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Physics for Mathematicians – Introduction

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Re: Physics for Mathematicians – Introduction

#11

Post by dang »

antognini wrote: post #8
Y_Y wrote: post #4

I've never gotten a satisfactory explanation of what sort of mathematical object a physical unit (meter, kilo, second etc) is. There are plenty of bones of contention between maths and physics, but this one bothers me the most. Anyone interested in coming at physics from a mathematics perspective should read Arnold's mechanics book.

Terence Tao wrote a nice blog post about this: https://terrytao.wordpress.com/2012/12/29/a-mathematical-for...

A couple of past discussions:

A mathematical formalization of dimensional analysis (2012) - https://news.ycombinator.com/item?id=37517118 - Sept 2023 (54 comments)

A mathematical formalisation of dimensional analysis - https://news.ycombinator.com/item?id=5018357 - Jan 2013 (19 comments)

Re: Physics for Mathematicians – Introduction

#13

Post by nyrikki »

random3 wrote: post #9
Y_Y wrote: post #4

I've never gotten a satisfactory explanation of what sort of mathematical object a physical unit (meter, kilo, second etc) is. There are plenty of bones of contention between maths and physics, but this one bothers me the most. Anyone interested in coming at physics from a mathematics perspective should read Arnold's mechanics book.

One concern is that measures, out of the box, have issues in 3+ dimensions. Concretely due to paradoxes such as Banach-Tarski, that arise from the Zermelo Fraenkel (ZF) + Axiom of Choice (AC) = ZFC axiomatic formulation for set theory. Since things need to conserve in pyhsics, one has to account for this issue and doing so is harder than it may seem as AC is part of the "fabric" of most mathematics which, at large, c…

IMHO that is the result of Gibbs style vectors and the cross product only being validated in R^3

Lie groups and geometric algebra remove a lot of problems.

It also applies to differential calculus and ML methods like back propagation and gradient decent.

Gibbs style vectors and the cross are convenient as they tend to match our visual intuitions.

But lots of the 'physics isn't real math' claims just don't understand how the algebra arises from the system.

Re: Physics for Mathematicians – Introduction

#15

Post by nomemory »

scionthefly wrote: post #12

Okay...I think this might be interesting. I've seen and read a lot of "math for dumb physicists" works, which as a physicist...yeah, I see their point. This could help me understand the math wizards a little better.

I was more math oriented during my studies, and I hated physics (couldn't openly admit that). I still don't get a lot of the physics I was taught, but I did juggle my way out of it using math, learning some formulas and getting a passing grade. Deep inside I admire physicist more, because for them the things that never clicked for me are natural.

Re: Physics for Mathematicians – Introduction

#16

Post by nyrikki »

Y_Y wrote: post #4

I've never gotten a satisfactory explanation of what sort of mathematical object a physical unit (meter, kilo, second etc) is. There are plenty of bones of contention between maths and physics, but this one bothers me the most. Anyone interested in coming at physics from a mathematics perspective should read Arnold's mechanics book.

A meter is a displacement vector with a basis vectorthe length of the path travelled by light in a vacuum during a time interval of 1/299,792,458 of a second.

In physics the length of the basis vector is set to 1 if possible which is called 'natural units'

But the SI system is the domain of Metrology, not physics.

Re: Physics for Mathematicians – Introduction

#17

Post by seydar »

> The presence of the negative signs in (1) may seem surprising at first, but this is due to the fact that (1) is describing the effect of a passive change of units rather than an active change of the object {x}.

This is where the limits of my brain were reached. Is there a translation of this into category theory terms? Is this where category theory could help formalize units in physics?

However, his paragraph after that is pretty interesting, which I read as sort of treating units as variables since you couldn't combine them, and he only has length, mass, and time for these examples. But then there's an exponent piece? Okay now I'm lost again.

Re: Physics for Mathematicians – Introduction

#18

Post by dawnofdusk »

seydar wrote: post #17

> The presence of the negative signs in (1) may seem surprising at first, but this is due to the fact that (1) is describing the effect of a passive change of units rather than an active change of the object {x}. This is where the limits of my brain were reached. Is there a translation of this into category theory terms? Is this where category theory could help formalize units in physics? However, his paragraph after…

>Is there a translation of this into category theory terms?

It's essentially the same as the relation between covariance and contravariance in category theory.

Re: Physics for Mathematicians – Introduction

#19

Post by mjburgess »

seydar wrote: post #17

> The presence of the negative signs in (1) may seem surprising at first, but this is due to the fact that (1) is describing the effect of a passive change of units rather than an active change of the object {x}. This is where the limits of my brain were reached. Is there a translation of this into category theory terms? Is this where category theory could help formalize units in physics? However, his paragraph after…

where on earth is this quote from?

Re: Physics for Mathematicians – Introduction

#20

Post by dawnofdusk »

Skimmed some of the articles, particularly those nearer to my field. Seems like a generally good set of informal notes.

Random comments:

>when the states evolve in time and the observables don’t we are using Liouville’s picture; when the observables evolve in time and the states don’t we are using Hamilton’s picture.

I have never heard this terminology, I have only heard Schrodinger's picture vs. Heisenberg's picture.

>This means that, very unlike on a Riemannian manifold, a symplectic manifold has no local geometry, so there’s no symplectic analogue of anything like curvature.

Perhaps the only enlightening comment I have ever heard about the tautological 1-form/symplectic approach to Hamiltonian mechanics.

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