7 ms·

"Mathematics does not give us truths, it gives us consequences." I'd like to point out that, as I stated in another comment in this thread, the author's suppos

"Mathematics does not give us truths, it gives us consequences." I'd like to point out that, as I stated in another comment in this thread, the author's supposed refutation of the inconsistency inherent in division by zero within fields is incorrect. Their refutation is as follows: The problem is in step (3): our division theorem is only valid for c ≠ 0, so you can’t go from 1/0 0 to 1 * 0/0. The “denominator is nonzero” clause prevents us from taking our definition and reaching this contradiction.* This is a strawman, and it's not the way a formal proof that division by zero in fields is undefined would proceed. First and foremost, the conceptual purpose of the proof is to demonstrate that you cannot define division by zero while still retaining the algebraic structure of a field. Trying to refute this point by stating that the proof cannot make use of division by 0 is begging the question. The whole point is that the proof shows you any way you define division by 0 is going to compromise your definition of a field, or it's going to make fields with nonzero elements impossible. What the author provided is a very contrived strawman for refutation that belies the actual point. You can define division by zero if you'd like, by using wheels, or extended number systems based on fields (i.e. positive and negativie infinities in the extended Real number system), but you cannot do it using fields. In fact, the modern, axiomatic definition of a field explicitly excludes the unit 0 from the otherwise sane rules of multiplicative inverses. More generally, I'd like to make a couple of observations from both a philosophical and a practical standpoint. First, mathematics does not give us true statements about the world, it gives us consequences that must follow if we accept various axioms or definitions. You can define division by 0 if you'd like, and you can even do so in a sane and useful way. But you will not have a field. But much more importantly, it's conceptually unsound to base an argument about the practical, programmatic behavior of an undefined operation based on imperfect arguments about abstract mathematics. Technically speaking, computers don't even deal with real numbers. If you find yourself mounting a defense of your programming language's behavior by running through the first lecture of a real analysis or linear algebra course, something has gone very wrong with your enterprise.

