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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

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.
  - skybrian 8y ago
    Maybe all this talk about fields is a distraction? Integer arithmetic isn't a field anyway. Other than 1 and -1, no integer has a multiplicative inverse that's an integer. Integer division just isn't the same operation as division on rationals or reals. The same laws don't apply. (For floats, division by zero can return Inf, -Inf or NaN and there's no reason to define it differently.) It looks like the theorem-proving languages that define 1/0 to be 0 tend to be using natural numbers as a fundamental type. Not only do they define division differently, they also don't have negative numbers, and so define 2-3 to be 0. https://coq.inria.fr/library/Coq.Init.Nat.html https://coq.inria.fr/library/Coq.Init.Nat.html
    - throwawaymath 8y ago
      Yes! Exactly, it's a distraction. There are reasonable, well-intentioned reasons to argue that division by 0 should be an acceptable operation in a computational setting. Computational settings need not admit all the rigor of theoretical math. I wish the author had not tried to involve field theory and programming like this because it detracts from the point :)
      - deleted 8y ago
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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.
- skybrian 8y ago
  Maybe all this talk about fields is a distraction? Integer arithmetic isn't a field anyway. Other than 1 and -1, no integer has a multiplicative inverse that's an integer. Integer division just isn't the same operation as division on rationals or reals. The same laws don't apply. (For floats, division by zero can return Inf, -Inf or NaN and there's no reason to define it differently.) It looks like the theorem-proving languages that define 1/0 to be 0 tend to be using natural numbers as a fundamental type. Not only do they define division differently, they also don't have negative numbers, and so define 2-3 to be 0. https://coq.inria.fr/library/Coq.Init.Nat.html https://coq.inria.fr/library/Coq.Init.Nat.html
  - throwawaymath 8y ago
    Yes! Exactly, it's a distraction. There are reasonable, well-intentioned reasons to argue that division by 0 should be an acceptable operation in a computational setting. Computational settings need not admit all the rigor of theoretical math. I wish the author had not tried to involve field theory and programming like this because it detracts from the point :)
    - deleted 8y ago
      [deleted]