It is inconsistent. If 1/0 = 0, then 1 = 0*0.
You can't logically refute that proof by saying, "well no, you have yet to define division by 0 according to the field axioms, so you can't use that division as part of your proof." That's the point! The proof does not demonstrate that division by 0 results in 1, it demonstrates that you cannot define division by 0 while maintaining the algebraic structure of a field.
If the author wants to talk about defining division by 0 in wheels or something esoteric they're more than welcome to. But among fields, and among the real numbers, it's not possible. This whole exercise of trying to refute what has been commonly accepted for over a century is frankly ridiculous for trying to justify undefined behavior in a programming language.
Mathematics is thoroughly pedantic about definitions for a reason. If those formalisms don't matter because what you've done is "close enough", then skip the song and dance about field definitions and stop trying to use it to justify the behavior of an undefined operation in a programming language. Just say you're defining 1/0 to be equal to whatever you want because the world doesn't break down. It actually detracts the author's point to so confidently (and incorrectly) refute something that is robustly proved in the first few weeks of an undergraduate analysis course. Why is this even in a blog post about a programming language?!
This is essentially the same point as the extended real (or complex) number systems. The sets of all real and complex numbers (respectively) form fields under the axioms of addition and multiplication. But you can define division by 0 and division by infinity in a way that works with familiar arithmetic (I explained how to do this in another comment barely two weeks ago [1]). But the key point here is that in doing this you sacrifice the uniqueness of real numbers.
The author tries to refute this observation by claiming the proof uses an undefined division operation, but that's a red herring. The real assertion is that you cannot define division as an inverse operation from multiplication to be inclusive of division by the unique unit (i.e. 0, in the real field) unless you are willing to state that every number is equal to every other number in the entire field. And you can do that, but it trivially follows that you no longer have a field without a nonzero element.
So really the actual proof is that division by 0 is undefined for any field with at least one nonzero element.
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1. Let F be a field containing an element x =/= 0.
2. Suppose we have defined division by zero in F such that, for all x in F, there exists an element y = x/0 (i.e. F adheres to the field axiom of multiplicative closure). Note that at this point it does not matter how we have defined division by 0, we will just generously continue and assume you've done it in a way that maintains the other field axioms.
3. Since y = x/0, it follows that the product of y and 0 is equal to x, because division is the inverse of multiplication. By the field axioms, division does not exist if there is no multiplicative inverse with which to multiply.
4. But by the field axioms this implies that x = 0, which contradicts our initial assumption. Likewise, since we can repeat this procedure with any element x in F, this demonstrates that there exists no nonzero element x in F, and in fact F = {0}.
The failure in the article's refutation is that this proof is designed to permit you to assume you have suitably defined division by zero, then proceed to demonstrate without any loss of generality that you could not possibly have unless 1) F is not a field, or 2) F contains only 0. The fundamental algebraic property you sacrifice by defining division by zero is uniqueness, and uniqueness is a hard requirement in fields with nonzero elements.
Can you explain how this follows? I thought division was only the inverse of multiplication for all nonzero denominators, which would mean we can't use that definition for deduction in x/0.
It might hinge on your next sentence:
>By the field axioms, division does not exist if there is no multiplicative inverse with which to multiply.
but I don't understand why that's necessarily true. I don't understand how the field axioms require division by x to require the existence of a multiplicative inverse of x when x is zero. Sorry to take a bunch of your Friday, but I'm very curious now. Explanation much appreciated.
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edit:
Come to think of it, couldn't I define x/y as cotton candy for all x,y in field F and still satisfy the field axioms? They just don't refer to division.
Any connection between x/y and y's multiplicative inverse is just a nice convention. That convention states that x/y = x * mult_inv(y) when y != 0, but nothing else. That definition has nothing to do with the field axioms and changing it doesn't require that I change anything about multiplicative inverses. That means I don't touch the field axioms and my field is still a field.
Yes. That's the thesis of the article. Make an arbitrary choice for 1 / 0 = ?, and if it helps you, use it. It's mathematically, rigorously fine.
Even if you could come up with another formalization that does cause a problem, e.g. `∀ x ∈ dom(1/t) . x(1/x) = 1` (and I would say that this is the only formalization that causes an issue, and it requires the use of a language with a dom operator, something that is absolutely not required for theories of fields), it won't matter because the question is not whether one could come up with a formalization that leads to contradiction, but whether there are reasonable formalizations of fields where this does not happen, and there are (in fact, most of them satisfy this, as they do not rely on a dom operator).
