[ndsipa_pomu]

Well, we can define mathematical objects for every gap (impossibility), but most of them will turn out to be inconsistent with our existing mathematical objects, and thus not very useful or interesting. I'd consider that mathematics is the study of consistency and what can be discovered using the simplest possible starting points (axioms).

The classic case would be if mathematicians wanted to assign a value to division by zero. It turns out that if you do allow that to take a value, then it becomes possible to "prove" that any number is equal to any other number. Quite simply, it makes maths less interesting to allow that, but instead having division by zero be undefined appears far more useful/interesting.

[nextaccountic]

> The classic case would be if mathematicians wanted to assign a value to division by zero. It turns out that if you do allow that to take a value, then it becomes possible to "prove" that any number is equal to any other number. Quite simply, it makes maths less interesting to allow that, but instead having division by zero be undefined appears far more useful/interesting.

There are multiple extensions to the real numbers that allow division by zero. One is a real projective line, which has only one infinity so that 1 / 0 = -1 / 0 = infinity

https://en.wikipedia.org/wiki/Real_projective_line

Another is the extended real number line which has positive infinity and negative infinity, so 1 / 0 = +infinity and -1 / 0 = -infinity and they are different from each other

https://en.wikipedia.org/wiki/Extended_real_number_line

Those are all perfectly fine but they still can't define 0 / 0, which is a harder problem.

[marcosdumay]

> There are multiple extensions to the real numbers that allow division by zero.

Well, the gotcha is that they redefine the operations so that none of addition, subtraction, multiplication or division are total. Those operations just break in a different number than zero.

[nextaccountic]

Addition can be total if you have a single infinity, just make infinity + n = infinity for all n

Subtraction, multiplication and division is harder

But making further operations partial isn't that of a big deal; in fields the division is already partial due to division by zero not being defined.

[comte7092]

1 / 0 = +infinity Implies that 0 * +infinity = 1, so it does run into make of the same issues.

There are instances that make it useful, but the extended real number line isn’t used heavily in practice.

[Grustaf]

You might not have much use for the real projective line when tallying up prices in the grocery store, but projective geometry is definitely very useful. https://en.wikipedia.org/wiki/Projective_geometry

[tshaddox]

Yeah, I don't really see what this gets you. With basic real number division you have to make the exception for zero in the definition:

    a/b = c if and only if a = c*b and b!=0

And with this infinity thing you just have to make essentially the same exception for multiplication and infinity:

    c*b = a if and only if a/b = c and b!=infinity and c!=infinity

[hgsgm]

What is the "issue"?

"/" means "* reciprocal of".

If "infinity" is defined as "reciprocal of 0", what is the problem?

Yes it is an exception to 0*n=0.

It won't work in every setting, but it works in some settings, like inversive geometry.

[scotty79]

In normal math 1/0 is undefined but in a math where 1/0 is defined to be inf the 0*inf is still undefined.

[iamerroragent]

Riemann Sphere:

https://en.wikipedia.org/wiki/Riemann_sphere

[Lichtso]

You don't even need the complex part for this. You can do the infinity-projection trick on the real numbers alone as well: https://en.wikipedia.org/wiki/Projectively_extended_real_lin...

A similar trick (point at infinity or ideal point) is used in projective geometry to distinguish between directions (vectors) and places (points) by using coordinates only: https://en.wikipedia.org/wiki/Projective_geometry

But if you actually want to do calculations with infinities and infinitesimals the surreal numbers might be better suited for that: https://en.wikipedia.org/wiki/Surreal_number

[zeroonetwothree]

Downside is that now 0⋅∞ is undefined so you’ve introduced a new ‘impossibility’

[hansvm]

They could have been more precise, but they probably shouldn't have to in the space of a comment. The Riemann Sphere defines a value for the expression x/0, and it's often useful, but it fails to uphold the most important property division should have -- that it undoes multiplication. Division by 0 (with some assumptions about not being in a trivially small space and how those operations behave with respect to addition) does lead to contradictions in that latter sense.

[iamerroragent]

"but it fails to uphold the most important property division should have -- that it undoes multiplication"

I'm not sure I follow that as it's most important property. I'm not sure if division could even be defined as an operation that undoes multiplication.

Number theory, fields, and rings I believe make it clear while subtraction and addition can be viewed as the same function; multiplication and division cannot.

Apologize if that's not clear as to why that is; it's been a while since I read up on those being defined.

However I recommend One, Two, Three: Absolutely Elementary Mathematics by David Berlinski that gives in my opinion pretty good layman understanding of these nuances and number theory.

[hansvm]

Take a look into division rings as a concept. The usual definition for division in rings and fields is via multiplicative inverses for some subset of the nonzero elements. Not all algebraic spaces have division, but that doesn't change what it is, especially from the "number theory, fields, and rings" point of view.

