“Unless a grain of wheat falls into the earth and dies, it remains alone; but if it dies, it bears much fruit.” – John 12:24
“The last enemy to be destroyed is death...Death is swallowed up in victory.” – 1 Corinthians 15:26, 54
"The greater the sin, the greater the mercy, the deeper the death and the brighter the rebirth.” - C. S. Lewis
"This story...has the very taste of primary truth." - J. R. R. Tolkien

Sunday, May 11, 2008

A Theology of Mathematics: Philosophies of Mathematics

For my Physics Seminar class, I had the opportunity to write a paper on the nature of mathematics, a topic that has interested me for some time. I argue for a "realist" perspective of mathematics and for the view that number is inherent to the nature of God.

A Theology of Mathematics

1. Introduction

What are numbers? This may seem like a simple question, and perhaps it is, but is a very significant question. After all, what is the “number” 2? What is 3, or e, or 239? A simple and accurate definition is elusive, and perhaps it is impossible to convey verbally the full idea of numbers. In any case, we perceive in numbers something seemingly universal and essential – an underlying structure to reality. Various thinkers throughout history have perceived something powerfully real in mathematics, even describing a sublime beauty to the structure of numbers. In this paper I will discuss the nature of mathematics and the idea of mathematical beauty from a distinctly Christian perspective. I will argue that numbers are real entities, independent of physical reality, and that our perception of things beyond the physical world poses problems for a naturalistic worldview. Furthermore, our perception of mathematical beauty points to the necessary existence of a deeper Reality that is the ontological foundation for the physical world. This Reality has been viewed by many of the greatest thinkers as an omnipotent and transcendent Being, and this view provides the most plausible account of reality and of the existence of mathematics in particular.

In speaking of abstract mathematical truths or realities or entities, I wish to refer to the field of mathematics in general, and to numbers in particular, leaving open the possibility that more abstract concepts like sets are not real abstract entities but merely useful concepts. Professor of mathematics Reuben Hersh points out that mathematicians have invented various different formulations, theories, or definitions in order to more accurately describe reality or to solve certain problems (Hersh 74). In this paper, though, I am concerned primarily with numbers, which have always been understood intuitively rather than having been formulated or invented as useful concepts.

2. Epistemology and Ontology

In considering the nature of mathematics, it is essential that we understand the nature of human knowledge and how it relates to actual reality. First, as Ben Carter states in “The Limits of Mathematics in Assessing Causality,” “the world we perceive is distinct from the world as it is” (Carter 283). What is true and real is a separate issue from what we think or perceive or believe to be true or real. However, in the words of John Polkinghorne, “epistemology models ontology.” That is, although we cannot have exhaustive knowledge of reality, what we do perceive should be taken as a generally reliable guide to truth. Our basic reaction to our perceptions and/or intuitions, whether sensory or otherwise, should be one of trust rather than suspicion. Of course, we cannot know with 100% certainty that we perceive correctly in every case (for example, there is a finite chance that the physical world which we perceive to exist does not exist), and each incidence of perception or intuition should be considered probabilistically. That is, how confident am I that what I perceive or think to be true is true? The answer may vary, and we must judge carefully and try to identify biases in our thought or feeling, recognizing that we have “no grounds for supposing that, by using our virtual world as a standard, we can model the actual world in any genuinely exhaustive way” (Carter 280, emphasis added). In general, though, we can do have some knowledge of what is out there and can increase our level of knowledge by further consideration of reality – epistemology models ontology.

3. Do Numbers Exist?

The basic question in philosophy of mathematics is whether purely mathematical entities, and numbers in particular, are in some sense real or are only useful inventions of man. That is, do numbers, as abstract entities, really exist, or are they merely tools that are helpful in describing the physical world (for example, in saying that there three cows)? Is mathematics discovered or invented? It should be noted that the written equations themselves are merely representations of the mathematical concepts. Other representations are possible, but we are concerned with the mathematical idea to which the invented representations refer.

