> 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? What mathematical axiom does this break?
> 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
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.
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.