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Mathematicians get annoyed at how physicists take beautiful formulas and clutter them up with 'useless' constants like
𝑐 - the speed of light
ℏ - Planck's constant
𝑘 - Boltzmann's constant
𝐺 - the gravitational constant
making it harder to see the essence of things. Mathematicians prefer units where all these constants are set equal to 1.
I used to be like that too - but right now I'm doing a project where I 𝑛𝑒𝑒𝑑 these constants to see the essence of things!
(Of course it's good to keep these constants around so you can use dimensional analysis to avoid mistakes: this is what computer scientists call a 'type discipline'. That's important, but it's NOT what I'm talking about now.)
When you're studying just one physical theory at a time, you can set dimensionful constants equal to 1 to simplify things. But often we like to study a whole 𝑓𝑎𝑚𝑖𝑙𝑦 of physical theories at once - a family where those constants take different values! We can't set them to 1 if we're interested in what happens when they approach 0. For example:
As 1/𝑐 → 0, special relativity reduces to Newtonian physics.
As ℏ → 0, quantum mechanics reduces to classical mechanics.
As 𝑘 → 0, statistical mechanics reduces to classical mechanics.
As 𝐺 → 0, general relativity reduces to special relativity.
And this is just the beginning of the story: various collections of constants can approach 0 at different rates, and so on.
When we do this, we're studying what mathematicians would call a 'moduli space' of theories - or even better, a 'moduli stack'. We may want to do 'deformation theory', where we expand answers in powers of some constant. And so on.
So don't scorn those constants!
@johncarlosbaez Hey, It's not our fault that humanity got stuck using arbitrary units of measurement that needed these insane constants to balance out the equation.
@AlexCorby - so you didn't read the post, huh?
@johncarlosbaez No, I did, and I very much appreciated it. I was just remarking that I always found it funny that we invented these constants to mitigate between otherwise unrelated units and then had to tack on bizzare units for the constants themselves to make the equations work (G in terms of N*m^2/kg^2 for example). Didn't mean to come off snarky.
@AlexCorby - okay, gotcha. Here's an example I love: it seems the pyramid-building Egyptians measured horizontal and vertical distances with different units! I don't know if there was ever an Egyptian Einstein who unified vertical and horizontal.
@johncarlosbaez @AlexCorby Okay, that's impressive if they built the pyramids using vw and vh to build the pyramids using CSS.
Why not have it both ways?
Replace the constants with 1, but also look at what happens as you vary from between 1 and 0?
(And if replacing some constants with a unit value forces other constants into non-unit values, that's interesting in itself, I'd say.)
@TruthSandwich - you can't continuously go from 1 to 0 without going through other numbers. So when we study the 'deformation theory' of physical theories we allow the relevant constants to take arbitrary values, and study how physics changes as we vary them.
(Yes, 'constants' is no longer the right name when we vary them, but people still call them that.)
In case I wasn't clear, I definitely did mean looking at all values between 1 and 0, and perhaps even higher values.
The point of using a unit value isn't to limit our options, just to establish 1 as the baseline, for clarity.
So, for example, when we see 3.313035075e-34 J⋅s, that's not quite as obviously half of Planck's constant as 0.5 J⋅s would be.
@TruthSandwich @johncarlosbaez
For some reason your post reminds me of the old quip: "1 + 1 = 3, for large values of 1." 
Definitely enjoying this discussion!
@johncarlosbaez @AlexCorby
I present to you: the survey foot!
@Hyolobrika - as you can see, we sorted it out and are happy again. 
I recall as an undergraduate a particle physics lecturer who insisted on using "natural units" - c = h-bar = e = G = (any other natural constant you care to name) = 1
The course was a nightmare!
I've also encountered "Mott's Law" (attributed to Neville Mott, though I can't find a source):
"Any definite integral which you cannot evaluate for whatever reason is equal to 1"
This actually works in most cases, at least to an order of magnitude which is often sufficient...
@Henrysbridge - being a mathematician I would have found it much easier to handle a physics course using natural units, and my book on gauge fields and gravity uses c = ℏ = G = e = 1. But when I recently wrote my little book on entropy I wanted to make contact with experiment, so I kept k around. And now I'm doing a project where I really need k and ℏ visible. So I think it's all a matter of what you're doing.
It makes tons of sense, though, for intro physics courses to keep all these constants visible!
The big problem this lecturer had was that not only were all constants = 1, but they were consequently ignored in equations and calculations - as you note, it made it impossible to keep track of what was physically going on in the formal mathematics!
I recall some politicians wanted to define the value of pi to be exactly 3.
Personally, I like that we calculate natural phenomena using symbols, though I now hesitate to call them "constants" because the values are often refined as our understanding of them improves.
