Ambiguity of Mathematics






by Bill Byers



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.





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.



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. 



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. 



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.



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.



                 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.



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? 



                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? 



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.




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.




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.” 



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.” 




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. 



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.



                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.



                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.



                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.



                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.




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. 



                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.



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. 






A matrix leads a double life.  On the one hand it is a collection of numbers arranged in a rectangular array, on the other, it is a function, a linear transformation.  (It also has many other interpretations but these two are sufficient for us to make our point.)  Whereas we add matrices as though they were collections of numbers, we multiply matrices in the way that we do because we are composing them as functions.  Often, in linear algebra, we jump back and forth between these two points of view.  Rank, for example, can be looked at from both points of view.  Or, think of the representation of a linear transformation, T, as a matrix relative to a certain pair of bases.  Since T is usually given by a matrix the whole situation is fraught with ambiguity.  It is the existence of this multiple perspective which gives the student so much trouble.  They often ask:  "When do you think of a matrix in one way, when in the other?  How do you know which way to think of a matrix in a given problem?"  However, it is precisely this ambiguous point of view which gives the concept of a matrix its depth.  The successful student has learned to alternate easily between these two ways of looking at a matrix.  In fact when we think of a matrix it has become a mathematical concept with an independent existence which can be looked in a multiplicity of ways.  No one of these ways is the exclusive or the correct way of understanding what a matrix is. 






The notion of a function is ambiguous.  There are many equivalent definitions but let us focus on two.  There is the ordered pair, graphical definition of a function.  This is a static definition: the function is a set (of ordered pairs) or a picture (the graph) or a table.  However there is also the mapping definition, which is related to the black box, input-output definition.  This latter is a dynamic definition.  Here the ‘x’ is transformed into the ‘y’.  This definition is the one that is used in thinking of a function as an iterative process or a dynamical system or a machine. 



Again mathematicians go back and forth from one of these representations to the other.  New developments in mathematics may entail looking at a concept in a new way.  The input-output model was crucial to looking at functions as the generators of iterative processes.  It came into its own with the development of computers.  The graphical representation of a function is of little value when one wishes to study the orbit structure that the function generates.



At a higher level one puts sets of functions together to form function spaces.  In fact one of the conceptual breakthroughs in analysis is the idea that a function may be considered a point in such a function space.   Here again the initial barrier to understanding, namely that a function could also be thought of as a point, turns into an insight.  That is, it is precisely the ambiguous way in which a function is viewed which is the insight.  Once a function is seen as a point in a metric space, we can talk about the distance between functions, the convergence of functions, about functions of functions, etc.  This sort of dual representation is often present in situations of mathematical abstraction.




Fundamental Theorem of Calculus


The Fundamental Theorem is a non-trivial application of the above discussion on ambiguity.  Differential Calculus and Integral Calculus can (and historically were) developed independently of one another.  The Fundamental Theorem says, of course, that these processes are inverses of one another.  This means that differentiation is not more fundamental than integration nor is the opposite true (at least for functions of one variable).  Actually the theorem says that there is, in fact, one calculus process which is integration when we look at it in one way and differentiation when we look at it in another.  That is, there is a multiple perspective that is essential to an understanding of calculus. 



How is this multiple perspective used?  Consider, for example, the proof of the existence theorem for the differential equation

There is a great proof that proceeds by rewriting the equation as an integral equation

and then seeing that the solution is a fixed point of the contraction mapping


This proof is possible because of the dual representation of the calculus as derivative/integral.  Mathematics is full of such dualities.  Each of them adds depth and power to mathematics.


Fermat’s Last Theorem

 A final and contemporary example involves the proof of Fermat’s Last Theorem.  The proof hinges on the validity of the Taniyama-Shimura conjecture.  This conjecture unifies the world of Elliptic Equations with that of Modular Forms.  This conjecture unifies the seemingly disparate worlds of elliptic equations and modular forms.  To understand the power of ambiguity to revolutionize mathematics one has but to read the comments on this conjecture by the Harvard number theorist Barry Mazur[vii].  He compared the conjecture to the Rosetta stone that contained Egyptian demotic, ancient Greek and hieroglyphics.  Because demotic and Greek were already understood, archeologists could decipher hieroglyphics for the first time. 


