That zero divided by zero may simply be **any quantity** is implied by the following chart. This would make zero/zero an indeterminable quantity in its own terms, but determinable in the context of other information. But on its face, the diagram raises a challenge to the foundations of mathematics. It is Euclid's First Axiom of geometry that "Things which are equal to the same thing are also equal to one another." Yet we see here that 0/0 = 1/4 and 0/0 = 3/4, which should mean that 1/4 = 3/4. But that is not the case. The axiom therefore should be false. If it is not, what does the diagram mean?

Here there is a regular array of fractions. The numerator increases by one with each column to the right, the denominator increases by one with each row down. All the columns and rows are equally spaced, and lines are drawn precisely through the middle of familiar fractions (i.e. 1/2, etc.) that are of equal value. It turns out that fractions of equal value are connected by *straight* lines. Red lines highlight values equal to 1/4, 1/2, and 3/4; green lines highlight values equal to 1/3 and 2/3. All the lines, however, converge on 0/0. If the lines connect equal values with all other fractions, then it would seem that 0/0 is equal to every fraction.

The chart does not extend into values larger than one, but it could. Indeed, the row with any number divided by zero would simply extend horizontally, which implies, in the same way as above, that 0/0 would be equal to all those values too.

Now, this demonstration has been criticized on the basis that x/0 is "undefined." "Undefined" can literally mean that it does not occur as an axiom, definition, or theorem in an axiomatic (i.e. set theoretical) number system. However, "undefined" is usually used casually to mean "meaningless," which would only be true if natural number problems outside of axiomatic systems are meaningless, which is not the case. There are axiomatic number systems only because numbers and number problems have natural and intuitive meanings. In those terms, "undefined" often really means "we don't want to think about it." A more relevant or honest answer would be, "No axiomatic number system has yet been able to deal with this, so I don't know what to say." Also, Gödel's Proofs mean that axiomatic systems *cannot capture* all of mathematical truth. Only someone like Wittgenstein, for whom mathematics is a closed and self-referential system, can deny this -- an awkward point that is avoided by Wittgensteinians, while mathematicians simply do not take it seriously. But if Wittgenstein is wrong, the problem of x/0 cannot be dismissed as "undefined." It is now somebody's job to "define" it, as the theory of mathematical truth progresses.

But it is really not too difficult to reason a bit about x/0. In a fraction x/y, if x is any integer, and y is an extremely small decimal, x/y is going to be very large. It will become larger as y gets smaller. As y approaches 0, x/y will become indefinitely large. The obvious step to take, which I think had already occurred to my friends and me in Junior High, would be to say that x/0 is infinite -- we had a natural impatience with this "undefined" business. This is similar to what we see in the tangent of 90 degrees, which sometimes is given as "undefined" and sometimes as infinite. In the trigonometric function, we can visualize the side of the triangle becoming longer and longer as the angle approaches 90 degrees. At 90 degrees it cannot have a finite length -- though a camera lens will still show the two sides of the triangle converging at "infinity." Infinite and other trans-finite numbers are something that have been dealt with in serious mathematics, thanks to Georg Cantor (1845-1918), though evidently x/0 hasn't made it into any results yet. But we have already seen, as with Fermat's Last Theorem, that some of the simplest traditional problems require the most sophisticated modern answers.

The device of using a diagram like this for a mathematical proof goes back to the Pythagoreans but is also familiar in 20th century mathematics, as in Cantor's demonstration that there are different sizes of infinity (i.e. there are actually *more* real numbers than integers) -- also a strange and counter-intuitive result, but not something that has been disputed, I gather, by any serious mathematicians (except for those, the "intuitionists," who don't like infinite quantities at all). The interpretation of such a 0/0 diagram is open to question, since it is not itself a conceptual deduction of a theorem. However, it does display a curious regularity -- it need not have been the case that fractions of equal value are connected by straight lines, or that the lines converge on 0/0 -- and it seems to me that anyone who would just dismiss the graphic result as of no significance is therefore simply not a person of much curiosity. The refuge of "undefined" for x/0 is certainly a way of simplifying one's problems, but it also means that an avenue of inquiry has been blocked off by a self-imposed know-nothing-ism. Now in science, simplifying one's problems can be a good thing, but neither mathematics nor philosophical meta-mathematics can be quite so content -- and it is especially the business of philosophy to question what everyone else thinks is obvious: though I still haven't heard anyone say that x/0 being "undefined" is "obvious" -- that would be a kind of category error, since "undefined" is about operations and decisions, while "obvious" is about insight and understanding.

What I think this diagram demostrates, is not that Euclid's First Axiom is false, but that there are **infinitesimals**. Indeed, we could say that the diagram is a *reductio ad absurdum* of the First Axiom; but this can be avoided by allowing that there are infinitesimals. *Without* infinitesimals, only the *ad hoc* prohibition of 0/0 prevents the contradiction, and the falsification, of the First Axiom. 0/0 = 1/4 is true only if 0/0 conceals of ratio of infinitesimals in which there is a determinant and definite quantity, namely 1/4. The 0/0 in "0/0 = 1/4" and the 0/0 in "0/0 = 3/4" are thus *not* actually the same quantity. This is a little awkward, but then we need only ask where the problem of 0/0 arises. Since it arises in relation to the derivative, and since the derivative provides a way of dealing with infinitesimals, we have the problem and the solution all at once.

