Axioms [LWN.net]

Axioms

Posted Feb 23, 2021 0:03 UTC (Tue) by Wol (subscriber, #4433)
In reply to: Axioms by mathstuf
Parent article: An introduction to lockless algorithms

Well, as far as light is concerned, the speed of light ... ISN'T. No time passes whatsoever between a photon leaving wherever it was created, and being absorbed by whatever it hits ...

Oh, and I believe it IS possible to calculate the speed of light based on pure logic ...

You'll have to look it up for yourself, but if you take a bunch of mathematical constants like e, pi, ln and so on, I think they somehow dictate a set of possible values for c, g, and other physical constants. It's too long ago, but I'm sure I've seen the physical constants defined in terms of the mathematical constants. And that's why astronomers are so unhappy with the cosmological constant. Because they can't define it in terms of anything else, it shouldn't exist ...

Cheers,
Wol

to post comments

Axioms

Posted Feb 23, 2021 0:38 UTC (Tue) by mathstuf (subscriber, #69389) [Link] (26 responses)

Eh, that sounds like numerology to me. Which is "more fundamental", e or pi? Is the Feigenbaum constant important? Gamma? The golden ratio? Silver ratio?

It'd also lay a blow to various theories of multiple universes which differ in the physical constants. Also, what is a "fundamental constant"? Is the radius of a proton a constant or a derived value? The size of a helium-4 nucleus? Ratio between the lepton sizes? Even physics is a *bit* arbitrary here.

Axioms

Posted Feb 23, 2021 1:19 UTC (Tue) by Wol (subscriber, #4433) [Link] (25 responses)

> Eh, that sounds like numerology to me. Which is "more fundamental", e or pi? Is the Feigenbaum constant important? Gamma? The golden ratio? Silver ratio?

Who said anything about fundamental? Both e and pi are mathematical constants, as is the golden ratio. They're all of equal importance.

Things like c, g, permittivity, that stuff, are all physical constants, which should be explainable in terms of maths.
Cheers,
Wol

Axioms

Posted Feb 23, 2021 13:10 UTC (Tue) by mathstuf (subscriber, #69389) [Link] (7 responses)

> Things like c, g, permittivity, that stuff, are all physical constants, which should be explainable in terms of maths.

But…why? What is this assumption based upon? Is it a bit of Platonic idealism?

Axioms

Posted Feb 24, 2021 0:27 UTC (Wed) by Wol (subscriber, #4433) [Link] (6 responses)

I don't know why, I just remember coming across it somewhere that most of the physical constants could be defined in terms of other things, and the reason they had the values they did was because the maths didn't make sense otherwise.

Cheers,
Wol

Axioms

Posted Feb 24, 2021 21:58 UTC (Wed) by mathstuf (subscriber, #69389) [Link] (5 responses)

Sure, but (as stated elsewhere) your numerical values are limited to our arbitrarily defined units. A mole is a number of atoms of a substance in a number of grams equal to its molecular weight. A meter is basically an arbitrary nice length, Seconds are based on a breakdown of the rotational speed of Earth. Sure, we've rebased them onto more stable footing now, but the counts of those things into the units are "arbitrary" based on what we had before.

If we wanted, we could say that the speed of light is pi/e Wol_lengths/Wol_time intervals. Doesn't mean that those units are any more useful than our numerical value in m/s.

Axioms

Posted Feb 25, 2021 7:48 UTC (Thu) by jem (subscriber, #24231) [Link] (4 responses)

One fun fact about the base unit for mass, kg, is that it is derived (indirectly) from the size of the Earth. On a bigger Earth, we would have a different "kilogram", with a bigger mass. (And I'm not talking about the gravitational force, which would increase even more, because of the bigger mass of both a kilogram and the Earth.)

Axioms

Posted Mar 16, 2021 21:09 UTC (Tue) by nix (subscriber, #2304) [Link] (3 responses)

Not any more it isn't. In 2019 it was redefined in terms of the Planck constant -- though, of course, its *actual value* comes down to the original French Revolutionary definition: but even that was the mass of a cubic centimetre of water at the melting point of ice. This, of course, depends on atmospheric pressure (since they didn't use the triple point), and thus indirectly on the mass of the Earth and a bunch of other contingent values -- but the current definition depends on no such things.

