**********************************************
Nothing I've found has disproved, or argued against, the S-K suggestion
that an incompleteness has occurred, at the level of arithmetical
procedure, in our mathematical representations of coupled physical
circumstances. (Coupled circumstances where several physical effects
are at play simultaneously in ways that are unavoidably tied together.)
I still believe that there is scale constraint on algebraic
simplification of crossterms in the measurement domain that has not
been understood.
**********************************************
The work is far more "thinkable" and much better contextualized, both
historically and culturally, because of the references suggested here.
In the last days I've learned, again, how hard it is to distinguish
the notions of the "axiomatic," the "reflexive," and the "certain" in
the mathematics of human beings, past and the present.
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.
I asked:
DOES ANYONE DOUBT (OR SUPPORT) THE NOTION THAT THE
MEASUREMENT DOMAIN IS BEYOND OUR ESTABLISHED AXIOMS?
I've seen no reason to believe that the measurement domain is governed
by established axioms. It is treated on the basis of assumptions based
on experience, and can be judged in that same empirical way.
George Hart, shows how muddled our culture's measurement domain
projections can be in MULTIDIMENSIONAL ANALYSIS: Algebras and
Systems in Science and Engineering, Springer-Verlag 1995,
especially in Chapters 0, 1. Hart cites the volumes Diana Kornbrot
recommended, and I'd missed them. I'm very glad she brought them
to my attention. They are:
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 books seem to me to be the best pieces on the foundations of
measurement (and its limitations) that I've found. Steve Kline's
SIMILITUDE AND APPROXIMATION THEORY (McGraw-Hill 1965, Springer-
Verlag 1984) assumes much that these books put in more solid
context. The first volume of the three above counts most for my
questions.
At the beginning of the Preface (p xvii) Krantz et al write:
"Scattered around the literatures of economics, mathematics,
philosophy, physics, psychology, and statistics are axiom
systems and theorems that are intended to explain why some
attributes of objects, substances, and events can reasonably
be represented numerically. These results constitute the
mathematical foundations of measurement. Although these
systems are of some mathematical interest, they warrant our
attention primarily as empirical theories - as attempts to
formulate properties that are observed to be true about
certain qualitative attributes. Some of the theories
appropriate to classical physics are so well accepted that
they are usually considered in the province of philosophy
rather than physics, but this should not be allowed to
becloud the basic empirical character of any theory that
purports for example to justify treating mass as an additive
numerical property. From time to time, the empirical nature
of basic measurement assumptions is forcibly brought to
everyone's attention.
......."
Chapter 10, "Dimensional Analysis and Numerical Laws" is a very good
treatment. Here is the first paragraph of that chapter (p.454):
"Taken together, then numerical measures of physics exhibit
a very simple algebraic structure which, although completely
familiar and therefore not surprising, tends to be mysterious
when given any thought. That is what we do in this chapter
- we think about it. We try to describe the structure
precisely, and to the extent that we can, to account for it."
Krantz et al set up some working assumptions, and call them axioms,
but these "axioms" are not very solid, nor were they claimed to be so.
I asked:
HAS ANYONE TRIED TO SHOW THAT THE ARITHMETIC OF THE REAL NUMBERS IS A
COMPLETE ARITHMETIC FOR THE MEASUREMENT DIMENSIONAL NUMBERS?
I've seen no reason to think that anyone has done so. Nor have I seen
any reason to think that anyone has tried to do so.
*****************
I've looked at all the books any of you have suggested to me that the
U.W. library has. That's almost all of them. I've looked at many
more besides, including some I've found of special interest because
they were written by distinguished members of Historia-Matematica.
I've been impressed with these books, and wish I'd had more time to
devote to reading them.
It will be a long time before I feel comfortable with all these
references. I'm working hard on them, and making headway with them.
I'll respond, by private correspondence, to the people who were kind
enough to suggest these references.
Some of these references, though they are different and interesting
in many other ways, raise a common issue that makes them look more
problematic than they are. These references assume Steve Kline and
my work suggests a change in PURE MATHEMATICS. We don't. ( If we do
so, the change is only indirect and peripheral.)
We suggest instead a change in THE WAY COUPLED PHYSICAL MODELS
ARE REPRESENTED IN EQUATIONS. We suggest that some representative
equations be changed. We are not suggesting a change in the
mathematical procedures themselves.
For example, this means that I've got no reason to argue with
Poincare's or Zeeman's or Thom's mathematics, on the equations they
treated. I may ask questions about whether they've dealt with the
right equations.
I responded privately as follows, and would like to repeat it here:
>
> (We are dealing with series of terms set out by a process that
> writes down explicitly what definitions imply. These series are not
> power series.)
