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Exactly. I've scrolled through hundreds of comments here now and it really is beyond me how the question of whether you can define x·y in terms of addition for
by codethief 6y ago
Exactly. I've scrolled through hundreds of comments here now and it really is beyond me how the question of whether you can define x·y in terms of addition for x and y being arbitrary reals is even a matter of debate.
- alentist 6y agoThe question is not whether you can define it in terms of addition in some abstract way, but whether you can define it in terms of repeated addition, i.e. something of the form x + x + … + x.
- codethief 6y ago> The question is not whether you can define it in terms of addition in some abstract way This is splitting hairs. The reals themselves are defined "in some abstract way". For instance, what does addition of reals even mean? How do you add two arbitrary real numbers? Exactly, by adding the elements of the Cauchy sequences. (And similarly for multiplication.) This is the definition of addition (multiplication) and the only abstraction involved here is the abstraction due to the way the reals are constructed in the first place.
- alentist 6y ago> This is splitting hairs. Nope. The question addressed by the article is precisely whether multiplication is repeated addition. > adding the elements of the Cauchy sequences. (And similarly for multiplication.) This is the definition of addition (multiplication) Nope. See https://news.ycombinator.com/item?id=26336108 https://news.ycombinator.com/item?id=26336108.