There's a joke in here somewhere. In fact, there's a whole series of them: https://users.cs.northwestern.edu/~riesbeck/mathphyseng.html
@Henrysbridge
That quote, whether apocryphal or not, is pure gold. Have a boost, sir!
@johncarlosbaez
@Henrysbridge @johncarlosbaez My professor for Quantum Mechanics was a student of Dirac. He didn't lecture from the book and used Dirac's notation for everything. I didn't understand much of what he taught and ended up taking the course again from another professor who I'd had before for Thermo. Great class and I got a B the 2nd time.
Sometimes it's a good idea to be at a distance from genius to learn stuff. A notable exception - Richard Feynman.
@Henrysbridge @johncarlosbaez
My physics lectures at the Uni were both the most fascinating and frustrating things at the same time. Remain to be somewhere in top 10 in my life.
Fascinating because you can calculate *everything* with a few (seemingly) simple techniques. Frustrating because I had no idea how the lector was doing it. The stuff written on the blackboard had absolutely no correlation with anything I could find in the textbooks. Dropped constants, different units, "let's say it's 0"
I know what you mean. As I was more of an experimental mindset, I got on with the "we can forget about those terms" idea pretty easily - first order approximations all the way!
"Fascinating because you can calculate *everything* with a few (seemingly) simple techniques. Frustrating because I had no idea how the lector was doing it."
It's sort of like learning to do gymnastics by watching videos of
@johncarlosbaez JFC you made me look and then read and then learn something about maths and physics. well played sir, well play. you are the real deal as a gentleman and a scholar 
@blogdiva @johncarlosbaez Same! A fairly large shift in my understanding of the relationships between math and physics, as well as several different layers of existence, just took place. And I haven't even had any coffee yet!
@SallyStrange - that's great! It's so fun when reality changes and suddenly makes *more* sense, not less. But for me that usually requires coffee.
@blogdiva - thanks, that means a lot to me!
@johncarlosbaez I think you still miss the most wierd constant of all (that’s my personal opinion) \alpha = 1/137. It keeps appearing in many calculations related to electroweek and strong force interactions. As it was defined by Sommerfeld in your book it would be equal to 1…
@roberossi - the fine structure constant is a dimensionless constant, so it can't be set to 1 without affecting the physical predictions. My post was about dimensionful constants, which can be set to 1 by choosing appropriate units, without changing any physical predictions.
But I agree that α is weird! We would all love to know why it has the value it does, but almost everyone has given up trying to answer that.
@johncarlosbaez I think that’s the same for dimensionful constants. We don’t know why they have that value and why it is so precise… thinking also about gauge coupling constants, but indeed also c,h,k…
I think I read about someone who theorised about alternative universes were these constants were different. Apparently in many cases the physics would break when the values were changed even by small percentages.
@roberossi - fundamental dimensionful constants have the values they do solely because we choose the units we do. For example, the speed of light is a description of how we've chosen to define the meter and second. If the speed of light were twice as big, and nothing else changed, the universe would be the same... except that people in that other universe would be defining meters and/or seconds differently.
@johncarlosbaez @roberossi Pretty sure that if you ‘just’ change the speed of light, only a little, and nothing else, you’ve fundamentally changed chemistry. Or so my solid state prof told me. So you probably need to tweak some/all of the constants just right to just have some scale changes and nothing else. (At which point I wonder if it is fundamentally different anyway, or you just have one redundant constant somewhere.)
@Wlm - if you take the fundamental laws of physics and just change the speed of light, everything else automatically changes in exactly the right way so that there are no observable changes at all. This is pretty well known.
Perhaps that difference is that I'm a physicist so I start with the basic laws of physics - the Standard Model and General Relativity - and I see how to map solutions of a version of these equations where the speed of light equals x to solutions of a version where the speed of light equals y, as long as x, y > 0.
If you start with the laws of chemistry, you might change the speed of light but not notice all the places where the speed of light actually shows up in the fundamental equations of physics.
@johncarlosbaez @roberossi Thanks for the quick reply. I got a masters of physics many years ago and haven’t used it much since, so you’re highly probably more right than me.
I was thinking along the lines of “if you just mess with c and nothing else, you’ll actually get a different fine structure constant out of [hold on, quick wiki search] 4πε_0ħcα = e^2”. Which is probably naive in a way I’ll have to think about and/or sleep on 
.
@Wlm - Yes indeed, you can do things that way if you want. It's not the way I'd do it.
@johncarlosbaez I wasn’t referring to changing the definition of the meter or the second. I agree that is our definition related to what humans observe (the earth or the average hearth beat length). I was referring to changing in the absolute value of fundamental constants: i.e. mass of the electron or quarks, the mixing angle of quarks or neutrinos, coupling constants… these are all fundamental dimensionful constants that doesn’t accepts any other value. Still I see your point.
