1. Krantz, D.H., Luce, R.D., Suppes, P. & Tversky, A. Foundations of
Measurement, Vol. 1 (Academic, London, 1971).
2. Suppes, P., Krantz, D.H., Luce, R.D. & Tversky, A. Foundations of
Measurement, Vol. 2 (Academic, London, 1989).
3. Luce, R.D., Krantz, D.H., Suppes, P. & Tversky, A. Foundations of
Measurement, Vol. 3 (Academic, London, 1990).
these are far from new ideas - there is a large literature
(some of it very mathematical!)
diana kornbrot
On Sun, 3 Jan 1999, M Robert Showalter wrote:
>
> *****************************
>
> 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.
>
>
> **********************************************
> **********************************************
>
> 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
>
>
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Dr. Diana Kornbrot
Reader in Mathematical Psychology
Associate Dean Research, Faculty of Health & Human Sciences
University of Hertfordshire
College Lane, Hatfield, Hertfordshire AL10 9AB, UK
voice: +44 0170 728 4626 fax: +44 0170 728 5073
email: d.e.kornbrot@herts.ac.uk
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