ideological vantage point of discovery (Platonism), we now turn our attention to mathematical creation. The formalist and intuitionist mathematical schools of thought, to a respectively stronger and weaker degree, consider mathematics to be an expression of creativity. Since I hold, with John Barrow,(17) that creative enterprise necessarily entails personal uniqueness-that is, that two creators will most likely not accidentally create the same work (what is the probability, after all, of two authors spontaneously and independently writing Crime and Punishment?)--I believe the implications of an understanding of physics' underlying mathematical structure as created are fairly substantial. The principal problem is one of foundations: that the validity of the inference (earlier described as mathematical manipulation) conform to the expectation that the final mathematical result will correspond to a meaningful notion of physical reality. If the assumption of a relationship between physical reality and its mathematical language is untenable. then this kind of mathematical hypothesizing so useful to physics is untenable as well. The study of physics is not the study of logic; mere coherence leads to no answers without empirical pinning points. In the same way that physicists would be uncomfortable with the claim that they merely create physical laws out of their imaginations instead of discovering independent truths about the universe, the use of a created (individually subjective) mathematical system to ascertain these truths calls into question the entire validity of the physicists' project.

Perhaps one of the most influential leaders of the formalist school of mathematics was David Hilbert, who realized after numerous paradoxes had been found that it was necessary to establish the consistency of arithmetic by certain indisputable logical principles. In a speech given in 1925 before a congress of the Westphalian Mathematical Society, Hilbert lamented,

In the joy of discovering new and important results, mathematicians paid

too little attention to the validity of their deductive methods.... The

present state of affairs where we run up against the paradoxes is intolerable.

Just think, the definitions and deductive methods which everyone

learns, teaches, and uses in mathematics, the paragon of truth and

certitude, lead to absurdities!(18) Earlier, Whitehead and Russell had attempted this project of clarification as well in the Principia Mathematica. Both projects ultimately fell prey to Godel's incompleteness theorem. As Paul Weiss has noted, the formalists ultimately were not creative, but functioned as a kind of cleanup crew for other mathematicians, inserting consistency and logical rationalizations for steps which were originally made with sometimes large intuitive gaps.(19) Although some axiomatic attempts, such as Hilbert's, aimed at starting from the very beginning, deriving all principles independently from metamathematical starting-points, most formalist enterprises concentrated exclusively upon establishing adequate proofs using only acceptable criteria for validity. Thus, they argued, mathematics would be grounded in absolute certainty.

There is, however, a very obvious problem with this formalist presupposition, which is that the "undisputed" metamathematical principles can hardly be assumed to be quite so clear-cut. The formalists are caught in a philosophical quagmire of question begging when asked to defend their criteria for an acceptable first principle. Even given a first principle, what then determines logical criteria for deriving one axiom from the next? The actual rules which the formalist uses to obtain rigorous consistency must themselves be justified by rigorously consistent analysis, whose rules must in turn be justified with rigorous consistency, resulting in either an infinite regress or a nonrigorously justified grounding point. The formalist Hilbert vehemently rejected Brouwer and Weyl's rather vague justification by virtue of "intuition"; yet ultimately one must ask what his metamathematical logical principles are based on if not intuition. It becomes apparent that both camps are arguing only about the boundaries of justification, while both make the same intuitionist assumptions.

It is actually more accurate to classify intuitionism as a mathematical method of discovery and not of creation; yet the discovery involved is internal psychological discovery rather than external physical discovery. Our intuitive knowledge is, in Kantian terms, correspondent to reality, but cannot be assumed to be a clear reflection of it. It is in many ways just as indicative of the structure of our minds as it is of the object that our minds seek to capture. As we have shown in our discussion of the formalist enterprise, when conducting a "rational reconstruction of knowledge"(20) starting from a single foundational principle, one will find that an infinite regress is inevitable, for one is always required to justify belief in the reasons which justify the final belief Otherwise, the fundamental truths become arbitrary. To avoid the infinite regress, it has been most commonly stipulated that the foundational beliefs be intuitive ones, requiring no rational justification. This too has complications. Even under a fairly broad interpretation of "requiring no reasons" and in spite of our use of intuitive metamathematical starting points. we must start with what is truly intuitive. Otherwise, we would base our entire construction of mathematics on a mere individual assertion which could be later disputed at whim. Unfortunately, there is no way of checking intuitive belief Even Quinton's "authoritative, laborious, and inferential way of justifying the beliefs in question"(21) is Meaningless without the assumption that the premises are correct in some sense. This caveat, of course, renders the process vicious.

A related problem with designating first beliefs as intuitive is that according to the psychological definition of "intuitive," all that is required for a belief to be so designated is the (again, intuitive) knowledge that one has neither a rational justification for the belief nor any way of providing such a justification. There always remains the possibility, however, that the intuitor is wrong, for he or she could be ignorant, or lacking in the appropriate insight. Perhaps another person might be able to justify those same beliefs rationally; in any case, it must always remain intelligible to imagine that there might exist some rational justification for the intuition of x, although no one may have thought of it yet. Lawrence Bonjour goes further along these lines and suggests that many beliefs, which are easily taken for granted as self-evident are in fact "depend[ent] for their justification on inferences which have not been explicitly formulated and indeed which could not be explicitly formulated without considerable reflective effort."(22) This belief would be supported by those formalists who have tried to go back and justify already existing mathematical proofs that are deemed useful and intuitively correct but were not rigorously derived. Bonjour's view is perfectly in line with our stated purpose, the "rational reconstruction of knowledge," and, like both our definitions of system closure, it also assumes the existence of an intelligible answer to every intelligible question.

Many epistemologists see the continuation of the debate surrounding foundationalism as being kept alive by the problem of epistemic regression. Foundationalism requires that there be some nonarbitrary grounding somewhere in the epistemic chain of justification in order that any belief have meaning. Bonjour distinguishes between two popular versions of foundationalism: "strong foundationalism," which assumes the existence of epistemically valid termination (grounding) points; and "weak foundationalism," which holds that beliefs may have a partial grounding, but may be additionally justified by inference. The larger the number of these hybrid beliefs, the greater the degree of knowledge. Thus, supposedly, the infinite regress is somehow magically avoided. As Bonjour himself points out, however, only a pseudoproblem has been solved, for one still has to explain how beliefs are ultimately grounded. If inference and coherence are supposed to lend validity to terminally grounded parts, why could they not be used as criteria alone? He sees the main concept of the "basic belief' as confused and fraught with contradictions, for the notion of intuition is itself illogical. He says, "Knowledge requires epistemic justification, and the distinguishing characteristic of this particular species of justification is, I submit, its essential or internal relationship to the cognitive goal of truth."(23) To accept such a belief without reason he calls "epistemically irresponsible."(24) Justification by its very definition implies the notion of reasonableness, to which personal intuition cannot answer.

Weak foundationalism also balks at the requirement of necessity and incorrigibility, characteristics which the mathematical intuitionist seems to assume. Though the refusal to require necessity and incorrigibility appears in some ways to be a reassuring compromise, ultimately it leads to a further problem with intuition: the consistency among the intuitions of different people that is required for intelligibility. If there is truly a first premise which is intuitive and correct, then everyone must agree on the choice of that particular premise, for it remains necessarily true, although not necessarily certain (that is, objectively and rationally true although it might not be known). Mathematical intuitionists must posit the existence of only one true mathematical system, or else admit to allowing the criteria for mathematical validity to be different for each person. This second option seems unacceptable. Assuming that everyone does agree on the designation of a particular foundational belief--a rather large assumption--how would one determine that the agreement is bona fide and not simply conventional? The theory of ostensive statements "argues that to avoid an infinite regress of explanation there must be a class of statements whose meaning is explained some other way, not by correlation with other statements but by correlation with the world outside language,"(25) as in the empiricist theory of mathematics. Here, then, foundationalism seems reduced to an empiricist correspondence theory of mathematical knowledge (external coherence). While the purely empiricist view of mathematics, which states that all mathematical structures are gleaned from experience with physical objects, has been suggested as an alternative by thinkers such as Jean Piaget,(26) it is a theory that is highly unpopular and serves to explain only a very small portion of mathematics. Yet if neither an empirical nor a strictly formalistic approach is possible, we are left with a still rather vague notion of what intuitionist discovery actually is.

By claiming necessity, intuitionism hints at the strong metaphysical assumptions about mathematical entities that characterize Platonism, but ultimately can go only far enough to claim a psychological rather than an ontological necessity. The underlying metamathematical rules which end the infinite regress that plagues intuitionism can only consistently be attributed to the structure of our thinking patterns. Thus, the discovery process of intuitionism is inward and analogous to the outward discovery process of Platonism. Both claim necessity and objectivity, but does intuitionism, like Platonism, require the possibility of just one right answer? As we have stated before, the answer to this question has crucial physical implications; for physics, in assuming a Platonistic application of mathematics, has no tools with which to discriminate between two equally useful systems that may, at least theoretically, yield radically different results. Only formalism openly allows for a plurality of valid yet mutually incompatible mathematical systems. Yet Morris Kline admits,

In defense of the formalist philosophy, one must point out that it is only

for the purposes of proving consistency, completeness, and other properties

that mathematics is reduced to meaningless formulas. As for

mathematics as a whole, even the formalists reject the idea that it is

simply a game; they regard it as an objective science.(27) This assumed objectivity provides a significant bridge between the seemingly opposed ideologies of mathematical discovery (Platonism, intuitionism) and creation (formalism). In the case of formalism, however, I do not believe that objectivity necessarily entails the presence of just one system if the "objectivity" is confined to the metamathematical rules of derivation only. Again, what will differentiate between different formalist systems will be simply the intuitions one accepts as starting points.