skybrian 8y ago
It seems like he covered that in the article. There is no multiplicative inverse of zero. But that isn't the same as defining a division operation.
- throwawaymath 8y ago
  He didn't cover that in the article, or at least not in a way that actually supports his point. Since there is no multiplicative inverse of 0, division by 0 is undefined behavior. Trying to remediate that by refuting the "proof" that it's undefined while still asserting that we can define it as something else is mathematically incoherent. If you want to see why it doesn't work from a number of different angles, read through the /r/math thread[1] or the Wolfram MathWorld blurb [2]. The formal problem comes down to one of uniqueness of elements. If you define division by zero in a field, you must end up in a place where it follows that every element is equal to every element. Then you only have one element - 0. So really we should be stating that division by zero is undefined behavior in fields with at least one nonzero element. But again - we can neatly sidestep all of this, because trying to defend your choice of undefined behavior in a programming language based on the abstractions of field axioms is silly. In the actual world we don't even deal with real numbers computationally, let alone the particular nuances of whether or not our programming language implements number systems composing a field or any other algebraic structure. __________________________ 1. https://www.reddit.com/r/math/comments/3b5i6p/can_you_divide_by_0/ https://www.reddit.com/r/math/comments/3b5i6p/can_you_divide... 2. http://mathworld.wolfram.com/DivisionbyZero.html http://mathworld.wolfram.com/DivisionbyZero.html
  - a_wild_dandan 8y ago
    > Trying to remediate that by refuting the "proof" that it's undefined He does not refute any proofs by citing that division by 0 is undefined. He refutes them by asserting that the multiplicative inverse doesn't exist. These are very different statements. Neither the standard field nor his modified field use the zero inverse, 0⁻. The proofs he's criticizing do erroneously use 0⁻. That's what he's calling out, I believe.
    - throwawaymath 8y ago
      > He refutes them by asserting that the multiplicative inverse doesn't exist. Yes - but in algebraic fields division by x is equivalent to multiplication by 1/x. This is precisely why you cannot have a field that admits division by 0: because 0 has no multiplicative inverse.
pron 8y ago
> The whole point is that the proof shows you any way you define division by 0 is going to compromise your definition of a field, or it's going to make fields with nonzero elements impossible. But it does neither. All field theorems pertaining to inverses/division take the form `x ≠ 0 ⇒ ... x⁻¹ ...` None of them is compromised. > In fact, the modern, axiomatic definition of a field explicitly excludes the unit 0 from the otherwise sane rules of multiplicative inverses. Exactly, which is why defining 1/0 = 0 (or 42) poses no problem to fields (BTW, the Lean proof assistant defines 0⁻¹ = 0 for fields, even though it's not the inverse and 0*0 = 0). The whole problem is one of formalization. "Undefined", as it's used in mathematics, is very much informal. It is used to say that a certain expression, 1/0, while "grammatically" correct, is meaningless. Formal systems simply cannot do that: the expression 1/0 must either be ill-formed (can be done with dependent types but is often inconvenient) or it must mean something in the semantic domain of the language. Different formal system define what that something is, but whatever it is, it is no longer "undefined" in the informal sense.
- throwawaymath 8y ago
  > The whole problem is one of formalization. "Undefined", as it's used in mathematics, is very much informal. It is used to say that a certain expression, 1/0, while "grammatically" correct, is meaningless. Formal systems simply cannot do that: the expression 1/0 must either be ill-formed (can be done with dependent types but is often inconvenient) or it must mean something in the semantic domain of the language. Different formal system define what that something is, but whatever it is, it is no longer "undefined" in the informal sense. We're substantially in agreement here. This circles back to my core thesis - there is no point in defending the implementation of something mathematically undefined in a programming language by retreating to the formal axioms of fields (or any other algebraic structure, for that matter). Your other point about 1/0 = x for whatever you'd like x to be is something I mentioned in my comment. The extended real number system does something similar by defining arithmetic of positive and negative infinities. But the extended real number system is not a field. It's a fine and useful system, but you insert pathologies into the real field by trying to accept infinity as a real number. Likewise if you want 1/0 to be equal 42, that's fine. But you're going to "infect" everything that 42 touches in the process. You won't compromise arithmetic, but you will compromise the definition of a field. What I'm getting at is that 1) mathematically speaking, the author's entire preamble about division by 0 is both incorrect and irrelevant, 2) we should not be trying to justify implementation behavior in systems which cannot use real numbers through the wizardry of field theorems.
antidesitter 8y ago
First, mathematics does not give us true statements about the world, it gives us consequences that must follow if we accept various axioms or definitions. Isn't a consequence in itself a true statement?
- throwawaymath 8y ago
  Allow me to clarify that point. The idea is that - conceptually speaking - mathematics cannot be used to tell you empirical axioms about the world in which we reside. All it can do is tell you what must be true given certain well-defined assumptions. It is a phenomenally powerful tool for proving things about the world from empirical axioms, but which empirical axioms we choose to rely on in engineering is mostly the domain of physics or the sciences. The underlying point here is mostly a philosophical one, but it has some bearing on the matter at hand. In effect, the definition (or lack thereof) for division by zero in fields is of no practical consequence for the real world impact of implementing an operation which admits division by zero. I can define an algebraic structure in which division by zero is sane (it is not in fields, despite what the article states!), or I can define an algebraic structure in which division by zero is insane. Both can be coherent, consistent and genuinely useful. But whether or not I can define something that works has nothing to do with what happens "when the rubber meets the road", so to speak. It was a mistake to open up an argument about programming division by 0 using the field axioms in the first place. There is no "one truth", there are only facts which must follow as consequences from assumptions. This is especially the case for programming, considering that computers are fundamentally incapable of working with real numbers in the first place.
- a_wild_dandan 8y ago
  They're true statements about the mathematical system. They might have no relevance to reality.

skybrian 8y ago

It seems like he covered that in the article. There is no multiplicative inverse of zero. But that isn't the same as defining a division operation.

throwawaymath 8y ago
He didn't cover that in the article, or at least not in a way that actually supports his point. Since there is no multiplicative inverse of 0, division by 0 is undefined behavior. Trying to remediate that by refuting the "proof" that it's undefined while still asserting that we can define it as something else is mathematically incoherent. If you want to see why it doesn't work from a number of different angles, read through the /r/math thread[1] or the Wolfram MathWorld blurb [2]. The formal problem comes down to one of uniqueness of elements. If you define division by zero in a field, you must end up in a place where it follows that every element is equal to every element. Then you only have one element - 0. So really we should be stating that division by zero is undefined behavior in fields with at least one nonzero element. But again - we can neatly sidestep all of this, because trying to defend your choice of undefined behavior in a programming language based on the abstractions of field axioms is silly. In the actual world we don't even deal with real numbers computationally, let alone the particular nuances of whether or not our programming language implements number systems composing a field or any other algebraic structure. __________________________ 1. https://www.reddit.com/r/math/comments/3b5i6p/can_you_divide_by_0/ https://www.reddit.com/r/math/comments/3b5i6p/can_you_divide... 2. http://mathworld.wolfram.com/DivisionbyZero.html http://mathworld.wolfram.com/DivisionbyZero.html
- a_wild_dandan 8y ago
  > Trying to remediate that by refuting the "proof" that it's undefined He does not refute any proofs by citing that division by 0 is undefined. He refutes them by asserting that the multiplicative inverse doesn't exist. These are very different statements. Neither the standard field nor his modified field use the zero inverse, 0⁻. The proofs he's criticizing do erroneously use 0⁻. That's what he's calling out, I believe.
  - throwawaymath 8y ago
    > He refutes them by asserting that the multiplicative inverse doesn't exist. Yes - but in algebraic fields division by x is equivalent to multiplication by 1/x. This is precisely why you cannot have a field that admits division by 0: because 0 has no multiplicative inverse.

