This should not be getting downvoted. The author of the article made an incorrect refutation of this point under the "Objections" heading. Division by 0 in fields is undefined precisely because it leads to contradictions like this. The full statement of a proof that 1/0 = 0 includes the temporary assumption that you have defined division by 0 such that it is no longer undefined.
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.
But the point is you keep the structure of a field for everything that already has it, right? No structure is taken away. All statements valid on any subset of the reals are still valid without modification under this new definition. The algebraic structure is still a field obeying the same axioms. There are just some new valid statements too.
No that's not correct, and this is why I think the author's entire point is pretty inane. If you're going to start off your argument with the full formalism of field axioms and consequent theorems, you need to be prepared to split hairs about whether or not your definitions constitute a field.
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.
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.
There is a mistake in your step 3, as it relies on an informal and imprecise "division is the inverse of multiplication". If you were to write that formally, you'd get `∀ x ≠ 0 . x(1/x) = 1`. This holds unchanged even if you define division by zero.
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.
>Since y = x/0, it follows that the product of y and 0 is equal to x, because division is the inverse of multiplication.
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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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.
This is explicitly talked about in OP about 2/3 of the way down the page. "If 1/0 = 0, then 1 = 0 * 0" is untrue. Your argument is "1/0 = 0, so multiply both sides by zero: 1/0 * 0 = 0 * 0, and then take 1/0 * 0 = 1 * 0/0." That last step isn't the case for reasons covered in the article we're ostensibly discussing.
Right. So the multiplicative inverse property _breaks_! He just points out that it breaks, and thus you need to use a more complicated property instead. That doesn't mean that the property doesn't break.
He's not saying that it breaks; quite the opposite, he repeatedly states that 0⁻ doesn't exist. He's just defining division in a specific way.
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.
The problem this and the other replies miss is that the standard definition of division is multiplication by the inverse. The entire argument rests on a notational slight of hand. The property that held before -- that _when defined_ division has the inverse property -- no longer holds. Thus many equational identities that otherwise would hold do not hold.
> The problem this and the other replies miss is that the standard definition of division is multiplication by the inverse.
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
You are absolutely correct that this equational theorem is broken by extending the domain of division. However, there is absolutely no way to say that the formalization of this theorem isn't actually
∀ x ≠ 0 . x(1/x) = 1
because the two are equivalent. You cannot then claim that something is necessarily broken if you choose to pick a formalization that is indeed broken, while an equivalent formalization exists, that is not broken (not to mention that the formalization that is broken requires a strictly richer language). All that means is that your formalization in this case is brittle, not that laws are broken.
I have to disagree -- this isn't sleight of hand. The standard definition isn't being violated here, because standard division isn't a total function. The denominator's domain in Hillel's function is a proper superset of the standard domain: when restricted to the standard domain, the two functions are precisely equivalent. Therefore, every standard identity still holds under Hillel.
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.
> 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.
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.
I know what he's doing. The problem is when you make it a different function (even by just extending it) then you change its equational properties. So equational properties that held over the whole domain of the function no longer hold over the extended domain. This is repaired by modifying the equational properties. But the modified equational properties mean that you now have a different system than before. So the whole thing is just playing around with words.
Multiplicative inverse is already broken. x / 0 * 0 is either 0 or ⊥ (aka exception). It wasn't true to begin with. I happen to like the exception/partial function version, so I'm playing devil's advocate, but it really was already broken when we're talking about zero.
There are many contexts where defining either 1/0 = -1/0 = ∞ or 1/0 = +∞ and –1/0 = –∞ is better than the alternatives, especially when working in an approximate number system like floating point.
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.
Look up the dirac delta function. It's a spike thats infinitely tall and infinitely narrow, with an area of 1. It's established now as a very useful tool in EE. But many people fought it tooth and nail because of this logic.
> This seems just as bizarre, since zero times anything shouldn't become 1, no matter how big or how many times you do it.
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.
You’re doing something different than real number arithmetic. Saying 1/0 = x, and then treating x like a real number is inconsistent. But just saying “we are going to augment the real numbers with an element x that is not a real number, and then define some properties of x and prove things about it” is not.
> You’re doing something different than real number arithmetic
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.
If division by zero is not undefined, yes. But how does this disagree with the assertion that it's mathematically inconsistent (as opposed to programmatically inconsistent)? If any case exists that leads to a logically false conclusion, then it's not logically consistent.
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.