Does the fact that we cannot divide by zero suggest to you that mathematics as a whole might be fallacious?

Question

Do you think we will discover one day (assuming humans survive that long) that most of mathematics, as a academic field (not your individual learning), is incorrect, a faulty model?

That, for example, arithmetic and algebra, while it seems to work with our current limited comprehension of the universe, at a later time they will be shown to cause other paradoxes and logical fallacies to exist if they were true?

Is that hole in math, dividing by zero, just one hint that something is wrong with this picture?

Answer 1

That is only a problem if you consider mathematical objects to be objects in the more commonly accepted way. This is the way Plato envisaged the world; there was a world of ideas which did not depend on humans for its existence but was merely discovered by us through its projections in an imperfect world. So if you view numbers as actual objects, and operations as functions applied on real objects, the division of zero becomes a problem. Why can't you divide by zero? Zero is a number - why can't it become a divider? One solution, of course, is to simply posit the answer "i" - some irrational number which we know nothing about, except it is the result of such a division. But this will cause you brand new problems - why can't you operate a proof with "i" - i.e. if 6/0 = i, then how come I can't posit that i x 0 = 6? is "i" the same "i" as that which is expressed in 8/0 = i?
But I take a more Wittgensteinian approach to the whole question. Mathematics, to Wittgenstein, is a language, just like French is a language. Numbers don't exist as such. They exist insofar as they express a reality which we communicate. As long as we both agree with what I mean by six dollar, we won't have an arguement. And that is the litmus test for the validity of numbers.
So then, we might ask the question: Well, when does division by zero actually occur? In a will, a man decides to divide his money between four descendants. So that's easy x/4 = what he leaves to each. What if the man had no descendants? Would it then get divided by zero? Well, no. The state then becomes the ward and it takes all the money, in which case it promptly gets divided by one.
Language can often cause us to imagine we divide by zero. Nobody owns any part of the planet Mars. So the surface of Mars is being divided between zero humans. Meanwhile, the planet continues on its merry way around the sun, unaware of this linguistic and logical connundrum. What it really means is that 6 billion humans each own a 0 km/sq share of the Mars for a total surface of 0 km/sq and in this form, it makes perfect sense.
So for the vast majority of humans, the question will never come up and our universe, as well as our social arrangements go on their merry way, unaware of the paradoxes mathematicians face in their ivory towers. As far as Wittgenstein, and the majority of humanity is concerned, there is no problem at all. And there will not be a problem, so long as a division by zero never occurs in which a dispute might arise in the handling of human affairs. And how could it? Who would complain? The owner of 0 stocks? The people awarded a territory of 0 acres? The person who has attended 0 hours of class? etc..., etc...
So as far as the majority of humanity is concerned, mathematics work just as well as they have been doing since Sumer and Babylon and counts the things they are interested in counting just fine. As for outer reality, it has no use for our models whatsoever and does what it does just fine with or without them.
So in conclusion, to suggest mathematics might be fallacious is a little like asking if French might be fallacious. An invention will occur in 20 years. The French don't yet have a word for it. How can French have the pretention of describing reality when it does not have a word for what will be? Well, the language will be updated in 20 years, when the French actually face that problem. Mathematics, to most people, works in the same way. Except mathematics tends to be more rigid. As a mathematician, your first instinct, faced with an unresolvable formula might be to think something is wrong with the language itself, but you might want to ask yourself whether such formulas even occur in cases where humans with no mathematical expertise might be tempted to see a problem. If not, then the problem remains academic and that will be your cross to bear as a mathematician.

Answer 2

The reason has to do with how we define rings. Rings are sets R with two operations: addition and multiplication. Each operation has an identity element, 0 and 1, respectively, such that a + 0 = 0 + a = a a * 1 = 1 * a = a for all a in R. (Addition is also commutative and shares a distributive property with multiplication, which is not necessarily commutative. In particular, R forms an abelian group under addition.) An element a has a multiplicative inverse a^(-1) if a * a^(-1) = a^(-1) * a = 1 Just using the fact that R is an abelian group under addition and that multiplication distributes over addition, you can show that 0 * a = a * 0 = 0 for all a in R. (Simply note that 0 * a = (0+0)*a = 0*a + 0*a. Then subtract 0*a from each side, which is guaranteed since R is abelian under addition.) This shows that, in particular, 0 has no multiplicative inverse. That is, there exists no element x in R such that 0 * x = 1. And division is defined by a/b = a* b^(-1) So division by 0 is not defined by virtue of it being the identity for addition.

Answer 3

No.

Learn a little calculus, and you'll find that the concept of dividing by 0 is not "a hole" and won't be near so hard to fathom.

Okay, I see that you have a degree in Math, so you must know calculus. Learning calculus, along with Zeno's paradox of Achilles and the tortoise, gave me a Eureka moment that made the concept of dividing by zero, which had always given me fits because it felt so wrong, feel comfortable.

I guess your question is just too deep for us common folk. :)

Answer 4

Look at euclidean vs. non-euclidean geometry. When you realize that 2 + 2 can equal 5, then yes, in some unknown twisted dimension, we may be able to divide by zero. The quest to solve that problem could take several millenia.
Lets go for it.

Answer 5

Math is not a "natural" system, rather it is one constructed by humans for humans. It is like any language that has rules and problems. Our arrogance is what will be our undoing.

My entire base of knowledge of an entire life is a tiny fragment of all the knowledge that exists now -- let alone what we will come to know in the thousands of years to come.

Answer 6

You can't divide by something that wasn't there to begin with.

But, for other reasons our mathematical system could, and probably is flawed. Like, the fact that the value of pi, goes on forever, without any pattern.

Answer 7

Currently there are only 10 digits used in math. There used to be 12. 2 are still missing. If / when those are recaptured humanity will then fill all the voids which traditional math exploits.

Answer 8

Why would you try to divide by zero when it isn't even necessary?