@roberossi - there are different ways to slice the pie, but the way many people in fundamental physics do it is this:
We use units where c, G, ℏ, k and the proton charge equal 1. This determines units called the Planck length, Planck time, Planck mass, and Planck temperature.
Then there are 26 fundamental dimensionless constants in the Standard Model and general relativity, which I've listed here:
https://math.ucr.edu/home/baez/constants.html
Note that in my list most these come from masses, but the corresponding *dimensionless* constants are these masses divided by the Planck mass.
It's the *dimensionless* mass ratios that we consider fundamental, since you can only measure a mass relative to another mass. Mixing angles and gauge field coupling constants are already dimensionless.
Of course you can play around with these 26 dimensionless numbers and get other collections of 26 dimensionless numbers that work just as well.
@roberossi @johncarlosbaez
You might be thinking of this argument for the anthropic principle?
@jsdodge @johncarlosbaez it’s behind a paywall, I’ll try Monday from the office where i should have access to be sure, but I think that was the paper I read!
Thanks a lot Steve
@johncarlosbaez
Clutter? I used to worry the opposite. Some of those equations hide complexity either with substitution, vectors, tensors, etc.
I seem to remember a metric fuck-ton of substitution being used to hide complexity from my physics classes.
Occasionally, I get confused in a calculation and, looking at an expression, ask: "What *is* this thing, anyway?"
Being able to put the units back temporarily to discover the type of object I'm looking at is useful. (Very much like types in computer languages, and for that matter, in some varieties of logic.)
Sure, taking the unit scales out makes stuff look more elegant, but being able to put them back in occasionally helps check that one hasn't made a mistake.
@johncarlosbaez
“I used to be like that too - but right now I'm doing a project where I 𝑛𝑒𝑒𝑑 these constants to see the essence of things!”
Mathematically speaking, you have expanded your limits in this project.
The process of calculating the realm of natural environments only makes me ask if our understanding of the forces at play is defined correctly. Should time and gravity be separated or do these invisible forces “flow” together?
I often find answers in nature when the math frustrates.
@johncarlosbaez
Have fun with your theory just leave me out of that box you seem to be constructing.
I’m an out the box type thinker….
@johncarlosbaez my question is what sort of scale do the graph axes have to get G = 1/c = hbar?
Thanks for the alt text reference to Bronshtein - they are all fascinating.
@johncarlosbaez That’s a beautiful graph/cube I hadn’t seen before, thanks.
There’s plenty physicists who take these constants as 1 too. And then there are cgs astrophysicists 
.
@Wlm - This cube is called Bronstein's cube. I'm glad you liked it! Here are some sources for it:
https://hsm.stackexchange.com/questions/14181/source-documents-for-bronsteins-cube-of-physics
@johncarlosbaez I’ve got to keep this in mind. I usually play in Natural units and set it all to 1, but sometimes I regret that. I can totally see wanting to connect it to more easily measurable things like spacetime curvature (1/Length^2)
@johncarlosbaez Don't forget to consider dimensionless constants such as fine-structure constants, and see where those fit into the picture!
@johncarlosbaez one is only one if the unit for one is one
@johncarlosbaez I find it useful to think of physical quantities like length, times, masses etc. as elements of a line (a one-dimensional vector space). Then a unit is a basis vector for the line. Equations with units become vector equations. So, for example, 1kg=1000g is a relation between the two vectors kg and g. Lines can be multiplied and divided, so c is a canonical element of L/T. You can put the units into equations with confidence and cancel them, e.g. km/m=1000.
@JohnBarrett - I agree! Lines are important!
Jim Dolan promulgated the idea of 'dimensional categories', which are symmetric monoidal k-linear categories where every object is a 'line object': an object with a tensor inverse, whose self-braiding is the identity. In applications to dimensional analysis, each object X is a 'dimension' (like length³ × time), and morphisms
f: I → X
are 'quantities' of dimension X.
A really great example is the category of complex line bundles over some variety. Here 'quantities' are global sections.
Todd Trimble explains the math better here:
All this is supposed to link up to what I mentioned about deformation theory. For example, special relativistic theories are isomorphic for different values of the speed of light 𝑐, except for 𝑐=∞ (Newtonian mechanics) and 𝑐=0 (which people now call Carrollian relativity). The quantity 𝑐 should be a partial section of a line bundle on ℂP¹ or at least ℝP¹. It becomes ill-defined when 𝑐=∞, but its inverse is defined there.
Anyway, sorry for talking your ear off, but what I say in my articles on Mathstodon is the intersection of what I want to say and what I think readers will follow, and you just massively expanded that intersection.