In addition, it is not true that "by the field axioms, division does not exist if there is no multiplicative inverse with which to multiply." It's just that the field axioms do not define what the meaning of division is in that case. Defining it, however, does not lead to contradiction with the axioms, at least not a contradiction you've point out. In fact, most common languages of mathematics cannot even explicitly express the statement "x is not in the domain of f." All they can do is define f(x) for values of x in the domain, and not define f(x) for values outside it. The "exist" in your statement does not refer to ordinary mathematical existence (usually formally expressed with the existential quantifier) but to an informal notion of definedness (discussed by Feferman) that has no formal counterpart in most formal systems, because it is very rarely needed; it is certainly not needed to state the field axioms.
You can't engage with the problem because it only exists as a syntactical annoyance. You seem to acknowledge this, but then continue to argue when I explicitly tell you I am in agreement on that point. Then you proceed to argue the theoretical basis all over again.
I'm not going to continue arguing this with you. You're presently the only one in this thread who isn't following and I've tried to direct you to other resources. You've alternated between saying those proofs are either incorrect outright or not applicable because they don't have relevance for programming. If you actually believe division by 0 is possible in fields you have an immediately publishable math paper waiting for you to submit it.
Otherwise we're just talking past each other because my whole point here has been that the author's discussion of fields is irrelevant for programming language theory in the first place.
But formalizing the statement "there is no division" poses challenges. If you agree that a formalization of fields that is acceptable to you exists, please write a formal axiom/theorem in that theory which would break if we were to extend the definition of division.
> just as is the case for subtraction and additive inverses.
No, because subtraction is not partial, and therefore poses no challenge for formalization.
> there is no what happens because nothing happens at all.
This is fine when working informally, but doesn't work for formalizations, and formalizations of fields do exist.
> You can't engage with the problem because it only exists as a syntactical annoyance.
But that is the heart of the problem. Either you say one cannot formalize fields at all, or you must admit that some acceptable formalizations do not break. No one is claiming that one should be able to divide by zero in informal mathematics.
> I'm not going to continue arguing this with you.
That's fine, but settling this argument is very easy: write a theorem of fields in a formal language that would break if we extend division, one that cannot be equivalently written in an acceptable formalization in a form that does not break. This can literally be done in one line. If you cannot write such a theorem, then there really is no point arguing, because you cannot provide a counter-argument. Repeating the same informal theorems over and over is indeed pointless.
> You've alternated between saying those proofs are either incorrect outright or not applicable because they don't have relevance for programming.
I've not alternated on anything. Your proofs are incorrect in formal systems that you yourself would find acceptable.
> and I've tried to direct you to other resources
Resources about informal mathematics, on which everyone agrees. That the informal concept of division cannot be extended to 0 is obvious. The only question is whether a formal definition of division would necessarily break formal field theorems if extended to division by zero. You seem to claim that's the case, yet have not stated a single formal theorem of the kind.
> If you actually believe division by 0 is possible in fields you have an immediately publishable math paper waiting for you to submit it.
I would if only that result (doesn't merit a paper) had not already been published by at least Paulson and Avigad, two of the world's best known logicians. The formal field theory in the Lean proof assistant explicitly allows division by zero. That division coincides with the informal division everywhere but at zero, and the extension introduces no inconsistencies to the theory.
> the author's discussion of fields is irrelevant for programming language theory in the first place.
It's not about programming, but about any formalization (which includes, but is not limited to programming).
Anyway, see my comment https://news.ycombinator.com/item?id=17738558, which distills the debate into a concise, precise form.
The MI property states that every element except 0 has a multiplicative inverse. He's defining division via two cases: If b≠0, then a/b = a*b⁻ (multiplicative inverse). If b=0, then a/b=0. This definition does not imply that 0⁻ exists, so there's no violation of MI.
Standard definition of division function, d:
d(x, y) = x * y⁻, for all x and y EXCEPT 0
Author's modified, piecewise (https://en.wikipedia.org/wiki/Piecewise) definition:
d(x, y) = x * y⁻, for all x and y EXCEPT 0
d(x, y) = 0, for y = 0
He's just adding 0 to the domain of d(x, y) to extend the definition, and deliberately not using xy⁻ for that particular element of the domain. No inverse needed.
Er... that's what mathematics is. It's a word game - we build systems from arbitrary rules and then explore the results.
Look through https://www.mathgoodies.com/articles/numbers for a bunch of uncommonly-defined numbers.