The logical problem of 0/0 has a real world application. Photons, subatomic particles that are the field quanta, the bosons, of electromagnetic radiation, have zero rest mass. However, they do have momentum, which ordinarily is **mass times velocity**. Since their mass is zero, and zero times anything is zero, one might think that, whatever their velocity, their momentum would be zero. Not so. And we can see why. Photons instantaneously travel at the velocity of light. Indeed, they *are* light. According to Einstein's theory of Special Relativity, any moving object approaching the velocity of light has its mass multiplied to the extent that its mass would be infinite *at* the velocity of light. This is the physical reason why things can't go faster than light, since it would take an infinite force to accelerate something that has an infinite mass. With light itself, however, it's already going the velocity of light. So in effect, we have zero mass times infinity. Zero times anything is zero, but then infinity times anything is infinite. So what happens to the photon? Well, its zero rest mass times the infinite multiplication of Relativity results in... a finite momentum, as though it had a non-zero rest mass times a velocity less than light. Since the reciprocal of zero (1/0) is infinite (or that "undefined" thing again), zero times infinity is equivalent to our problem of 0/0.

A correspondent recently complained that the momentum of a photon is derived from its frequency and is not simple result of velocity times mass, and that zero times infinity is actually zero.

First of all, while we should indeed wonder about the application of the definition of momentum, as mass times velocity, to photos, nevertheless, the point here was the application of the equations of Relativity, where a factor derived from velocity is multiplied by rest mass. Since, at the velocity of light, this factor is infinite, we have the problem that I am addressing.

The de Broglie wavelength for particles is given by the equation **λ = h/p** -- with "h" here for Planck's Constant and "p" for momentum. Since the wavelength of light is **= c/ν** (the velocity of light divided by the frequency), the equation can be rewritten **ν = cp/h**. So it looks like we can derive the frequency from the momentum, rather than the other way around. Since these are equations, however, we can go either way: **p = νh/c** is just as good. So this really tells us nothing about the *origin* of the momentum for a massless particle. In fact, the ordinary meaning of momentum and the ordinary Relativistic effect of velocity on mass are only a puzzle here because we do end up with particles with zero rest mass but finite momentum.

As for the assertion that zero times infinity is zero, this would mean that a particle going at the velocity of light, with zero rest mass, would have a momentum of zero, according to the equations of Special Relativity. Also, if we multiply infinity by any number, and get infinity, this result does not change as the number becomes smaller and approaches zero. If an infinitesimal times infinity is still infinity, it would be surprising if the number suddenly became zero *at* infinity. Instead, we can consider a definition of infinity as the reciprocal of zero: **1/0 x 0**. Any number multiplied by its reciprocal should result in unity, which is one of the results we could get from **1/0 x 0**. The ordinary rules of algebra enable us to evaluate **1/0 x 0/1** either as **1/1** or **0/0**, depending on which factors we regard as canceling each other out. In the diagram above, **1/1** is one of the results of **0/0**, but of course there are others. Dividing zero by zero for photons enables us to have finite results that are not unity, but as in Calculus it allows the same thing. What that is going to be is indeed, in the former case, specified by the de Broglie wavelength or frequency, just as the derivative of particular equation gives us the value of 0/0 for that particular equation.

There is a 1991 science fiction story by Ted Chiang, "Division by Zero," which begins with the statement:

Dividing a number by zero doesn't produce an infinitely large number as an answer. The reason is that division is defined as the inverse of multiplication; if you divide by zero, and then multiply by zero, you should regain the number you started with. However, multiplying infinity by zero produces only zero, not any other number. There is nothing which can be multiplied by zero to produce a nonzero result; therefore, the result of a division by zero is literally "undefined." [Stories of Your Life and Others, Vintage Books, 2002, 2016, p.71]

The problem with this reasoning is the statement, "multiplying infinity by zero produces only zero." This is true if we apply the rule "anything multiplied by zero is zero," but it encounters problems if we also want to apply a comparable rule, "anything multiplied by infinity is infinity." This is the irresistable force meeting the immovable object. But the resolution is the conclusion that zero multiplied by infinity is a finite number, any finite number. This is what the diagram for **0/0** demonstrates. What the expression "literally 'undefined'" is supposed to mean is itself something that is left (literally) undefined. Instead, "undefined" means either that it *cannot* be defined, or just that "we" haven't yet defined it. The former would need a proof, which Chang does not supply, and the latter means that perhaps we have just not gotten around to it yet, which proves nothing.

As it happens, the story references a trick proof, without showing it, that results in "a=2a." It is explained that the trick turns on dividing by zero. From the diagram, we can see how this is possible, if 0/0 = 1/4 and 0/0 = 3/4. The rest of the story is how a mathematician discovers a proof of "1=2," demonstrating that arithmetic is inconsistent -- a possibility we have already seen if 0/0 is allowed to falsify Euclid's First Axiom. As Aristotle knew, if you allow a contradiction, you can prove anything. So the mathematician of the story realizes that she has destroyed mathematics: "I've just disproved most of mathematics: it's all meaningless now" [p.81]. The mathematician contemplates suicide, but the story ends without any particular resolution. This may have been how Frege felt when confronted with Russell's Paradox.

While the consistency of arithmetic is a good question, a story with an imaginary disproof of it can only be of limited interest. Division by zero is of much greater and more real interest, yet the story does not really get into it, beginning with the questionable assertions quoted above. The interest there is just how Ted Chiang is tripped up.

Return to Philosophical Problems with Calculus

Philosophical Problems with Calculus

Philosophy of Science, Mathematics