In response to the problem of competing mathematical systems, much has been made of the impact of non-Euclidean geometry upon the Kantian ideal of geometry, which can be described as containing elements of both Platonism and intuitionism. If Plato believed the realm of the mind and the realm of objective ontological reality to be one and the same, Kant adapted this strong metaphysics to include at least a nod to epistemological concerns. He lays out the foundations of his theory in the Critique of Pure Reason. Kant believes that it is absurd for us to be able to assume that we can understand reality in the pure ontological sense; what we do understand, however, is in some fundamental way connected (that is, integrally linked) with reality. This view has particular implications for his theories of space and time, and Kant's theory of space has great importance for the development of geometry in the nineteenth century. Gauss, in fact, reportedly refused to publish his paper on non-Euclidean geometry immediately because the philosophical air at the time was saturated by Kantians.(28) I shall attempt to demonstrate, however, that non-Euclidean geometry and Kant's philosophy are actually quite compatible. The connection between an objective reality and our own intuitionist principles is firmly entrenched in Kant's philosophy, and this is what links him closely to Plato, if only to Plato's more imperfect way of knowing described by the cave metaphor.

In spite of Gauss's nervousness, it is not so much the presence of an alternate theory (or mathematical system) that Kant would have found so disturbing, but rather the presence of two equally logically valid (internally coherent) and equally true physically interpretive theories. Kant's theory of geometry, Platonistic in the sense of holding that our mind's intuition corresponds to the outside world of reality, has no room for competing mathematical systems. Can physics allow for this kind of flexibility without striking a death blow to the overall coherence of the grand unification that is sought? This problem, thought initially to arise from the geometry of Gauss, Bolyai, and Lobatchevsky, has not made such unification impossible because their geometry is not truly competing with the standard Euclidean geometry. To date, the issue of competing mathematical systems has not yet been dealt with in physics, and thus has not yet been resolved.

In the "Transcendental Aesthetic" of the Critique of Pure Reason, specifically, in section 1 on "Space") Kant uses the subject of pure mathematics as the most perfect example of his synthetic a priori, consisting not of a strict formalism but rather of absolute logical consistency coupled with what he calls "rational intuition," a leap of understanding, or "insight" (in contrast to an "empirical intuition"). Although Kant claims that this knowledge is "necessary," it is different from the necessity of the tautologies produced by the analytic a priori. What the nature of its difference actually is, we will examine later. It is, in any case, a heading which denies any kind of empirical contribution. Kant makes it clear that

we shall understand by a priori knowledge, not knowledge independent

of this or that experience, but knowledge absolutely independent of all

experience.(29) While it might be supposed that this method applies to arithmetic, its claim is more tenuous for geometry, which seems to have a strong dependence on our empirically based conception of space. Kant would agree with the claim that there is an implicit connection between geometry and our understanding of space, yet he strongly denies our conception of space any epistemic grounding in the sensory world. This idea, though radically counterintuitive at first glance, proves crucial for his conviction that geometry and arithmetic fulfill the two requirements of the synthetic a priori, namely, necessity and universality.

When Kant wrote the Critique in the late eighteenth century the two characteristics of necessity and universality in mathematics seemed firmly entrenched and permanent in the Leibnizian literature. With the advent of a non-Euclidean geometry (and later, other alternate systems, such as Hilbert's), however, their foundations were shaken. If geometry and arithmetic are truly a priori, then there is the implication that there can be only one absolute form for their construction. To determine which form is mathematically correct (which may not necessarily mean physically accurate), a firm understanding of the basis for judgment must be established.

Given the reality of competing mathematical systems, each claiming to be free of any internal contradictions, we must again note that there can arise a multiplicity of systems based on the a priori as long as each is grounded on different premises, that is, on different primary axioms. This explanation is insufficient, however, given Kant's dependence on intuition as the sole starting point for our understanding of space and time, which, he claims, are the bases for any mathematical genesis. For intuition to have any kind of epistemological validity, it must lead only to true insight; therefore, a single insight into space and time must bring forth a single definitive insight into a single (non-arbitrary) mathematical structure. Thus, it is clear that to save the validity of the a priori as a meaningful process, one must be able to prove that a certain system of arithmetic and geometry (either [A.sub.1][G.sub.2] or [A.sub.2][G.sub.2], for instance) is the correct form or, alternatively, that our understandings of space and time are indeed grounded empirically--that is, that Kant was wrong and that [A.sub.n][G.sub.n] are not derived a priori at all, leaving no problem with a possible multiplicity. To save Kant we must agree that the true test for the validity of a mathematical structure is internal consistency and elegance rather than physical applicability. We must be able to decide which of the two, [A.sub.1][G.sub.2] or [A.sub.2][G.sub.2], is wrong and why, while maintaining that the ultimate basis of our knowledge of space and time is not empirical but rather intuitive. Perhaps the only way to save both the a priori and Kant is to be able to prove that one AG is more internally consistent than the other. Unfortunately, this is no small task, and may ultimately be inconclusive if the problem lies not with the a priori but rather with the synthesis.

Another approach is to suggest that the two geometries, while remaining synthetic a priori truths, are not mutually exclusive. We know that relativistic physics is not entirely incompatible with the laws of Newtonian physics because for most velocities, which are extremely small compared to the speed of light (c = 3 x [10.sup.10] m/s), in the Lorentz transformation (L' = L/[(1 - [v.sup.2]/[c.sup.2].sup.1/2]) the denominator goes to one, and L' = L.(30) The derivation of this equation, which indirectly created the necessity for non-Euclidean geometry, is, in fact, based on visible (Euclidean) geometry, a thought experiment constructed in ordinary three-dimensional space. It thus seems logical that the two geometries be related in some way, since for some circumstances at least (for instance, the everyday macroscopic world at slow velocities), Euclidean geometry remains accurate and useful. It seems absurd simply to dismiss Euclid's geometry so easily in light of the fact that it remains intuitive. Kant would see intuitivity as the final test of a mathematical system's veracity, but again we come to the question of how strong is the necessity of intuition?

When Kant says that "all mathematical inferences proceed in accordance with the principle of contradiction,"(31) he seems to be establishing its necessity through its a priori nature. But does the fact that it is noncontradictory mean that it is necessarily true? Of course not. I could say, for example, "My third child will be a daughter." This statement is not logically contradictory, but it is obviously not a statement of necessity, in contrast to an alternative statement, "My third child will be a four-sided triangle," which is necessarily impossible. Thus, it would perhaps be more accurate to say that the a priori process derives or creates all possible mathematical systems. What would be further needed to establish its necessity is the second step, namely, the synthetic or "intuition."

While the analytic a priori has the necessity of a tautology, the Kantian intuition appeals to a different kind of necessity. It seems contradictory to speak of more than one kind of necessity in this context; the only way that one could modify this modal category would be to assume that one type of necessity was stronger or weaker than the other, in which case they could not both be necessary in the true definition of the word. Part of the problem is that Kant at times seems to be flirting simultaneously with the two claims that spatial intuition is subjective, in the sense that it "underlies all outer appearances" (intuitions); and that it not only can but must correspond to the supposed objective domain." In other words, although empirical evidence could never bring about our understanding of space (Kant would say that it is our intuition of space that makes empirical data intelligible), neither should they contradict one another. Why is this?

The answer lies in the fact that Kant's intuition of geometry is inextricably tied to visualization, which, of course, is at least in part dependent on our ability to see. The fact that we are able to abstract and idealize geometric forms, however, shows us that we must have some underlying understanding of such fundamental properties as extension, limit, and symmetry with which to categorize and synthesize our random sensory perceptions.33 These fundamental properties are some of the rules by which our understanding is bound simply by being human; Kant alludes to these limitations when he begins his Introduction to the Critique with the sentence, "There can be no doubt that all our knowledge begins with experience."(34) Therefore, even when we are able through greater and greater abstraction and mathematical representation to derive scientific views of reality which are in conflict with our normal modes of perception, such as the Minkowski space-time in four dimensions, they are always conceptually explained in three-dimensional Euclidean space, with lines, warps, twists, and so forth. Thus it must be questioned whether the scientific reality that has been derived mathematically presumes to describe actual space independently of how we perceive it, at last free from our sensory limitations; or whether what it concludes is a mathematical analogy of our same three-dimensional understanding, but presented in a different way for mathematical simplicity in describing observed empirical phenomena. For while the theories, manipulations, and explanations may be possible in any dimension mathematically, experiments are actually always constructed and their results physically measured or interpreted by human beings with these conceptual limitations. Therefore physics, by virtue of being physics (requiring empirical verification subject finally to our senses, no matter how augmented they may be) and not solely mathematics (requiring no empirical verification), must always have one foot firmly grounded in our intuition of space, understood as Euclidean.(35) Since Kant recognizes correspondence with human intuition of space as the final test of a true geometry, the emergence of a non-Euclidean geometry should not affect his position. Furthermore, the emergence of this particular non-Euclidean geometry does not affect our position either; for since the two geometries are not actually competing to explain the same phenomena (not only because they are restricted to different parameters of magnitude, as with quantum versus classical mechanics, but also and more crucially because the structure of relativity theory allows for a connection with Euclidean geometry at some common level), we as of yet have no model with which to test the hypothesis of physically useful yet contradictory mathematical systems.

