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I’d like to describe the length of the question text. I’m fairly certain that is not outside the realm of physics, however if I measure it in “bytes” it’s not clear to me how to convert that to SI units.

My best guess would be to find the temperature $T$ of the hard drives where copies of this string are (it gets sent from the Stack Exchange server over EM waves to the antenna on my phone, then stored locally on the hard drive, then transmitted via light in the visual range from my screen to the cameras in my head [my eyes!], then stored locally in my brain), and get a bound using Landauer’s limit $kT\ln(2)$ for the minimum energy it takes to erase one of those bits.

If the length of this string can’t be measured in SI units, is it outside the realm of physics? Another way to phrase this question is why, exactly, isn’t “bit” an SI unit directly, and does that imply that physicists are not yet ready to admit that information is part of the physical world?

Qmechanic
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Jackson Walters
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  • @noah Almost, yes. However, any CS student can count the number of bits in the question text, and any high school physics can perform a conversion using $kT\ln(2)$, however the bit still isn’t an SI unit. It’s not analogous to the unit “people” as it’s directly comparable to SI units. The fact that they’re discrete really doesn’t discount their reality, as many quantities in physics are discrete (angular momentum or energy in QM w/ boundary conditions!). Finally, the “bit” is only one rep’n of info; other bases yield trit, etc. so the argument that info isn’t a real number like mass seems off. – Jackson Walters May 19 '21 at 22:35
  • Well, first step would be to define "length" of text. What do you mean by it, what's the measure? Is it number of characters? Do you include spaces? Or all whitespace? If you measure bits, do you measure the underlying representation (and which representation do you pick)? Or do you want to characterize the information content of the statement itself, etc. – Filip Milovanović May 19 '21 at 22:35
  • The whole point is that you don't need a unit if the quantity is naturally discrete. For a, e.g., length, we have no natural way to describe it. So we compare it to some reference, and then give the prefactor of how many times that reference we mean. 2 m is twice as long as our reference of the meter. If the unit is inherently discrete and not divisible, we don't need a unit for it. I find the assumption that anything that doesn't have an SI unit dedicated to it is not recognized as part of the physical world... odd. – noah May 19 '21 at 22:58
  • @FilipMilovanović Sure. "The text" exists on the server, client, in the user, or in an EM wave/fiber/copper cable, etc. Let's focus on my desktop. "The text" I see in a browser (so translated by JS, CSS, etc.), and likely in RAM or a temp folder (so my SSD). It's 19cm on my monitor, but this will vary; length should be invariant under hardware. We could zoom in on the SSD, and would probably find a grid with potential wells holding some # of electrons. We could count those and divide by electrons/cell, but still, not invariant. Let's agree on 7-bit ASCII encoding. I measure then 476 bits. – Jackson Walters May 19 '21 at 23:28
  • @noah Well, true, length does appear to exist on a continuum, which we model with real numbers (we've yet to find a minimum). The bit, the resolution of the uncertainty inherent in two states, rests pretty heavily on the number "two". Let's say I have a 3 sided die (a fat cylinder). Knowing which state it's in is one trit of information. How do I count that in terms of bits? Well, that's $\frac{\ln(3)}{\ln(2)}=\log_2(3) \approx 1.585, bit$ of information. Doesn't feel so different from length. I feel that physics is all about what you can measure, so one must reduce complex systems to that. – Jackson Walters May 19 '21 at 23:37

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SI units are specified for physical measurement, and are 'international' partly because of trade practicalities.

Bytes, and bits, measure information; the "length of this question" might be a count of symbols, which would be an integer. Or, could be an information measure (bits) after compression with a language-aware compression algorithm. Both are meaningful length measures, and other than mathematical precision, do not have to respond to any authority (such as Systeme Internationale).

The "length of this question" could also be the pixel count of a 7x9 array for each of the characters in that text. As I sit here reading the question, that's the minimum info required to render it on a screen; the actual info for a color screen is higher by a factor of 24, 30, or 36 depending on the DVI system version.

SI offers no guidance on which of these measures is correct.

Whit3rd
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  • Right, it did need to be said that SI itself is a human construct, so really if the "international community" agreed that we need to measure everything in terms of pink elephants, I suppose we could but a different group of people may choose to use other constructs, yada. Compression-wise, yes there's always Kolmogorov complexity and measures like that. Ah true, I guess I'd call pixel count the CSS/JS level invariant. – Jackson Walters May 19 '21 at 23:41