1. The unreasonable effectiveness of mathematics:

Numerous mathematicians and philosophers have entered into discussions about the unreasonable effectiveness of mathematics, and it is easy to see why. From the time of the Egyptians, mathematics has been used to describe physical reality. A classic example is geometry. We all know that the area of a triangle equals one half the base times the height, so if we want to know the square footage of a triangularly shaped plot of land with one corner being a right angle, we simply measure the lengths of the two sides at that corner and multiply appropriately. But why do we get the "right" answer (if in fact we do)? Why does geometry, which proceeds from points (objects which have no dimension), lines ("breadthless lengths", ie, one-dimensional objects), and planes (two-dimensional objects) have to do with our three-dimensional world? Everything in our physical reality is three-dimensional. There are no such things as points, lines, and planes. So why does geometry (Euclidean, in this case) provide such an effective model of our physical universe?

If you find this simple example less than earth-shattering (or model-shattering), many other more sophisticated examples exist. Perhaps the most compelling lie in the fields of astronomy and physics. Since ancient times, man has striven to understand his place in the universe. For both philosophical and practical purposes, one aspect of this quest has been to find a unifying theory for the mysterious motions of heaven and earth. Early attempts in this direction were guided by sense perceptions (the daily "rotation" of the sun and moon across the horizon, ie, "around the earth"), aesthetic judgments (that circles are the most perfect of shapes), rational predispositions (toward movement at constant speeds and equal distances as the most "natural" motion), and religious beliefs (in God's design of the universe with man and Earth at its center) to produce and justify geocentric models. Such models, based on "epicycles", do exist and do describe the movement of the planets quite accurately. In fact, they are capable of the same accuracy as the heliocentric models advanced by Copernicus, Galileo, and Kepler. Despite the fact that we have been educated to accept the latter model, the modern view is that either theory will do. The only true advantage to the heliocentric model is its simplicity, a topic to which we will return later. The larger question at the moment is why either one of them works. More generally, why does any model based on ideal geometric shapes, exact algebraic equations, and purely logical deduction have anything whatsoever to do with the movement of physical matter through physical space, let alone describe this movement to within any experimentally verifiable degree of accuracy?

Even more impressive examples lie in the domain of mathematical physics. I will focus on the "simple" notion of gravity. Most of us learned somewhere along the line that gravity is "the attraction between two bodies at a distance", or more precisely, the force causing this attraction. But have you ever stopped to wonder what a "force" is, how it can operate "at a distance", what the mechanism behind such an entity might be? The fact is that physicists do not have answers for these questions. They know how gravity works, but not why it works or what it is. It is just a name. Yet we have a precise mathematical formula, , which describes its effects (attractive force?) on any two masses in the universe. And then, if we skip ahead a century or so to the early investigations into the relationship between electricity and magnetism, we find that two unlike electrical charges q and Q at a distance r attract each other according to a law of identical form, , although of course the constants G and K may differ. Again, why do these exact mathematical formulae apply to inexact physical phenomena? Moreover, why would two such disparate phenomena "obey" formulae of identical form?

Extending this, the investigations into the nature of gravity and electromagnetic phenomena bring us to an even more puzzling issue. Early scientists as far back as the Greeks sought qualitative explanations of why things happen. They wanted rational explanations, cause and effect, mechanistic models. Even during the late Medieval Ages and Renaissance, when the why's were attributed to a divine plan for the universe, scientists still sought to discover the underlying principles of this rational design. Taking their lead from mathematics, they evolved a methodology of systematic investigation and logical deduction from first principles. With Decartes and Galileo, scientific methodology became even more mathematicized, both in form and content. So that when Newton quantified the Law of Gravitation and then extended the related Laws of Motion to incorporate not only the motion of the planets but also a unifying theory applying to all matter in the universe, scientists were so pleased that many barely noticed that these laws were descriptions, not explanations. They provided no physical mechanisms to account for the actions and "forces". Thus the history of the theory of gravity contains a major turning point in the history of science. Mathematical descriptions replaced explanations. And gravity was just the beginning. Similar results followed for electricity, magnetism, heat, light, and a multitude of other physical phenomena. In any qualitative sense, these phenomena were and remain unexplained. All we really "know" are the mathematical formulae describing their quantitative aspects.

Incidentally, to complete this tale, Einstein's Theory of Relativity, the ultimate mathematization of science, has now outdated whatever intuitive grasp of gravity (as a "force") remained. According to the relativistic model, the world in which we live is a four-dimensional space-time continuum and "gravity" is merely one of the properties of its non-Euclidean geometry. More accurately, since acceleration is a natural consequence of the curvature of space (and force equals mass times acceleration, making force and acceleration essentially equivalent), the "force of gravity" is no longer a necessary or even relevant concept. In fact, since it is mass that accounts for the curvature, matter itself can be incorporated into the geometric model. Note that it is now a non-Euclidean geometry that effectively models space. Roughly speaking, non-Euclidean geometry is to Einstein's universe as Euclidean geometry is to Newton's.

But, how can all of this be possible? Why is the physical universe so open to mathematical modeling? There would seem to be some validity to Galileo's contention that "the laws of nature are written in the language of mathematics." However, mathematical formulae are idealizations, composed of abstract symbols representing equally abstract notions such as exact numbers and ideal geometric entities. Physical reality on the other hand is composed of material objects. Exactness is at best improbable; equality is impossible. How can there possibly be any meaningful isomorphism between the two? Yet, it is hard to ignore the accomplishments of science and hence the effectiveness of the mathematical models on which it is built. At the moment, our original inquiry about the unreasonable effectiveness of mathematics would seem to be not so much a question about its effectiveness as about whether or not this effectiveness is in fact unreasonable.