In summation, we review the possibilities for physics. The formalist seeks to give a clear account of mathematical structure, and to define which moves are acceptable and which are not. This would seem to be beneficial for physics; yet by allowing for a multiplicity of equally valid systems and by concentrating upon tautological consistency, the formalist introduces elements of incoherence into the larger structure of physics and limits enormously the scope of mathematics which is already used. The mathematical intuitionist shares the Platonistic assumption of a single true mathematics by requiring only moves that are intuitively obvious (being logically unambiguously specified by the previous step) and by postulating single true principles as starting points. Kronecker, for instance, referred to whole numbers as the "works of God," although the origins of the mathematical entities derived from these whole numbers were less well articulated.(36) There is a question as to whether the mathematical intuitionist is involved in an epistemological or psychological endeavor, or whether the objectivity for which he or she strives goes beyond his or her own existence. We also must further inquire as to the possibility of more than one valid mathematical system derived on intuitionist principles by mathematicians with different intuitions. This would, again, introduce physical incoherence. The mathematical formalist evaluates seemingly without foundations--an approach that initially bypasses the problem of infinite regress--yet we find that the formalist's evaluative structures of derivation have what are ultimately intuitionist foundations. Even the empiricist must demonstrate by what virtue different manifestations fall under one abstraction coherently. To use the language with which we began, the physicists' project requires a mathematics which possesses both internal and external coherence; that is, it must be internally consistent, and its propositions must not be reduced to "meaningless formulae." Rather, they should be able to describe the physical world.

It is my argument that the primary physical assumption made about mathematics is one that is essentially Platonistic in form, and that the absence of this assumption would have a profound influence upon the structure of physics. Physics would then have to discriminate between two equally useful or elegant systems, while the adoption of either system could alter the course of physical examination because each system possesses differing processes of derivation. This is a discrimination for which physics is as of yet ill-equipped, and which it has not yet had to make. Even accepting the Platonic view, physicists will still need to determine their own boundaries for the Principle of Totality, and thus to decide which mathematical entities and processes they will understand as valid. Although the old, more empirically based physics also left us problems concerning the fittingness of its mathematical language, it is the fundamental change in method, with a renewed emphasis (harkening back to the Greeks) on the aesthetic and the theoretical, that makes the investigation of hitherto unquestioned physical assumptions about the mathematics it utilizes mandatory.(37) (1) Morris Kline, Mathematics: The Loss of Certainty (Oxford: Oxford University Press, 1980), 7. (2) For an example of a physical question to which there are two answers that are equally valid mathematically, one of which is physically invalid, see the "coconut puzzle" in John D. Barrow, The World Within the World (Oxford: Oxford University Press, 1988). 254. (3) Eugene P. Wigner, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," Communications in Pure and Applied Mathematics 13, no. 1 (February 1960): 13-14. (4) The discovery of chaos theory and quantum physics does not in any way invalidate this project. The indeterminancy or "uncertainty" has not spread like a plague, engulfing old secure areas of physics and rendering them useless. Our uncertainty, as a matter of fact, is rather well-behaved, found consistently and constrained between very tightly mathematically defined limits. (5) Nicholas P. White, Plato on Knowledge and Reality (Indianapolis: Hackett Publishing Company, 1976), 7. (6) Cf. White, Plato on Knowledge and Reality, 38. (7) Meno 80d, trans. W. K C. Guthrie in Plato: The Collected Dialogues, ed. Edith Hamilton and Huntington Cairns (Princeton: Princeton University Press, 1987). (8) Republic 510d7, trans. Paul Shorey in Collected Dialogues. (9) Immanuel Kant, Critique of Pure Reason, trans. Norman Kemp Smith (New York: St. Martin's Press, 1965). A27. (10) Richard Robinson, Plato's Earlier Dialectic (Ithaca: Cornell University Press, 1941), 207. (11) Ibid., 196. (12) Robert S. Brumbaugh, Plato's Mathematical Imagination (Bloomington: Indiana University Press, 1954), 5-6. (13) I am referring here to an inconsistency in set theory as related to self-referential sets, first proposed by Bertrand Russell and Alfred North Whitehead. While it has been held that "the set of objects is not that object itself," in the case of self-referential sets the sets of some objects (such as "not-objects" or "concepts") are often also their objects. Hence the "paradox. For example: the set of not-objects is a not-object, the set of concepts is a concept, both cases breaking the rule. See Bertrand Russell, Principia Mathematica (George Allen and Unwin, 1903), 101. In 1908 Ernst Zermelo expanded upon Russell's paradox to include paradoxes in all of classical analysis; Ernst Zermelo, "Untersuchungen uber die Grundlagen der Mengenlehre I," Mathematische Annalen 65 (1908): 261-81. See also Paul Bernays, "Sur le platonisme dans les mathematiques," trans. C. D. Parsons as "On Platonism in Mathematics," in Philosophy of Mathematics: Selected Readings, ed. Paul Benacerraf and Hilary Putnam (Cambridge: Cambridge University Press, 1987), 261. (14) Bernays, "On platonism in mathematics," 262. (15) David Hilbert, "On the Infinite," trans. E. Putnam and G. J. Massey in Philosophy of Mathematics: Selected Readings, ed. Paul Benacerraf and Hilary Putnam (Cambridge: Cambridge University Press, 1987), 188. (16) See Kant, Critique of Pure Reason, A26/B42. (17) See John D. Barrow, "What is Mathematics?" in John D. Barrow, The World Within The World (Oxford: Oxford University Press, 1988), 553. (18) Hilbert, "On the Infinite," 190-1. (19) Paul Weiss, Creative Ventures (Carbondale: Southern Illinois University Press, 1992), 105. (20) The phrase is that of Anthony Quinton in his "The Foundations of Knowledge," in British Analytic Philosophy. ed. Bernard Williams and Alan Montefiore (London: Routledge and Kegan Paul. 1971), 56. (21) Ibid., 59. (22) Lawrence Bonjour, "Can Empirical Knowledge Have a Foundation?" American Philosophical Quarterly 15 (1978): 2. (23) Bonjour, "Can Empirical Knowledge Have a Foundation?" 5. (24) Ibid., 5. (25) Quinton, "The Foundations of Knowledge," 60. (26) See Jean Piaget, The Child's Conception of Number, trans. C. Gattegnot and F. M. Hodgson (London: Routledge and Kegan Paul, 1965); and Jean Piaget, Barbel Inhelder, and Alina Szeniska, The Child's Conception of Geometry, trans. E. A. Lunzer (London: Routledge and Kegan Paul, 1960). (27) Morris Kline, Mathematical Thought From Ancient to Modern Times (Oxford: Oxford University Press, 1972), 1208. (28) Kline, Mathematical Thought from Ancient to Modem Times, 871. (29) Kant, Critique of Pure Reason, A2. (30) Lorentz Transformation: L = length of object at zero velocity (the length normally observed), L' = observed length at high velocity, v = velocity, c = speed of light, the highest velocity possible--at which point matter becomes energy in accordance with the equation E = m[c.sup.2]. (31) Kant, Critique of Pure Reason, A10/B14. (32) Kant, Critique of Pure Reason, B39. (33) So far we have been using three-mensional perceived space and Euclidean geometry almost interchangeably. It would be an error to presume that everything included in Euclidean geometry can be sensed empirically, however. As defined by Euclid in his primary definitions, a point (position without extension) and a line (extension without breadth) would be impossible to "see," since vision requires extension in all three dimensions. An idealization is a visualization modified by intellectual understanding of given postulates that describe it, while a generalization is a mere averaging of seen objects, for instance, imagining the "perfect" circle after having seen a thousand "imperfect" ones. The reason that the view that our knowledge of geometric forms is obtained by generalization is incorrect is that we would never be able even to identify the imperfect circle as imperfect, much less see the connection between different imperfect circles, unless we already had the mental idealization of that circle. " Kant, Critique of Pure Reason, B1. This is a response, then, to our earlier question about the respective necessities of the analytic and the synthetic. The former contains an objective necessity: a tautology is necessary by virtue of its being known. The latter is a subjective necessity which arises from the relation of the knower to the proposition and makes no claims about the object's status independent of our knowledge of it. Even "objective" experimentation ultimately takes into account our perspective and relates the object back to us. Einstein's relativity theory demonstrated this idea graphically for the scientific world when it proved the absence of absolute space, that is, of a perspectiveless point of reference for all observers. (36) See Kline, Mathematical Thought from Ancient to Modern Times, 1197. (31) 1wish to express my thanks to Dr. Paul Weiss, Dr. Timothy Eastman, and Mr. Baruch Kfia for their helpful comments and criticisms.