Try to state this definition formally. The statement: ∀ x,y . x/y = xy⁻¹ is not a theorem of fields or a definition of division. However, ∀ x, y . y ≠ 0 ⇒ x/y = xy⁻¹ is, but is completely unaffected by defining division at 0. Those who think they see a problem rely on informal and imprecise definitions. Could you formally state a theorem that is affected? That would help you get around issues that are merely artifacts of imprecision.
But let's entertain you, and state that what we really mean by the informal and vague statement, "division is the inverse of multiplication," could be stated formally as:
∀ x ∈ dom(1/t). x(1/x) = 1
∀ x ≠ 0 . x(1/x) = 1
The hole that he is filling here isn't one that he bored into the standard definition, but a hole that the standard definition already admitted. If something is explicitly undefined, there's nothing mathematically wrong with defining it, as long as the definition doesn't lead to inconsistency.
The definition does lead to inconsistency...you can't look at the field axioms, observe that 0 has no multiplicative inverse, then proceed to define a special, one-off division rule that doesn't involve multiplicative inverses for that one element. Either your division rule is pathological and breaks a fundamental field property or you've introduced a division rule which is just a syntactical sugar, not a real operation (in the latter case you've introduced confusing notation, not a new division function). Why do you think mathematicians explicitly state that the real field with the augmentation of positive and negative infinity (which allow division by 0) is not a field?
I don't understand why there is so much resistance to this idea in this thread, but the simple fact remains that if you define division by an additive identity (0) in any way, the field containing that unit ceases to be a field. This is because all elements cease to be unique. You can quickly prove that every element is equal to every other element, including (critically) the additive and multiplicative identity elements. Fields are defined by closure under the operations of addition and multiplication, and that closure requires uniqueness of their respective identities. Upend that and your entire field structure breaks down, because all you're left with is a field with a single element 0.
Stating that you've defined division by 0 using a one-off case that permits all other field identities to remain consistent is like saying you've turned the complex field into an ordered field using lexicographic ordering. You haven't, because i admits no ordering, much like 0 admits no multiplicative inverse.
Onlookers reading these comments probably think those of us harping on this point are anal pedants with a mathematical stick up our ass. But this thread is increasingly illustrating my central point, which is that the author shouldn't have tried to justify numerical operation definitions in a programming language using field axioms of all things.
Why not? What mathematical axiom does this break?
> Either your division rule is pathological and breaks a fundamental field property
It doesn't, or you could show one here: https://news.ycombinator.com/item?id=17738558
> or you've introduced a division rule which is just a syntactical sugar, not a real operation
Is this math? We define symbols in formal math using axioms. I am not aware of a distinction between "syntactical sugar" and a "real operation."
> Why do you think mathematicians explicitly state that the real field with the augmentation of positive and negative infinity (which allow division by 0) is not a field?
That's simple: because unlike the addition of the axiom ∀x . x/0 = 0, adding positive and/or negative infinity does violate axioms 2, 7 and possibly 11 (depending on the precise axioms introducing the infinities).
> I don't understand why there is so much resistance to this idea in this thread, but the simple fact remains that if you define division by an additive identity (0) in any way, the field containing that unit ceases to be a field.
Because a field is defined by the axioms I've given here (https://news.ycombinator.com/item?id=17738558) and adding division by zero does not violate any of them. If you could show what's violated, as I have for the case of adding infinities and as done in actual math rather than handwaving about it, I assume there would be less resistance. I don't understand your resistance to showing which of the field axioms is violated.
> You can quickly prove that every element is equal to every other element, including (critically) the additive and multiplicative identity elements.
So it should be easy to prove using the theory I provided, which is the common formalization of fields. Don't handwave: write proofs, and to make sure that the proofs aren't based on some vagueness of definitions, write them (at least the results of each step) formally. The formalization is so simple and straightforward that this shouldn't be a problem.
> Stating that you've defined division by 0 using a one-off case that permits all other field identities to remain consistent is like saying you've turned the complex field into an ordered field using lexicographic ordering. You haven't, because i admits no ordering, much like 0 admits no multiplicative inverse.
Stop handwaving. Show which axioms are violated.
You can't use your own stubbornness to justify itself.
I'm waiting for you to justify your claim that this extension to division breaks any of the field axioms. pron even made you a nice list of them.
Just name one equation/theorem that the new axiom would break.
I'm completely open to being convinced! But so far you've only given arguments about giving a multiplicative inverse to zero. Everyone agrees on that. It's the wrong argument.
This is getting to be Kafkaesque...it breaks the field axioms themselves. How many different explanations and external resources do I need to provide in this thread for you to be convinced that this is not a controversial point in modern mathematics? I just explained it in the comment you responded to.