3.1. Philosophies of Mathematics

In his book Philosophy of Mathematics, professor of philosophy Stephen Barker lays out three main interpretations of abstract objects, and of numbers in particular: nominalism, which states that numbers have no ontological status independent of physical reality, conceptualism, which holds that numbers are merely inventions of the mind, and realism, which maintains that numbers exist independent of physical reality or human thought (Barker 69). The mainstream interpretation, taken by many of the greatest philosophers and mathematicians throughout history (Descartes, Kant, Russell, Gödel, etc.) has been the realist interpretation; “mathematics is superhuman – abstract, ideal, infallible, eternal” (Hersh 92). This view of mathematical truth as real, eternal, absolute, and almost divine has its roots in Pythagoras and Plato, and it was embraced by Augustine, perhaps the most influential Christian thinker, who stated that “this incorruptible truth of number is common to me and to any reasoning person whatsoever” (cited in Hersh 104). Despite the rise of naturalism and materialism in the last few centuries, most mathematicians continue to see abstract numbers as real (Howell and Bradley 3). The non-realist interpretations of mathematics are much younger in their origins and have been less popular throughout the history of mathematics.

3.2. Non-realist Approaches

Nominalism and conceptualism share a good deal of common ground with one another, subjecting number to dependence on mind and/or matter, whereas the mainstream approach of realism is strikingly different. Conceptualism is less far from realism, though, in granting the reality of abstract universal concepts within the mind (Holmes 4). Both perspectives, however, fail to provide an adequate account of reality, and for similar reasons. In general, the restriction of mathematics to dependence on either the mind or on matter invalidates too much of what we actually perceive in mathematics. Certainly the mathematician would say that 3410000 is no less valid a number than, say, 13; indeed, there are infinitely many valid natural numbers. But since the human mind is finite, it is at least difficult to imagine how it could “create” an infinity of numbers.1 One might object that “It’s not the infinite that our minds/brains generate, but notions of the infinite” (Hersh 75). But Barker argues that number theory implies an actually infinite number of successive natural numbers (Barker 70) – if our minds do not generate this infinity, but only notions of it, then “number theory cannot be given any thoroughly nominalistic interpretation under which it will come out literally true” (Barker 72). That number theory is false is hard to swallow, though, given (at the very least) the extensive usefulness of mathematics in describing the physical world. One might respond that we create a finite number of axioms, which in turn imply an infinity of numbers, but this answer still implies that our minds generate, at least indirectly, an infinity of numbers, and it is hard to fathom infinity coming from something finite. In distinguishing between actual infinity and notions of the infinite, Hersh makes a valid point which demonstrates the subtlety and complexity of the issue and lessens the tension for nominalism and conceptualism somewhat. Still, there is a definite tension here which reckons nominalism and conceptualism less plausible.

Furthermore, in claiming that number is real only as a creation of the mind, conceptualism fails to give a realistic (no pun intended) account of the practice of mathematics. First, it struggles to explain the universality of mathematics, as understood by humans, across time and space. Mathematicians in different cultures and throughout history have arrived at the same mathematical ideas and worked from the same intuitive understanding of numbers. For example, mathematicians in pre-modern China, ancient Greece, and medieval Islam arrived at many of the same theorems or equations despite the fact that their mathematical formulations or methods differed; there is a “universality of content” (Howell and Bradley 62) throughout different cultures and worldviews. Can this universality be explained if mathematics is a mere invention of the mind? If number is created in the mind, then why do mathematicians feel constrained to “create” only certain intuitively sensible axioms or definitions? One might reply that a “number sense” is built into humans genetically. While this may explain the cultural universality of mathematics to an extent, it is not a philosophical answer and offers no solution to the numerous other problems faced by non-realist interpretations. Furthermore, there still remains the fact of our strong intuitive understanding of mathematical truth, which is no less reliable if there are biological descriptions of how it exists.2 Mathematicians have always done their work and arrived at conclusions as if they are trying to discover an absolute reality that is already there (for example, in the finding of larger and larger prime numbers), rather than merely finding an efficient way to describe the world.