Axioms

Posted Mar 17, 2021 10:37 UTC (Wed) by jem (subscriber, #24231) [Link]

>Not any more it isn't. In 2019 it was redefined in terms of the Planck constant

Well, I said derived, not defined.

Axioms

Posted Mar 18, 2021 10:14 UTC (Thu) by mgedmin (guest, #34497) [Link] (1 responses)

> mass of a cubic centimetre of water

a cubic *deci*meter (which is also one liter) of water.

Axioms

Posted Mar 20, 2021 1:24 UTC (Sat) by nix (subscriber, #2304) [Link]

I changed that repeatedly before posting... and picked the wrong one, despite the metric system being the only system I actually know. Sigh.

(And obviously you're right, unless water has suddenly become much, much denser than lead :) )

Axioms

Posted Feb 23, 2021 15:24 UTC (Tue) by rschroev (subscriber, #4164) [Link] (16 responses)

π² m/s² is an approximation for g, but only because of our definitions for the meter and the second, and because we happen to live on Earth. I really don't see why there should be any connection between mathematical constants and the values of constants that govern physical phenomena. Mathematics is purer than that: it can with equal ease describe a universe with completely different constants or even different physical laws.

Axioms

Posted Feb 24, 2021 0:23 UTC (Wed) by Wol (subscriber, #4433) [Link] (1 responses)

Are you confusing g and G, or am I?

I think the equation is G=g.m1.m2/d^2

G is 9.8m/s^2, while g is the gravitational constant, which is believed to be the same everywhere in the universe.

Cheers,
Wol

Axioms

Posted Feb 24, 2021 0:47 UTC (Wed) by rschroev (subscriber, #4164) [Link]

G is the gravitational constant (or indeed the universal gravitational constant), approximately 6.674×10^−11 m³/(kg⋅s²).
g is the gravity of earth, approximately 9.81 m/s².
The equation is F = G⋅m1⋅m2/r². In case of Earth, that's F = G⋅m⋅m_earth/r_earth², with g = G⋅m_earth/r_earth². So in that case we get F = m⋅g.

Axioms

Posted Feb 24, 2021 0:31 UTC (Wed) by Wol (subscriber, #4433) [Link] (11 responses)

> Mathematics is purer than that: it can with equal ease describe a universe with completely different constants or even different physical laws.

It can *de*scribe it, yes. But I think you'll find it also *pre*scribes it and says "completely different constants and laws doesn't make logical sense".

Cheers,
Wol

Axioms

Posted Feb 24, 2021 1:01 UTC (Wed) by rschroev (subscriber, #4164) [Link] (10 responses)

I think you're mistaken in this, but I would be happy to be proven wrong because that would lead to very interesting insights in physics and the relationship between physics and mathematics.

BTW there's a video of Richard Feyman's lecture about that relationship here: https://youtu.be/obCjODeoLVw
And some comments about it: https://medium.com/cantors-paradise/richard-feynman-on-th...

Interesting stuff, if you're interested in that kind of stuff.

Axioms

Posted Feb 24, 2021 18:28 UTC (Wed) by Wol (subscriber, #4433) [Link] (9 responses)

Whoops. I didn't mean to say there aren't other solutions for the equations. Just that the possible universes are seriously constrained by the maths.

Cheers,
Wol

Axioms

Posted Feb 24, 2021 21:54 UTC (Wed) by mathstuf (subscriber, #69389) [Link] (3 responses)

That we had such a firm grasp on physics at that level to tell a "broken" from a "working" universe given a set of "knob settings" is news to me. That we even know all of the relevant knobs is news even. I thought we were still at the "we have a working universe on our hands…what makes it tick" stage about the fundamentals. Do you have any journal papers about this you could cite?

Axioms

Posted Feb 24, 2021 23:22 UTC (Wed) by Wol (subscriber, #4433) [Link] (2 responses)

How many dimensions does the Universe have? We know it's not three, Einstein proved that.

We think it may be four - that's what Einstein thought but that has a whole bunch of problems. If it *is* four, I believe that means string theory is correct as it's the explanation for relativistic singularities.

Or it could be ten or eleven. Anything betwen five and nine inclusive just doesn't work because we get an explosion of infinities - infinity itself isn't a problem, but there are different sorts of infinity and for reality to work they need to cancel out. For those dimensions they don't. (These universes, if I remember correctly, define mass as the fifth dimension ...)