You have exactly two options here.
If you define x/0, that definition must interact with the rest of the definitions and elements of the field. To maintain multiplicative closure (a field axiom!) there must be a unique y element equal to x/0. So tell me how you will define x/0 such that x is not equal to the product of 0 and y. Regardless of what you think the author has shown, the burden of proof is not on me at this point to show that you can't do it, because it follows directly from the field axioms. Trying to impose a one-off bizarro divisor function defined only on {0} is not only mathematically inelegant, it immediately eliminates the uniqueness of all field elements. Therefore your "field" just becomes {0}, and since it lacks a multiplicative identity it ceases to be a field. There is your contradiction. Why don't you tell me how you're going to prove any equation defined over a field that relies on the uniqueness or cancellation properties of fields?
On the other hand, let's say you tell me you want define x/0 so that the definition doesn't interact with any of the field definitions or elements. Then you haven't actually introduced any new operation or definition, you've just designed a notation that looks a lot like division but has no mathematical bearing on the field itself (i.e. absolutely nothing changes, including for 0). That's not a divisor function, it's just a confusing shorthand. You can't just add another axiom to a field and call it a field.
If you believe I'm stubborn, that's fine. I might be a poor teacher! There are ample resources online which will patiently explain this concept in mind numbing detail. It boggles my mind that there are people in this thread still fighting an idea in earnest which has been settled for over a century.
So there are two separate issues here. One is whether we can extend the definition of the "/" operator, and the other is whether we call it "division".
I'm not interested in what we call it. I'm interested in the claim that extending "/" will break the field.
The dichotomy you're talking about is wrong. The two options are not "multiplicative inverse" and "does not interact with anything". "1/0 = 0" interacts with plenty! If I make a system where it's an axiom, I can calculate things like "1/0 + 5" or "sqrt(1/0)" or "7/0 + x = 7". I can't use it to cancel out a 0, but I can do a lot with it.
> It boggles my mind that there are people in this thread still fighting an idea in earnest which has been settled for over a century.
Remember, the question is not "should this be an axiom in 'normal' math?", the question is "does this actually conflict with the axioms of a field?"
> You can't just add another axiom to a field and call it a field.
Yes you can. There is an entire hierarchy of algebraic structures. Adding non-conflicting axioms to an X does not make it stop being an X.
• Our first function f(x, y) is defined for all x in F, and all nonzero y in F. That is, the domain of this function f is (F × F\{0}). The definition of this function f is:
f(x, y) = x times the multiplicative inverse of y
(Note that in the definition of f(x,y) we could say "if y≠0", but whether we say it or not the meaning is the same, because the domain of f already requires that y≠0.)
• Our second function g(x, y) is defined for all x in F, and all y in F. That is, the domain of this function g is (F × F). To define this function g, we pick a constant C in F, say C=0 or C=1 (or any C∈F, e.g. C=17 if 17 is an element of our field). And having picked C, the definition of this function g is:
• g(x, y) = x times the multiplicative inverse of y, if y ≠ 0, and C otherwise [that is, g(x, 0) = C].
Note that this function is defined for all (x, y) in F×F, and when y≠0 it agrees with f, i.e. g(x,y) = f(x,y) when y≠0.
Would you agree that both of these are functions, on different domains?
Next, we have the notation x/y. To assign meaning to this notation (and to the word “division”), there are two conventions we could adopt:
• Convention 1: When we say "x/y", we will mean f(x,y) (as defined above) — that is, x * y^{-1}.
• Convention 2: When we say "x/y", we will mean g(x,y) (as defined above).
The point of the post is that we can well adopt Convention 2: with such a definition of "division", all the properties that were true of f (on the domain F×F\{0}) continue to be true of g, except that g is defined on a wider domain.
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Now, maybe Convention 2 offends you. Maybe you think there is something very sacred about Convention 1. In all your posts in this thread, you seem to be insisting that "division" or "/" necessarily have to mean Convention 1, and the provided justification for preferring Convention 1 seems circular to me — here are some of your relevant comments, with my comments in [square brackets]:
> Since there is no multiplicative inverse of 0, division by 0 is undefined behavior [This seems to be saying: Because of Convention 1, we cannot adopt Convention 2.]
> Since y = x/0, it follows that the product of y and 0 is equal to x, because division is the inverse of multiplication [Here, your reasoning is "because we adopt Convention 1"]
> 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. [Again, you're stating that Convention 1 has to hold, by fiat.]