Intuitionism is a more recent interpretation, which Barker describes as a variation on conceptualism that states that mathematical truth is no more than what we intuitively sense to be true in our minds. In other words, the intuitionist restricts mathematical truth to the intuitively obvious. Possibilities such as Fermat’s last theorem and Goldbach’s conjecture, which are unproven but may well be true, Barker argues, are rejected by the intuitionist (Barker 76-77). This kind of approach, I would argue, flies in the face of logic and is incompatible with the philosophical distinction between epistemology and ontology made at the beginning of this paper. It is an essential logical premise that a given proposition is either true or false whether we know so or not.

3.3. Logicism

There are several other philosophical interpretations of mathematics which vary in the extent to which they are realist or non-realist. Here I will only briefly touch on one of them. Logicism is closer to realism, but is not necessarily in contradiction with non-realism. Formed primarily by Gottlob Frege and expanded by Bertrand Russell and A. N. Whitehead, all of whom were mathematical realists, logicism is the idea that mathematical concepts can be reduced to logic and deduced from the basic laws of logic. According to Barker, this idea was formulated be realists in an attempt to ground the reality of mathematics in the laws of logic (Barker 81). Although it is a more plausible theory than nominalism or conceptualism, logicism faces several problems. According to professor of mathematics Paul Zwier, “the additional axioms created by Russell and Whitehead…are widely held not to be purely logical in character” (Howell and Bradley 25). Even in Zermelo-Fraenkel Set Theory, the favored system for presenting mathematics, “there are axioms…generally acknowledged as not purely logical” (Howell and Bradley 29). Furthermore, Kurt Gödel’s incompleteness theorems demonstrated that it is impossible to formulate a set of axioms that are both consistent and give a complete description of mathematics. In other words, “mathematics ultimately rests on intuitions that cannot be proved” (Carter 280). Thus, it is highly doubtful that mathematical truth could be proven from logic alone, at least in any complete sense. Our intuition of mathematical truth then seems to be that of something irreducible and fundamental – related to logic yet not logically verifiable. It is all the more remarkable that we have such a strong “sense” of numbers.

3.4. Yes, Numbers Exist

Even if it requires further axioms beyond logic, there are numerous reasons to think of mathematical objects, and numbers in particular, as real abstract objects. One of the strongest reasons for realism is the fact of our strong intuitive perception of mathematical truth. When we consider the idea “2+3=5” we realize intuitively that it is a true statement. Similarly, we understand that “2+3=7” is a false statement, and this intuitive perception is of number in a general and abstract sense. If epistemology models ontology, as we have assumed, then it would seem our knowledge of pure numbers and of a distinction between true and false statements describing numbers indicates some sort reality for pure numbers.

In summary, realists are free to look at the infinite range of numbers in mathematics as a reality that exists independent of the restrictions of the finite physical world or finite human mind. The realist perspective also offers the best account of the culturally universal content of mathematical conclusions, the practical approach of mathematicians which aims to discover rather than invent, and of our intuitive understanding of mathematical truth. The most significant piece of evidence for the abstract reality of mathematics, though, is perhaps our perception of mathematical beauty, and it is this perception that we will consider next.

1 Similarly, the physical universe is finite, but it is difficult to imagine the infinite range of numbers depending on a finite physical reality.
2 Our thoughts cannot be dismissed as unreliable because our brains may be affected by the physical world – this would cause us to doubt our perception of the physical world, which is the very premise of the train of thought. Evolutionary biologist J.B.S. Haldane wrote “If my mental processes are determined wholly by the motions of atoms in my brain, I have no reason to suppose my beliefs are true...and hence I have no reason for supposing my brain to be composed of atoms” (Byl 6).

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