“It is as if you know one language and this Rosetta stone is going to give you an intense understanding of the other language.  But the Taniyama Shimura conjecture is a Rosetta stone with a certain magical power. The conjecture has the very pleasant property that simple intuitions in the modular world translate into very deep truth in the elliptic world, and conversely.  What’s more, very profound problems in the elliptic world can et solved sometimes by translating them into the modular world, and discovering that we have insights and tools in the modular world to treat the translated problem.  Back in the elliptical world we would have been at a loss.”




Those who feel that mathematics is a formal system whose fundamental characteristic is consistency and thus whose mission is to banish contradiction do not appreciate the audacity of mathematics. We saw in the case of ambiguity how mathematics manages to incorporate ambiguities within its logical structures.  This is one source of its power and effectiveness.  Now contradiction may be taken as an extreme form or limiting case of ambiguity.  It is interesting that the idea of “proof by contradiction” is already present in Greek mathematics. Euclid needs to use it because he expresses the famous Parallel Postulate in a negative way.  Arguing by contradiction is already building contradiction into mathematics.  One proves that Ö2 is irrational by contradiction.  Cantor uses contradiction to show that most real numbers are transcendental when we scarcely know for certain that any particular numbers are transcendental.



The Number 0[viii]


                A particular, though not popularly appreciated, use of contradiction in mathematics involves the notion of zero.   Those who feel that mathematics is merely a logical enterprise do not understand the subtlety of many mathematical concepts such as the notion of 0.  Now isolating the idea of 0 happened relatively late in the history of mathematics—much later than the concepts of the small positive integers.  Why was 0 such a difficult idea to isolate?  Could it be that the difficulty was that in a certain way 0 is a contradiction in terms?  0 is a something that stands for nothing.  It is a presence that indicates an absence.  0 is a much more difficult idea than 1 or 2 or 3 although these too have their subtleties.  At the heart of the idea of 0 there is the idea of naming that which is unnamable.  We are creating something out of nothing, literally.  Now isolating and naming any concept involves a contradiction in the sense that the name is not the thing named but has the circular relation to it in that it both evokes what is named and is in turn evoked by it.  Be this as it may, the creation of the concept of 0 is a particularly audacious idea—one of the triumphs in the intellectual history of mankind.  It is a triumph of Hindu civilization, related to their fascination with the idea of “nothingness” or sunyata.[ix]


                There are those who believe it is the mission of mathematics to eliminate contradiction from human discourse and their preferred instrument is the computer.  How mortifying to see contradiction firmly planted in the foundations concepts of mathematical science—inside the number 0 itself.  This is the very same number 0 that is the basis for all machine language for this 0-1 machine.




                 To demonstrate that 0 is not an isolated case, that other concepts are built upon a foundation, not of logic, but of contradiction, consider the case of infinity.  It will not be possible here to do justice to the vast topic of infinity.  We shall just stress that, like zero, to discuss infinity is to enter into contradiction and paradox.  The philosopher A. W. Moore begins his book on Infinity[x] with a list of paradoxes associated with the idea of infinity.  He stresses that the word infinity is itself a paradox since it is a word, therefore finite and bounded, purporting to stand for that which is not finite and which is beyond bounds.  To conceptualize infinity, many people have thought, was impossible in principle.  This concern was behind attempts by numerous philosophers and mathematicians to distinguish between potential and actual infinity.  Gauss, the greatest mathematician of his time and perhaps of all time stated:

 . . .I protest above all against the use of an infinite quantity as a completed one, which in mathematics is never allowed.  The Infinite is only a manner of speaking. . .[xi]


Yet Cantor succeeded in creating a mathematical theory involving actual infinities, infinite cardinal and ordinal numbers.  Controversial though his theory was at the time it has now found a home at the heart of mathematics.  It is a valid and useful mathematical theory, logically consistent yet based on a contradiction.  To give an astronomical metaphor, it like a black hole that are reputed to live at the centre of galaxies, invisible (to the eye of logic or computers) yet immensely powerful.  Formal mathematics is well able to deal with infinity once it has been domesticated, that is given a formal definition and placed within a consistent theoretical development.  Yet a formal development is blind to tensions that this concept is holding together.  To get a glimpse of these tensions one has but to read a biography of Cantor, his repeated hospitalizations for psychiatric reasons, and his conflicts with the mathematical community of his time.[xii]