Unless you're talking about some higher-order concept?

Edit: For a bit of completeness, what's happening with the Riemann Sphere is that the algebraic definition is being extended in a way that has some useful analytic, topological, and quality-of-life properties, but which is no longer wholly compatible with the underlying algebra. The algebraic issues are isolated to the extra point at infinity, so they're not terrible to work around, but the operation in question is a proper extension of the underlying algebraic definitions -- much how the gamma function in no way can be defined as multiplication of integers but is a useful extension of the factorials nonetheless.

[syzarian]

Division is multiplication by the multiplicative inverse. Subtraction is addition by the additive inverse. Both division and subtraction undo their corresponding operation. Multiplying by a (provided it’s not zero) is undone by dividing by a. Adding a is undone by subtracting a.

In a ring the elements form a group under addition and thus every element has an additive inverse. The additive identity element, let’s call it e, has the property that ea = e and ae = e. For this reason we use 0 instead of e. In a nontrivial ring 0 can’t have a multiplicative inverse because if it did then every element would be equal to the multiplicative identity (which is unique).

[iamerroragent]

Okay so if you can get to a ring without a multiplicative inverse and then applying that operation to the ring forms it into a field then wouldn't it be fair to say that division is not really the opposite of multiplication the same way that subtraction absolutely is for addition?

[syzarian]

The definition of division is multiplication by the multiplicative inverse. It may be the case that some elements don’t have such an inverse but the definition is analogous to that of subtraction. The analogy is not perfect because every element has an additive inverse while not every element had a multiplicative inverse.

[ndsipa_pomu]

That's a good example of where defining division by zero leads to interesting maths, but it ends up sacrificing some of the usual rules of arithmetic, so it comes down to a choice of which is more useful in the relevant circumstance.

[2muchcoffeeman]

This just goes to show that you really have to be careful when slinging out math facts. I've done some under grad maths and the only line on that page that I understand is

"The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as 1 / 0 = ∞ 1/0=\infty well-behaved."

It clearly does not satisfy a primitive understanding of 1/0.

[hgsgm]

1/0 is the limit of x/y as x approaches 1 and y approaches 0. It works fine if you choose to put a a point there called \infinity, with an appropriate notion of nearness.

[antognini]

An example where this does work quite nicely has to do with Bring radicals or "ultraradicals [1]. One of the most important results from Galois theory is that the quintic equation has no solution using standard radicals. But the introduction of "Bring radicals" allows quintic equations to be formally solved. As far as I'm aware though, Bring radicals only work for quintic equations in general and don't work for 6th order or higher polynomials, so your bang for the buck is a somewhat limited.

[1]: https://en.wikipedia.org/wiki/Bring_radical

[amitport]

"it becomes possible to "prove" that any number is equal to any other number."

There are multiple ways to define what division by zero means. Which definition leads to this outcome? How?

[afiori]

let ϴ = 0/0 then 1*ϴ = ϴ = 0/0 = (0*0)/0 = 0*(0/0) = 0*ϴ it follows 1 = 0 and thus x = x * 1 = x * 0 = 0 = y * 0 = y * 1 = y for all x and y

[xigoi]

This is assuming that Θ interacts with arithmetic operations the usual way (that is, ℝ ∪ {Θ} is a field), which the person you're replying to did not say.

[afiori]

True, but the point of giving a "value" to 0/0 is to use it somehow.

For example in the context of limits you define a whole lot of number like values like 0+ or 0- that are useful wrt operations on limits.

I was trying to give an example of how ℝ ∪ {Θ} has almost no advantages compared to just ℝ

[tshaddox]

Sure, but the whole "problem" we were trying to solve was that zero doesn't interact with arithmetic operations the usual way.

[ndsipa_pomu]

The most common definition of division being the inverse of multiplication.

if b ≠ 0 then the equation a/b = c is equivalent to a = b × c. Assuming that a/0 is a number c, then it must be that a = 0 × c = 0. However, the single number c would then have to be determined by the equation 0 = 0 × c, but every number satisfies this equation, so we cannot assign a numerical value to 0/0

[amitport]

Thanks, this definition does seem problematic. In any case, it is not the only possible definition and in a/0=c, c does not have to be defined as a real number. We can define it as similarly to complex number with new rules that do not collide with existing reals.

[ndsipa_pomu]

There's a couple of mentions in other comments about the Riemann Sphere (https://en.wikipedia.org/wiki/Riemann_sphere) which does define division by zero, but sacrifices the numbers forming a field under addition and multiplication.

[henry2023]

Division by zero is not defined anywhere on math.

The closest thing you'd get to it is to

1. define a limit (lim x->a of ƒ(x) exists if and only if given any ε > 0 there exists a δ > 0 such that ...)[1].