2. The nature of mathematics:

In order to properly consider this question, we must first look a bit more carefully at the nature of mathematics. For example, the perennial problem of whether one discovers or creates mathematics seems to be central. If we discover mathematics, then we are by definition discovering something that already exists. And if it exists, it is, again by definition, a part of the same physical universe it is being used to model. In other words, the structures of mathematics and physical reality spring from the same source, so that the existence of isomorphisms between the two and the effectiveness of one to model the other seems much less likely to be unreasonable. If, on the other hand, we create mathematics, then we are creating abstract mathematical structures to model already existing physical ones and then claiming that there exists an isomorphism between the two. This scenario seems to be much more vulnerable to the charge of being unreasonable. Therefore, we cannot avoid the discover versus create controversy. This is a long standing debate penetrating to the core of the philosophy of mathematics, and will not be resolved in the next few pages. I will, however, toss my hypothesis onto the heap.

The issue arose, for us, out of the relationship between mathematical models and the physical universe. The natural question to ask therefore is whether we discover or create these mathematical models. On the surface, the answer is fairly easy. Most (although not all) scientists and mathematicians would admit that these models are human inventions; we create them. We assume certain axioms which we feel match the physical phenomena closely enough to be deemed self-evident and then we reason deductively from there. If our deductions, however, contradict our experience or disagree sufficiently with our expectations, we go back and change the assumptions! This circular process can go on and on. The axioms are not sacrosanct and never will be. In fact the original selection was as much an art as a science. A manageably finite set of axioms can never capture all of the complexity and subtlety of physical reality. In determining axioms, the scientist or mathematician must make judgments and choices with regard to such matters as empirical consistency, self-evidence, functional import, and computational practicality. A model is by definition an over-simplification for the sake of simulating determinism, the strict necessity of one event following from another. And, if somehow we do manage to construct a model whose predictions match both our expectations and our experiments, it need not even be unique. There is not necessarily one "right" model. For example, both the epicycle geocentric model and the heliocentric model accurately describe the motions of the planets. The initial choice between the two was actually an aesthetic (and religious) judgment, based on a belief in the simplicity of the laws of nature. The current choice also weighs coherence with the current scientific paradigm, but, as Kuhn has noted, paradigms are not sacrosanct either. Hence, by most, mathematical models are seen to be human inventions. We create them. They are subject to change and revision. When there is more than one satisfactory model, we simply choose the most useful.

But, what about under the surface, i.e., the mathematics used to build the model? Is it discovered or created? Do the abstract structures that mathematical symbols "define" and "represent" already exist or are they created by our definitions and axioms, i.e, by us? And even more fundamentally, what about the logic which supports the entire edifice? What is its origin? Clearly we need to go a bit deeper into the nature of mathematics to consider these questions.

Although the origins of counting and the concept of number can be traced back to the Old Stone (Paleolithic) Age, the origins of mathematics as a formal discipline began with the classical Greeks. Up to that point, mathematics had been largely an empirical science, concerned with the question "how." The primary criterion was empirical consistency, i.e., whether mathematics accurately described the real world. The Greeks, beginning with the Ionians and culminating with Euclid and the Alexandrians, introduced a new criterion, logical consistency, i.e., whether conclusions were strict logical consequences of assumptions. They emphasized rigor and proof. They insisted upon the axiomatic method. Although their ultimate goal may have been "truth" about the workings of nature, they advocated that conclusions follow rationally from clearly stated assumptions. The Greeks were not satisfied with approximate answers and empirical proofs, limited by the accuracy and precision of our senses and our instruments; they sought exact answers with deductive proofs. They may have wanted these answers to be empirically valid, but this alone was no longer sufficient. To empirical consistency, they added logical consistency. The most famous example of this new emphasis was Euclid's Elements, which derives 465 propositions (theorems) from 5 common notions and 5 axioms and contains no practical applications. Although later discovered to be flawed, it was hailed as the model of mathematical reasoning for over 2000 years.

The Elements includes both number theory and geometry. However, I will concentrate on the geometry, for it is events within the history of geometry that eventually reshaped the history and even the definition of mathematics as a whole.

Although the Elements represented a significant advance in the right direction, the axiomatic method demands total rigor. One must lay out all assumptions as explicit axioms and use only logic to prove resulting theorems. There can be no unstated assumptions. Nothing can follow from diagrams. Everything must follow from the axioms. Hilbert captured it beautifully when he said, "One must be able to say at all times -- instead of point, line, and plane -- table, chair, and beer mug." In fact the danger in using the word "point," instead of "table" or, better yet, a completely empty symbol such as "A," is that "point" comes laden with preconceptions, hidden empirical content. And it is easy to let these hidden assumptions slip in unnoticed, which is exactly what happened in the Elements. For example, although Euclid included "any two points determine a unique line" as an axiom, he let slip "any line contains at least two points." He also used a number of other empirically consistent but unstated assumptions, so that the Elements was not quite the perfect model of pure reasoning as had been believed.