That there is a metaphysically strong presupposition of the sort to which I refer is not immediately obvious, and I would like here to address three common refutations of this position initially given. Perhaps the most immediate is the insistence that mathematics serves a purely descriptive function in the sciences, that it acts only as a kind of language. Although this characterization is certainly applicable in some cases, it cannot possibly justify the present use of mathematics to make hypotheses and predictions in physics. It cannot explain the prescriptive use of mathematics to verify and suggest physical possibility.

Assuming the prescriptive use of mathematics, another argument can be made that mathematics is simply logic, in its most absolute, noncontroversial tautological sense. Thus, the use of mathematics in physics simply ensures the same consistency, although in a much more easily manipulatable form, that would occur by our following out the implications of theories using what amounts to common sense reason, for it is obvious that our knowledge of physical reality (physics) must be limited by, or at least not be inconsistent with, our own mental principles of logic. Unfortunately, this tautological view of mathematics too is untenable; for besides its rather narrow view of the role of mathematics, it makes the mistaken assumption that mathematics as used in physics is in fact logical, never mind tautological. One need only think of the prominent use in physics of complex numbers and common surds such as the exponential function and pi to realize how many mathematical inconsistencies have been wholeheartedly embraced without question and with success. The intuitionist school of mathematics, very much concerned with consistency and solid grounding, deems the use of infinity as unacceptably anti-intuitive, yet where would its absence leave calculus, a veritable cornerstone of the foundations of physics? A merely tautological system would severely limit the present scope of the physical and even social sciences.

The final and most common argument against inherent assumptions of mathematical Platonism in physics is simply that of cold pragmatism, which claims that we use the mathematical systems that we use not because we endow them with any real ontological status, but because they are effective. They make possible certain coherent explanations of the world around us; if they did not, we would not hesitate to adopt the next expedient mathematical system that did. This view is problematic on many different levels. First, the adoption of more than one mathematical system within one body of knowledge threatens to make any kind of coherence impossible. The movement in modem physics is currently toward simplification, the search for the grand unification theory that will tie all of the area-specific theories together, yielding consistency within the large mass of knowledge known as physics. While quantum mechanics uses four-dimensional Riemann-space, the theory of relativity operates in infinitely dimensional Hilbert space. Most of classical mechanics functions with Euclidean geometry. For the grand unification theory to be successful, it will be necessary to reconcile the three geometries so they fit within one large coherent and structurally sound system. The basis for the physicist's optimism in this project lies in his or her usually unexamined assumption of a Platonistic ideal of mathematics, that is, that there is only one true system, and that it corresponds to reality as he or she knows it. Differing competing mathematical processes that describe one area of physics or set of physical phenomena, even more so than differing processes describing areas on opposite ends of the size spectrum (as in the case of quantum mechanics and relativity theory), would wreak havoc on any attempt to synthesize the whole of this vast field of knowledge.

Second, "effective" does not mean "true." This in itself would not be such a large problem (we can, after all, be more modest in our expectations) except for the presence of still a third problem, namely, that there exists the possibility that one is simply limiting one's array of physical hypotheses by the kind of mathematical questions one chooses to pose, for these questions are a product of the particular mathematical system being used. If one poses a physical question in strictly mathematical terms, one should see the assumptions inherent within that particular mathematical description reflected in the final physical outcome (an outcome, in fact, which is supposed to validate its mathematical vehicle). Since there is the potential for one's mathematically defined hypothesis to determine the result as much as the scientific reality does, one finds that the term "effective" is not particularly reassuring in this context. The issue is being approached backward. As the mathematician Morris Kline has stated:

The disagreements concerning what correct mathematics is and the variety

of differing foundations affect seriously not only mathematics

proper but most vitally physical science. As we shall see, the most well-developed

physical theories are entirely mathematical.... Hence scientists,

who do not personally work on foundational problems, must

nevertheless be concerned about what mathematics can be confidently

employed if they are not to waste years on unsound mathematics.(1) Physicists must first judge which kinds of mathematical questions are acceptable and which are not before they can even begin to evaluate the answers to these questions.

My argument is that the mathematical assumptions upon which much of modem theoretical physics is grounded are characteristically Platonic, in the sense that the language of mathematics is thought to correspond to the actual physical order of the universe. By positing an actual connection between the external world of physics and the seemingly internal mental world of mathematics, we can deduce certain truths about that external world by the internal method. Thus, both physical and mathematical truths are implicitly understood as "discovered," and are meant to reflect more than merely the discoverer's own private creative process.

The problem is that while physics implicitly makes certain assumptions regarding the objectivity of its mathematical vehicle, allowing for only one true description, mathematicians are not at all unanimous in their support of the Platonic school. In fact, there are currently four general schools of thought. of which the Platonic school is but one. The scenario of equally valid yet conflicting mathematical systems (leading to conflicting results or predictions) has the potential to create a conflict which physics is at this time ill-equipped to face.(2) The Nobel prize-winning physicist Eugene Wigner worried that eventually,

The argument could be of such abstract nature that it might not be possible

to resolve the conflict, in flavor of one or of the other [mathematically

derived] theory, by an experiment. Such a situation would put a

heavy strain on our faith in our theories and on our belief in the reality

of the concepts which we form. It would give us a deep sense of frustration

in our search for what I called the "ultimate truth." The reason

that such a situation is conceiving is that, fundamentally, we do not

know why our theories work so well.(3) Physicists must subject their choice of mathematics to foundational scrutiny in order to be secure in their own foundations of method.

This kind of scrutiny requires an evaluation of the different mathematical schools of thought, and the implications of each for the method of physics. The most fundamental ideological division between them is whether they understand their project as the creation of mathematical systems or their discovery. Is the order that we see, the symmetry, a product solely of our minds, or is it rather a real reflection of the fabric of the universe? For purposes of clarity, I propose first that we define three basic standards for judging the validity of mathematical, and by extension, physical systems. The first is internal coherence. This litmus test for validity merely prohibits self-contradiction: one part of the theory may not contradict another part, nor may different theories within one physical system be mutually exclusive. Initially, we may use as examples created mathematical systems, such as Hilbert's nonstandard analysis (which as of yet has no physical applications), or any other formalistic system; its only requirement is that it be internally consistent. The second candidate is external coherence (or "correspondence"), by which I mean that the theory must be consistent with observed (empirical) data, as well as be able to ground predictions with regard to that physical data. Absolute internal consistency assumes a secondary role, and is not as rigorous. Thus, a particular mathematical system, such as non-Euclidean geometry, must have practical applications to be valid by external coherence. The third candidate for criterion of mathematical validity, which I am describing loosely as "Platonic," requires both internal and external coherence, bridging the gap between human intuition and experience.

In addition, there are two different senses of which we may speak of the "closure" of mathematical systems. The first is the more open of the two (as well as the more useful), and states: Given x finite assumptions, every intelligible question has an intelligible answer with respect to x. The second, more restrictive closure entails that given x finite assumptions, there can be drawn y finite intelligible questions, as well as the corresponding intelligible answers with respect to x. The only difference between the two is the restriction on the number of questions; the second closure requires not only a finite number of assumptions, but also that there will be a limited number of questions arising from these assumptions. The second type of closure, being axiomatic, seems to be restricted by Godel's proof; the first type does not. Likewise, the first two general candidates of internal and external coherence allow for the possibility of competing systems, while the third requires that the existence of only one particular system be necessary, thereby excluding all others. Obviously, one can see that a method which yields necessary results as opposed to merely possible ones would be deemed more useful by anyone seeking knowledge, if for no other reason than that it provides more clues. We must, however, question to what extent our security of necessary results may actually be a false security. By virtue of what, other than pure pragmatism at best, or arbitrariness at worst, do we decide that one method is superior to another?

We may very well ask whether a Platonistic understanding of mathematics, even if merely a myth, is a necessary myth for physics. Put another way: even if a Platonistic understanding is not correct, can the method of physics function without at least postulating the Platonistic understanding hypothetically, and then correcting for the hypothetical element by later empirical verification of the results? What are the limitations imposed by the scientific method on physics and physical knowledge? For mathematical derivations to be of use in predicting empirical phenomena, we must make certain assumptions not only about the validity of mathematical coherence, but about the coherence, or order, of the physical world. The latter assumption is evident in the physicists' project itself; if physical properties changed constantly and randomly, then there would be no laws of nature to discover.(4) A pragmatic definition of "order" requires a reference to an ability to predict. If mathematics (intuitively rational) represents order, and the physical world is in fact ordered, then mathematical order should be able to be used to predict occurrences and properties in the physical world. The essence of Platonism (with a metaphorical rather than the literal interpretation of his theory of Forms) is the necessary joining of the intuitive and the empirical.