2. chose a function ƒ(x) such that on a given "a", ƒ(a) = ƒ(a)/0.

3. prove that the limit exists and is finite.

Now if we defined division by zero it would look like this:

Axiom: For every element x of the real numbers there exists a x' in the real numbers such that x/0 = x'

I advise you to play with this new "rule" to see if it leads to something interesting. Hint: try to prove that 1/0 = 2/0

[1]: https://en.wikipedia.org/wiki/Limit_of_a_function#(%CE%B5,_%...

[topaz0]

"_____ is not defined anywhere in math"

is a kind of sentence that is almost never true, and even if it were, it would be impossible to prove that someone hadn't jotted a valid definition on a napkin somewhere. In this case it is certainly not true (as others have mentioned: https://en.wikipedia.org/wiki/Riemann_sphere ). Now, specifying a definition for division by zero does require you to be careful about how the other operations extend to this new number, but there are perfectly consistent (and useful!) ways to do so.

[momentoftop]

The Isabelle/HOL theorem prover assigns 0 to x/0 for all x, without contradiction.

[ndsipa_pomu]

Thanks - I was not aware that theorem provers often allow "division" by zero.

Looking at https://xenaproject.wordpress.com/2020/07/05/division-by-zer... I see that they don't use mathematical division, but define a slightly different operator with an additional condition for handling zero. This appears to be far more convenient for theorem provers.

The trade-off would be that "division" is no longer the inverse of multiplication.

[momentoftop]

Ah, thanks for the link. I suggested the reason that Isabelle/HOL does this is because it requires total functions and you don't have a convenient way to do refinement types. But that's not an adequate explanation, because Lean does allow such refinements, but it still turns out to be inconvenient for division.

I will note that setting a - b = 0 for a <= b is pretty standard, and is often called "partial subtraction."

[brookst]

I believe you but that’s kind of mind blowing. How do they avoid the seemingly-obvious corollary that 0*0 = X, for all values of X? That is, just multiplying both sides of “x/0 = 0” by zero.

[xigoi]

By specifying that x/y*y is only equal to x if y≠0, I guess?

[momentoftop]

Exactly.

Functions in these logics are total, so if you want division to be a function (and you probably do), it has to assign something to division by 0.

It would be acceptable to assign an unspecified object from the domain, for which you have no non-trivial theorems, and so all your real theorems must have a precondition about the denominator being non-zero. But if you specify a candidate like 0, you can get some theorems which don't have the precondition. Consider:

a/b * c/d = ac/bd.

This now holds even if one of b or d is 0.

[brookst]

I appreciate the explanation and I’m in no position to disagree, but ugh. Seems like it would work just as well to define x/0 as 6, or e, or -15. I’m sure that’s not the case. But as a long time tech person who’s always considered underflow/overflow to be a hack to get around limitations of hardware, it offends be a bit to find conditionals in abstract math. Undefined seems cleaner, like null, since it implicitly says “don’t treat this as a normal value that you can operate on”.

I suspect the real math people know what they’re doing more than I do, though.

[momentoftop]

The theorem a/b * c/d = ac/bd doesn't hold if x/0 = 6, though.

The theorem prover HOL Light is a close cousin of Isabelle/HOL and doesn't adopt this, and just says that x/0 is some unspecified number. You can't prove much interesting about it. You can prove, say, that x/0 * 0 = 0, but you can't prove whether or not x/0 is, say, positive or not.

If you prefer null, there was a logic that allowed for undefined terms and partial functions that became the basis of the IMPS theorem prover. I found it most notable for the fact that it doesn't have reflexivity of equality: 1/0 = 1/0 is false in IMPS.

[hgsgm]

Division is already defined conditionally in regular old elementary school field theory.

[poizan42]

It's not making a multiplicative inverse of 0 exist though, it just defines a '/' operator that is slightly different from our usual one (i.e. a/b = a*b^(-1))

[Grustaf]

On the contrary, the extensions can be very useful and interesting. You do typically have to sacrifice something, like commutativity in the case of quaternions, but it will often be worth it.

[renewiltord]

Yep, an extension is only interesting if it is a true extension, i.e. retains the properties of the thing being extended. So complex numbers are interesting as an an extension of reals since reals are isomorphic to the subring. Likewise with quaternions and reals / complex numbers.

[JumpCrisscross]

> we can define mathematical objects for every gap (impossibility), but most of them will turn out to be inconsistent with our existing mathematical objects

Is the short answer it's not parsimonious or useful?

[paulddraper]

You're right, though your example is weak.

Infinity and negative infinity can make a lot of sense. You can even allow imaginary infinities.

The caveats:

* You cannot multiply zero by infinity, or divide infinity by infinity

* You cannot add different infinities