However, this flaw proved not to be fatal. Numerous axiomatizations of Euclidean geometry have been advanced, the most popular of which was provided by Hilbert. He had to enlarge the number of axioms from five to eighteen, but logical consistency (as we have defined it) was established. All necessary assumptions were now contained within the axioms and all theorems could be derived from the axioms, using only logic. Empirical knowledge was still represented, but it was appropriately consigned to its proper place, the axioms. In the end, there were two criteria, empirical consistency and logical consistency. Euclidean geometry accurately described the real world (at least according to Newton), but was logically consistent in the process.

However, at this point, an interesting question arises. If we are not allowed to use empirical knowledge in our proofs, why should it be so critical in defining our axioms, especially since we don't "really" know what it is? All we really "know" is the latest paradigm, the currently accepted model of empirical reality. And as we began to see with Copernicus and Kepler, Newton and Einstein, the most certain thing about a paradigm is that someday it will be superseded.

And, once one questions empirical knowledge (or our current hypothesis about what that might be) as a criterion for axiom selection, whole new worlds open up. In the case of geometry, the result was non-Euclidean geometry (both hyperbolic and elliptic). Ever since Euclid's time, mathematicians had questioned the Parallel Postulate, attempting to either replace it with a more self-evident axiom or to prove it from the remaining axioms. Attempts to replace it inevitably ended in the replacement axiom being shown to be its logical equivalent. Attempts to prove it generally took the form of assuming its negation and searching for a contradiction, whereby one could conclude by the Law of the Excluded Middle that the Parallel Postulate must have been true in the first place. The problem was that the apparent contradictions always proved to be contradictions of the empirical paradigm instead of axioms or other logical deductions. So that if one has already relinquished empirical consistency as a given, there are in fact no contradictions at all, only a logically consistent collection of strange seeming geometrical theorems. As Bolyai concluded, "Out of nothing, I have created a strange new universe." This strange new universe was non-Euclidean geometry.

What's more, since Euclidean and hyperbolic non-Euclidean geometries are absolute geometries (i.e., descriptions of space as a whole) and contain contradictory results, they are not empirically compatible. However, as Beltrami later proved, logically they stand or fall together. If Euclidean geometry is logically consistent, then so is hyperbolic, and vice versa. This would seem, by the way, to provide the capstone for our earlier argument that mathematical models are created, not discovered.

But, returning to our inquiry with respect to mathematics itself, bear in mind that the reigning paradigm for space in the 19th century was Euclidean geometry. Thus, with non-Euclidean geometry, geometry (hence mathematics) and empirical consistency parted company. Logical consistency became its only criterion. Empirical consistency might make certain mathematics more interesting to certain people, but it was no longer decisive with regard to validity as mathematics -- a good thing since, as we have seen, scientists have historically been less than decisive about what was in fact empirically consistent and with the advent of the Theory of Relativity, Quantum Theory, and the Uncertainty Principle, they do not appear to be moving rapidly in that direction. Thus, by the end of the 19th century, mathematics was no longer constrained by the scientific paradigms of reality. If you recall Hilbert's "tables, chairs, and beer mugs" and assign to "true" its most commonly used meaning of "empirically consistent," then Bertrand Russell's quote, "Mathematics is the subject in which we know neither what we are talking about nor whether what we say is true," sums the situation up quite nicely.

Therefore, the bottom line in mathematics is logical consistency. We assume axioms and derive theorems. Naively, it might appear that now that we are no longer confined with respect to empirical consistency, we are free to choose any axioms, i.e., to arbitrarily "create" axiom sets. But this is not quite the case. We require, first and foremost, that both our axioms systems and our reasoning be logically consistent. Thus, although we haven't rigorously defined mathematics, e.g., as logicism, formalism, or intuitionism, we have isolated what most would agree to be its most essential characteristic, logical consistency.

3. Mathematics is anthropomorphic.

Before we take our point about the fundamental import of logical consistency to the nature of mathematics and incorporate it back into our discover versus create inquiry, perhaps it is time to define our terms a bit better. Up to now, I have been using "logical consistency" in a fairly loose and not totally accurate manner for the sake of juxtaposing it against empirical consistency. Roughly speaking, empirical consistency meant consistency with respect to physical reality and logical consistency meant consistency with respect to the axiomatic method or the rules of logic, or more simply deducibility. The term, however, has a very specific mathematical meaning. For a mathematical system to be "logically consistent" it must be contradiction-free in the sense that not only must the axioms themselves be free from contradiction but also all of the logically deduced theorems. In other words, assuming the axioms and using only the rules of logic, it must be impossible to deduce both a proposition P and its negation Ø P. The only significant way in which my previous usage was inaccurate was when I said that Euclidean geometry had been shown to be logically consistent. What had actually been shown historically was that no contradiction did arise, not that no contradiction could arise. The difference will become clear as we go along. But, why is this freedom from contradiction such a fundamental criterion in the first place? Because without it, deduction is meaningless. If we can deduce a contradiction, then from there, we can deduce anything. For example, suppose that we have derived both P and Ø P. From P we can immediately derive P OR Q, for any Q, but then since we have also derived Ø P, we can use this together with the definition of OR to deduce Q. Therefore from a logically inconsistent axiom system, anything and hence everything follows. The system is useless. Empirical consistency may have been a value. Logical consistency is a necessity.

