[varsigma (OP)]

I've been having a stoush with someone recently, about a simple enough thing which is, how physical units are defined.

It seems that, at first we don't bother defining distance, other than in terms of a unit thereof. That is, we just assume that distances are somehow axiomatic. This is clearly not the case when we define units of time; currently defined accurately in terms of a single, 'central' frequency, and the constant speed of light.

But physics doesn't stop and ask, "what really is a unit of distance", or "what is a unit of time", etc. Physics just assumes that units can be defined in terms of each other--speed, in terms of time and distance e.g.--and doesn't bother with the philosophy.

Distance "is what it is"; you measure some and just get on with it. In fact humans measure distances all the time, walking or driving is a 'measurement' of distance; when you move any part of your body it 'moves through' a distance and this constitutes a measurement (of distance). Accurate measurements aren't a rule as such, but are assumed to be a requirement if a theory is being supported by experiments.

Or, why don't we have definitions of time and distance, etc, in other more fundamental terms?

[varsigma (OP)]

I'll add that there are also definitions of more abstract kinds of 'distance'; one obvious one is the distance between two primes in number theory.

Another kind is the distance of a path through a graph in graph theory, or the distance between two permutations of a permutation group in group theory. An affine distance is encountered sometimes in lectures on black holes and the like, but what is an affine distance? In any manifold distance is given by a metric; the surface of our planet qualifies as a manifold because it can be given coordinates, because it has distances 'built in' between different locations. It's just a fact of nature that planets have a finite volume and a finite surface, so finite distances are just a kind of trivial subset.

But a topological distance can be nontrivial, right?

[evan_au]

we just assume that distances are somehow axiomatic

A lot of people (most people?) are confused when they first run into concepts from Special Relativity whereby observers in different frames of reference will will come up with a different distance and time between the same same two events.

[Origin]

Or, why don't we have definitions of time and distance, etc, in other more fundamental terms?

I'm not sure what your asking.  Fundamentally, distance is "from here to there" and time is "from now to then".  We developed tools (rulers and clocks) to accurately measure these dimensions.  So what do you think is missing?

[Deecart]

Or, why don't we have definitions of time and distance, etc, in other more fundamental terms?

I think that we are just using the ancient greeks system when it comes to "measure" the "earth" things.
Geometry mean "the measurment of earth" (literaly).
When "we" invented science, so greek science, we initialy used the so called "philosophy".
Within the philosphy we had the "geometry" (this portion of philosphy ended up creating mathematic).
The principe was : I am a man, i try to investigate the relation (bigger or smaller and in some quantitativ way what is the relational mathematical proportion) between things "i" can measure ("i" is the one who measure, so "the man").

Therefore the fundamental terms are in some ways "anthropomorphic", so related to men, and if you want to be less human related, you can say "macroscopical" (this term is better accepted within modern science but it is the same).

Now, you can effectivly do some "other" (dimensional) definition of space (i will not speak of time, this is too complicated since you cant understand the simple space problem).
The second dimension is "who" is measuring the distance.
The first, primitiv way to quantitize distance is "greek related", so macroscopical.
The second is to add some dimension, saying : "For who" is the distance related ? (for who is the second dimension).
The "distance", so the proportion between your unit and your measure depends on what you are.
If your own scale (what you are) is around plancks lower limit (that exists for macroscopic things), the result of the distance is not the same as the distance for some other being at other scale.
Distance = f(scale).


 .


 

[Origin]

The second dimension is "who" is measuring the distance.
The first, primitiv way to quantitize distance is "greek related", so macroscopical.
The second is to add some dimension, saying : "For who" is the distance related ? (for who is the second dimension).

I don't believe that 'who' is a dimension by any scientific definition.

[Deecart]

I don't believe that 'who' is a dimension by any scientific definition.

Thats just because you never heard of the scale realtivity of Laurent Nottale.
https://luth.obspm.fr/~luthier/nottale/ukrechel.htm

[Origin]

I don't believe that 'who' is a dimension by any scientific definition.

Thats just because you never heard of the scale realtivity of Laurent Nottale.
https://luth.obspm.fr/~luthier/nottale/ukrechel.htm

Even in the lunatic fringe, 'who' is not a dimension.

[Deecart]

Even in the lunatic fringe, 'who' is not a dimension.

No, "who" is the scale.

If you consider the length of ths some coast you can have :
1. The classical length (dimension 1).
2. The fractal length (dimension higher).

So the length is dependant of the scale resolution, length = f(scale).
Typicaly the scale of man is 1 m (the "who").

[Bored chemist]

It's sort-of true that if two people walked round the coastline of the UK counting their paces and then multiplied the number of paces by the length of their stride they would disagree on the length of the perimeter (and an ant would have a much longer trip).

But that's not a "who" thing, it's "a how long is their pace?" thing- that's a distance, for which the unit is the metre.

[alancalverd]

counting their paces and then multiplied the number of paces by the length of their stride they would disagree on the length of the perimeter

Surely the disagreement would be of the order of (the uncertainly of stride length multiplied by the number of paces)  plus (one average stride multiplied by the uncertainty of counting) for each participant. You can take the root sum of squares for your best prior estimate of the limit of disagreement.

[alancalverd]

Typicaly the scale of man is 1 m (the "who").

