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The author describes a process called "ontological remodeling", which is when a change in viewpoint radically simplifies a previously intractable problem. This

The author describes a process called "ontological remodeling", which is when a change in viewpoint radically simplifies a previously intractable problem. This is the story, not just of Sudoku, but all of mathematics.

barathr 5y ago
Chapman's writings on the subject are worth a read: https://metarationality.com/remodeling https://metarationality.com/remodeling
marmshallow 5y ago
Indeed. After reading the title of the post, I thought it may be about the Cook-Levin theorem, which is about how all NP problems (of which Sudoku is one) can be reduced to one another.
rhdunn 5y ago
3Blue1Brown has a few videos where he discusses and uses this technique to solve some complex problems. One of those was for a mathematical competition. It's also common in audio manipulation, e.g. change to the frequency domain in order to modify pitch, then change back to the time domain.
- orlp 5y ago
  > It's also common in audio manipulation, e.g. change to the frequency domain in order to modify pitch, then change back to the time domain. Funnily enough, yes but actually no. For understanding and mathematical proofs the Fourier Transform is obviously essential. But when you first get into audio DSP programming it might seem that the FFT is crucial as well. But virtually all digital audio filters directly operate on the on the time domain. The problem is that we always work with a sampled signal. And while the Nyquist theorem tells us that as long as our sample frequency is at least twice as high as the highest frequency in our input that our sampling doesn't lose any information, we still have to be aware of it. When you convert a fixed frequency sampled signal into the frequency domain, you get a sum of a fixed set (co)sine functions, because we're still sampled. Now some operations you can do in this frequency domain exactly and simply. E.g. if 100Hz is part of our fixed set of functions and we wish to subtract a 100Hz signal, we can directly do that on our coefficient. A pitch shift by an exact multiple of the frequency sample delta is possible exactly too, assuming the lowest/highest frequencies shifted off are inaudible. But you almost never want to do these operations. A classic example of something you might want to do is a low-pass filter. Simple right? FFT to frequency domain, zero out the coefficients above the cutoff frequency, and convert back. No! Zeroing out coefficients is equivalent to subtracting those specific sine waves. But that is not a low-pass. As an example, suppose our sample frequency is such that the FFT's sine wave coefficients are 10Hz apart, and we wish to do a 30Hz low-pass filter. This means that a 25Hz signal should be completely unaffected, but a 25Hz signal can't be represented just using the 10Hz and 20Hz coefficients - you need higher order terms! Thus if we zero those out, we distort our 25Hz signal. So to solve this in the real-world you get into the difficult topic of filter design, or other audio DSP algorithms that are vastly more involved than a simple FFT.
  - rhdunn 5y ago
    That's fascinating. Thanks for the information.
  - 10000truths 5y ago
    Is there any scenario where the discrepancy between FFT and its analog cousin cannot be resolved by upping the sampling density? Distortion is always there, whether it comes from an imprecise sensor/instrument or from extrapolating sampled data. In the audio space, nothing above 20 kHz is audible anyways, so even a bog-standard 44.1 kHz sampling rate should do "good enough" for most DSP operations there.
    - orlp 5y ago
      > Distortion is always there, whether it comes from an imprecise sensor/instrument or from extrapolating sampled data. Specifically regarding the latter part of "extrapolating sampled data", I would highly recommend watching this video: https://xiph.org/video/vid2.shtml https://xiph.org/video/vid2.shtml. As long as your input signal is low-passed to below 22kHz, the 44.1kHz sampling is perfect. No information is lost, no distortion. I however am not qualified enough to tell you how the naive FFT filter approach changes in distortion as you raise the sampling frequency.
    - 112233 5y ago
      Transients. FFT is time-symmetrical, so signals that start because of an event (like a mallet hitting something) have a lot of "nothing audible "above 20kHz", because of the abrupt start. Forcing nyquist limit on such signal alway causes pre-ringing, that breaks causality - you have sound starting before the event happened. (edit) e.g. bandwidth limited signal with only a single non-zero sample does not represent a rectangular function, but a sinc. So, FFT is a lie. But very useful one.
  - skybrian 5y ago
    I'm just using simple IIR filters because it's one float of storage and one line of code, but I'm curious: which filtering algorithms are you talking about?
- ykonstant 5y ago
  Since you mention it, 3b1b has a splendid video juxtaposing two attitudes to problem solving via an example here: https://www.youtube.com/watch?v=ltLUadnCyi0 https://www.youtube.com/watch?v=ltLUadnCyi0 . I highly recommend watching it, I think it is one of his best.
fsckboy 5y ago
> "ontological remodeling", which is when a change in viewpoint radically simplifies a previously intractable problem. This is the story, not just of Sudoku, but all of mathematics. it's such a powerful tool that the only problem it leaves is, what benefit is there to calling it ontological remodeling instead of "a change in viewpoint"?
- deleted 5y ago
  [deleted]

barathr 5y ago

Chapman's writings on the subject are worth a read: https://metarationality.com/remodeling https://metarationality.com/remodeling

marmshallow 5y ago

Indeed. After reading the title of the post, I thought it may be about the Cook-Levin theorem, which is about how all NP problems (of which Sudoku is one) can be reduced to one another.