                 Mathematics is not merely algorithmic.  It is not merely about formal, deductive thinking.  If you think of mathematics in that way you will never understand its subtlety and power.  Mathematics derives its power from an intricate and complex marriage of logic with the creative power of something that is akin to metaphor.  0 and ¥ are metaphors.  They capture something that does not exist on one level and create something absolutely new and original that describes basic properties of the natural world.  Any deep mathematical definition is ambiguous and so metaphoric.  Understanding the definition means no less than grasping the ambiguity, that is, being able to move freely from one of the many viewpoints that is covered by the single definition.  Equations, though often considered as statements of fact, could better be thought of as metaphors.  I even think of 2+3 = 5 as a metaphor for surely 2+3 is not identical to 5.  2+3 is an action, 5 is an object.  Children say that 2+3 “makes” 5.  The word = should be understood exactly as the word “is” is understood in a metaphor. The equation E = mc2 says that what looks like too different ideas; matter and energy are, in fact, one ambiguous idea.  And we all know the consequences of that metaphor.



                Think of mathematics as having these two dimensions.  The first is the logical-deductive. The second is the ambiguous-metaphoric.  To return to the definition of Koestler, “a single situation or idea is perceived in two self-consistent but mutually incompatible frames of reference.” Each of these two dimensions is a self-consistent frame.  Mathematics itself is the “single situation or idea” that is the creative resolution of these two points of view.  Thus not only are the elements of mathematics—the concepts and the theorems—ambiguous.  Mathematics itself is ambiguous. 



Bill Byers

Professor, Dept. of Mathematics and Statistics

Concordia University

7141 Sherbrooke St. W.

Montreal, Quebec, Canada, H4B 1R6


[ This paper was presented at the Montreal Inter-University Seminar, 22 January 2003, at Concordia University ]

[i] Bill Byers, The Ambiguity of Mathematics, Proceedings of the International group for the Psychology of Mathematics Education, Vol.2, pp.169-176(1999)


[ii] Wigner E.,  [1960]  'The Unreasonable Effectiveness of Mathematics in the Natural Sciences',  Comm. Pure & App. Math.13, 1.


[iii] Arthur Koestler, On Creativity


[iv] William Dunham, Journey through Genius: the great theorems of mathematics, Penguin, New York, (1991)


[v] Kieren, C. [1981].  'Concepts associated with the Equality Symbol'.  Educational Studies in Mathematics, 12, 317-326.


[vi] Gray, E. and Tall, D. [1994] ‘Duality, Ambiguity and Flexibility:  A “Proceptual” View of Simple Arithmetic’. Journal for Research in Mathematics Education,                 25, 2, 116-40.


[vii] Quoted in Simon Singh, Fermat’s Enigma, Penguin Books (1998)


[viii] cf. John D. Barrow, The Book of Nothing, Vintage, London  (2000).  Robert Kaplan, The Nothing That Is: A Natural History of Zero, Oxford University Press (1999).


[ix] Barrow pp.35-45.


[x] A. W. Moore, The Infinite, Routledge, London and New York (1990)


[xi] Quoted in William Dunham, Journey Through Genius: The Great Theorems of Mathematics, Penguin, New York, pg.254.


[xii] Amir D. Aczel, The Mystery of the Aleph, Simon & Schuster, New York  (2000)




Bill Byers is a Professor in the Department of Mathematics and Statistics at Concordia University in Montreal.  He received his Ph.D.

in Mathematics from the University of California at Berkeley under the direction of Stephen Smale.  He has published  in the areas of

Dynamical Systems, ergodic theory and differential geometry as well as in mathematics education.  The present paper is a revised version of a

paper given to the international Group for the Psychology of Mathematics Education.  At present he is writing a longer work on the

philosophy of mathematics for which this paper will form the nucleus of one section.