(That's why I was moaning about computers crashing when you divide by zero. If you declare zero and infinity as non-numbers for which arithmetic doesn't work, you're in trouble. If you say "to make arithmetic work, they swap places on division", then you can do this sort of maths and come up with something that makes sense.)

At the end of the day, we have loads of maths that describes what we see. And that *constrains* what is a plausible universe. We have a local maximum or minimum, don't know which. By adjusting some values, we can force others to impossible values. Plausible universes must have all these values at maximum or minimum, not off the scale or impossible or at some non-equilibrium value.

Cheers,
Wol

Axioms

Posted Feb 24, 2021 23:36 UTC (Wed) by Cyberax (✭ supporter ✭, #52523) [Link]

> Or it could be ten or eleven. Anything betwen five and nine inclusive just doesn't work because we get an explosion of infinities - infinity itself isn't a problem, but there are different sorts of infinity and for reality to work they need to cancel out. For those dimensions they don't. (These universes, if I remember correctly, define mass as the fifth dimension ...)
You can construct a universe with multiple time axes, with many more dimensions, and so on. The math will be internally consistent.

Axioms

Posted Feb 25, 2021 0:05 UTC (Thu) by nybble41 (subscriber, #55106) [Link]

> That's why I was moaning about computers crashing when you divide by zero.

To say that division by zero is undefined is mathematically correct and not just an arbitrary computer limitation. There is no proper answer to the question "What number, when multiplied by zero, gives the non-zero product X?", at least not in any system that would uphold basic idioms such as the product of a number and its reciprocal being equal to one. "Infinity" times zero is not equal to any particular finite number X, so that isn't a solution. Depending on the particular forms of the equations which gave rise to the infinity and the zero (or infinitesimal) the product could be another infinity or any real number; it depends on how you phrase the question. $\lim_{x \to 0+} x ln \frac{1}{x} = 0$, but $\lim_{x \to 0+} x \frac{1}{x} = 1$. In both cases you're multiplying an infinitesimal ($\lim_{x \to 0+} x$) by an infinity ($\lim{x \to 0+} ln \frac{1}{x}$ or $\lim{x \to 0+} \frac{1}{x}$, respectively) but the specific form of the equation changes the result.

Axioms

Posted Feb 24, 2021 22:24 UTC (Wed) by Cyberax (✭ supporter ✭, #52523) [Link] (4 responses)

Not really. Math is just a language, it basically restricts nothing.

If you want to read something mind-blowing, try the "Clockwork Rocket" series by Greg Egan. It's accompanied by a thesis-sized exploration of its (fictional) physics: http://www.gregegan.net/ORTHOGONAL/ORTHOGONAL.html

Axioms

Posted Feb 24, 2021 23:26 UTC (Wed) by Wol (subscriber, #4433) [Link] (3 responses)

Maths is just a language, true. But if we try to define a - six-dimensional universe, say - the maths just doesn't add up. So yes the maths does restrict our universes - it says six dimensions just won't work.

Cheers,
Wol

Axioms

Posted Feb 24, 2021 23:34 UTC (Wed) by Cyberax (✭ supporter ✭, #52523) [Link]

> Maths is just a language, true. But if we try to define a - six-dimensional universe, say - the maths just doesn't add up. So yes the maths does restrict our universes - it says six dimensions just won't work.
Who told you that? A six-dimensional classic (Newtonian) universe works just fine. Sure, you won't have stable orbits but apart from that it's OK.

You can also construct a quantum field theory for such a universe, it also would work just fine.

Axioms

Posted Feb 25, 2021 15:16 UTC (Thu) by mathstuf (subscriber, #69389) [Link]

I think you're confusing the physics of our universe versus those of any possible universe. Sure, *ours* might not make sense with 5-9 dimensions, but that makes no conclusion about *any possible universe* having such a number of dimensions. Maybe you're just being imprecise with your language in places?

Axioms

Posted Mar 8, 2021 14:59 UTC (Mon) by LtWorf (subscriber, #124958) [Link]

I think you are very confused about the dimensions thing in physics.

Physics tries to model measurements.

So for example you measure a planet that is orbiting, make an equation, and see if tomorrow the equation and the position are the same (within a certain range of precision).

