Summary Statement, to provide context for the questions:
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.
Our measurements in the measurement domain are expressed as "measurement
dimensional numbers." These measurement dimensional numbers have
numerical parts that correspond to the real numbers, but also have
dimensional parts related to specific measurement procedures.
Our physical laws are expressed by a special kind of measurement
dimensional number. These numbers are ratios of measurable quantities
that encode linear relations between these quantities.
Who has proved that the arithmetic for the real numbers is a COMPLETE
set of arithmetical rules for the measurement dimensional numbers?
So far as I know, this arithmetical correspondence has not been questioned,
but only assumed.
It has been assumed that the measurement real numbers, which are
structurally more complicated than the real numbers, are subject
to just the same arithmetical rules as the reals, with no
additional rules.
Sometimes more complicated entities have more complicated rules than
simpler ones. Checking of the arithmetic mechanics of the measurement
dimensional numbers seems justified. We are beyond "proof" here,
because the measurement dimensional numbers are not axiomatic constructs.
However, we CAN check arithmetical rules for consistency by taking
computations around loops that should close, and seeing whether closure
does or does not occur. (This loop testing is logically analogous to
the loop testing used in surveying and used to perfect measurement
instruments of all kinds.)
When the arithmetic of measurement dimensional numbers is evaluated by
loop tests, an additional arithmetical rule is found to be necessary.
Algebraic simplification of terms involving products or ratios of
measurement dimensional numbers MUST occur at "unit scale" (which
may also be interpreted as "point scale.")
After equations are algebraically simplified at unit scale, they can be
mapped, term for term, into the domain of abstract mathematics (domain
of the algebra). Not before. No usage in the domain of the algebra
changes, but the form of some equation-representations being manipulated
does change.
The fixed unit scale requirement for algebraic simplification of cross
terms has not been known before. In some instances, terms have been
called infinities that are finite, and other terms have been called
infinitesimals that are finite. Mistakes due to the mislabeled
infinitesimals are typically negligible, but can be enormous.
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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