Does Manifold think that 0 is divisible by 0?
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resolved Mar 16
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NO

To be resolved via poll at close time

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Well, Manifold is disappointingly bad at math...

I'm disappointingly bad at predicting how good Manifold is at math...

Manifold has spoken! Thanks to everyone who voted or traded!

@koadma did I miss the poll? Was it on manifold or elsewhere?

@koadma It would have been good to post it in the comments here.

@NcyRocks Maybe, but I deliberately didn't post it here to not bias the results as the question was about what Manifold thinks

a is divisible by b, iff there is an integer c, such that a = b * c

in our case, a = 0, b = 0.

we see that for c = 69, b * c = 0 = a

therefore, 0 is divisible by 0.

I tried it on a calculator and it said undefined

I tried it on an abacus and it said bitch, u unrefined.

nvm

The formalized mathematics Lean library Mathlib has defined that 0/0 = 0, so the answer should be yes. 🙃

@Pazzaz "Does Manifold think"

@Pazzaz Lean library Mathlib's definition isn't widely accepted.

Mathematically it is right? An integer a divides b if there exists an integer k such that a*k = b. 0*1 = 0, so 0 divides 0.

@Arky Inclined to agree but I gather it depends on what definition for divisibility you use, some definitions I've seen require that a≠0, or that k must be unique; eg

@Nat recently, wikipedia made me aware of definitions other than the one given by Arky and I saw that Manifold has already determined 0^0 to be 1, so I'm hoping it will also choose the better definition in this case!

@Arky I’m curious what you would say if I asked what is the result of 0 divided by 0.

@themightysalmon 0/0 is undefined. However, the statement “a divides b” is actually slightly different than saying “b/a is an integer.”

@Arky Also, the ideal generated by 0 is perfectly well defined, and it's just {0}.

@themightysalmon As Arky says it's undefined. But if you insist, then the most consistent answer is n / 0 = 0, since that allows defining the "integers modulo zero" as just the integers themselves (via the fundamental homomorphism theorem).

@Arky Alright, then if this issue is the wording, the question doesn’t actually ask if “0 divides 0”. The question asks if 0 is divisible by 0. Is 0 able to be divided by 0?

You agree that even if 0 “divides” 0, the operation of 0 divided by 0 is undefined - one is not able to carry out this operation.

Does this word divisible - “able to divide” - not imply that the operation of dividing can actually be carried to completion, yielding a unique answer? If so, how can we say 0 is divisible by 0? We are not able to carry out such an operation.

@themightysalmon English has nothing to do with mathematical convention.

@themightysalmon I believe “b is divisible by a” is generally considered to be exactly equivalent to “a divides b.” So it doesn’t matter that 0/0 is undefined.

@tfae To translate (somewhat lossily) the former for nonmathy types: For every positive integer "is divisible by" is synonymous with "is a multiple of", so we might like that to also hold for 0 and, as 0 is a multiple of 0, then we could say that 0 is divisible by 0.

@Arky Perhaps I agree with you that “b is divisible by a” is equivalent to “a divides b”. But then you’ve dodged the question, because the equivalence runs both ways.

If 0/0 is undefined, then it is an impossible

operation: 0 is not divisible by 0. Then shouldn’t it follow that 0 doesn’t “divide” 0, either? If these two words are truly the same?

You can refer back to your definition of “divides” if you like, but I notice that nowhere does your definition actually use the operation of dividing. It’s entirely based on multiplication.

@themightysalmon The mathematical definition of divisible does not use division because this generalizes to rings where division doesn’t always exist.

@DottedCalculator What use does the concept of “divisible” have in a structure where division doesn’t exist? And what possible bearing can it have on the question at hand?

@themightysalmon The same reason it makes sense to split integers into odd or even. Even though division doesn't exist within the integers, it still makes sense to say that even numbers are divisible by 2.

Sure, the definition doesn't involve division, but there is a connection to division in the broader realm of rational numbers. The even numbers are the ones where if you divide by 2, you still get an integer. That's why the concept was named 'divisible', but then later defined using multiplication as explained above.

In general, concepts shatter into different pieces when you generalize. Equivalent definitions in a specific basic case very often end up picking out very different parts of the broader concept space once you generalize. Number theory is a great example of this, where the concept of 'prime number' has many distinct generalizations that are all important, such as prime element, irreducible element, prime ideal, maximal ideal, and absolute value (yes, this one also generalizes the standard absolute value you've seen - the normal absolute value is sometimes called "the prime at infinity" (and strictly speaking, it's really equivalence classes of absolute values that correspond to primes)). No matter how well you try to name the concepts, it's nearly inevitable that you'll have some slightly counter-intuitive edge cases somewhere.

@themightysalmon In the integers modulo 12, you can say the "multiples of 2" are `{0, 2, 4, 6, 8, 10}`. You can say that 2 divides all of these. However, it would be inconsistent to actually do the division, as `6 = 12/2 = 0/2 = 0`.

In mathematical language, we say that 2 generates a principal ideal, but it is a zero divisor.

@themightysalmon since any number c is a valid solution of c = 0/0 (meaning c * 0 = 0), this means that 0 is not just divisible by 0, but it's infinitely divisible.