The Platonic understanding of mathematics can be characterized as having mathematical entities linked to perfect templates of their imperfect representations; it is therefore desirable first to discuss what is meant by Plato's perfect templates. the Forms. They first appear as well-defined entities in the Phaedo and the Republic, although even in his earlier dialogues, where the theory of Forms is not yet full-blown, Plato refers to nonsensible "things," such as the better (Gorgias 489d) and holiness (Euthyphro 6d). Plato makes the strong claim that we do not simply require the Forms in a kind of metaphorical sense in order to understand particulars, but that these particulars actually do participate in the Forms. White explains that "objects in the world somehow fall out into "natural kinds," and that certain ways of collecting things together yield genuine groups and others do not, quite independently of the way in which we ourselves happen to classify things."(5)

The Platonic dialectic, by which we are able to get closer to understanding the Forms, requires that we seek definitions. This process involves moving from the particular to the more general. Admittedly, there is a circular aspect to this attempt: the way to understand the Forms is to come closer to their true definitions. This requires one to engage in a continual process of clarification, yet the only way that one knows if a particular definition is correct is through a comparison of it with the Forms. We can resolve the issue at least partially by moving from an intuitive (remembering) knowledge to intellectual (creative) knowledge. Even after we have moved past the problem of finding a correct definition, however, the role of the definition remains ambiguous, for it is unclear whether, once a definition is found, one can logically deduce truths about the Forms. Or, once one comes close enough to the Forms through definitions, can one then intuit information independently of that definition?(6)

Perhaps the characteristic of the Forms most crucial to mathematical Platonism is the theory of recollection presented in the Meno. Socrates tries to explain to Meno that knowledge, particularly knowledge about virtue, cannot be taught, but rather is remembered. In Meno 80-86 Socrates uses a slave boy as an example of how a person without instruction "knows" certain mathematical truths; in this case, the fact that the "square on the diagonal of the original square is double its area." This assumption is also that made by modern Platonic mathematicians, who believe that the purpose of mathematics is to discover an originally existing mathematical structure rather than to create it, and that any person possessing reason, regardless of time, place, or culture, may find it. Recollection provides an answer to the paradox of Meno's question,

But how will you look for something when you don't in the least know

what it is? How on earth are you going to set up something if you don't

know what is the object of your search? To put it another way, even if

you come right up against it, how will you know that what you have

found is the thing you didn't know?(7) Plato presents his theory of mathematics within the context of the Forms. One of the most explicit presentations of the Forms is given in Plato's discussion of the divided line (Republic 509d-511c). Ideas are described as eternal and one, while things are characterized as temporal and multiple. Mathematical entities are somewhere in the middle, being both eternal and multiple.

According to Plato, mathematicians seek to understand at the abstract level of the Forms: not the individual imperfect manifestations of a man-made square here or a diagonal appearing in nature there, but rather the "square and the diagonal itself."(8) The relationship of mathematics to the Forms stands in contrast with that of the dialectic, which also attempts to take one to a final Form or Idea. In comparison to the dialectic, the mathematical method is somehow less pure, for although both begin with hypotheses, the dialectic aims at establishing irrefutable premises, while mathematical premises always remain at the level of hypotheses (Republic 510c). The second major difference between dialectic and the mathematical method is that, unlike dialectic, the mathematical method is dependent on sensibles, although mathematics itself is not about sensibles but rather about the Forms. In other words, although "square itself' is not actually a square, we begin to conceive the former through visual contact with the latter's imperfect exhibition of the Forms. This connection between geometry and vision is later expanded on by Kant in the Critique of Pure Reason,(9) on which I will elaborate later. In both cases, the fact that we are able to abstract and idealize geometric forms (which should not be confused with generalization from observed objects), shows us that we must already have some underlying understanding of extension, limit, and symmetry so as to be able to categorize and synthesize our sensory perceptions. It is this "underlying understanding" that is remembered by the slave boy in the Meno. Robinson makes the claim that this process is "analogous to grasping a visible object by direct vision [as compared to] grasping it through its shadow or reflection."(10) Within the line's division of the intelligible and the visible, mathematics therefore straddles the boundary between the two divisions. It is dependent on both the intelligible (since it deals with Ideas) and the visible (since it requires vision at least to initiate the inquiry). Unlike geometry, however, other areas of mathematics, such as number theory and set theory, do not seem to be dependent on vision or any other sense. It could be argued that number theory is grounded in the same division of one and two that the dialectic depends upon for its binary disjunctions, while the dialectic's dependence on categorization is also the basis of set theory. To my knowledge, however, Plato does not discuss the problem except in the context of geometry.

In the metaphor of the Cave, mathematics floats in an intermediate stage. Robinson states:

In the figure of the cave everything from the first moment of the prisoners'

release to the last moment of looking at shadows and reflections

outside the cave is mathematics, and everything from the first look at

the real things outside the cave to seeing the sun is dialectic. It follows

that the state of the unreleased prisoners is everything below mathematics.(11) Because of the two criticisms given of geometry--of its uncertified hypotheses and its dependence on sensation--Plato makes the implicit claim at the end of his discussion of the line that mathematics is preferable to science (because science depends even more on the empirical, which provides not only its method, but also its subject matter), and also that mathematics itself would have a higher status as knowledge if it were to give a logos to its hypotheses (Republic 511c). How it is to do so is unclear, but perhaps Plato would for this reason put the less visually intuitive arithmetic and set theory on a higher rank than geometry.

It is important to note that Plato's understanding of mathematics was highly influenced by Pythagoras.(12) Unlike Aristotle, both Plato and Pythagorus believed that mathematics and its laws can act as a basis for other kinds of knowledge, such as morality or aesthetics. In his dialogues, Plato examines the question of what constitutes mathematics; he also uses it as a tool, providing illustrations, analogies, and even direct correlations with his treatment of other subjects. In the Republic, he compares temperance to harmony (books 2-4), uses the divided line's mathematical proportions to explain different kinds of knowledge (book 6), and enumerates the desirability of lives of each type in the tyrant's number (books 8-9). As in the Meno, Plato uses geometrical diagrams frequently. This conflation of mathematics with other kinds of knowledge is crucial because it is a precursor to the interrelatedness of mathematical perfection and science that would later be taken almost for granted, from the Ptolemaic system's pure circular orbits to the elegant simplicity of Maxwell's equations for the electromagnetic field. The thinkers of the Enlightenment also had a peculiar fascination with mathematics that went beyond the realm of science, and used mathematical models to explain both philosophy and society.

Today, Platonism in a modified form occupies a place among the major competing theories about the nature of mathematics, although absolute Platonism has proved to be untenable under the Russell-Zermelo paradox.(13) Unlike the formalists, such as Whitehead and Russell, who found the need to dismantle all of Platonism, some realized that they could keep most Platonic underpinnings simply by imposing a few restrictions. The chief restriction was to reject the so-called principle of totality, the assumption that there exists a totality of mathematical entities (the second type of closure earlier defined). Subsequent modifications along these lines include those of Kronecker and Brouwer, who proposed the further elimination of the assumption of a presupposed totality of integers. The rejection of the principle of totality would seem to have widespread physical implications, for it in fact rejects the notion on which infinite series problems are based, as well as the foundations of calculus. Without it, one cannot look at the "end" of an infinite process, but only continue with a potentially infinite number of finite repetitions (of the n + 1 variety). Bernays explains,

A general theorem about numbers is to be regarded as a sort of prediction

that a property will present itself for each construction of a number;

and the affirmation of the existence of a number with a certain property

is interpreted as an incomplete communication of a more precise proposition

indicating a [particular] number having the property in question

or a method for obtaining such a number; Hilbert calls it a "partial

judgement."(14) In Bernays' statement we can see two different interpretations of infinity: as a closed system (Platonic) and as an open, repetitive system (formalistic). Hilbert also describes them respectively as "actual infinity" versus the "potential infinite."(15) Put in the context of Plato's theory of recollection of the Forms, we can say that while Euclid's system of axioms is described by Euclid himself as constructed (that is, we can construct a line from point A to point B), Hilbert's axiom system is set in the language of discovery (that is, between any two points there will be found one straight line).

So far we can see that there are evidently at least three distinct strains in contemporary mathematics, each assuming a greater degree of necessity, and each claiming a stronger independent ontological position. The first is that of the formalist, who believes that we can produce many mathematical systems, none more correct than the other, as long as they all remain logically consistent. The second postulates that, given our existence, mind structure, and so forth, we can conceive of mathematical entities in only one way. This Kantian(16) approach I will classify as "intuitionist." although it contains elements from both the conceptualist and intuitionist schools. The third approach is Platonic, holding that mathematical entitles exist in only one way, before and independently of our existence. perhaps within the very pattern of the universe. This. as mentioned before, can be said to be the most important assumption made by theoretical physicists in using mathematical representations to discover nonempirically tested physical phenomena. Plato would undoubtedly be pleased to see the increasing dependence on mathematics in contemporary theoretical physics.

