KARL JASPERS FORUM
TARGET ARTICLE 59
THE AMBIGUITY OF MATHEMATICS
by Bill Byers
[1]
Abstract: This paper deals with mathematical rigor and the notion of ambiguity in mathematics. It takes the counter-intuitive position that ambiguity is of central importance to the mathematical endeavor—that it is essential and cannot be avoided. In our view, rigor and ambiguity form two complementary dimensions of mathematics—what we characterize as the surface versus the depth dimensions of the subject.
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Introduction
In this paper I shall attempt to get the reader to think about mathematics in a different and slightly unusual way[i], one that is closer to the way mathematics is seen by mathematicians when they are doing mathematics. I hope that this new point of view will suggest a new approach to some of the key questions that any philosophy of mathematics must inevitably deal with. Is mathematics discovered or invented? Is it objectively true or is it constructed by human beings? Why has mathematics had such a profound influence on human thought both in the humanities and in the sciences? Finally, the question of whether one can imagine a computer doing mathematics.
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Now mathematics is usual approached in one of two ways. The first way is instrumentally as a body of useful results, techniques, formulas, etc. that are assumed to be valid and can be applied to solve various kinds of problems. This is how engineers use mathematics, for example, or psychologists use statistics. One doesn’t delve too deeply into the derivation of the techniques, the question of why it works. One accepts that it works and moves on.
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The question of why mathematics works is generally answered in theoretical mathematics, the approach you will find in courses taught by mathematicians. Courses in pure mathematics are characterized by a certain approach—the deductive, axiomatic approach. Thus a system is developed involving a consistent set of axioms, the definition of a number of concepts and logical inferences, theorems or propositions, that can be shown to follow.
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The prototype for such a deductive system was, of course, Euclid’s Elements. Even though subsequent thinkers have discovered many places where the work of Euclid is incomplete nevertheless the Elements represents a paradigm of what we call rigorous mathematics. That is, results are proved and the entire body of valid results is organized into a large system of thought. This is the way in which mathematics is taught to this day.
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Why was Euclid so influential in the history of ideas? It was as though the Greeks of that era discovered a new way of using the human mind. A window had been opened that looked out at Truth, Absolute Truth. It is true that this way of thinking is very powerful but, as is the case with many a great discovery, people tended to get a little carried away with potential of logical, deductive thought. Some people even imagined that human beings would one day create a systematic body of thought that was so vast that it would encompass all of reality. We get echoes of thinking to this day with the so-called “theories of everything” that one heard about so frequently in Physics a few years ago.
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There have, of course, been those who have opposed these claims. Today they would include postmodernists who disbelieve in principle in grand theories and constructivists who believe that meaning is constructed. Those who feel that this debate is irrelevant should be reminded that in a sense a computer programme is an example of a logical system. Thus in thinking about the correct way to position deductive thought in the ecology of the mind we are, in particular, discussing the possibilities of machine intelligence, a very active area of research.
[8]
We intend to get into this debate but only insofar as it involves mathematics. Though theoretical mathematics is usually presented as a deductive system in which all results are rigorously developed, does such a description adequately describe what is really going on in the mathematical enterprise? One way of putting it is “Since logical results are ultimately tautological, how does anything new arise in mathematics?” In particular, why does mathematics work as well as it does in describing the physical world?
[9]
In the famous paper on the “Unreasonable Effectiveness of Mathematics in the Natural Sciences” Wigner[ii] claims that the power of mathematics lies in the ingenious definitions that mathematicians have developed. These definitions, he feels, capture some very deep aspects of reality and are therefore the secret as to why mathematics works as well as it does. I have no problem with that description as far as it goes. However I intend to probe a little more deeply. For one thing what is the nature of these ingenious definitions? What is it, exactly, about mathematical concepts that makes them so fruitful?
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It is my contention that to really understand how and why mathematics works it is necessary to go back to reconsider the very things that mathematical reasoning seem to be delivering us from, namely ambiguity and contradiction. After having done so I hope that you will agree with me that the logical structure of mathematics is a necessary but only one dimension of mathematics. At the level at which one does mathematical research, mathematics could be seen to be an art form that relies on something that is akin to metaphor when it attempts to unify the human mind with the objective world.
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Ambiguity and Depth
Anyone who has done some creative work in mathematics will agree that some pieces of mathematics are "deeper" or more profound than others. Often in a piece of mathematics or in a proof one asks questions like, "What is really going on here?" or "What is the basic idea?" These questions go in the direction of depth. The most complimentary thing that one can say about a mathematical idea is that is "deep." So mathematics has more than one dimension. On the one hand there is the dimension of the logical structure, what we will call the "surface structure", (which we will take to include instrumental or algorithmic aspects) but on the other there is the dimension of depth. Of course the division between the two is not so simple but for the purposes of this discussion the distinction is clear enough to talk about. When one says that mathematics is basically tautological or that logic is the essence of mathematics one is referring to the surface structure (which mathematicians usually take for granted). The power of mathematics clearly comes from the other dimension, that of depth. When we talk about the ambiguity of mathematics we are trying to get a handle on the phenomenon of depth.
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What is ambiguity?
People often take ambiguity to be synonymous with incomprehensibility. However we shall primarily focus on the following part of the dictionary definition of ambiguity: "admitting more than one interpretation or explanation: having a double meaning or reference."(Oxford 1993). The writer Arthur Koestler[iii], in his book on creativity, proposed a definition of creativity. He said that creativity arises in a situation where
“a single situation or idea is perceived in two self-consistent but mutually incompatible frames of reference.”
