For example? Although, if we agree on the definition below, there’s maybe no point.
A very early appearance of infinity is the proof that there are infinitely many primes. It is most certainly not a proof that there is a very large but finite number of primes.
I can agree with “there are infinitely many primes” if I interpret it as something like “if I ever expect to run out of primes, that belief won’t pay rent.”
In this case, and in most cases in mathematics, these statements may look and operate the same—except mine might be slower and harder to work with. So why do I insist on it? I’m happy to work with infinities for regular math stuff, but there are some cases where it does matter, and these might all be outside of pure math. But in applied math there can be problems if infinity is taken seriously as a static concept rather than as a process where the expectation that it will end will never pay rent.
Like if someone said, “Black holes have infinite density,” I would have to ask for clarification. Can it be put into a verbal form at least? How would it pay rent in terms of measurements? That kind of thing.
Like if someone said, “Black holes have infinite density,” I would have to ask for clarification.
Actually, the way I learned calculus, allowable values of functions are real (or complex), not infinite. The value of the function 1/x at x=0 is not “infinity”, but “undefined” (which is to say, there is no function at that point); similarly for derivatives of functions where the functions go vertical. Since that time, I discovered that apparently physicists have supplemented the calculus I know with infinite values. They actually did it because this was useful to them. Don’t ask me why, I don’t remember. But here is a case where the pure math does not have infinities, and then the practical folk over in the physics department add them in. Apparently the practical folk think that infinity can pay rent.
As for gravitational singularities, the problem here is not the concept of infinity. That is an innocent bystander. The problem is that the math breaks down. That happens even if you replace “infinite” with “undefined”.
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.
As for gravitational singularities, the problem here is not the concept of infinity. That is an innocent bystander. The problem is that the math breaks down.
I never really got why the math is said to ‘break down’. Is it just because of a divide by zero thing or something more significant? I guess I just don’t see a particular problem with having a part of the universe really being @%%@ed up like that.
I guess I just don’t see a particular problem with having a part of the universe really being @%%@ed up like that.
What I think is more likely is that the universe does not actually divide by zero, and the singularity is a gap in our knowledge. Gaps in knowledge are the problem of science, whose function is to fill them.
A very early appearance of infinity is the proof that there are infinitely many primes. It is most certainly not a proof that there is a very large but finite number of primes.
I can agree with “there are infinitely many primes” if I interpret it as something like “if I ever expect to run out of primes, that belief won’t pay rent.”
In this case, and in most cases in mathematics, these statements may look and operate the same—except mine might be slower and harder to work with. So why do I insist on it? I’m happy to work with infinities for regular math stuff, but there are some cases where it does matter, and these might all be outside of pure math. But in applied math there can be problems if infinity is taken seriously as a static concept rather than as a process where the expectation that it will end will never pay rent.
Like if someone said, “Black holes have infinite density,” I would have to ask for clarification. Can it be put into a verbal form at least? How would it pay rent in terms of measurements? That kind of thing.
Actually, the way I learned calculus, allowable values of functions are real (or complex), not infinite. The value of the function 1/x at x=0 is not “infinity”, but “undefined” (which is to say, there is no function at that point); similarly for derivatives of functions where the functions go vertical. Since that time, I discovered that apparently physicists have supplemented the calculus I know with infinite values. They actually did it because this was useful to them. Don’t ask me why, I don’t remember. But here is a case where the pure math does not have infinities, and then the practical folk over in the physics department add them in. Apparently the practical folk think that infinity can pay rent.
As for gravitational singularities, the problem here is not the concept of infinity. That is an innocent bystander. The problem is that the math breaks down. That happens even if you replace “infinite” with “undefined”.
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.
I might call engineers “practical folk”; astrophysicists I’m not so sure. I’d like to see their reason for doing so.
I never really got why the math is said to ‘break down’. Is it just because of a divide by zero thing or something more significant? I guess I just don’t see a particular problem with having a part of the universe really being @%%@ed up like that.
What I think is more likely is that the universe does not actually divide by zero, and the singularity is a gap in our knowledge. Gaps in knowledge are the problem of science, whose function is to fill them.