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Understanding what mathematical objects are can lead to insanity. Much better to focus on what you can do with them. Or to quote John von Neumann, "In mathemat
by T_S_ 15y ago
Understanding what mathematical objects are can lead to insanity. Much better to focus on what you can do with them.
Or to quote John von Neumann, "In mathematics you don't understand things, you just get used to them."
- harshreality 15y agoYou have to know whether your definition of "numbers" makes them into a field or not, before you can go applying field operations to them and expecting usual results. Including infinite ordinals in the set of numbers is legitimate, as parent points out that "number" is not strictly defined, but if you include infinity, numbers are no longer a field, and you have to cut someone off when they try to use field axioms in theorems, the uniqueness of multiplicative and additive inverses, which must be how people end up with nonsense like 1=0.