|
John Carlos Baez
@
johncarlosbaez
Riverside, CA
|
|
I do math, physics, network theory at U. C. Riverside and the Centre for Quantum Technologies in Singapore.
|
|
|
16.377
Tweetovi
|
187
Pratim
|
26.426
Osobe koje vas prate
|
| Tweetovi |
|
|
John Carlos Baez
@johncarlosbaez
|
6 h |
|
That'd be great. How old will you be then? 🙃
|
||
|
||
|
|
John Carlos Baez
@johncarlosbaez
|
6 h |
|
I'd never heard of "UMAP". This paper defines fuzzy simplicial sets as simplicial objects in the category of fuzzy sets. That's a completely obvious maneuver if one knows category theory, and a lot of theorems instantly hold. Category theory gives you power.
|
||
|
|
||
|
|
John Carlos Baez
@johncarlosbaez
|
10 h |
|
The sensitivity on initial conditions, and the impossibility of taking all relevant effects into account, means that all we can do is study probable scenarios... and maybe estimate their probabilities.
|
||
|
|
||
|
|
John Carlos Baez
@johncarlosbaez
|
10 h |
|
No, I've given up on fundamental physics. I think I had some good ideas, but not enough to add up to a theory. Other subjects are so much easier to make progress in! I'm more productive now.
|
||
|
|
||
|
|
John Carlos Baez
@johncarlosbaez
|
12 h |
|
I guess it's mainly a question of how much it interests you! Some historian of mathematics should have written about the evolution of the concept of function. Someday somebody will, that's for sure. If nobody has yet, maybe that person will be you. 🙃
|
||
|
|
||
|
|
John Carlos Baez
@johncarlosbaez
|
14 h |
|
Whoops. I always set c = 1 so I tend to screw up when sticking the c's back in.
|
||
|
|
||
|
|
John Carlos Baez
@johncarlosbaez
|
14 h |
|
I think Minkowski is great! I learned how great only when I started studying what he *really* worked on: number theory. He invented "the geometry of numbers":
en.wikipedia.org/wiki/Geometry_…
which is all about lattices, and secretly quadratic forms.
|
||
|
|
||
|
|
John Carlos Baez
@johncarlosbaez
|
14 h |
|
I agree. That's why I said Einstein's discovery "boils down to" this. Einstein's old math professor Minkowski did some of the crucial boiling here!
(Though I believe Minkowski still used "i t", imaginary time, which is a bit distracting - especially when we get to GR.)
|
||
|
|
||
|
|
John Carlos Baez
@johncarlosbaez
|
15 h |
|
There are different ways to try to mesh this with other mathematical structures, but quaternions turn out not to be very good for this. It's better to just learn about "Minkowski spacetime":
khanacademy.org/science/physic…
|
||
|
|
||
|
|
John Carlos Baez
@johncarlosbaez
|
15 h |
|
Einstein's big discovery - special relativity - boils down to this. Spacetime is 4-dimensional, and vector have this sort of dot product:
(t,x,y,z) ⋅ (t',x',y',z') = tt' - xx' - yy' - zz'
Everything else follows from this, basically. (1/n)
|
||
|
|
||
|
|
John Carlos Baez
@johncarlosbaez
|
15 h |
|
In the modern understanding, mass is the Lorentz-invariant "length squared" of the energy-momentum vector (E,p). That's what the formula
m^2 = E^2 - p^2
says, working now in units where c = 1. Namely:
m^2 = (E,p) · (E,p)
(2/n, n = 2)
|
||
|
|
||
|
|
John Carlos Baez
@johncarlosbaez
|
15 h |
|
In the old days people talked about "rest mass" m_0 and "relativistic mass" m_0/sqrt(1-v^2/c^2).
A better understanding treats mass as invariant under Lorentz transformations, defined by
m^2 = E^2 c^4 - p^2 c^2
(an equation which looks nicer in units with c = 1).
(1/n)
|
||
|
|
||
|
|
John Carlos Baez
@johncarlosbaez
|
16 h |
|
Good!
|
||
|
|
||
|
|
John Carlos Baez
@johncarlosbaez
|
16 h |
|
I was trying to say that all of physics, except for some very simple parts, fits into the field of nonlinear dynamics.
|
||
|
|
||
|
|
John Carlos Baez
@johncarlosbaez
|
16 h |
|
Okay. Almost any physical system is nonlinear. There are just a few where the dynamics is linear, like the harmonic oscillator.
|
||
|
|
||
|
|
John Carlos Baez
@johncarlosbaez
|
16 h |
|
I think Bourbaki should have called them "jections".
|
||
|
|
||
|
|
John Carlos Baez
@johncarlosbaez
|
16 h |
|
"Is special relativity a phenomenon of nonlinear dynamics?"
I can't answer that question because I don't know what it means, especially the "a phenomenon of" part.
|
||
|
|
||
|
|
John Carlos Baez
@johncarlosbaez
|
16 h |
|
They're called free, but you'll pay dearly in the end.
|
||
|
|
||
|
|
John Carlos Baez
@johncarlosbaez
|
16 h |
|
So you have to get inside their head... which is sometimes an unpleasant place to be. 🙃
|
||
|
|
||
|
|
John Carlos Baez
@johncarlosbaez
|
17 h |
|
|
||