You don’t need any theory about infinity, hierarchical or otherwise, to get Fourier transforms working. Everything can be done using approximations with bounded error in finite steps. It’s just a pain in the butt to write proofs that way.
(And indeed, in practice, we can’t perform any infinite processes when we are computing Fourier transforms of real data, in either analog or digital systems. All real-world systems are based on approximations and imperfect models, full of measurement error, bugs in edge cases, etc.)
You have a very strange definition of “real”... or rather, it’s like the use in “real numbers” (i.e. a pure thought experiment / set of abstract manipulations of an abstract formal system with no physical embodiment), not the use in “physical reality”.
Every mathematical use of completed infinities inherently involves “mysticism”, and you seem to have embraced the absolution you mentioned. [Which is fine... accepting Cantor’s paradise on faith and not worrying about whether it is “real” or not has been very productive for mathematics, whether or not most of the same results could have been found in a more cumbersome way otherwise.]
thank you, you write what I was trying to say but was ultimately not clear enough in expressing my thoughts. I don’t understand why people think that because something is good at approximating values that it necessitates its truth or existence.
It's not about them, nobody wished them. They came as necessity.
> Everything can be done using approximations
Approximations are analytical animals.
> with bounded error
I don't know how to bound errors without analysis.
> in finite steps.
By usage I didn't mean occasionally computing. (This is not free of problems. BTW you'd be horrified to know even finite purely combinatorial mathematics leads to irrational polytopes, ending up with infinities.) But using as a cornerstone of theory that allows for reasoning of epistemic value i.e. one affording symbolic manipulations that behave in a guaranteed way. Taking differences instead of differentials doesn't matter (why people bother if it were that easy). Whereas Finitism just shifts infinities elsewhere ending up with ultrafilter.
> You have a very strange definition of “real”
I didn't realize I used one. Or that reality has a definition (old problem in philosophy).
Or that it matters. Irrespective of what reality is mathematics can be ① real, ② have different being still connected to reality, ③ or be unreal with some mechanism for reasoning to cross the boundary to reality. First and last are troubling. Issues with the middle one is why there is a philosophy of mathematics at all.
> “real numbers”
Real numbers of course are called so for being perversion that they are. Do try get rid of them. Two centuries people have failed and still try intensely. They need a cheer.
> Every mathematical use of completed infinities inherently involves “mysticism”
I don't see what you could possibly mean. Any specifics?
Other than a lazy trollish attempt at obverting my position. To remind it is that Fictionalism absolves one from studying internal mechanics of fictional things. (Other than stop such worries, what use of invoking it?) You say mathematics is the same. In fact worse: mysticism gets in at every other point, inherently. (Why do I even bother replying…) Is it a cult? I assure you whenever a garden variety idol has deep feelings about eternity and numbers, there is a pushback https://arxiv.org/abs/1211.0244
Mathematics is a study, deals in mysticism as much as accounting: when infinities arise, their need and utility are studied, employed and scrutinized as generously as quantitative easing. If you find baffling mysticism plaguing your understanding, there is no shortage of philosophical equivalents of Bitcoin.
(And indeed, in practice, we can’t perform any infinite processes when we are computing Fourier transforms of real data, in either analog or digital systems. All real-world systems are based on approximations and imperfect models, full of measurement error, bugs in edge cases, etc.)
You have a very strange definition of “real”... or rather, it’s like the use in “real numbers” (i.e. a pure thought experiment / set of abstract manipulations of an abstract formal system with no physical embodiment), not the use in “physical reality”.
Every mathematical use of completed infinities inherently involves “mysticism”, and you seem to have embraced the absolution you mentioned. [Which is fine... accepting Cantor’s paradise on faith and not worrying about whether it is “real” or not has been very productive for mathematics, whether or not most of the same results could have been found in a more cumbersome way otherwise.]