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Infinity and moreover a hierarchy of infinities is very real and has real consequences. You want your Fourier transform to work, you have to quantify infinities

Infinity and moreover a hierarchy of infinities is very real and has real consequences. You want your Fourier transform to work, you have to quantify infinities (this was in fact what prompted development of set theory: how to make Fourier analysis precise). Supertasks are real too. If you take Turing machine something like the lamplighter group pops up at the other end and you have no way to handwave that away. Unless of course you subscribe to Fictionalism as your philosophy of mathematics, in which case stuff still has consequences just your epistemological stance changes from studying them in favour to some kind of mysticism that absolves you from the effort.

cpsempek 9y ago
I want Fourier transforms to work, but I don't need an abstraction for how they work to be apart of my reality in order for them to be efficacious.
- thraway180306 9y ago
  Mathematics is tautological, by definition. In what sense anything within a tautological system can be an abstraction? It's a whole package. There is no à la carte Fourier transform without philosophical commitments.
  - cpsempek 9y ago
    Sure there is. I can use Fourier transforms because they are efficacious but this does not force ZFC upon my reality. Could they be apart of reality? Sure, and, if they are they produce efficacious tools. Could they not be apart of my reality and efficacious tools still be available? I don't see why not.
    - thraway180306 9y ago
      It depends on your reality of course. It might be the physical reality, whatever that is. Or your own-er reality which might not be logical for example. There are things invariant between realities of course, like the physical experiment and mathematics. But if your reality upholds logic, you'd have to be very careful as not to take any consequence of your usage of the Fourier transform to be a part of your reality or its description. In which case why use it at all? Using it is, like I said, an ontological commitment inviting infinities into your system. You may not ascribe them physical meaning, or may renormalize them out, but you can't deny them (do try! though attempts at mathematical ultrafinitism are plagued by problems, whereas finitism, or various less purifying forms of constructive mathematics lead to infinities in just slightly different places). What you may deny is the reality of the description (that is of you perceiving your reality) whatsoever as Fictionalists do.
      - cpsempek 9y ago
        I just mean to say mathematical objects do not have to exist in order for them to be efficacious. We may just disagree on the ontology of mathematical objects. Or, I may just be misunderstanding your argument.
        deleted 9y ago
        [deleted]
- deleted 9y ago
  [deleted]
- EtDybNuvCu 9y ago
  Why did your ear, the outer part and the inner bones and the canal and the fluids and the brain, all evolve together in a way that grants Fourier analysis for free? Coincidence? Maybe maths could help us understand biology or other non-maths fields better! (Extra mystery: Why did we evolve phase-invariance?)
  - cpsempek 9y ago
    I believe you missed my point entirely. "Why did your ear, the outer part and the inner bones and the canal and the fluids and the brain, all evolve together in a way that grants Fourier analysis for free? Coincidence? " No, we model physical systems using math, so you should expect that a physical system involving dynamic systems to use Fourier transforms to effectively model that system. But the efficacy of math does not imply that mathematical objects exist. "Maybe maths could help us understand biology or other non-maths fields better!" Um, clearly. Math is extremely successful at describing many aspects of many sciences. But this does not imply mathematical objects exist and are describing the reality of what is occurring, despite providing accurate outputs.
  - dboreham 9y ago
    Phase invariance is a side-effect of doing the transform to the frequency domain, isn't it?
  - comex 9y ago
    It’s not a coincidence, but it is only an approximation. Fourier analysis of sound assumes a continuous field of air pressure at each point in space, which evolves according to a continuous function; in reality, air consists of a large number of discrete particles, and “pressure” approximates the result of a large number of discrete, random collisions. We can prove mathematically that the more particles are in the system, the closer its behavior gets (with overwhelming probability) to the continuous ideal; and sound happens to operate at a sufficiently large scale that the discrepancy is far too small to make a difference, so we - and evolution - can just use the continuous ideal for our calculations. But that doesn’t require it to have any physical meaning. On the other hand, at a lower level of abstraction, the most fundamental theories of physics known do tend to involve real numbers and continuous functions – from my limited understanding, that applies even to quantum mechanics in many cases, even though it’s known for discretizing quantities that were continuous in classical mechanics. However, it’s unknown – unknowable, even, at some point – whether these infinities are “real”, or themselves approximations of even more fundamental laws.
jacobolus 9y ago
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.]
- cpsempek 9y ago
  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.
- deleted 9y ago
  [deleted]
- thraway180306 9y ago
  > You don’t need any theory about infinity 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 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.

