"I think that each culture has certain blind spots. There are
some concepts that, though not necessarily difficult, nevertheless
get overlooked by a culture for many years. ...................
What really intrigues me is thinking: What might be the blind
spots of *our own* culture?"
On 17 Oct, G.C. Rota spoke of a particular kind of blind spot:
"........mathematics advances ........... whenever some commonsense
notion that had heretofore been taken for granted is discovered to
be wanting, to need clarification or definition."
There is an error of long standing in the procedures we use to map
coupled physical circumstances into abstract mathematical representations.
That error is at the level of arithmetical procedure. There is a scale
constraint on algebraic simplification of crossterms in physical models
being inferred directly from physical circumstances. For many, the idea
that an error can be buried so deep in our usages is not just "hard" or
"unwelcome." It is unbelievable. One can think of this error as a blind
spot.
We learn habitually that the arithmetic of the real numbers is all
the arithmetic there is that can apply to numerical entities. More
complicated kinds of numerical entities are typically and reflexively
considered to have arithmetic no more complicated than the arithmetic of
the real numbers. So far as I can tell, people don't check this assumption,
even when problems such as problematic infinities and infinitesimals
persist. That's a blind spot.
What if the arithmetic were more complicated, in a way that usually
made no detectable quantitative difference, but a way that could produce
large errors under some conditions? Would we check? Given our conceptual
patterns, could we check? Or would the problem be "unthinkable"?
Here are some "commonsense" questions that may be "unthinkable" because of
our blind spots.
Would it not seem commonsensical to say that before we apply
limiting arguments to terms in a domain logically outside the
domain of the algebra, we need to be able to explain in a
consistent way what the terms mean at finite scale? Would it
not seem commonsensical to ask that we use arithmetical rules
that assure us that both the dimensional and the numerical
parts of all such terms act self-consistently?
There is an error - an "obvious" but wrong assumption made in Newton's
time and not changed since. It is at the level of arithmetic. There is
an unrecognized but important constraint on permitted scale of algebraic
simplification on certain crossterm quantities. I believe that this
constraint must be recognized if we are to understand brain function, and
some related issues, some plain matters of life and death.
At a formal level, the question is fairly easy. At a practical and
professional level, it is not easy. The error is based on an unexamined
belief that the arithmetic of the real numbers can be projected, without
modification or augmentation, onto physical circumstances. The error
can generate false infinities, and false infinitesimals. The error is
as old as our models of coupled physical circumstances, which is to say,
at least as old as Newton. The stakes are sufficiently high that I have
to get past the question
"Is this THINKABLE?"
before I can make the detailed, technical case that can be peer reviewed.
I discuss work involving that technical case in
"A Modified Equation for Neural Conduction and Resonance"
( http://xxx.lanl.gov/html/math-ph/9807015 3 Sept 1998)
Save for a typo now corrected, this paper has been uncriticised.
I've discussed the logic of the math, and its background, in George
Johnson's MYSTERIES OF THE UNIVERSE forum at THE NEW YORK TIMES
(http://www.nytimes.com) over about two years time. George Johnson
practically never signs his name to a posting in these forums, and did
not do so on any of the postings referenced here.
Even so, some of the anonymous discussants on the forums seem to me to
combine outstanding writing ability, deep scientific background, and, in
my view, much insight. The math was discussed in #584-#641 of the BLACK
HOLES AND THE UNIVERSE site.
http://forums.nytimes.com/webin/WebX?14@^1475934@.ee74d5b/649
That discussion is of about 80 typed pages. In Johnson's PI IN THE SKY
forum, which gave much attention to Gregory Chaitin's work, I submitted my
brain model, in postings #817 to #823.
http://forums.nytimes.com/webin/WebX?14@^1476127@.eeb3cdd/886
This is about 8 pages of text. In my view, a great deal of "the code of
the brain" is in breakable condition, and that section illustrates my
reasons for thinking so compactly. I believe that my participation in
these New York Times forums may have gotten me the invitation into Historia
Matematica - that gives me the honor of addressing you here.
I'm saying that an error, analogous to the Y2K error, has existed
in our physical models for three centuries, and has propagated through the
system, patched but not detected. (See PROPOSED ROUGH HISTORY, which
follows.) The stakes involved are conceptual and practical, and are
relevant to many fields, from the history of mathematics to the most
applied parts of engineering. There are good reasons for people NOT to want
to look at this. In an ideal world, math and passion might always be
separable. However, my experience has been that when I question the
completeness of the arithmetical foundations of our physical models, I get
quite understandable but nevertheless violent aversive responses, but
no counterexamples or logical arguments. The work needs to be THINKABLE
before it can be seriously considered, and tested by mathematical and
laboratory investigation. The question of THINKABILITY involves insight
about math as a human construct in a historical context. That's why I'm
here.
