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by throwawaymath
2872 days ago
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I follow everything you're saying. As I have said many times already, I have no problem with convention 2. But don't use field theory to justify convention 2, because it's mathematically incoherent and unnecessary. I broadly agree - there is no reason to be involving field theory in programming language design. 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. |
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> 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.