> If F is a field with elements x, y, then the quotient x/y is equivalent to the product x(1/y). If 0 has no multiplicative inverse, there is no division by 0. The two concepts are one and the same [This is just stating repeatedly that we have to adopt Convention 1.]
> A multiplicative inverse is a division. [This is merely insisting that Convention 1 has to be adopted, not Convention 2]
> divisor cannot exist unless it is a multiplicative inverse [Again, stating Convention 1.]
> you can't look at the field axioms, observe that 0 has no multiplicative inverse, then proceed to define a special, one-off division rule that doesn't involve multiplicative inverses for that one element. [Why not?] ...you've introduced a division rule which is just a syntactical sugar [yes the same is true of Convention 1; what's the problem?]
All these comments, which emphatically insist on Convention 1, seem to ignore the point of the article, which is that Convention 2 has no more mathematical problems than Convention 1, because the function g is no less a “valid” function than the function f.
In mathematics when we use words like “obvious” or insist that something is true because it just has to be true, that's usually a hint that we may need to step back and consider whether what we're saying is really mathematically justified. What we have here is a case of multiple valid definitions that we can adopt, and there's no mathematical reason for not adopting one over the other. (There's a non-mathematical reason, namely “it breaks convention”, but the entire point is that we can break this convention.)
I don't take issue with division by 0 - you can do that in mathematics just fine! I take issue with defining that division and calling the consequent system a field when it's not a field, and acting as though everyone else is wrong. The author invited this criticism when they loaded in the full formalism of field theory without needing to.
If the author had just stated they wanted to define division by zero that wouldn't be a problem. I have no idea why they felt the need to pull in abstract mathematics. I'm not disagreeing with their point, I'm taking issue with the strange and incorrect way they defended it.
Note that in my top level comment I specifically said, "Mathematics does not give us truths, it gives us consequences." I will happily agree with you that there is usefulness and coherence in a definition of 0. There is no canonical truth about the world regarding that matter. But a direct consequence of defining any division by 0 is that you cease to have an algebraic field.
Therefore, using field theory to defend a system which defines division by 0 doesn't make sense. It's not that the system is "wrong" for some meaning of wrongness. It's that you shouldn't be trying to pigeonhole field theory to make it work, because you don't need to.
> But don't use field theory to justify convention 2, because it's mathematically incoherent
> defining that division and calling the consequent system a field when it's not a field
> a direct consequence of defining any division by 0 is that you cease to have an algebraic field
If you go back to my comment (the one you're replying to), both the functions f and g assume a field F, and they are well-defined functions on F×F\{0} and on F×F respectively. (Do you agree?) For example, F may be the field of real numbers. Forget about the word “division” for a moment: do you think there is something about the function g, that makes F not a field?
To put it differently: I agree with you that it is a direct consequence of defining a multiplicative inverse of 0 that you cease to have an algebraic field. But the only way this statement carries over when we use the word “division”, is if we already adopt Convention 1 (that “division” means the same as “multiplicative inverse”).
Again, I think you are implicitly adopting Convention 1: you're saying something like “if we adopt Convention 2, then x/y means the function g and includes the case when y=0, but [something about multiplicative inverses, implicitly invoking Convention 1], therefore there's a problem". But there's no problem!
It is not a direct consequence of defining the function g(x,y) that something ceases to be a field: it is a consequence only if you also insist on Convention 1, namely if you try to assign a multiplicative inverse to 0 (which everyone here agrees is impossible).
Let me emphasize: whether we adopt Convention 1 or Convention 2, there is no problem; we still have the same field F.
This seems just as bizarre, since zero times anything shouldn't become 1, no matter how big or how many times you do it.
In geometric modeling kinds of applications, I would say that these definitions are typically desirable, with 1/0 = undefined only better in unusual cases. As a simple example, it is typically much more useful for the “tangent” of a right angle to be defined as ∞ than left undefined.
But anyhow, there are no “facts” involved here. Only different choices of mathematical models, which can be more or less convenient depending on context / application.
It isn't bizarre, because there's an equal and opposite argument that anything times infinity is infinity, no matter how small the thing you multiply by.
If you actually do infinity * 0 you get NaN since there's no way to determine (without more information) whether the result should be 0, infinity, or anything in between.
1 != 0*0
I don't see the inconsistency. Abstract math vs practical application.
Computer languages execute on rules that are not utilizing real number arithmetic. I didn't want to mention it, but there's these things called floats...
Edit: Pony took out the "normal" version of division by zero and suggest to write a wrapper to check beforehand.