pron 8y ago

> The whole point is that the proof shows you any way you define division by 0 is going to compromise your definition of a field, or it's going to make fields with nonzero elements impossible. But it does neither. All field theorems pertaining to inverses/division take the form `x ≠ 0 ⇒ ... x⁻¹ ...` None of them is compromised. > In fact, the modern, axiomatic definition of a field explicitly excludes the unit 0 from the otherwise sane rules of multiplicative inverses. Exactly, which is why defining 1/0 = 0 (or 42) poses no problem to fields (BTW, the Lean proof assistant defines 0⁻¹ = 0 for fields, even though it's not the inverse and 0*0 = 0). The whole problem is one of formalization. "Undefined", as it's used in mathematics, is very much informal. It is used to say that a certain expression, 1/0, while "grammatically" correct, is meaningless. Formal systems simply cannot do that: the expression 1/0 must either be ill-formed (can be done with dependent types but is often inconvenient) or it must mean something in the semantic domain of the language. Different formal system define what that something is, but whatever it is, it is no longer "undefined" in the informal sense.

throwawaymath 8y ago
> The whole problem is one of formalization. "Undefined", as it's used in mathematics, is very much informal. It is used to say that a certain expression, 1/0, while "grammatically" correct, is meaningless. Formal systems simply cannot do that: the expression 1/0 must either be ill-formed (can be done with dependent types but is often inconvenient) or it must mean something in the semantic domain of the language. Different formal system define what that something is, but whatever it is, it is no longer "undefined" in the informal sense. We're substantially in agreement here. This circles back to my core thesis - there is no point in defending the implementation of something mathematically undefined in a programming language by retreating to the formal axioms of fields (or any other algebraic structure, for that matter). Your other point about 1/0 = x for whatever you'd like x to be is something I mentioned in my comment. The extended real number system does something similar by defining arithmetic of positive and negative infinities. But the extended real number system is not a field. It's a fine and useful system, but you insert pathologies into the real field by trying to accept infinity as a real number. Likewise if you want 1/0 to be equal 42, that's fine. But you're going to "infect" everything that 42 touches in the process. You won't compromise arithmetic, but you will compromise the definition of a field. What I'm getting at is that 1) mathematically speaking, the author's entire preamble about division by 0 is both incorrect and irrelevant, 2) we should not be trying to justify implementation behavior in systems which cannot use real numbers through the wizardry of field theorems.

antidesitter 8y ago

First, mathematics does not give us true statements about the world, it gives us consequences that must follow if we accept various axioms or definitions. Isn't a consequence in itself a true statement?

throwawaymath 8y ago
Allow me to clarify that point. The idea is that - conceptually speaking - mathematics cannot be used to tell you empirical axioms about the world in which we reside. All it can do is tell you what must be true given certain well-defined assumptions. It is a phenomenally powerful tool for proving things about the world from empirical axioms, but which empirical axioms we choose to rely on in engineering is mostly the domain of physics or the sciences. The underlying point here is mostly a philosophical one, but it has some bearing on the matter at hand. In effect, the definition (or lack thereof) for division by zero in fields is of no practical consequence for the real world impact of implementing an operation which admits division by zero. I can define an algebraic structure in which division by zero is sane (it is not in fields, despite what the article states!), or I can define an algebraic structure in which division by zero is insane. Both can be coherent, consistent and genuinely useful. But whether or not I can define something that works has nothing to do with what happens "when the rubber meets the road", so to speak. It was a mistake to open up an argument about programming division by 0 using the field axioms in the first place. There is no "one truth", there are only facts which must follow as consequences from assumptions. This is especially the case for programming, considering that computers are fundamentally incapable of working with real numbers in the first place.
a_wild_dandan 8y ago
They're true statements about the mathematical system. They might have no relevance to reality.