The crucial question now seems to be the nature of logical consistency. Is it discovered or created? Its origins are definitely empirical. As we have seen, logical consistency is a requirement imposed on reasoning by the rules of logic. These rules were first formulated by Aristotle and known as the Laws of Thought. They modeled human reasoning. Eventually they evolved into what we know as Predicate Calculus. Probably the two most fundamental rules in this calculus are the Law of the Excluded Middle (P OR Ø P, the assertion that any statement must be true or false) and the Law of Contradiction, (Ø (P AND Ø P), the assertion that a statement cannot be both true and false at the same time, i.e., a contradiction). Without such rules, implicit or explicit, Socratic dialogues would have self-destructed into a sequence of ineffectual dead ends and the currently accepted notion of mathematical proof would require total revision. Predicate Calculus has in fact proven so effective that it now permeates the very core of scientific reasoning and even western intellectual thought. It is difficult to imagine rational discourse without it. But, what is even more incredible than the fact that there appears to be some sort of isomorphism between the rules of logic and our reasoning processes is that this isomorphism also seems to extend into the physical domain. For instance, if we begin with axioms that we accept as empirically consistent ("true") and proceed to apply the rules of logic, we amazingly arrive at conclusions that are also empirically consistent!

Why do these "laws of thought" also apply to nature? Is there something empirically consistent about them? Is it possible that they "exist" in some sense -- for instance, a la Kant, "in the structure of the mind," or even as a part of some "order of the universe"? Numerous hypotheses for the apparent relationship have been advanced. They range from the assertion that since man is a part of the physical universe and his mind and logic have been the most successful (a la Darwin) in understanding and controlling nature, they must therefore share some structural commonality with nature, all the way to the assertion of a divine mathematical design for the universe. It is difficult to see how these types of speculations will ever be resolved, especially since they are being conducted within the confines of our minds and the constraints of our logic? The inherent problem here brings to mind Godel's theorem that it is impossible to prove the consistency of a system from within the system. Of course, Godel's theorem does not really apply here, but it does help delineate the issue. The fact that we discern order and structure does not mean that it is really there. As Morris Kline said, "The order of the universe may be the order of our own minds. We are not merely observers of reality, we are participants." [5] He may have been referring to empirical reality, but if there is such a thing as a logical reality, his statement would apply equally well there. In any event, it is highly unlikely that we will (or even can) ever be certain regarding either the existence of order or its relationship to our logic. The more useful question therefore might be whether or not the rules of logic are unique and to what degree they mandate logical consistency.

Until this century, Aristotelian logic and its offspring Predicate Calculus were taken almost as a given. In fact, most still accept them as such. However, there now exist clear alternatives. For example, there are three-valued logics, many-valued logics, and even infinite-valued logics. Predicate Calculus, as one can see from its Law of the Excluded Middle and Law of Contradiction, is a two-valued logic. A statement is true or false. In three-valued logics, statements are true, "undecidable", or false. In infinite-valued logics, statements may take on infinitely many truth values, say the real values between 1 (true) and 0 (false). Note that the Law of the Excluded Middle no longer holds. However, not only are these logics equally viable as logical systems, but they even have applications to probability, fuzzy sets, and artificial intelligence.

Another alternative logic is that advanced by the Intuitionist school of mathematicians. For them, mathematical existence depends on self-evident constructibility. Since completed infinite processes can neither be inspected nor introspected, constructibility is limited to certain finitist methods. This has substantial consequences, especially for the Law of the Excluded Middle. A classic example is the definition of a number x equal to , where k is the position of the first digit of the sequence 0123456789 in the decimal expansion of pi if such a k exists, and x equal to 0 if no such k exists. [1] But since we do not know any finitist construction which would show whether the 0123456789 sequence occurs in pi, there is no way to say, using intuitionist criteria, whether the proposition "x = 0" is true or false. Again, this non-Aristotelian logic fails to admit the Law of Excluded Middle. Incidentally, since the assumption of non-existence leading to a contradiction does not necessarily imply existence, neither does Intuitionist mathematics admit the method of indirect proof or any proofs or theorems that require it.

If you recall, the Law of the Excluded Middle played a vital role in our proof that from a contradiction anything follows, which implied the necessity of logical consistency. Bear in mind that this was a proof conducted within Predicate Calculus, i.e., our logic. It does not necessarily hold in other logics or outside the domain of our logical assumptions. Take Zen, for instance. In Godel, Escher, Bach, Hofstadter describes Zen as "transcending dualism." Its very essence, one-ness, rejects the conceptual division of the universe into categories. It therefore also rejects the Law of the Excluded Middle and even the very notion of contradiction. For Zen, all is one. There are no mutually exclusive, let alone exhaustive, categories P and Ø P. "Contradictions" are not fatal, but then neither are reason and logic the road to knowledge and truth.

It is therefore clear that Predicate Calculus is in no way unique, i.e., the only model of logic. It is merely a convention, within western intellectual thought, within select schools of mathematics, within certain axiom systems. But this does not necessarily imply that it is an invention. The fact that there exist alternative models for logic may be analogous to the fact that there exist alternative models for physical reality. And, as we have seen in science, emerging theories and paradigms tend to be increasingly macro in their perspective, often encompassing previously conflicting micro views. There is nothing to say that there may not someday emerge some super-macro-logic.