rhdunn 5y ago

3Blue1Brown has a few videos where he discusses and uses this technique to solve some complex problems. One of those was for a mathematical competition. It's also common in audio manipulation, e.g. change to the frequency domain in order to modify pitch, then change back to the time domain.

orlp 5y ago
> It's also common in audio manipulation, e.g. change to the frequency domain in order to modify pitch, then change back to the time domain. Funnily enough, yes but actually no. For understanding and mathematical proofs the Fourier Transform is obviously essential. But when you first get into audio DSP programming it might seem that the FFT is crucial as well. But virtually all digital audio filters directly operate on the on the time domain. The problem is that we always work with a sampled signal. And while the Nyquist theorem tells us that as long as our sample frequency is at least twice as high as the highest frequency in our input that our sampling doesn't lose any information, we still have to be aware of it. When you convert a fixed frequency sampled signal into the frequency domain, you get a sum of a fixed set (co)sine functions, because we're still sampled. Now some operations you can do in this frequency domain exactly and simply. E.g. if 100Hz is part of our fixed set of functions and we wish to subtract a 100Hz signal, we can directly do that on our coefficient. A pitch shift by an exact multiple of the frequency sample delta is possible exactly too, assuming the lowest/highest frequencies shifted off are inaudible. But you almost never want to do these operations. A classic example of something you might want to do is a low-pass filter. Simple right? FFT to frequency domain, zero out the coefficients above the cutoff frequency, and convert back. No! Zeroing out coefficients is equivalent to subtracting those specific sine waves. But that is not a low-pass. As an example, suppose our sample frequency is such that the FFT's sine wave coefficients are 10Hz apart, and we wish to do a 30Hz low-pass filter. This means that a 25Hz signal should be completely unaffected, but a 25Hz signal can't be represented just using the 10Hz and 20Hz coefficients - you need higher order terms! Thus if we zero those out, we distort our 25Hz signal. So to solve this in the real-world you get into the difficult topic of filter design, or other audio DSP algorithms that are vastly more involved than a simple FFT.
- rhdunn 5y ago
  That's fascinating. Thanks for the information.
- 10000truths 5y ago
  Is there any scenario where the discrepancy between FFT and its analog cousin cannot be resolved by upping the sampling density? Distortion is always there, whether it comes from an imprecise sensor/instrument or from extrapolating sampled data. In the audio space, nothing above 20 kHz is audible anyways, so even a bog-standard 44.1 kHz sampling rate should do "good enough" for most DSP operations there.
  - orlp 5y ago
    > Distortion is always there, whether it comes from an imprecise sensor/instrument or from extrapolating sampled data. Specifically regarding the latter part of "extrapolating sampled data", I would highly recommend watching this video: https://xiph.org/video/vid2.shtml https://xiph.org/video/vid2.shtml. As long as your input signal is low-passed to below 22kHz, the 44.1kHz sampling is perfect. No information is lost, no distortion. I however am not qualified enough to tell you how the naive FFT filter approach changes in distortion as you raise the sampling frequency.
  - 112233 5y ago
    Transients. FFT is time-symmetrical, so signals that start because of an event (like a mallet hitting something) have a lot of "nothing audible "above 20kHz", because of the abrupt start. Forcing nyquist limit on such signal alway causes pre-ringing, that breaks causality - you have sound starting before the event happened. (edit) e.g. bandwidth limited signal with only a single non-zero sample does not represent a rectangular function, but a sinc. So, FFT is a lie. But very useful one.
- skybrian 5y ago
  I'm just using simple IIR filters because it's one float of storage and one line of code, but I'm curious: which filtering algorithms are you talking about?
ykonstant 5y ago
Since you mention it, 3b1b has a splendid video juxtaposing two attitudes to problem solving via an example here: https://www.youtube.com/watch?v=ltLUadnCyi0 https://www.youtube.com/watch?v=ltLUadnCyi0 . I highly recommend watching it, I think it is one of his best.

fsckboy 5y ago

> "ontological remodeling", which is when a change in viewpoint radically simplifies a previously intractable problem. This is the story, not just of Sudoku, but all of mathematics. it's such a powerful tool that the only problem it leaves is, what benefit is there to calling it ontological remodeling instead of "a change in viewpoint"?

deleted 5y ago
[deleted]