When I tell undergraduate students they can’t I just go with the cleanest and simplest explanation. It is not defined under division. a/b where b is not 0.

Many years ago, when I was told I couldn’t divide by zero, I thought, “Why not?” Never getting a good answer, I continued to think about it from time to time. Then, when I ran into derivatives in college, I thought, “What a crock! They’re dividing by zero!” And, while they define things in tortuous ways to make the system self-consistent, I remain unconvinced. The idea of taking the limit of an infinite sum should make anyone who hears it say, “Hey, wait a minute…” But it doesn’t.

It was due in no small part to reflection on this hypocrisy that a powerful realization eventually came to me: that propositions aren’t useful because they’re true; rather, they’re true because they’re useful. This applies, not just to mathematical reasoning, but to all forms and contexts of rational thought.

Therefore, I love this statement from the above-referenced webpage concerning division by zero: “Strictly speaking, it is allowed, but we just don’t get a useful answer when we do.” This statement rightly isolates the critical factor in denying division by zero: it isn’t useful. One might even say it’s anti-useful. After all (as Dr. Arsham points out), if we can divide by zero, then 1=2(=3=4=…). That would be horrible.

Wouldn’t it?

All models are incomplete. No matter how inclusive you try to make them, they cannot help but leave something out. Not only can they be incomplete by leaving out details, they can be incomplete in their very methodology, choosing to exclude whole aspects of experience (and therefore reality). Who’s to say that at least some of those aspects aren’t important? Even vital? Such that their exclusion renders the model warped and twisted, such that re-applying that model to experience warps and twists our evaluation thereof?

I ask myself: is zero real? Does it exist? It must exist, if only as the point of equilibrium or balance between positive and negative quantitivity. Therefore, the idea that an operation like division would have meaning at every other point on the number line, yet not at this one, suggests strongly that someone is trying to hide a red-headed stepchild. It’s inconvenient, and they don’t know where it comes from; but ignoring it won’t make it go away.

I think Zeno shows us the way. Zeno posited several paradoxes designed to refute the idea that continua are divisible, such as that of Achilles and the tortoise. The argument is pretty simple: if Achilles and a tortoise have a race, and the tortoise runs only a tenth as fast as Achilles, if the tortoise is given a head start Achilles can’t catch him. This is because, by the time Achilles catches up to the tortoise’s starting position, the tortoise has moved ahead–only one-tenth the distance, but ahead nonetheless. By the time Achilles reaches this next point, the tortoise as moved on again…and so it goes, ad infinitum. Achilles never catches up. Since we know that Achilles will always leave the tortoise behind in a cloud of dust, experience refutes the proposition that continua are divisible.

This paradox stood for about 2,000 years before mathematicians found a way around it. (Note: I did not say they *solved* it; they didn’t. They found a way around it.) They did this by creating the notion of the limit of an infinite sum. The proposition that such a creature exists is preposterous on its face; the sum of an infinite addition problem cannot exist, because the problem is never over. They attempt to get around this by defining a limit as something never reached, only approximated; but watch what they do with them when it comes to calculus.

Calculus is based on the concept of the derivative of a function, which is defined as the limit of the change in the dependent variable (y) divided by the change in the independent variable (x) as the difference between two independent variables (x, x+dx) shrinks to nothing (zero). To perform their calculations, they assign a variable to this difference (dx), factor it out of the denominator, and then assign it a value of zero.

You really have to ask yourself about the intellectual integrity of people who poop on division by zero and yet do this, all the time with a straight face.

As I said, I think Zeno gives us some indication of what division by zero might mean. What if division by zero is incomprehensible because it is a point where our model *breaks down* in its reflection of reality? (Don’t say math is artificial; there are no purely artificial constructs. Besides, such a criticism is disingenuous; usefulness is what drives almost all mathematical inquiry.)

And, if so, what is it that has broken down, but quantitivity itself?

Quantification requires divisibility; and, without quantification, there are no classes or groups. Now, it’s interesting that, besides blurring individual distinctions, classification is ineradicably depersonalizing; but, without quantitivity, each thing is unique, appreciated for itself and not because it is like or unlike something else. Is this why people rebel at being quantified? Is this why people feel alienated in this culture to the extent that they do–because *everything* is grist for this mill, because *everything* is viewed through this lens?

Quantitivity isn’t real, and the creatures it creates aren’t real. It’s only a model, an approximation; and something gets lost in the translation. We need to be reminded of this; for, when it’s people who are being quantified, it’s our humanity, our individuality, our value as unique, individual persons that is fractured.

Not only this, the problem posed by division by zero is a reminder of our limitations, and of what a cosmic joke it is to try to cram all of reality within the limits of the human mind.

I think this is valuable stuff. Too valuable, in fact, to be dismissed with a thwack and “because I said so.”

But it is not the promise of math to live up to those kinds of expectations. I also, as you, do not know as much about math as I’d like to, but a certain “willing suspension of disbelief” (up until a certain point) will give you the insight to accept math for what it is worth, and then work to get the wrinkles out. Math is constantly developing, and simply saying you “don’t like how it’s done,” is meaningless. Until you make a contribution, you need to work with what you’ve got.

Don’t fall into the trap of being a kind of denying science because it’s “not perfect.” It isn’t supposed to beperfect, it’s supposed to investigate, describe, and correct itself. These things take time. There are lots of people who are working on things you can’t even imagine. We can’t invalidate their system because it offends our proprieties.

The point of math is that it requires no “belief.” You just need to suspend your disbelief as you learn something new. If it doesn’t work, prove it (not because you “believe” it is this way or that – you must put your proof up to the same standards that you demand of others). Once you learn and understand something, it requires no further suspension of disbelief.

I hope this helped.