Having discussed mathematics from the ideological vantage point of discovery (Platonism), we now turn our attention to mathematical creation. The formalist and intuitionist mathematical schools of thought, to a respectively stronger and weaker degree, consider mathematics to be an expression of creativity. Since I hold, with John Barrow,(17) that creative enterprise necessarily entails personal uniqueness-that is, that two creators will most likely not accidentally create the same work (what is the probability, after all, of two authors spontaneously and independently writing Crime and Punishment?)--I believe the implications of an understanding of physics' underlying mathematical structure as created are fairly substantial. The principal problem is one of foundations: that the validity of the inference (earlier described as mathematical manipulation) conform to the expectation that the final mathematical result will correspond to a meaningful notion of physical reality. If the assumption of a relationship between physical reality and its mathematical language is untenable. then this kind of mathematical hypothesizing so useful to physics is untenable as well. The study of physics is not the study of logic; mere coherence leads to no answers without empirical pinning points. In the same way that physicists would be uncomfortable with the claim that they merely create physical laws out of their imaginations instead of discovering independent truths about the universe, the use of a created (individually subjective) mathematical system to ascertain these truths calls into question the entire validity of the physicists' project.

Perhaps one of the most influential leaders of the formalist school of mathematics was David Hilbert, who realized after numerous paradoxes had been found that it was necessary to establish the consistency of arithmetic by certain indisputable logical principles. In a speech given in 1925 before a congress of the Westphalian Mathematical Society, Hilbert lamented,

In the joy of discovering new and important results, mathematicians paid

too little attention to the validity of their deductive methods.... The

present state of affairs where we run up against the paradoxes is intolerable.

Just think, the definitions and deductive methods which everyone

learns, teaches, and uses in mathematics, the paragon of truth and

certitude, lead to absurdities!(18) Earlier, Whitehead and Russell had attempted this project of clarification as well in the Principia Mathematica. Both projects ultimately fell prey to Godel's incompleteness theorem. As Paul Weiss has noted, the formalists ultimately were not creative, but functioned as a kind of cleanup crew for other mathematicians, inserting consistency and logical rationalizations for steps which were originally made with sometimes large intuitive gaps.(19) Although some axiomatic attempts, such as Hilbert's, aimed at starting from the very beginning, deriving all principles independently from metamathematical starting-points, most formalist enterprises concentrated exclusively upon establishing adequate proofs using only acceptable criteria for validity. Thus, they argued, mathematics would be grounded in absolute certainty.

There is, however, a very obvious problem with this formalist presupposition, which is that the "undisputed" metamathematical principles can hardly be assumed to be quite so clear-cut. The formalists are caught in a philosophical quagmire of question begging when asked to defend their criteria for an acceptable first principle. Even given a first principle, what then determines logical criteria for deriving one axiom from the next? The actual rules which the formalist uses to obtain rigorous consistency must themselves be justified by rigorously consistent analysis, whose rules must in turn be justified with rigorous consistency, resulting in either an infinite regress or a nonrigorously justified grounding point. The formalist Hilbert vehemently rejected Brouwer and Weyl's rather vague justification by virtue of "intuition"; yet ultimately one must ask what his metamathematical logical principles are based on if not intuition. It becomes apparent that both camps are arguing only about the boundaries of justification, while both make the same intuitionist assumptions.

It is actually more accurate to classify intuitionism as a mathematical method of discovery and not of creation; yet the discovery involved is internal psychological discovery rather than external physical discovery. Our intuitive knowledge is, in Kantian terms, correspondent to reality, but cannot be assumed to be a clear reflection of it. It is in many ways just as indicative of the structure of our minds as it is of the object that our minds seek to capture. As we have shown in our discussion of the formalist enterprise, when conducting a "rational reconstruction of knowledge"(20) starting from a single foundational principle, one will find that an infinite regress is inevitable, for one is always required to justify belief in the reasons which justify the final belief Otherwise, the fundamental truths become arbitrary. To avoid the infinite regress, it has been most commonly stipulated that the foundational beliefs be intuitive ones, requiring no rational justification. This too has complications. Even under a fairly broad interpretation of "requiring no reasons" and in spite of our use of intuitive metamathematical starting points. we must start with what is truly intuitive. Otherwise, we would base our entire construction of mathematics on a mere individual assertion which could be later disputed at whim. Unfortunately, there is no way of checking intuitive belief Even Quinton's "authoritative, laborious, and inferential way of justifying the beliefs in question"(21) is Meaningless without the assumption that the premises are correct in some sense. This caveat, of course, renders the process vicious.

A related problem with designating first beliefs as intuitive is that according to the psychological definition of "intuitive," all that is required for a belief to be so designated is the (again, intuitive) knowledge that one has neither a rational justification for the belief nor any way of providing such a justification. There always remains the possibility, however, that the intuitor is wrong, for he or she could be ignorant, or lacking in the appropriate insight. Perhaps another person might be able to justify those same beliefs rationally; in any case, it must always remain intelligible to imagine that there might exist some rational justification for the intuition of x, although no one may have thought of it yet. Lawrence Bonjour goes further along these lines and suggests that many beliefs, which are easily taken for granted as self-evident are in fact "depend[ent] for their justification on inferences which have not been explicitly formulated and indeed which could not be explicitly formulated without considerable reflective effort."(22) This belief would be supported by those formalists who have tried to go back and justify already existing mathematical proofs that are deemed useful and intuitively correct but were not rigorously derived. Bonjour's view is perfectly in line with our stated purpose, the "rational reconstruction of knowledge," and, like both our definitions of system closure, it also assumes the existence of an intelligible answer to every intelligible question.

Many epistemologists see the continuation of the debate surrounding foundationalism as being kept alive by the problem of epistemic regression. Foundationalism requires that there be some nonarbitrary grounding somewhere in the epistemic chain of justification in order that any belief have meaning. Bonjour distinguishes between two popular versions of foundationalism: "strong foundationalism," which assumes the existence of epistemically valid termination (grounding) points; and "weak foundationalism," which holds that beliefs may have a partial grounding, but may be additionally justified by inference. The larger the number of these hybrid beliefs, the greater the degree of knowledge. Thus, supposedly, the infinite regress is somehow magically avoided. As Bonjour himself points out, however, only a pseudoproblem has been solved, for one still has to explain how beliefs are ultimately grounded. If inference and coherence are supposed to lend validity to terminally grounded parts, why could they not be used as criteria alone? He sees the main concept of the "basic belief' as confused and fraught with contradictions, for the notion of intuition is itself illogical. He says, "Knowledge requires epistemic justification, and the distinguishing characteristic of this particular species of justification is, I submit, its essential or internal relationship to the cognitive goal of truth."(23) To accept such a belief without reason he calls "epistemically irresponsible."(24) Justification by its very definition implies the notion of reasonableness, to which personal intuition cannot answer.

Weak foundationalism also balks at the requirement of necessity and incorrigibility, characteristics which the mathematical intuitionist seems to assume. Though the refusal to require necessity and incorrigibility appears in some ways to be a reassuring compromise, ultimately it leads to a further problem with intuition: the consistency among the intuitions of different people that is required for intelligibility. If there is truly a first premise which is intuitive and correct, then everyone must agree on the choice of that particular premise, for it remains necessarily true, although not necessarily certain (that is, objectively and rationally true although it might not be known). Mathematical intuitionists must posit the existence of only one true mathematical system, or else admit to allowing the criteria for mathematical validity to be different for each person. This second option seems unacceptable. Assuming that everyone does agree on the designation of a particular foundational belief--a rather large assumption--how would one determine that the agreement is bona fide and not simply conventional? The theory of ostensive statements "argues that to avoid an infinite regress of explanation there must be a class of statements whose meaning is explained some other way, not by correlation with other statements but by correlation with the world outside language,"(25) as in the empiricist theory of mathematics. Here, then, foundationalism seems reduced to an empiricist correspondence theory of mathematical knowledge (external coherence). While the purely empiricist view of mathematics, which states that all mathematical structures are gleaned from experience with physical objects, has been suggested as an alternative by thinkers such as Jean Piaget,(26) it is a theory that is highly unpopular and serves to explain only a very small portion of mathematics. Yet if neither an empirical nor a strictly formalistic approach is possible, we are left with a still rather vague notion of what intuitionist discovery actually is.

By claiming necessity, intuitionism hints at the strong metaphysical assumptions about mathematical entities that characterize Platonism, but ultimately can go only far enough to claim a psychological rather than an ontological necessity. The underlying metamathematical rules which end the infinite regress that plagues intuitionism can only consistently be attributed to the structure of our thinking patterns. Thus, the discovery process of intuitionism is inward and analogous to the outward discovery process of Platonism. Both claim necessity and objectivity, but does intuitionism, like Platonism, require the possibility of just one right answer? As we have stated before, the answer to this question has crucial physical implications; for physics, in assuming a Platonistic application of mathematics, has no tools with which to discriminate between two equally useful systems that may, at least theoretically, yield radically different results. Only formalism openly allows for a plurality of valid yet mutually incompatible mathematical systems. Yet Morris Kline admits,

In defense of the formalist philosophy, one must point out that it is only

for the purposes of proving consistency, completeness, and other properties

that mathematics is reduced to meaningless formulas. As for

mathematics as a whole, even the formalists reject the idea that it is

simply a game; they regard it as an objective science.(27) This assumed objectivity provides a significant bridge between the seemingly opposed ideologies of mathematical discovery (Platonism, intuitionism) and creation (formalism). In the case of formalism, however, I do not believe that objectivity necessarily entails the presence of just one system if the "objectivity" is confined to the metamathematical rules of derivation only. Again, what will differentiate between different formalist systems will be simply the intuitions one accepts as starting points.

