This isn’t really correct. Allowable values of functions are whatever you want. If you define a function on R-{0} by “x goes to 1/x”, it’s not defined at 0; I explicitly excluded it from the domain. If you define a function on R by “x goes to 1/x”… you can’t, there’s no such thing as 1⁄0. If you define a function on R by “x goes to 1/x if x is nonzero, and 0 goes to infinity”, this is a perfectly sensible function, which it is convenient to just abbreviate as “1/x”. Though for obvious reasons I would only recommend doing this if the “infinity” you are using represents both arbitrarily large positive and negative quantities. (EDIT: But if you want to define a function on [0,infty) by “x goes to 1/x if x is nonzero, and 0 goes to infinity” with “infinity” now only being large in the positive direction, which is likely what’s actually under consideration here, then this is not so dumb.)
All this is irrelevant to any actual physical questions, where whether using infinities is appropriate or not just depends on, well, the physics of it.
Yes, and of course which theory will be appropriate is going to be determined by the actual physics. My point is just that your statement that “pure math does not have infinities” and physicists “added them in” is wrong (even ignoring historical inaccuracies).
But here is a case where the pure math does not have infinities
That is not a statement that the field of mathematics does not have infinities. I was referring specifically to “the way I learned calculus”. Unless you took my class, you don’t know what I did or did not learn and how I learned it. My statement was true, your “correction” was false.
Ah, sorry then. This is the sort of mistake I that’s common enough that it seemed more obvious to me to read it the that way rather than the literal and correct way.
This isn’t really correct. Allowable values of functions are whatever you want. If you define a function on R-{0} by “x goes to 1/x”, it’s not defined at 0; I explicitly excluded it from the domain. If you define a function on R by “x goes to 1/x”… you can’t, there’s no such thing as 1⁄0. If you define a function on R by “x goes to 1/x if x is nonzero, and 0 goes to infinity”, this is a perfectly sensible function, which it is convenient to just abbreviate as “1/x”. Though for obvious reasons I would only recommend doing this if the “infinity” you are using represents both arbitrarily large positive and negative quantities. (EDIT: But if you want to define a function on [0,infty) by “x goes to 1/x if x is nonzero, and 0 goes to infinity” with “infinity” now only being large in the positive direction, which is likely what’s actually under consideration here, then this is not so dumb.)
All this is irrelevant to any actual physical questions, where whether using infinities is appropriate or not just depends on, well, the physics of it.
They are limited by the scope of whatever theory you are working in.
Yes, and of course which theory will be appropriate is going to be determined by the actual physics. My point is just that your statement that “pure math does not have infinities” and physicists “added them in” is wrong (even ignoring historical inaccuracies).
Selective quotation. I said:
That is not a statement that the field of mathematics does not have infinities. I was referring specifically to “the way I learned calculus”. Unless you took my class, you don’t know what I did or did not learn and how I learned it. My statement was true, your “correction” was false.
Ah, sorry then. This is the sort of mistake I that’s common enough that it seemed more obvious to me to read it the that way rather than the literal and correct way.