Our more immediate concern is whether we discover or create Predicate Calculus, along with which comes logical consistency. Without complete knowledge about the existence and nature of some order in the universe, we cannot know for certain. As for me, I share the modern view (of mathematicians, if not philosophers) that we create it. Predicate Calculus seems to me to involve, as all models are prone to do, too many idealizations. The requirement that propositions be either true or false seems at odds with the current state of science, to say nothing of our minds. Logical deduction that proves implication is certainly an idealization of the much less certain notion of causation. It is also a less than perfect fit for human reasoning. Take for example our little proof that (P AND Ø P) Þ Q, for ANY Q. This certainly violates the more (humanly) acceptable notion of relevant implication. Once again, the model appears to be an over-simplification for the sake of determinism.

It is therefore my contention that we create mathematics. There may or may not exist some order in the universe. That is not our question. It is not even our question whether mathematics is isomorphic to such an order, if it does exist. Our question revolves around the nature of mathematics and the relationship between this nature and the effectiveness of mathematics. And whether we refer to mathematical models, or to underlying structures and systems delineated by a set of axioms, or the logic used to draw conclusions, mathematics is a human creation. More than that, it is fundamentally anthropomorphic. It resembles our mode of thinking. Reasoning, at least within the western intellectual tradition, takes place in the context of categories and rules. We rationalistically compartmentalize "reality" into existence and non-existence, equality and inequality, and a whole host of other definitive conceptual divisions. Likewise with mathematics. The fact that mathematics has often been described as an "exact science" is quite revealing. As we noted earlier, nothing empirical is ever quite so exact, so cleanly delineated and deterministic. It is only our conceptual and mathematical compartments that create the illusion. If there exists any isomorphism at all, it is between our conceptualization of the universe and the structures of mathematics. In both cases, we select the axioms, delineate the structures, and derive the conclusions, all according to rules that we ourselves have invented. And these rules are ultimately just conventions. They vary from culture to culture, school to school, and to some extent, person to person. But, they are necessary. At some point, we must have agreed-upon rules. Otherwise, in order to legitimize our rules, we would need rules (metarules) for determining rules, and then metametarules for determining the metarules. Unless we agree at some level to conventions, we have infinite regress and get nowhere. In the final analysis, the whole rationalistic enterprise is designed to accommodate human reasoning. "Reality" does not demand this conscious act of modeling; we do. And whether reality can or will ever be captured isomorphically by it is an open question. At the moment, our anthropomorphic attempt is less than a perfect fit. At best, the results can be used to simplistically model reality. At worst, they are a total figment of our imagination, useful only within the conceptual framework we impose upon the universe.

4. Effectiveness reconsidered:

So we are back to the question that if mathematics is merely a human invention, how can it be so unreasonably effective? Or, since the effectiveness seems difficult to deny, whether or not it is in fact unreasonable? Here, "unreasonable" can be taken to mean either "unexplainable" or "extreme". In either event, I intend to argue that mathematics is neither as effective nor as unreasonably so as it may appear. There will be two components to this argument, one empirical, the other logical. Most of the discussion, here and elsewhere, regarding the unreasonable effectiveness of mathematics has centered around its remarkable empirical abilities. Although impressive, the examples cited nevertheless present a rather myopic view and they almost always lose sight of mathematics' primary criterion, logical consistency. This seems particularly shortsighted given the move away from empirical consistency as a value, leaving, in my opinion, the ability of mathematics to establish logical consistency lying at the very heart of any proper consideration of its effectiveness.

We have already seen compelling evidence of the effectiveness of mathematics in the area of physics and astronomy, the "hardest" of the sciences. Mathematics has not proven nearly so effective in biology, psychology, or the behavioral sciences, which incidentally makes them "softer." The simple fact is that we are very selective about where and how we use mathematics and claim effectiveness. Even within the hard sciences, we must be careful. For example, it we "add" one raindrop to another raindrop, we do not get two raindrops. One might think that addition would be less ambiguous for "hard" bodies, such as rocks or balls. But it was this very question ("How closely must two bodies be joined to be considered one?") which led Galileo to challenge the accepted theory that heavier bodies fall to earth faster than lighter bodies. If you find these examples more philosophical than physical, try the fact that one quart of alcohol added to one quart of water produces about 1.8 quarts, not two. In other words, we only apply integer arithmetic where it is effective. The same can be said for other arithmetical and algebraic systems. In short, we consciously fit our mathematics to the world. And even when this fit appears to be truly isomorphic, there is still the question of whether the "world" our mathematics fits is actually physical reality or merely our conceptualization of it. From this perspective, the effectiveness appears a bit less mystical and all powerful, a bit more reasonable.

An analagous situation arises in geometry. As we have aready noted, we now have not one but several geometries, all of which can be effectively applied to model space and none of which has any definitive claim on truth (empirical consistency). Our choice is governed more by practicality than by necessity. The existence of an isomorphism between any given set of abstract geometrical structures and reality seems improbable if not impossible. Here and throughout mathematics, mathematicians develop alternative models and structures and scientists apply whichever one seems useful and convenient. In the case of geometry, we are not sure if any of them exactly fit space, much less which one. Again, "unreasonably effective" hardly seems an appropriate descriptor.

Also relevant here is the issue of simplification versus determinism. As we have discussed, models by definition sacrifice subtlety for simplicity. Using a figure-ground metaphor, in selecting the figure, models ignore the ground. It is this very act that renders them deterministic and hence "effective." But it is also this act that causes the resulting effectiveness to fall short of total determinism. Although the general principle is clear, there is a rather simple example that may help drive the point home. It involves the heliocentric model for planetary motion. In this model, a solitary planet revolving about a stationary sun follows the path of an ellipse. But the moment even a third body is introduced (let alone the heavens), our elliptical model fails and although we can approximate closely enough to put men on the moon, no exact solution exists. Also recall that at best, mathematics describes how, but does not even address why. Newton was very clear on this when he said, before laying out his Law of Gravitation and its consequences, that it was his "design only to give a mathematical notion of these forces, without considering their physical causes and seats." Clearly, mathematics functions only within carefully delineated boundaries. Its effectiveness is not unfettered and free-ranging.

