Re: [HM] Questions for historians of mathematics
Gordon Fisher (gfisher@shentel.net)
Sun, 03 Jan 1999 21:48:49
At 02:06 PM 1/3/99 -0600, M Robert Showalter wrote:
> ...
> Here are some key questions I'd like to ask historians of mathematics.
>
> I've said: Consider the notion of a "measurement domain" of physical
> things that we can measure. We do not derive the "measurement domain"
> from axioms, but learn of it from physical experience.
>
> DOES ANYONE DOUBT (OR SUPPORT) THE NOTION THAT THE MEASUREMENT
> DOMAIN IS BEYOND OUR ESTABLISHED AXIOMS?
>
>
> I've said: Our measurements in the measurement domain are expressed as
> "measurement dimensional numbers."
>
> HAS THE QUESTION OF THE *COMPLETENESS* OF THE REAL NUMBER ARITHMETIC
> RULES FOR THE MEASUREMENT DIMENSIONAL NUMBERS BEEN ADDRESSED?
> (A reference I know, that goes some of the way toward answering
> this question is MULTIDIMENSIONAL ANALYSIS: Algebras and Systems
> in Science and Engineering by George Hart, Springer-Verlag 1995,
> especially Chps 0, 1. Hart argues strongly, and with extensive
> bibliography, that the mathematical nature of our dimensional
> numbers has been neglected.)
>
> HAS ANYONE TRIED TO SHOW THAT THE ARITHMETIC OF THE REAL NUMBERS IS
> A COMPLETE ARITHMETIC FOR THE MEASUREMENT DIMENSIONAL NUMBERS?
>
>
> HAVE OTHERS USED LOOP CONSISTENCY TESTS AS TESTS OF ARITHMETIC? (The
> procedure seems natural, and is widespread in instrumentation.)
>
> HAS ANYONE ARRIVED AT SUCH A SCALE CONSTRAINT FOR ALGEBRAIC
> SIMPLIFICATION OF CROSSTERMS IN THE MEASUREMENT DOMAIN BEFORE?
>
>
> Answers, comments, or bibliography related to any of these questions, or
> any related questions that may occur to you, would be much appreciated.
>
> M. Robert Showalter
>
Are you concerned about something more than a marriage of abstract axioms
for the real numbers with what physicists and engineers call dimensional
analysis, involving such calculations as
(ft/sec^2) * (sec*2) = ft ???
If this sort of thing has any relation to what you're after, you may want
to look at an elementary calculus book published by Karl Menger in which
such a marriage was effected. Menger felt strongly that calculus books of
his time neglected to incorporate physical dimensions with numerical
calculations in using calculus. His work, however, had no effect to speak
of on the way calculus is taught by mathematicians up to the present, and
physicists and engineers customarily meld together dimensional analysis and
real number arithmetic without benefit of explicit axioms, in the manner of
mathematicians.
An exact reference for the Menger book is:
Karl Menger, *Calculus, a modern approach*, 2nd edition, revised and enlarged,
Illinois Institute of Technology, Chicago IL, 1953.
Gordon Fisher gfisher@shentel.net