In response to the problem of competing mathematical systems, much has been made of the impact of non-Euclidean geometry upon the Kantian ideal of geometry, which can be described as containing elements of both Platonism and intuitionism. If Plato believed the realm of the mind and the realm of objective ontological reality to be one and the same, Kant adapted this strong metaphysics to include at least a nod to epistemological concerns. He lays out the foundations of his theory in the Critique of Pure Reason. Kant believes that it is absurd for us to be able to assume that we can understand reality in the pure ontological sense; what we do understand, however, is in some fundamental way connected (that is, integrally linked) with reality. This view has particular implications for his theories of space and time, and Kant's theory of space has great importance for the development of geometry in the nineteenth century. Gauss, in fact, reportedly refused to publish his paper on non-Euclidean geometry immediately because the philosophical air at the time was saturated by Kantians.(28) I shall attempt to demonstrate, however, that non-Euclidean geometry and Kant's philosophy are actually quite compatible. The connection between an objective reality and our own intuitionist principles is firmly entrenched in Kant's philosophy, and this is what links him closely to Plato, if only to Plato's more imperfect way of knowing described by the cave metaphor.

In spite of Gauss's nervousness, it is not so much the presence of an alternate theory (or mathematical system) that Kant would have found so disturbing, but rather the presence of two equally logically valid (internally coherent) and equally true physically interpretive theories. Kant's theory of geometry, Platonistic in the sense of holding that our mind's intuition corresponds to the outside world of reality, has no room for competing mathematical systems. Can physics allow for this kind of flexibility without striking a death blow to the overall coherence of the grand unification that is sought? This problem, thought initially to arise from the geometry of Gauss, Bolyai, and Lobatchevsky, has not made such unification impossible because their geometry is not truly competing with the standard Euclidean geometry. To date, the issue of competing mathematical systems has not yet been dealt with in physics, and thus has not yet been resolved.

In the "Transcendental Aesthetic" of the Critique of Pure Reason, specifically, in section 1 on "Space") Kant uses the subject of pure mathematics as the most perfect example of his synthetic a priori, consisting not of a strict formalism but rather of absolute logical consistency coupled with what he calls "rational intuition," a leap of understanding, or "insight" (in contrast to an "empirical intuition"). Although Kant claims that this knowledge is "necessary," it is different from the necessity of the tautologies produced by the analytic a priori. What the nature of its difference actually is, we will examine later. It is, in any case, a heading which denies any kind of empirical contribution. Kant makes it clear that

we shall understand by a priori knowledge, not knowledge independent

of this or that experience, but knowledge absolutely independent of all

experience.(29) While it might be supposed that this method applies to arithmetic, its claim is more tenuous for geometry, which seems to have a strong dependence on our empirically based conception of space. Kant would agree with the claim that there is an implicit connection between geometry and our understanding of space, yet he strongly denies our conception of space any epistemic grounding in the sensory world. This idea, though radically counterintuitive at first glance, proves crucial for his conviction that geometry and arithmetic fulfill the two requirements of the synthetic a priori, namely, necessity and universality.

When Kant wrote the Critique in the late eighteenth century the two characteristics of necessity and universality in mathematics seemed firmly entrenched and permanent in the Leibnizian literature. With the advent of a non-Euclidean geometry (and later, other alternate systems, such as Hilbert's), however, their foundations were shaken. If geometry and arithmetic are truly a priori, then there is the implication that there can be only one absolute form for their construction. To determine which form is mathematically correct (which may not necessarily mean physically accurate), a firm understanding of the basis for judgment must be established.

Given the reality of competing mathematical systems, each claiming to be free of any internal contradictions, we must again note that there can arise a multiplicity of systems based on the a priori as long as each is grounded on different premises, that is, on different primary axioms. This explanation is insufficient, however, given Kant's dependence on intuition as the sole starting point for our understanding of space and time, which, he claims, are the bases for any mathematical genesis. For intuition to have any kind of epistemological validity, it must lead only to true insight; therefore, a single insight into space and time must bring forth a single definitive insight into a single (non-arbitrary) mathematical structure. Thus, it is clear that to save the validity of the a priori as a meaningful process, one must be able to prove that a certain system of arithmetic and geometry (either [A.sub.1][G.sub.2] or [A.sub.2][G.sub.2], for instance) is the correct form or, alternatively, that our understandings of space and time are indeed grounded empirically--that is, that Kant was wrong and that [A.sub.n][G.sub.n] are not derived a priori at all, leaving no problem with a possible multiplicity. To save Kant we must agree that the true test for the validity of a mathematical structure is internal consistency and elegance rather than physical applicability. We must be able to decide which of the two, [A.sub.1][G.sub.2] or [A.sub.2][G.sub.2], is wrong and why, while maintaining that the ultimate basis of our knowledge of space and time is not empirical but rather intuitive. Perhaps the only way to save both the a priori and Kant is to be able to prove that one AG is more internally consistent than the other. Unfortunately, this is no small task, and may ultimately be inconclusive if the problem lies not with the a priori but rather with the synthesis.

Another approach is to suggest that the two geometries, while remaining synthetic a priori truths, are not mutually exclusive. We know that relativistic physics is not entirely incompatible with the laws of Newtonian physics because for most velocities, which are extremely small compared to the speed of light (c = 3 x [10.sup.10] m/s), in the Lorentz transformation (L' = L/[(1 - [v.sup.2]/[c.sup.2].sup.1/2]) the denominator goes to one, and L' = L.(30) The derivation of this equation, which indirectly created the necessity for non-Euclidean geometry, is, in fact, based on visible (Euclidean) geometry, a thought experiment constructed in ordinary three-dimensional space. It thus seems logical that the two geometries be related in some way, since for some circumstances at least (for instance, the everyday macroscopic world at slow velocities), Euclidean geometry remains accurate and useful. It seems absurd simply to dismiss Euclid's geometry so easily in light of the fact that it remains intuitive. Kant would see intuitivity as the final test of a mathematical system's veracity, but again we come to the question of how strong is the necessity of intuition?

When Kant says that "all mathematical inferences proceed in accordance with the principle of contradiction,"(31) he seems to be establishing its necessity through its a priori nature. But does the fact that it is noncontradictory mean that it is necessarily true? Of course not. I could say, for example, "My third child will be a daughter." This statement is not logically contradictory, but it is obviously not a statement of necessity, in contrast to an alternative statement, "My third child will be a four-sided triangle," which is necessarily impossible. Thus, it would perhaps be more accurate to say that the a priori process derives or creates all possible mathematical systems. What would be further needed to establish its necessity is the second step, namely, the synthetic or "intuition."

While the analytic a priori has the necessity of a tautology, the Kantian intuition appeals to a different kind of necessity. It seems contradictory to speak of more than one kind of necessity in this context; the only way that one could modify this modal category would be to assume that one type of necessity was stronger or weaker than the other, in which case they could not both be necessary in the true definition of the word. Part of the problem is that Kant at times seems to be flirting simultaneously with the two claims that spatial intuition is subjective, in the sense that it "underlies all outer appearances" (intuitions); and that it not only can but must correspond to the supposed objective domain." In other words, although empirical evidence could never bring about our understanding of space (Kant would say that it is our intuition of space that makes empirical data intelligible), neither should they contradict one another. Why is this?

The answer lies in the fact that Kant's intuition of geometry is inextricably tied to visualization, which, of course, is at least in part dependent on our ability to see. The fact that we are able to abstract and idealize geometric forms, however, shows us that we must have some underlying understanding of such fundamental properties as extension, limit, and symmetry with which to categorize and synthesize our random sensory perceptions.33 These fundamental properties are some of the rules by which our understanding is bound simply by being human; Kant alludes to these limitations when he begins his Introduction to the Critique with the sentence, "There can be no doubt that all our knowledge begins with experience."(34) Therefore, even when we are able through greater and greater abstraction and mathematical representation to derive scientific views of reality which are in conflict with our normal modes of perception, such as the Minkowski space-time in four dimensions, they are always conceptually explained in three-dimensional Euclidean space, with lines, warps, twists, and so forth. Thus it must be questioned whether the scientific reality that has been derived mathematically presumes to describe actual space independently of how we perceive it, at last free from our sensory limitations; or whether what it concludes is a mathematical analogy of our same three-dimensional understanding, but presented in a different way for mathematical simplicity in describing observed empirical phenomena. For while the theories, manipulations, and explanations may be possible in any dimension mathematically, experiments are actually always constructed and their results physically measured or interpreted by human beings with these conceptual limitations. Therefore physics, by virtue of being physics (requiring empirical verification subject finally to our senses, no matter how augmented they may be) and not solely mathematics (requiring no empirical verification), must always have one foot firmly grounded in our intuition of space, understood as Euclidean.(35) Since Kant recognizes correspondence with human intuition of space as the final test of a true geometry, the emergence of a non-Euclidean geometry should not affect his position. Furthermore, the emergence of this particular non-Euclidean geometry does not affect our position either; for since the two geometries are not actually competing to explain the same phenomena (not only because they are restricted to different parameters of magnitude, as with quantum versus classical mechanics, but also and more crucially because the structure of relativity theory allows for a connection with Euclidean geometry at some common level), we as of yet have no model with which to test the hypothesis of physically useful yet contradictory mathematical systems.

