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This essay needs a type system. Every time it says “log” it should say: log of what, into what? It’s like audio where people say "dB" as if it answers the next

This essay needs a type system. Every time it says “log” it should say: log of what, into what? It’s like audio where people say "dB" as if it answers the next question. Relative to what, measured how, and weighted for whom? Author should brush up on https://en.wikipedia.org/wiki/Lie_theory https://en.wikipedia.org/wiki/Lie_theory

jmyeet 3mo ago
The first section details how the author thinks of "log N" with no base as an abstract object rather than a number. Or what are you referring to?
- badlibrarian 3mo ago
  The first section is the good part. The later reuse of “log” across valuations, dimension, vector fields, orders of vanishing is not so good. Those may be related ideas, but each needs a type signature: from what, to what, and preserving which operation?
  - exmadscientist 3mo ago
    Or, to say a little more explicitly what you're getting at: when you take a logarithm of some quantity, log x, x absolutely must be unitless. There's no way whatsoever to take a logarithm of something with a unit attached. (This is an important and useful dimensional analysis check in formulas and long calculations!) So what do you do in practice? You have to normalize: you don't calculate log x, but instead log x/U for some scaling unit U. It's typical for U to be something like 1 mV or 1 W in electrical engineering, for example. This is completely legitimate, but it does mean that the thing that comes out needs a corresponding unit attached to it: dBmV, dBW, et cetera. And it's really kind of important to be careful about that.
rq1 3mo ago
The important properties of the logarithm are structural: we usually do not care about units or bases, except when carrying out an actual numerical computation. As developed in the article, informally, but somewhat sufficiently, the change of base formula shows that the choice of base is largely irrelevant: different bases give equivalent logarithms up to a constant factor. The Taylor expansion of exp gives a more intrinsic and general definition of the exponential function. This allows exp to be generalised structurally to many algebraic settings, provided the relevant convergence conditions are met: for example, the complex exponential and its many possible logs, the matrix exponential, and so on…
- eru 3mo ago
  > The important properties of the logarithm are structural: we usually do not care about units or bases, except when carrying out an actual numerical computation. Units are important as a sort-of type system, even at the conceptual level. You are right that bases are not as important conceptually.
jfengel 3mo ago
I still don't understand why audio dB are negative. That's relative to what? What happens at 0dB?
- kevin_thibedeau 3mo ago
  That is dB full scale where 0 is an absolute ceiling and you can deduct from there.
- deepspace 3mo ago
  0db is usually defined as the loudest sound that the audio system can produce. Hence, everything else must be negative.
  - mitthrowaway2 3mo ago
    Of course, typical of the wonderful ambiguity of decibels, 0 dB is also usually defined as the quietest sound that the human ear can perceive. https://en.wikipedia.org/wiki/Absolute_threshold_of_hearing https://en.wikipedia.org/wiki/Absolute_threshold_of_hearing
    - ianburrell 3mo ago
      That's why important to give the scale. dBfs is full scale level, and db SPL is sound pressure level.
      - rramadass 3mo ago
        Yep. "Sound Power Level SWL", "Sound Pressure level SPL", and "Sound Intensity Level SIL" are different quantities which should not be confused. - https://sengpielaudio.com/calculator-soundpower.htm https://sengpielaudio.com/calculator-soundpower.htm A sound source produces sound power and this generates a sound pressure fluctuation in the air. Sound power is the distance independent cause of this, whereas sound pressure is the distance-dependent effect. Sound pressure p is a "sound field quantity" and sound intensity I is a "sound energy quantity". In teachings these terms are not often separated sharply enough and sometimes are even set equal. But I ~ p2.

jmyeet 3mo ago

The first section details how the author thinks of "log N" with no base as an abstract object rather than a number. Or what are you referring to?

badlibrarian 3mo ago
The first section is the good part. The later reuse of “log” across valuations, dimension, vector fields, orders of vanishing is not so good. Those may be related ideas, but each needs a type signature: from what, to what, and preserving which operation?
- exmadscientist 3mo ago
  Or, to say a little more explicitly what you're getting at: when you take a logarithm of some quantity, log x, x absolutely must be unitless. There's no way whatsoever to take a logarithm of something with a unit attached. (This is an important and useful dimensional analysis check in formulas and long calculations!) So what do you do in practice? You have to normalize: you don't calculate log x, but instead log x/U for some scaling unit U. It's typical for U to be something like 1 mV or 1 W in electrical engineering, for example. This is completely legitimate, but it does mean that the thing that comes out needs a corresponding unit attached to it: dBmV, dBW, et cetera. And it's really kind of important to be careful about that.

rq1 3mo ago

The important properties of the logarithm are structural: we usually do not care about units or bases, except when carrying out an actual numerical computation. As developed in the article, informally, but somewhat sufficiently, the change of base formula shows that the choice of base is largely irrelevant: different bases give equivalent logarithms up to a constant factor. The Taylor expansion of exp gives a more intrinsic and general definition of the exponential function. This allows exp to be generalised structurally to many algebraic settings, provided the relevant convergence conditions are met: for example, the complex exponential and its many possible logs, the matrix exponential, and so on…

eru 3mo ago
> The important properties of the logarithm are structural: we usually do not care about units or bases, except when carrying out an actual numerical computation. Units are important as a sort-of type system, even at the conceptual level. You are right that bases are not as important conceptually.

jfengel 3mo ago

I still don't understand why audio dB are negative. That's relative to what? What happens at 0dB?

kevin_thibedeau 3mo ago
That is dB full scale where 0 is an absolute ceiling and you can deduct from there.
deepspace 3mo ago
0db is usually defined as the loudest sound that the audio system can produce. Hence, everything else must be negative.
- mitthrowaway2 3mo ago
  Of course, typical of the wonderful ambiguity of decibels, 0 dB is also usually defined as the quietest sound that the human ear can perceive. https://en.wikipedia.org/wiki/Absolute_threshold_of_hearing https://en.wikipedia.org/wiki/Absolute_threshold_of_hearing
  - ianburrell 3mo ago
    That's why important to give the scale. dBfs is full scale level, and db SPL is sound pressure level.
    - rramadass 3mo ago
      Yep. "Sound Power Level SWL", "Sound Pressure level SPL", and "Sound Intensity Level SIL" are different quantities which should not be confused. - https://sengpielaudio.com/calculator-soundpower.htm https://sengpielaudio.com/calculator-soundpower.htm A sound source produces sound power and this generates a sound pressure fluctuation in the air. Sound power is the distance independent cause of this, whereas sound pressure is the distance-dependent effect. Sound pressure p is a "sound field quantity" and sound intensity I is a "sound energy quantity". In teachings these terms are not often separated sharply enough and sometimes are even set equal. But I ~ p2.