And, finally, transcending all these reservations regarding the effectiveness of mathematics in the empirical realm hangs the question, "Even if it were effective, how would we know? How could we be sure, either way?" Empirical measurements are not exact. If they were, determining the "true" geometry would be as simple as measuring the angle sum of a triangle. To make things worse, experimentation inevitably takes place within some paradigm. For this reason, and more deeply rooted physical ones, we should recall Kline's admonition that we are not merely observers of reality, but participants. Hence, empirical consistency itself is subject to question and arguments asserting unreasonable effectiveness with respect to it are doubly vulnerable. Perhaps the last word on the subject should go to Einstein: "As far as mathematical theorems refer to reality, they are not sure; and as far as they are sure, they do not refer to reality."

Before turning to the effectiveness of mathematics with respect to logical consistency, it will be useful to examine further the historical origins of the problem. As we saw, geometry had been organized into a formal deductive system by 300 B.C. Euclid's presentation may have been flawed, but its axiomatic-deductive approach was essentially sound. Subsequent axiomatizations demonstrated that all of the theorems of Euclidean geometry can be rigorously deduced from a small number of self-evident axioms. Algebra, however, was another matter. It evolved from arithmetic and, despite the work of Vieta, remained essentially empirical in nature until the 19th century. With respect to numbers, the literals eventually introduced to represent them, and the "logical" rules that governed both their manipulations, mathematicians reasoned more by informal induction and analogy than by deduction. There were no self-evident axioms followed by deductive proofs. The primary criterion for effectiveness was pragmatic success. Early on, the Greeks had attempted to establish secure foundations for their algebra within the rigors of geometry, but this "geometrical algebra" involved severe limitations. For example, since a cube necessarily represented volume and a square area, adding the two made no geometrical sense, hence was prohibited. The Hindu, Arab, and later the Renaissance mathematicians, however, did not let such esoteric considerations stand in their way. Although troubled by the absence of a logical foundation, they were consoled by the utility and effectiveness of their final results. For most, the end did in fact justify the means. So they continued on. However, as their sophistic manipulations led to stranger and less intuitive entities, e.g., irrational, negative, and complex numbers, they felt a greater need to supply arithmetic and algebra with a rigorous logical foundation. This growing need eventually reached critical proportions with the development of calculus and analysis. Here, mathematicians found themselves in need of infinitely large and infinitely small numbers. Neither were well understood, let alone rigorously formulated. Particularly perplexing were infinitely small numbers called infinitesimals (or differentials) which seemed to demand in certain circumstances within calculus to be treated as non-zero entities and in others as zero. Berkeley derided them as "ghosts of departed quantities." In short, by 1800, everything from number to analysis was in serious need of a solid logical foundation.

The early 19th century only managed to confound matters further. Despite their dissimilar origins, one notes some parallels between arithmetic and geometry at this stage. The "exemplary" logical foundations of Euclidean geometry had been found to be flawed, but the body of knowledge itself was so useful and comfortably intuitive that few doubted the ability of mathematicians to establish the proper axiomatic account. Similarly, the concepts of arithmetic, although increasingly less intuitive, had proved eminently useful and consequently gained familiarity and acceptance. However, not only did the first half of the century see the arrival of non-Euclidean geometries, but also the advent of quaternions. As whole (natural) number arithmetic evolved, encompassing the rationals, reals, and complexes, mathematicians had been lulled into taking field properties for granted. They had even coined the Principle of Permanence of Form which, more or less by fiat, allowed them to assume that that the properties of the real numbers were universal. Quaternions, which failed to have the property of multiplicative commutativity, challenged this complacency. They destroyed the belief in one unique algebra in much the same way that non-Euclidean geometry dethroned Euclidean geometry as the reigning monarch of truth. Thus, by mid-century, little in mathematics seemed secure except its utility. Work had recently begun on the foundations of calculus, but the logical underpinnings of both geometry and algebra were anything but secure. Empirical consistency was no longer the name of the game and logical consistency was nowhere in sight.

Mathematicians needed to put their house in order, to shore up the foundations. Interestingly, instead of starting in the basement with the natural numbers, they commenced on the "thirteenth floor" with calculus, largely because they regarded the lower levels as comfortable and intuitive, whereas calculus at the time was riddled with fuzzy thinking, opposing schools of thought, and outright errors. Roughly speaking, mathematicians remodeled top-down, establishing the logical consistency of calculus on the basis of the reals, the consistency of the reals on the rationals, until they got down to the naturals. And who could doubt the logical consistency of the natural numbers? They were certainly as intuitive as points, lines, and planes, if not more so. Recall, however, that the logical consistency of geometry had never really been proven. Beltrami's proof that Euclidean and non-Euclidean geometry stand or fall together was only a relative proof. No absolute proof existed, merely conviction. But, with the aid of analytical geometry, it was now possible for mathematicians to annex Euclidean geometry to their foundational structure, so that its consistency too was based on that of the natural numbers. And then, to assure absolute quality control, they began work on the substrata underlying the entire edifice, logic and set theory. However, as Frege, a leading logician, put it, "Just as the building was finished, the foundation collapsed."

