[Y_Y]

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.

[antognini]

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

[dang]

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)

[random3]

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, chooses to ignore the problem.

[eigenket]

This is very very strongly not a concern for physics. Banach-Tarski and similar "paradoxes" require dividing things up into unmeasurable sets. These sets are pretty wild and aren't ever going to turn up when you're doing physics.

Speaking as a physicist, we don't care at all about stuff like Banach-Tarski, and there is essentially zeo expectation that stuff like ZF vs ZFC will have any impact on physics.

[nyrikki]

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.

[eigenket]

Could you explain what you think Gibbs style vectors have to do with Banach-Tarski or the axiom of choice?

As an aside, I'd like to emphasise that geometric algebra gives exactly the same physics outcomes as doing the maths with vectors or tensors or whatever else you like. The difference is essentially just notation. Some things look prettier.

[nyrikki]

Specific to building _intuitions_ for why the Banach-Tarski arises in ZF+AC.

GA gets rid of the external conventions for coordinate and chirality and also uses SU(2) which is simply connected vs SO(3) which is not. Rotors in GA can be used as elements of the algebra like any number avoiding the complexity of Euler angles, gimbal lock, etc....

GA's rotors are geometrically intuitive and can do rotations around an arbitrary axis, where quaternions are limited to an axis through the origin.

As Banach-Tarski is not physically realizable and because physics uses the computable reals, rationals and other aleph naught numbers it doesn't cause a problem there.

Lots of important work resulted _from_ the Banach-Tarski paradox but really it is just a cautionary tail about ZF+AC and on-measurable sets as far as physics goes.

What I was talking about is tools about building intuitions on why it arises.

Note that the maths aren't exactly the same, as an example Maxwells equations require four separate formula to express in Vector Calculus vs just one in GA. I don't think I fully comprehended the connection before learning GA.

This is also digging deep into the implications of your chosen groups and resulting algebra but as an example:

A Tensor can't represent a spinor but an even multivector can. A pure grade multivector can only completely represent antisymmetric tensors.

You can look into Dirac's belt trick as a physical example showing that SO(3) isn't simply connected but it arises in E(3) in that particular case too.

I wish this site had latex support, so I apologize for the above which is probably of little value in reality.

[eigenket]

I don't want this to come across as insulting, but this message sounds more like someone trying to sell me something than an objective scientist. You can just say no, there is no link (intuitive or not) between geometric algebra and Banach-Tarski.

To answer some of the other points

1. No physicist is particuarly confused by Euler angles, and gimbal lock is not a problem in physics.

2. I'm not sure I agree that physics even uses computable reals or rationals. I would say in reality we use fuzzy confidence intervals mostly and not exact numbers.

3. Maxwell's equations look different when written in geometric algebra style, you get one neat looking equation rather than the traditional 4, but its just a difference of notation, the same stuff is happening just written in a slightly different way.

4. A tensor can represent a spinor if you let it transform under the correct transformation rule. A basic spinor just looks like (https://en.wikipedia.org/wiki/Dirac_spinor) a complex vector which you let the Clifford algebra act on.

More generally everything I've seen from Geometric Algebra enthusiasts is just a weird way of doing fairly standard stuff in special cases of Clifford algebras in slightly weird old-fashioned notation. Pretty much everyone I've seen doing real work just does stuff in the Clifford algebra.

[scrubs]

Was there an answer in there somewhere?

[cycomanic]

Why is the measure not a satisfactory answer?

https://en.m.wikipedia.org/wiki/Measure_(mathematics)

[Koshkin]

Unfortunately, even though it is said to be a "generalization" of these things, mathematical measure theory has nothing to do with physical units of measure or dimensional analysis.

[elbear]

Why?

[Koshkin]

Because its focus is mostly on measures defined on "non-physical" sets, such as various functional spaces (with applications to integrating differential equations, for instance).

[nyrikki]

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.

[4ad]

Meters, seconds, joules, etc, are torsors.

[bsdpufferfish]

A unit is nothing more than a a relative comparison. A “unit meter” is the length of a stone in Europe.

[adrian_b]

The physical quantities are of 2 kinds, as already classified by Aristotle, discrete quantities and continuous quantities.

Examples of discrete quantities are the amount of substance and the electric charge.

The discrete quantities are just counted, so their values are integer numbers. They have a natural unit. Nevertheless, for those that are expressed in very large numbers it may be convenient to choose a conventional unit that is a big multiple of the natural unit, for instance the mole and the coulomb in the SI system of units.

All the continuous physical quantities are derived in some way from the measures of space and time, which is the reason for their continuity. For instance the electric charge is discrete, but the electric current is continuous, because it is the ratio between charge and time and time is continuous.

In order to measure a continuous physical quantity, a unit must be chosen. The unit may be chosen arbitrarily or it may be chosen in such a way as to eliminate universal constants from the formulae that express the relationships between physical quantities.

In either case, the value of a measurement is the result of a division operation between the measured value and the chosen unit, which is a real number, though it is normally approximated by a rational number.

In order to be able to define a division operation on the set of values of a physical quantity that has as a result a scalar, the minimum algebraic structure of that set of values is an Archimedean group.

That means that it must be possible to add and subtract and compare the values of the physical quantity and given two values it is always possible to add one of them with itself multiple times and eventually there will be a multiple greater than the second value (which will determine that the second value lies between two consecutive multiples of the first).

Based on the axioms of Archimedean groups it is possible to devise an algorithm that can multiply a value by a rational number and which can determine that a second value lies between two rational multiples that are as close as desired, producing by passing to the limit a real scalar. Thus any value can be divided by another value chosen as unit.

In practice, all the continuous physical quantities have richer algebraic structures, they are vector spaces over the real numbers, so the division of two collinear vectors is the scalar that multiplies one to give the other.

Nevertheless, the fact that the continuous physical quantities form vector spaces over the real numbers can be demonstrated based only on the supposition that they are Archimedean groups.

So the units of continuous physical quantities are just arbitrarily chosen values of those physical quantities, which are normally vector spaces with one dimension or with more dimensions, while the measured values are just rational approximations of the scalars obtained by division.

This division process is very obvious in the structure of the analog-digital converters used to measure voltages. These ADCs have two inputs, the voltage to be measured and the reference voltage, which is the arbitrarily chosen unit. The ADCs produce a rational number that is the approximate result of the division of the measured voltage by the reference voltage. If the reference voltage is not equal to the conventional unit, i.e. 1 V, the measurement result will be converted by multiplying with an appropriate conversion factor. The division operation can be done in the ADC for example by successive approximation, i.e. by binary search of the two multiples of a fraction of the reference value between which the measured value lies. The fraction of the reference voltage may be generated by a resistive or capacitive divider, while its multiples can be generated by a multiplying DAC.