From: nospam@de-ster.demon.nl   
      
   Thomas Koenig  wrote:   
      
   > Are there mathematical axioms for physical units and dimensional   
   > analysis?   
      
   Of course.   
   The subject of dimensional analysis   
   can be easily and trivially axiomatised,   
   since it is all about finite dimensional algebras.   
      
   > We know that you cannot just add a velocity to a time.   
      
   Before going on it is necessary to understand   
   that dimensions have no physical reality.   
   We can assign dimensions in any way that is suitable,   
   subject only to the requirement of consistency.   
   Physical quantities do not -have- a dimension.   
   So two physical quantities can have the same dimension in one system,   
   and different dimensions in another. (E and B for example)   
   Do not mistake the conventional MKSA dimensions   
   for -the- dimensions.   
      
   > Everybody just applies the rules of mathematics and calculus   
   > to variables with units without giving it a second thought.   
      
   There is no need for much thought.   
   log (pressure) for example   
   is automatically understood as   
   log (pressure/conventional unit of pressure)   
      
   > For example, "clearly", the derivative with respect to t of   
   > a*sin(omega*t) is a*omega*cos(omega*t).   
      
   Of course, and dimnsionally correct, since [d/dt] = [omega] = [T]^-1   
      
   > And the indefinite integral of 1/T over T, where T is a themperature,   
   > is ln(T) + C, correct?  Well, not really, since you cannot have   
   > the logarithm of anything with a physical unit, and it is best   
   > for your piece of mind if you just rewrite that as ln(T/T0),   
   > where T0 takes the place of the arbitrary constant in integration.   
      
   Any way you want, nothing but a matter of notation.   
      
   > Mathematicians have developed a very rigid theoretical system around   
   > (dimensionless) numbers, but seem quite uninterested in units.   
      
   If you have units you are doing physics, not mathematics.   
      
   > There is the Buckingham pi theorem with its associated proof.   
   > This is based on what I learned at university as the "Bridgman's   
   > axiom", that physical relationships cannot depend on the unit   
   > system (although that name appears to be uncommon).   
      
   It is self-evident and does not need a named axiom.   
      
   > So, did anybody do further work on formalizing this?   
      
   Have you looked at the wikipedia articles on dimensional analysis?   
      
   Jan   
      
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