4 ms·

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
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deleted 9y ago
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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.

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.