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.
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 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."
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, though evidently x/0 hasn't made it into any results yet. But we have aleady 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 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.
Return to Philosophical Problems with Calculus