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.   
      
   The system of types form an Abelian group with the identity 1   
   standing for the type of dimensionless quantities   
      
   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).   
      
   Additional structure is required to do calculus. The system of types   
   becomes localized at each space-time point: a copy for each point.   
   One then needs a kind of "connection" to be able to compare quantities   
   of a given type between different points. This leads to the potential   
   of the comparison, itself being path dependent.   
      
   In such a case, hypotheses phrased like "c is constant" would be   
   utterly devoid of any meaning. Instead, one would need to replace   
   it with an hypothesis that makes no reference to constancy or   
   variability (e.g. that the chronogeometry representing the space-time   
   manifold is locally Lorentzian).   
      
   I'll have more to say, time permitting, on the issue of the   
   electromagnetic field and its units -- particularly in the context   
   of a metric like that given by the time element   
   ds^2 = dt^2 - K t (dx^2 + dy^2 + dz^2), where K > 0.   
      
   This geometry is locally Lorentzian ONLY for t > 0. At t = 0, there   
   is a so-called "null surface" (i.e. a 3D slice of Newton-Cartan   
   geometry, where c is infinite!) and for t < 0, it is locally Euclidean   
   (i.e. a 4-dimensional timeless space).   
      
   Such a geometry is radiation dominant near t = 0 and having a null   
   surface at t = 0 is also an essential condition to accord with the   
   ability to predict Inflation (which arises as a consequence of the   
   junction conditions required to join the t < 0 with the t > 0   
   sector); a' la Mansouri. This is something that is normally only   
   "written in by hand", rather than predicted.   
      
   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.   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)