6 ms·

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

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.

amitport 4y ago
"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 4y ago
  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 4y ago
    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 4y ago
      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 4y ago
      Sure, but the whole "problem" we were trying to solve was that zero doesn't interact with arithmetic operations the usual way.
- henry2023 4y ago
  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,_%CE%B4)-definition_of_limit https://en.wikipedia.org/wiki/Limit_of_a_function#(%CE%B5,_%...
  - deleted 4y ago
    [deleted]
  - topaz0 4y ago
    "_____ 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 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.
- ndsipa_pomu 4y ago
  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 4y ago
    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 4y ago
      There's a couple of mentions in other comments about the Riemann Sphere (https://en.wikipedia.org/wiki/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.
iamerroragent 4y ago
Riemann Sphere: https://en.wikipedia.org/wiki/Riemann_sphere https://en.wikipedia.org/wiki/Riemann_sphere
- hansvm 4y ago
  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 4y ago
    "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 4y ago
      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.
momentoftop 4y ago
The Isabelle/HOL theorem prover assigns 0 to x/0 for all x, without contradiction.
- brookst 4y ago
  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 4y ago
    By specifying that x/y*y is only equal to x if y≠0, I guess?
    - momentoftop 4y ago
      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 4y ago
        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.
nextaccountic 4y ago
> 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 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 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.
- comte7092 4y ago
  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.
  - tshaddox 4y ago
    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
  - Grustaf 4y ago
    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 https://en.wikipedia.org/wiki/Projective_geometry
  - scotty79 4y ago
renewiltord 4y ago
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.
antognini 4y ago
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 https://en.wikipedia.org/wiki/Bring_radical
Grustaf 4y ago
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.
JumpCrisscross 4y ago
> 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 4y ago
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

amitport 4y ago

"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 4y ago
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 4y ago
  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 4y ago
    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 4y ago
    Sure, but the whole "problem" we were trying to solve was that zero doesn't interact with arithmetic operations the usual way.
henry2023 4y ago
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,_%CE%B4)-definition_of_limit https://en.wikipedia.org/wiki/Limit_of_a_function#(%CE%B5,_%...
- deleted 4y ago
  [deleted]
- topaz0 4y ago
  "_____ 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 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.
ndsipa_pomu 4y ago
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 4y ago
  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 4y ago
    There's a couple of mentions in other comments about the Riemann Sphere (https://en.wikipedia.org/wiki/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.

iamerroragent 4y ago

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

hansvm 4y ago
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 4y ago
  "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 4y ago
    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.

momentoftop 4y ago

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

brookst 4y ago
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 4y ago
  By specifying that x/y*y is only equal to x if y≠0, I guess?
  - momentoftop 4y ago
    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 4y ago
      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.

nextaccountic 4y ago

> 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 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 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.

comte7092 4y ago
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.
- tshaddox 4y ago
  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
- Grustaf 4y ago
  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 https://en.wikipedia.org/wiki/Projective_geometry
- scotty79 4y ago

renewiltord 4y ago

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.

antognini 4y ago

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

Grustaf 4y ago

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.

JumpCrisscross 4y ago

> 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 4y ago

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