Logic and set theory contained paradoxes. A classic example is Russell's Paradox. It goes like this. Some sets belong to (are members of) themselves; some are not. The set of words on this page is not a word on this page, hence cannot belong to itself. However, the set of abstract ideas is an abstract idea and therefore belongs to itself. Now consider the set R of all sets which do not belong to themselves and ask yourself whether R belongs to itself or not. By the Law of the Excluded Middle, one of these two possibilities must be true, but both lead to contradictions. Euphemistically we call this a paradox. For logical consistency, it is a disaster. And the walls came tumbling down.

It's only fair to mention that these paradoxes can be resolved, for instance by introducing such contrivances as the theory of types, which serves to forbid the sets, such as R above, which lead to the paradoxes. However, as suggested by "contrivances," these purported solutions are anything but intuitive and self-evident. In fact, their role in precluding paradoxes is the only real justification for their inclusion. Therefore, they are not only suspect, but their inclusion at this axiomatic level is deemed by many to be wholly inappropriate. Remember that the logical consistency of virtually the entire structure of mathematics rests upon the axioms of logic and set theory. If they cannot be accepted as self-evident, nothing derived on their basis carries much weight. And logical consistency remains as remote as ever.

So, where does all of this leave us with regard to the effectiveness of mathematics in establishing logical consistency. Logical consistency guarantees that axioms will not lead to a contradiction. Given that one cannot foresee all the possible logical consequences derivable from a set of axioms, this is certainly not an intuitive property. However, if we assume our axioms to be "true" (whatever that means, since empirically consistent is no longer apropos), and also assume the validity of logic and the Law of Implication, then whatever we can derive from these axioms must also be true, and free from contradiction. But, what do we mean by "true"? In the axiomatic domain, the standard answer is "self-evident." But self-evident to whom? All too quickly, this becomes an immense quagmire. Another tack would seem advisable.

Basically, there are several approaches to establishing the logical consistency of an axiom system. Since the issue for us is the logical consistency of the natural numbers (and hence the entire mathematical edifice erected upon it), let us use their axiom system as an example. One can attempt to prove its logical consistency relative to some simpler system. But then, of course, one must somehow establish the consistency of this simpler system. This either leads to infinite regress or to some simplest-of-the-simple system. For the logicists, this bottom level was logic. However, this raises some rather sticky questions about what in fact qualifies as an axiom of logic. Also, even if agreement were to be reached here and the success of the logicist program of deriving all of arithmetic from the agreed upon axioms assumed, we would still be faced with establishing either the truth or logical consistency of these axioms of logic. Some might argue, given the anthropomorphic nature of logic, that to question logic is to question our own mode of thought, therefore our own sanity, but this is far from a proof. For an opposing group, the intuitionists, the truth and consistency of mathematics came down to intuition and finite constructability, although agreement regarding the exact meaning of these axiomatic-level concepts proved illusive. In addition, the inevitable restrictions led, as we saw, to some rather severe limitations with respect to existence, the Law of the Excluded Middle, indirect proof, etc. which sacrificed significant portions of classical mathematics, and still did not mandate any necessary truth attached to the fundamental intuitions.

The set-theorists, who probably came the closest to success, sought to base everything on the consistency of set theory. Using such devices as the theory of types to restrict admissable sets, they were in fact able to generate all of mathematics while at the same time avoiding Russell's Paradox and the like. Note, however, that merely avoiding the predictable contradictions does not guarantee that others will not appear later, hence does not guarantee logical consistency. Establishing the logical consistency of set theory again comes down to establishing some kind of self-evidence or truth of the axioms. And, given the restrictive set-theoretic axioms necessary to avoid the paradoxes, self-evidence can become somewhat self-serving. In short, none of the attempts to move to the logical consistency of a simpler system have proven satisfactory.

Another approach is to argue the consistency of an axiom system from a metasystem. This was essentially the strategy employed by the formalists. They derived the consistency of the natural numbers from a formal system composed of meaningless (in order to assure absolute rigor) symbols. However, this approach poses two problems. The first is establishing the appropriate relationship between the (meaningless) formal system and the (intuitively meaningful) natural numbers. The second is the by now perennial problem of establishing the logical consistency of the formal metasystem. One can, of course, appeal to a metametasystem, but, as before, this either leads to infinite regress or some metameta...metasystem which would surely be immeasurably less intuitive and self-evident than the naturals were in the first place.

Therefore, since attacking the problem from the outside seemed doomed to failure, the last hope was an attack from within, i.e., to prove the logical consistency of an axiomatic system (in this case for the naturals) from within the system itself. Enter Godel. Godel's Incompleteness Theorem says essentially that for any consistent formal system F extensive enough to generate the natural number system, there are undecidable propositions in F, i.e., propositions P in F such that neither P or Ø P can be derived from the axioms of F. Apparently the price of consistency is the existence of unprovable propositions. To a discipline whose business is proving, this was hardly welcome news. A corollary to the Incompleteness Theorem then showed that for any consistent formal system F adequate to generate the natural numbers, the consistency of F cannot be proved within F. Clearly this declared futile the entire early 20th century effort by Russell, Whitehead, et al to establish an exhaustive axiomatization of mathematics and rigorously prove its logical consistency. And without such a proof, there is no guarantee that a contradiction was not just around the corner. Mathematics could be rendered meaningless tomorrow. Naturally, no one believes that this will happen, but conviction is no substitute for proof.