cpsempek 9y ago

I want Fourier transforms to work, but I don't need an abstraction for how they work to be apart of my reality in order for them to be efficacious.

thraway180306 9y ago
Mathematics is tautological, by definition. In what sense anything within a tautological system can be an abstraction? It's a whole package. There is no à la carte Fourier transform without philosophical commitments.
- cpsempek 9y ago
  Sure there is. I can use Fourier transforms because they are efficacious but this does not force ZFC upon my reality. Could they be apart of reality? Sure, and, if they are they produce efficacious tools. Could they not be apart of my reality and efficacious tools still be available? I don't see why not.
  - thraway180306 9y ago
    It depends on your reality of course. It might be the physical reality, whatever that is. Or your own-er reality which might not be logical for example. There are things invariant between realities of course, like the physical experiment and mathematics. But if your reality upholds logic, you'd have to be very careful as not to take any consequence of your usage of the Fourier transform to be a part of your reality or its description. In which case why use it at all? Using it is, like I said, an ontological commitment inviting infinities into your system. You may not ascribe them physical meaning, or may renormalize them out, but you can't deny them (do try! though attempts at mathematical ultrafinitism are plagued by problems, whereas finitism, or various less purifying forms of constructive mathematics lead to infinities in just slightly different places). What you may deny is the reality of the description (that is of you perceiving your reality) whatsoever as Fictionalists do.
    - cpsempek 9y ago
      I just mean to say mathematical objects do not have to exist in order for them to be efficacious. We may just disagree on the ontology of mathematical objects. Or, I may just be misunderstanding your argument.
      - deleted 9y ago
        [deleted]
deleted 9y ago
[deleted]
EtDybNuvCu 9y ago
Why did your ear, the outer part and the inner bones and the canal and the fluids and the brain, all evolve together in a way that grants Fourier analysis for free? Coincidence? Maybe maths could help us understand biology or other non-maths fields better! (Extra mystery: Why did we evolve phase-invariance?)
- cpsempek 9y ago
  I believe you missed my point entirely. "Why did your ear, the outer part and the inner bones and the canal and the fluids and the brain, all evolve together in a way that grants Fourier analysis for free? Coincidence? " No, we model physical systems using math, so you should expect that a physical system involving dynamic systems to use Fourier transforms to effectively model that system. But the efficacy of math does not imply that mathematical objects exist. "Maybe maths could help us understand biology or other non-maths fields better!" Um, clearly. Math is extremely successful at describing many aspects of many sciences. But this does not imply mathematical objects exist and are describing the reality of what is occurring, despite providing accurate outputs.
- dboreham 9y ago
  Phase invariance is a side-effect of doing the transform to the frequency domain, isn't it?
- comex 9y ago
  It’s not a coincidence, but it is only an approximation. Fourier analysis of sound assumes a continuous field of air pressure at each point in space, which evolves according to a continuous function; in reality, air consists of a large number of discrete particles, and “pressure” approximates the result of a large number of discrete, random collisions. We can prove mathematically that the more particles are in the system, the closer its behavior gets (with overwhelming probability) to the continuous ideal; and sound happens to operate at a sufficiently large scale that the discrepancy is far too small to make a difference, so we - and evolution - can just use the continuous ideal for our calculations. But that doesn’t require it to have any physical meaning. On the other hand, at a lower level of abstraction, the most fundamental theories of physics known do tend to involve real numbers and continuous functions – from my limited understanding, that applies even to quantum mechanics in many cases, even though it’s known for discretizing quantities that were continuous in classical mechanics. However, it’s unknown – unknowable, even, at some point – whether these infinities are “real”, or themselves approximations of even more fundamental laws.

jacobolus 9y ago

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.]

cpsempek 9y ago
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.
deleted 9y ago
[deleted]
thraway180306 9y ago
> You don’t need any theory about infinity 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 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.