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We shall take the above to be a definition of ambiguity. The sense in which we use the term "ambiguity" will be further clarified by the examples that follow. The key thing here is that there exist two self-consistent frames of reference, and that these frames of reference appear, from the initial point of view, to be incompatible. However I wish to point out that the situation is a dynamic one. There may be a single or unified viewpoint that may be looked at in two different ways. On the other hand there may be the two inconsistent points of view that are reconciled by the creative act of producing the “single situation or idea.”
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Examples of Ambiguity in Mathematics
Ambiguous situation are to be found everywhere in mathematics so perhaps a few examples will illustrate and explain further what we are talking about.
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Square Roots
The square root of 2, Ö2, is one single idea. It is a number with an interesting history. It appears in Euclidean Geometry as the length of the hypotenuse of a right-angled triangle with sides of unit length. Thus Ö2 existed for the Greeks as a concrete geometric object. On the other hand they were able to prove that this (geometric number) was not rational, that is, it could not be expressed as the ratio two (positive) integers, like 2/3 or 127/369. Such non-fractions came to be called irrational numbers and the name “irrational” indicates the kind of emotional reaction that the demonstration of the existence of non-rational numbers produced.
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Now William Dunham[iv] says that the irrationality of Ö2 is one instance of “a continuous feature of the history of mathematics. . .the prevailing tension between the geometric and the arithmetic.” There are two primordial sources of mathematics: counting which leads to arithmetic and algebra and measuring which leads to geometry. These are the two consistent contexts that appear in the definition we proposed for ambiguity: the arithmetic and the geometric. Ö2 poses no problems when considered as a geometric object. It created a major problem when this geometric number is considered as an arithmetic object. The statement of the irrationality of Ö2 is a statement that this geometric number is incompatible with the arithmetic world. Thus the two contexts, the geometric and the arithmetic, appear to be inconsistent or mutually exclusive. In a word, Ö2 is ambiguous. It is this ambiguity that caused the problem.
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There are two possible reactions to the sort of ambiguous situation that we have described above. One can either abandon one of the seemingly inconsistent contexts or one can build a new context that is general enough to reconcile the two contexts. The Greeks chose the former and essentially abandoned algebra for geometry. Even so the irrationality of Ö2 was a great blow to those, like the Pythagoreans, whose entire world-view was based on the rationality (in the sense of rational numbers or fractions) of the natural world. In fact most of Greek geometry had been developed on the assumption that any two lengths are commensurable which amounts to saying that the ratio of their lengths is rational. Thus all of the proofs that depended on this assumption had to be done again in a different way.
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The Greeks never resolved the ambiguity of Ö2. It was only after Descartes had arithmetized geometry that the real number system was rigorously developed. The real numbers provided a context within which the geometric and arithmetic properties of Ö2 could be reconciled and understood.
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This important example from the history of mathematics is relevant to our discussion in many ways. It shows that ambiguity exists in mathematics and that is important. It is not that the geometric context is right and the arithmetic is wrong nor even that they are both right. The importance of the story lies precisely in the fact that Ö2 is ambiguous and that this ambiguity was a spur to the development of mathematics.
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Decimal Numbers
Our second example comes from the world of real numbers. Consider decimal notation for real numbers. For example, we are all taught in school that the fraction 1/3 when written as a decimal number is .333… where the dots indicate that the sequence of 3’s has no end. Thus we might write the equation
1/3 = .3333…
Multiplying both sides by 3 we get
1 = .999…
Now we ask what is the meaning of these equations? What is the precise meaning of the “=” sign? It surely does not mean that the number 1 is identical to that which is meant by the notation .999… . The latter notation stands for an infinite sum. Thus
.999… = 9/10 + 9/100 + 9/1000 + ….
Now an infinite sum is a little more complicated than a finite sum and this complexity is revealed by the fact that the notation is deliberately ambiguous. Thus this notation stands both for the process of adding this particular infinite sequence of fractions and for the object, the number that is the result of that process. The two contexts here (in the above definition of ambiguity) are precisely those of process and object. Now the number 1 is clearly a mathematical object, a number. Thus the equation 1 = .999… is confusing because it seems to say that a process is equal (identical?) to an object. This appears to be a category error. How can a process, a verb, be equal to an object, a noun? Verbs and nouns are “incompatible contexts” and thus the equation is ambiguous. Similarly all infinite decimals are ambiguous.
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We hasten to add once more that this ambiguity is a strength not a weakness of our way of writing decimals. To understand infinite decimals means to be able to freely move from one of these points of view to the other. That is understanding involves the realization that there is “one single idea” that can be expressed as 1 or as .999…, that can be understood as the process of summing an infinite series or an endless process of successive approximation as well as a concrete object, a number. This kind of creative leap is required before one can say that one understands a real number as an infinite decimal.
[22]
The difficulty here is similar to the difficulty that has been pointed about by various authors (e.g. [Kieren][v]) concerning children's propensity to understand the equality sign in simple sums like '2 + 3 = 1 + 4' in operational terms. Gray and Tall[vi] have discussed these "process-product" ambiguities in mathematical notation. They stress that the learner's grasp of these ambiguities is central to their success or failure in mathematics.