Therefore the effectiveness of mathematics in the logical domain seems considerably less "evident" than it did in the empirical. Mathematics, at least in its axiomatic-deductive form, certainly appears inadequate, if not totally incapable, in dealing with its most critical criterion, logical consistency. Without this, effectiveness depends on "truth." It seems a fair conclusion that mathematics will never be able to establish truth in its usual empirically consistent sense. Any other sense of truth would seem to necessitate first some order to the universe and second our ability to demonstrate isomorphisms between this order and mathematical entities. Whether or not there exists such an order in the first place is a question as old as time (whatever time is). For Plato, ideal forms were reality. He clearly believed in order. Nietzsche, on the other hand, argued that "the general character of the world ...is to all eternity chaos" and asserted "the absence of order, structure, form,...". [8] The fact that mathematicians are now discerning some order within chaos does not alter the basic question here. The existence of order is clearly an article of faith. And even if such an order exists, the hope of our being able to properly axiomatize or model it seems also to have been dealt a fatal blow by Godel and his successors. For Godel's undecidable P, neither P or Ø P is a theorem, yet by the Law of the Excluded Middle, one of them must be true. It therefore follows that theoremness (what can be derived from the axioms) cannot capture truth. In addition, since undecidable propositions are independent of the axioms, either P or Ø P is acceptable as an additional axiom. This suggests the existence of equally acceptable (from a logical point of view) yet fundamentally distinct axiom systems. The situation is reminiscent of the Parallel Postulate in relation to Euclidean and non-Euclidean geometries, but turns out to be much more disturbing. The implication (a la Lowenheim-Skolem) is that any axiom system rich enough to encompass the natural numbers will permit multiple and diverse interpretations or models. This means ultimately that no matter how carefully one designs an axiom system to exactly fit a particular collection of mathematical entities, and only that collection, it will always fit another dramatically different collection as well. Even if there is order out there, we cannot categorically axiomatize it; hence the axiomatic-deductive approach of mathematics cannot isomorphically capture it.

5. Faith ...

Therefore, in my opinion, much of the effectiveness of mathematics comes down to a kind of faith. By "faith," I am in no way harking back to that Renaissance belief in some divine order. I am not even espousing the more secular belief in order itself. My notion of faith does however entail certain rational (or perhaps irrational) convictions, which I will specify.

In the empirical realm, we saw that within limited domains mathematics was indeed remarkably effective. Our question was therefore why, even there, our anthropomorphic constructs modeled reality so unreasonably well. If "unreasonably" well is taken to mean "extremely" well, the answer comes partially from the fact that we are measuring these constructs not against physical "reality," but against our conceptualization of that reality. Extreme effectiveness here should not come as a great surprise. However, that still does not explain why either of these anthropomorphically imposed orders, mathematics or our scientific conceptualization, has anything whatsoever to do with reality itself, hence the "unexplained" reading of unreasonable. In the logical domain, the issues are different, but basic similarities still exist. Toward establishing its own logical foundations, mathematics has again been remarkably effective, deriving essentially all of mathematics from the natural numbers, and in the eyes of some, from set theory. It was only when asked to deal with a fundamentally metamathematical question, its own consistency, that unresolvable problems arose. Therefore, both empirically and logically, it's only when we ask mathematics to exceed its own limits that its effectiveness is called into question and unreasonableness becomes an issue. It's only when we ask this essentially anthropomorphic construction to capture "truth", with respect to either physical reality or some sublime order of the universe, that it falls short. Why we ask this of ourselves and our "heroes" is a separate question. Even in ancient times, one of the essential components of a Greek tragedy was man, the hero, pushing the limits of his destiny, his portion. Since the modern day hero is science, and its heroine mathematics, we should not be surprised to find some parallels. And when we push beyond the limits, there is little recourse, except faith. With respect to empirical effectiveness, the very act of applying mathematics to model reality requires faith in the existence of some isomorphism or quasi-isomorphism between inexact reality and exact mathematical ideality. With respect to logical effectiveness, a whole litany of convictions come into play. The most obvious and most crucial is our faith in logical consistency, and without proof, faith is all it will ever be. Then there are also a host of second order convictions, for instance, in the applicability of anthropomorphic logical principles to such un-anthropomorphic entities as infinite sets. In the past, I have always emphasized to my classes that "the bottom line in mathematics is proof." In the future, perhaps it would be more apropos to say that "the bottom line in mathematics is faith."

References

1. Howard Eves and Caroll Newsom, An Introduction to the Foundations and Fundamental Concepts of Mathematics, Holt, Rinehart, and Winston, 1965.

2. Marvin Jay Greenberg, Euclidean and Non-Euclidean Geometries, W. H. Freeman & Co., l980.

3. R. W. Hamming, "The Unreasonable Effectiveness of Mathematics", American Mathematics Monthly, (Feb., l980).

4. Douglas Hofstadter, Godel, Escher, Bach: An Eternal Golden Braid, Vintage Books, l980.

5. Morris Kline, Mathematics and the Search for Knowledge, Oxford University Press, l985.

6. Morris Kline, Mathematics, the Loss of Certainty, Oxford University Press, l980.

7. Steven Korner, The Philosophy of Mathematics, Dover Publications, l968.

8. Friedrich Nietzsche, Joyful Wisdom, Gordon Press, l974.

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