From: nospam@de-ster.demon.nl
wrote:
> On Wednesday, July 19, 2017 at 12:13:12 AM UTC-5, rockbr...@gmail.com wrote:
> > It is not so much the question of what *additional* axioms are
> > needed, but which ones need to be removed. Physical quantities are
> > not numbers, so that not all the infrastructure of a field (or more
> > generally: of the number system) has physical meaning with them;
> > both rather only a substructure thereof.
>
> On subsequent thought it's suddenly clear what the nature of those
> restrictions are. It's something that's clear to any programmer.
> Physical quantities form a TYPED ALGEBRA.
Just a finite dimensional algebra.
Given a system of physical equations,
you can choose an arbitrary number
of of arbitrarily chosen physical quantities
as base elements, and build your algebra from there.
(provided you do it consistently, and that the system of equations
you start out with is consistent)
Systems of dimensions are agreed upon conventions.
There is no way of singling out one system of dimensions
as -the right one-, (and consequently all others wrong)
except by burning heretics at the stake.
You can't objectify a dimension by measuring it.
It must be decided upon by majority vote.
> The system of types form an Abelian group with the identity 1
> standing for the type of dimensionless quantities
No need to drag in group theory,
it is just a finite dimensional algebra.
> A type judgement e: T means quantity e has type T, which in dimensional
> analysis means [e] = T.
>
> Addition and subtraction are subject to type-restriction: e + f and
> e - f are only defined if e:T and f:T, in which case (e +/- f): T.
>
> For comparison (if applicable): the same restriction applies, e and
> f can only be compared if they have the same type: e < f and e = f
> are only defined if e: T and f: T.
>
> The 0 is a polymorphic constant (0: T for all types T). Alternatively
> one may have a zero 0_T for each type.
>
> Dimensionless numbers all have type 1.
>
> Multiplication, division and reciprocals have the expected types:
> if e: T and f: U then ef: TU and e/f: T/U; and 1/e: 1/T (and 1/f:
> 1/U).
Unfortunately physical quantities
do not -have- inherent units or dimensions.
Quantities which have different dimensions in one system
may have the same dimension in another system.
(and the other way round)
[snip unnecessary complications]
> For this geometry, the notion of c being used as a "constant" for
> eliminating dimensions is completely undercut (and this also shows
> by way of example why it's wrong for the literature to Sapir-Worf
> away such physically interesting cases like this by its wrong
> convention of taking "c = 1"). One does not Sapir-Worf away physical
> postulates. That leads to dogma.
c being 1 (or some other number) is metrologicaly inevitable.
For c being anything else -in a physically meaningful way-
you must be able to measure it.
This implies having length and time standards
that can be realised to a greater accuracy and reproducibility
than the c that is to be measured with them.
Such is not the case, and it will not be the case
for the forseeable future, so giving c a defined value is inevitable.
(this is physics, Sapir and Worf really have nothing to do with it)
So you had better get used to it.
And worse (for you) \hbar is set to obtain a defined value too,
in the very near future, (one hopes 2018, to be implemented in 2019)
So after that hbar may be taken to be 1 too also on the practical level.
Our system of units has evolved into a time only system. [1]
Anyway, we agree presumably that the conventional system of dimensions
for MKSA is a valid system of dimensions.
So it follows immediately that the contraction obtained
by equating [L] and [T] (which makes c dimensionless)
is a valid system of dimensions too,
Jan
[1] With c, hbar, Josephson and Von Klitzing having defined values
every physical quantity is measured in terms of time.
--- SoupGate-Win32 v1.05
* Origin: you cannot sedate... all the things you hate (1:229/2)
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