In summation, we review the possibilities for physics. The formalist seeks to give a clear account of mathematical structure, and to define which moves are acceptable and which are not. This would seem to be beneficial for physics; yet by allowing for a multiplicity of equally valid systems and by concentrating upon tautological consistency, the formalist introduces elements of incoherence into the larger structure of physics and limits enormously the scope of mathematics which is already used. The mathematical intuitionist shares the Platonistic assumption of a single true mathematics by requiring only moves that are intuitively obvious (being logically unambiguously specified by the previous step) and by postulating single true principles as starting points. Kronecker, for instance, referred to whole numbers as the "works of God," although the origins of the mathematical entities derived from these whole numbers were less well articulated.(36) There is a question as to whether the mathematical intuitionist is involved in an epistemological or psychological endeavor, or whether the objectivity for which he or she strives goes beyond his or her own existence. We also must further inquire as to the possibility of more than one valid mathematical system derived on intuitionist principles by mathematicians with different intuitions. This would, again, introduce physical incoherence. The mathematical formalist evaluates seemingly without foundations--an approach that initially bypasses the problem of infinite regress--yet we find that the formalist's evaluative structures of derivation have what are ultimately intuitionist foundations. Even the empiricist must demonstrate by what virtue different manifestations fall under one abstraction coherently. To use the language with which we began, the physicists' project requires a mathematics which possesses both internal and external coherence; that is, it must be internally consistent, and its propositions must not be reduced to "meaningless formulae." Rather, they should be able to describe the physical world.

It is my argument that the primary physical assumption made about mathematics is one that is essentially Platonistic in form, and that the absence of this assumption would have a profound influence upon the structure of physics. Physics would then have to discriminate between two equally useful or elegant systems, while the adoption of either system could alter the course of physical examination because each system possesses differing processes of derivation. This is a discrimination for which physics is as of yet ill-equipped, and which it has not yet had to make. Even accepting the Platonic view, physicists will still need to determine their own boundaries for the Principle of Totality, and thus to decide which mathematical entities and processes they will understand as valid. Although the old, more empirically based physics also left us problems concerning the fittingness of its mathematical language, it is the fundamental change in method, with a renewed emphasis (harkening back to the Greeks) on the aesthetic and the theoretical, that makes the investigation of hitherto unquestioned physical assumptions about the mathematics it utilizes mandatory.(37) (1) Morris Kline, Mathematics: The Loss of Certainty (Oxford: Oxford University Press, 1980), 7. (2) For an example of a physical question to which there are two answers that are equally valid mathematically, one of which is physically invalid, see the "coconut puzzle" in John D. Barrow, The World Within the World (Oxford: Oxford University Press, 1988). 254. (3) Eugene P. Wigner, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," Communications in Pure and Applied Mathematics 13, no. 1 (February 1960): 13-14. (4) The discovery of chaos theory and quantum physics does not in any way invalidate this project. The indeterminancy or "uncertainty" has not spread like a plague, engulfing old secure areas of physics and rendering them useless. Our uncertainty, as a matter of fact, is rather well-behaved, found consistently and constrained between very tightly mathematically defined limits. (5) Nicholas P. White, Plato on Knowledge and Reality (Indianapolis: Hackett Publishing Company, 1976), 7. (6) Cf. White, Plato on Knowledge and Reality, 38. (7) Meno 80d, trans. W. K C. Guthrie in Plato: The Collected Dialogues, ed. Edith Hamilton and Huntington Cairns (Princeton: Princeton University Press, 1987). (8) Republic 510d7, trans. Paul Shorey in Collected Dialogues. (9) Immanuel Kant, Critique of Pure Reason, trans. Norman Kemp Smith (New York: St. Martin's Press, 1965). A27. (10) Richard Robinson, Plato's Earlier Dialectic (Ithaca: Cornell University Press, 1941), 207. (11) Ibid., 196. (12) Robert S. Brumbaugh, Plato's Mathematical Imagination (Bloomington: Indiana University Press, 1954), 5-6. (13) I am referring here to an inconsistency in set theory as related to self-referential sets, first proposed by Bertrand Russell and Alfred North Whitehead. While it has been held that "the set of objects is not that object itself," in the case of self-referential sets the sets of some objects (such as "not-objects" or "concepts") are often also their objects. Hence the "paradox. For example: the set of not-objects is a not-object, the set of concepts is a concept, both cases breaking the rule. See Bertrand Russell, Principia Mathematica (George Allen and Unwin, 1903), 101. In 1908 Ernst Zermelo expanded upon Russell's paradox to include paradoxes in all of classical analysis; Ernst Zermelo, "Untersuchungen uber die Grundlagen der Mengenlehre I," Mathematische Annalen 65 (1908): 261-81. See also Paul Bernays, "Sur le platonisme dans les mathematiques," trans. C. D. Parsons as "On Platonism in Mathematics," in Philosophy of Mathematics: Selected Readings, ed. Paul Benacerraf and Hilary Putnam (Cambridge: Cambridge University Press, 1987), 261. (14) Bernays, "On platonism in mathematics," 262. (15) David Hilbert, "On the Infinite," trans. E. Putnam and G. J. Massey in Philosophy of Mathematics: Selected Readings, ed. Paul Benacerraf and Hilary Putnam (Cambridge: Cambridge University Press, 1987), 188. (16) See Kant, Critique of Pure Reason, A26/B42. (17) See John D. Barrow, "What is Mathematics?" in John D. Barrow, The World Within The World (Oxford: Oxford University Press, 1988), 553. (18) Hilbert, "On the Infinite," 190-1. (19) Paul Weiss, Creative Ventures (Carbondale: Southern Illinois University Press, 1992), 105. (20) The phrase is that of Anthony Quinton in his "The Foundations of Knowledge," in British Analytic Philosophy. ed. Bernard Williams and Alan Montefiore (London: Routledge and Kegan Paul. 1971), 56. (21) Ibid., 59. (22) Lawrence Bonjour, "Can Empirical Knowledge Have a Foundation?" American Philosophical Quarterly 15 (1978): 2. (23) Bonjour, "Can Empirical Knowledge Have a Foundation?" 5. (24) Ibid., 5. (25) Quinton, "The Foundations of Knowledge," 60. (26) See Jean Piaget, The Child's Conception of Number, trans. C. Gattegnot and F. M. Hodgson (London: Routledge and Kegan Paul, 1965); and Jean Piaget, Barbel Inhelder, and Alina Szeniska, The Child's Conception of Geometry, trans. E. A. Lunzer (London: Routledge and Kegan Paul, 1960). (27) Morris Kline, Mathematical Thought From Ancient to Modern Times (Oxford: Oxford University Press, 1972), 1208. (28) Kline, Mathematical Thought from Ancient to Modem Times, 871. (29) Kant, Critique of Pure Reason, A2. (30) Lorentz Transformation: L = length of object at zero velocity (the length normally observed), L' = observed length at high velocity, v = velocity, c = speed of light, the highest velocity possible--at which point matter becomes energy in accordance with the equation E = m[c.sup.2]. (31) Kant, Critique of Pure Reason, A10/B14. (32) Kant, Critique of Pure Reason, B39. (33) So far we have been using three-mensional perceived space and Euclidean geometry almost interchangeably. It would be an error to presume that everything included in Euclidean geometry can be sensed empirically, however. As defined by Euclid in his primary definitions, a point (position without extension) and a line (extension without breadth) would be impossible to "see," since vision requires extension in all three dimensions. An idealization is a visualization modified by intellectual understanding of given postulates that describe it, while a generalization is a mere averaging of seen objects, for instance, imagining the "perfect" circle after having seen a thousand "imperfect" ones. The reason that the view that our knowledge of geometric forms is obtained by generalization is incorrect is that we would never be able even to identify the imperfect circle as imperfect, much less see the connection between different imperfect circles, unless we already had the mental idealization of that circle. " Kant, Critique of Pure Reason, B1. This is a response, then, to our earlier question about the respective necessities of the analytic and the synthetic. The former contains an objective necessity: a tautology is necessary by virtue of its being known. The latter is a subjective necessity which arises from the relation of the knower to the proposition and makes no claims about the object's status independent of our knowledge of it. Even "objective" experimentation ultimately takes into account our perspective and relates the object back to us. Einstein's relativity theory demonstrated this idea graphically for the scientific world when it proved the absence of absolute space, that is, of a perspectiveless point of reference for all observers. (36) See Kline, Mathematical Thought from Ancient to Modern Times, 1197. (31) 1wish to express my thanks to Dr. Paul Weiss, Dr. Timothy Eastman, and Mr. Baruch Kfia for their helpful comments and criticisms.

Printer friendly Cite/link Email Feedback | |

Author: | Kfia, Lilianne Rivka |
---|---|

Publication: | The Review of Metaphysics |

Date: | Sep 1, 1993 |

Words: | 9423 |

Previous Article: | The science we are seeking. |

Next Article: | Thomists and Thomas Aquinas on the foundation of mathematics. |

Topics: |