>
> ************************
>
> Here are some things that it took Steve and I a long time to think
> through, and become comfortable with. I hope they'll be easier for
> you than they were for us.
> The ideas are simple enough, and might be easy to teach to a kid
> just starting out:
>
> People have taken equation-representations, and thought them
> complete, when they've been incomplete.
>
> Sometimes (indeed, most often) the incompleteness is entirely
> trivial numerically, and the incomplete representation is the
> right approximate representation in any case.
>
> But under circumstances where extreme accuracy is needed, or
> under circumstances where the values of the parameters are very
> large or very small, the neglected terms can be important, and
> the physical system can be badly represented by the incomplete
> equation-representation of it.
>
> THIS IS A PROBLEM WITH REPRESENTATIONAL PROCEDURE, NOT WITH PURE,
> AXIOMATIC MATHEMATICS ITSELF.
>
> Some (not all, and not most) representative equations INPUT into
> pure axiomatic math will change. That is all that is needed.
>
>
> The S-K formulation says that coupled physical relations based on a
> continuum assumption typically contain an endless series of crosseffect
> terms. The magnitude of these crosseffect terms can be determined by
> arithmetic for the values of the parameters being modeled in a case.
> For very different values of the parameters, different terms will
> predominate, and different terms will matter.
>
> ( In the case of a conducting (electrical or passive neural)
> line, crosseffects that are far too small to measure and
> consider for a wire become the dominant terms in the case of
> a small neural line. For a comparison of the wire and neural
> case, with crosseffect magnitudes compared, see
> http://www.wisc.edu/rshowalt/ranges )
>
> A crosseffect term that is trivial and properly neglected in electrical
> engineering is a huge effective inductance term in neurophysiology.
> See "A Modified Equation for Neural Conductance and Resonance"
> http://xxx.lanl.gov/html/math-ph/9807015/ . Appendix 1 shows how the
> crossterms occur by definition of the interactions in the wire. Appendix
> 2 shows how the operations on the natural law operators are constrained
> by experimental math.
>
> ************
>
> Going from models based directly on measurable physics to
> representative equations, we have to be more careful than we have
> been. We've been using the same symbols for different things in the
> same logical sequences, and not noticing.
******************************************************************
The data that most motivated Steve Kline and I to re-examination
inference of physical equations from coupled circumstances is only a
few pages. It is the data of David Regan, the psychologist and
electrophysiologist, in 1989.
http://www.wisc.edu/rshowalt/regandat/
That data, and a good deal of data since, is now unexplained, explanation
requires effective neural line inductances many billions of times the
inductances now predicted. The S-K equations predict those inductances.
That difference, if it is true as I believe, has moral consequences in
many places in neural medicine.
******************************************************************
The late S.J. Kline and I have suggested something that we feel makes
logical sense. We've believed that it makes historical and technical
sense, as well. Even so, the suggestion has violated our own conceptual
reflexes, and the reflexes of others. And these reflexes have been
essential parts of the discourse we've had to deal with.
Here is Lipman Bers:
"What is the strength of mathematics? What makes mathematics
possible? It is symbolic reasoning. It is like "canned thought."
You have understood something once. You encode it, and then
you go on using it without each time having to think about it.
.................
Without symbolic reasoning, you cannot make a mathematical
argument."
(MORE MATHEMATICAL PEOPLE, D.Albers, G.L.Alexanderson, C. Reid,
eds. p. 16)
This reflexive strength of human mathematics function makes the
distinction between the axiomatic and the reflexive hard to maintain,
and harder and harder to maintain as reflexes become more deeply
embedded in thought processes of individuals and groups. People can
even feel, if you ask them to change a deep reflex, that you are
proposing injury to them. Anger and other strong emotions can attach
to reflexive commitments. People find it easy to react as if changes
in reflex notions are "unthinkable." Perhaps, in an animal sense,
that's right. It would seem so to me. I reacted that way myself.
************************************
None of the references that I've found so far has said "the S-K
position is right." Nor can that be expected. But it does seem,
so far, that there is "logical room" for our position, and that
EXPERIMENTAL MATHEMATICS, in the spirit of Godel and Chaitin, is the
way to test it. Because of kindness here, I'll be able to put the
S-K work in a far sharper, much deeper historical context than I had
known how to do before.
If I'm wrong, I'm that much closer to finding that out. Anyone who
can help me see that I'm wrong, if in fact I am, will be doing me a
favor.
I've been enormously impressed with HISTORIA-MATEMATICA, and believe
that this group, constitutes a contribution to the culture. The
Internet is making intellectual discourse more flexible and
sophisticated than ever before. Historia-Matematica is a prime
example. I appreciate your kindness and your corrections.
Thanks,
M. Robert Showalter