Before Galileo saw that Jupiter had satellites, they had perfectly fine equations that predicted where everything would be in the sky. The problem arose because new data could not fit the model.

You can absolutely model an orbit of a planet using 3 dimensions, or you can model it in an higher space with an equation of a lower degree. Both work. We can't really know which is "exact" if both work.

You can keep adding dimensions and make equations that work, but we don't really know what the "truth" is.

Axioms

Posted Feb 24, 2021 1:24 UTC (Wed) by SiB (subscriber, #4048) [Link] (1 responses)

The earliest definition of a meter is the length of a seconds pendulum. By that definition, g=π² m/s² exactly.

Axioms

Posted Feb 24, 2021 8:53 UTC (Wed) by mpr22 (subscriber, #60784) [Link]

Jean Picard's proposed toise universelle was twice the length of a seconds pendulum. Unfortunately, if you set up a seconds pendulum in (say) Cayenne, French Guiana, you would find that it was 0.3% longer than one set up in the Paris Observatory, because the acceleration due to gravity at sea level varies with a number of factors including latitude.

The first formally adopted definition for the metre itself, proposed in 1791 by the French Academy of Sciences and adopted in 1793 by the National Assembly, was one ten-millionth of the distance from the North Pole to the Equator along the Paris meridian.

Axioms

Posted Feb 23, 2021 1:06 UTC (Tue) by mpr22 (subscriber, #60784) [Link] (3 responses)

It is not possible to calculate the speed of light by pure logic, because the numerical value of the speed of light is a consequence of your units of measurement.

There is one particular physical constant, µ_0 (the permeability of free space) which was historically defined (as a consequence of the relationship between electrical current and induced magnetic fields) as exactly 4π × 10^-7 H/m ... but that value is no longer strictly valid (although it's close enough for the vast majority of practical purposes), because the ampere has been redefined in terms of the elementary charge and the second.

Axioms

Posted Feb 24, 2021 1:00 UTC (Wed) by SiB (subscriber, #4048) [Link] (2 responses)

Physical constants that carry units do not carry any physics. The universe does not care about the value of the speed of light.

Some years ago in our coffee room at the institute there was a discussion about the possibility that the fine-structure constant α may not have been constant in the past after all. The question was raised, which of the constituents of α=e²/(4πε₀ħc) may be the reason. Well, the answer is e, if at all. α is a constant without units. It is pure physics. It defines the strength of electromagnetic interactions. The universe cares a lot about the value of the fine-structure constant α. In our experimental units it is the charge of the electron e that represents that strength, but in the end, it also just defines units of measure.

Theoretical physics defines the units differently than experimental physics. They mostly use c=4πε₀=ħ=k=1. Everything is measured in eV or eV¯¹.

From a theoretical point I am a bit unhappy about the new definition of the SI units. They fixed the value of e, although it indirectly represents a quantity the universe cares about. The cost is that µ₀ and ε₀ are now _not_ fixed by definition in the SI, as they were before, like Wol said.

Ideally, all units were defined by fixing the value of some fundamental constant. We are now almost there. The second is still defined by an artifact. At least it is an atom, so it can be reproduced everywhere in the universe. To fix that we need to fix the gravitational constant. But metrology defines units by what they can measure with the best precision. The gravitational constant does not apply, it has to catch up at least eight orders of magnitude in precision.

Axioms

Posted Feb 24, 2021 1:16 UTC (Wed) by SiB (subscriber, #4048) [Link]

s/Wol/mpr22/

Axioms

Posted Feb 25, 2021 3:46 UTC (Thu) by NYKevin (subscriber, #129325) [Link]

> Physical constants that carry units do not carry any physics. The universe does not care about the value of the speed of light.

The universe may not care about the speed of light, but the stuff in the universe surely does. To borrow from your "theoretical physics normalizes everything to 1" example, the mass of the proton is much, much (numerically) smaller than the charge of the proton, and surely there's some physical significance of that? Or, alternatively, we can say that there's nothing wrong with the proton, and instead it's gravity which is too weak, but that's just reframing the same question in different units. I'm not sure you can ask that question without using units at all.

Ultimately, I suppose the "proper" framing of this question is some nonsensically complicated question about why the relevant quantum fields happen to interfere in exactly the way that they do to give the proton the properties that it has. But you can't even ask that question until you know what gravity is, at a quantum level, and to the best of my knowledge, nobody does.