That is where the French revolutionaries got it wrong. The Romans had sensible natural scales, that still persist in the last backward country - the USA.

Thumb - inch
Foot - foot (12 inches)
Arm - yard (36 inches thumb to nose)
Adult humans are mostly between 5 and 6 feet tall (1.5 - 1.8m)
Mile - milia passuum - 1000 paces (a Roman pace being together, left, right, together - like a proper dance script) - 1760 yards for the average soldier, who covered 4 - 6 miles per hour on a road (the Romans made the roads, and invented standard shoe sizes to keep the army marching efficiently on them)
Pound - 2 day's bread ration
Gallon - 10 pounds (2 days ration) of water

though the Yanks have screwed up the system by having a US pint equal a pound, and keeping 8 pints to the US gallon.

Defining  the meter as a fraction of the earth's polar circumference  through Paris was all very well politically, but not directly measurable. Navigators generally use the sexagesimal system  with a nautical mile equalling one minute of latitude (directly measurable) and being adequately close to 1000 fathoms (hand-to-hand span when pulling up your sounding weight).
   
 

[Bored chemist]

counting their paces and then multiplied the number of paces by the length of their stride they would disagree on the length of the perimeter

Surely the disagreement would be of the order of (the uncertainly of stride length multiplied by the number of paces)  plus (one average stride multiplied by the uncertainty of counting) for each participant. You can take the root sum of squares for your best prior estimate of the limit of disagreement.

Are you deliberately missing the point?
It's not about the inaccuracy involved; it's about teh effect of scale when measuring a fractal object.

[alancalverd]

So it's about the interpretation of "round the coastline". The continuous smooth ovoid that just encloses Great Britain is almost certainly shorter than the high tide contour.

[varsigma (OP)]

Ok. So it looks like the uncertainty problem has come up. I suppose it's a rule of thumb that any measurement is uncertain.

And I've realised that the problem of 'fundamental' units is more or less resolved (depending on I also guess a realist or positivist stance) by the constant speed of light.  So the metre and second are defined in terms of this constant plus a constant frequency of light. Nonetheless the measurement problem doesn't go away--in wave mechanics there is always a classical uncertainty when measuring frequencies or counting wavefronts. (It's in Modern Physics by Kenneth Krane too)

So if, axiomatically, the speed of light is constant in  a vacuum, then there is a finite number of wavefronts in a unit distance, passing a fixed point, in a unit time. Uncertainty has to apply to this 'fixed' point, though.

[varsigma (OP)]

Just checking I'm on the same page as most people, when I try to define physical stuff like distance.

Or rather, why I assume distances exist and that measuring them isn't a problem with any special difficulty. Distance, area and volume are such a part of our experience that we don't ask, usually, about why distances, or why spaces, exist.
Beyond measurement and theory, there is philosophy. I guess the philosophy of distance isn't interesting.
I remember though, an interesting question about time in an electronics lecture; why is time a real number (of seconds) in formulas and equations? The answer given was pretty much "it just is".

[varsigma (OP)]

My best attempt at explaining what distance is, is that it turns out to be a kind of information.
Or at least it does when we introduce a unit of distance and use it to measure another distance.

So it's like an alphabet with one character if you choose units that evenly divide a physical distance--the information per se is then literally the length of a string. Is information physically real or just conceptual? Can I really think of a real distance in metres as having entropy?

[evan_au]

in wave mechanics there is always a classical uncertainty when measuring frequencies or counting wavefronts

As I recall, one earlier definition of the meter specified:

The metre is the length equal to 1,650,763.73 wavelengths in vacuum of the radiation corresponding to the transition between the levels 2p10 and 5d5 of the krypton 86 atom.

https://en.wikipedia.org/wiki/History_of_the_metre#Krypton_standard

So it's not like they are limited to measuring an integer number of wavelengths (here they measured 2 decimal places of a wavelength)...

There are fundamental limits to measurement due to Heisenberg uncertainty, but you can usually overcome these limits by taking the measurement over a very long time period, which allows reduced uncertainty in position or momentum.

You have to be very patient when making fundamental physical measurements...

[alancalverd]

You have to be very patient when making fundamental physical measurements...

The UK National Physical Laboratory used to budget 20 person-years per decimal point. Turned out to be a remarkably good estimate as my team spent almost exactly that amount of time reducing the uncertainty in radiation dose measurement from 1% to 0.1%.

Pedant mode: Heisenberg described the inherent indeterminacy of a phenomenon, not the uncertainty of its measurement. Lots of people use the mistranslation because many textbooks use a hypothertical  "reflected photon" measurement to derive the equation.

[evan_au]

Pedant mode: Heisenberg described the inherent indeterminacy of a phenomenon, not the uncertainty of its measurement.

It is possible to weight the odds in Heisenberg indeterminacy, by using squeezed quantum states. You reduce the uncertainty in one dimension (which you care about), and accept that it will be less certain in another dimension (that you don't care so much about).
- So if you were measuring a distance accurately (eg the length of the standard meter, or the length of a gravitational wave detector), you may care more about uncertainty in distance than you care about uncertainty in photon momentum.
- Both LIGO and VIRGO use squeezed quantum states to improve sensitivity in detection of gravitational wave events.
- They both use real reflected photons to make the measurement
See: https://en.wikipedia.org/wiki/Squeezed_states_of_light