My background is in engineering, and application of mathematical
techniques to engineering and invention. I've worked with my close
friend, the late Professor Stephen J. Kline, of Stanford and the NAE, at
the foundations of mathematical modeling of physical circumstances. We
were motivated by difficulties in engineering modeling of coupled systems,
difficulties in computational fluid mechanics, and especially difficulties
in neurophysiology.
Steve Kline wrote SIMILITUDE AND APPROXIMATION THEORY
(McGraw-Hill, 1968, Springer-Verlag, 1984) a much respected book in
physical modeling. We found that this book contained an error in its first
paragraph. Steve had assumed that the accepted differential equations
could be trusted, and proceeded on that basis. We've found that, sometimes,
the standard differential equations cannot be trusted. We've found that in
some cases workable representative equations involving coupled physical
circumstances could not be derived. In neurophysiology, we had reason to
believe that the inductance that was being assumed was in error by more
than
1,000,000,000,000,000:1.
We found this error in neurophysiology morally compelling. The neural
physics involved is a matter of life and death (and the difference between
understanding and muddle) in neural modeling. And so we looked very
hard at the procedures people have been using for inference of differential
equations from coupled physical circumstances. (Steve died last year,
but left me a letter of recommendation that I can supply on request.)
Here's what we've found.
The dimensional numbers used to represent measurements have numerical
parts that correspond to the real numbers, but also dimensional parts that
correspond to measurement procedures. George Hart has shown, in some detail,
that these dimensional numbers are different from real numbers, and have
additional rules that constrain addition and multiplication among these
numbers.
(Hart's book MULTIDIMENSIONAL ANALYSIS: ALGEBRAS AND
SYSTEMS FOR SCIENCE AND ENGINEERING, Springer-Verlag 1995, largely
devoted to working out the consequences of the following restriction
in dimensional numbers, as it applies to linear algebra
"You can't add apples and oranges..... But you can multiply them!"
Hart's Chapters 0 and 1 describe in much detail how muddled and extra
axiomatic our dimensional numbers are.)
The modeling situation with dimensional numbers is yet more complicated
than this. The physical laws by which we relate one measurable quantity
to another are expressed in dimensional numbers that are RATIOS of measured
quantities. Steve and I have called these ratios natural law operators,
to call attention to their special character. These natural law operators
are coefficients in the finite increment equations or differential equations
we define from physical models.
We know NOTHING from axioms about the arithmetical rules we can apply
to these natural law operators.
The correct derivation of differential equations from coupled physical
models depends on what the arithmetical rules applied to natural law
operators happen to be. When one represents a coupled physical
circumstance by means of coupled equations, the coupled equations are each
implicitly defined in terms of each other.
Each of these equations, if fully written out to express the model being
represented, would include an endless series of crosseffect terms.
http://www.wisc.edu/rshowalt/lanap1_2
(the first appendix of the LANL paper, op.cit.)
Now, these crosseffect terms are dismissed, on the basis of a limiting
argument that hinges on an assumption about the arithmetic that governs
those terms. In some other cases, a limiting argument applied to
crosseffect terms yields infinities.
Some infinitessimals and some infinities derived in this way have been
problematic for a long time.
Steve and I decided that we could not trust these derivations, and
had to test them for internal consistency, according to the same sort of
logic used in design of instruments, the logic of loop tests. This is
experimental mathematics, applied to a domain beyond the axioms, where
experimental mathematics seems to be all we have.
Steve and I have found, using experimental mathematics, that there
is a restriction on the numerical scale for defined multiplication or
division of these crosseffect terms, which are made up of natural law
operators, in combination with spatial variables (x,y, z or t). The
restriction requires that the multiplication or division be done at unit
numerical scale (or, by a closely related and numerically identical
interpretation, point scale.) (Any physical scale is unit scale in some
unit system.) Unless this restriction is conformed to, arithmetical
operations that look correct yeild absurd results when tested for self
consistency.
http://www.wisc.edu/rshowalt/lanap2
(the second appendix of the LANL paper, op.cit.)
An equation now central in neurophysiology, is absurd by that test.
There's long been reason to doubt this equation, and many neuroscientists
would welcome a better equation. But the equation is justified as based
on "certain mathematics." That "certain mathematics" is based on a
mistaken belief that the arithmetic of the real numbers can be projected,
without modification or augmentation, onto our descripiton of physical
circumstances.
The restriction Steve and I have found is mechanically easy, and
once equations are algebraically simplified at unit scale, mapping into
the ordinary domain of the algebra is straightforward. The only difficulty
is that this has not been done in the past, and some mistakes have been
built into our applications of mathematics.
"A Modified Equation for Neural Conduction and Resonance"
( http://xxx.lanl.gov/html/math-ph/9807015 3 Sept 1998)
The gift of peer review occurs for propositions that are "believable" -
not for propositions that can be rejected out of hand, without attention
or need of contradiction. The Showalter-Kline work has to be THINKABLE.
I submit three related, freestanding pieces with this submission:
QUESTIONS FOR HISTORIANS OF MATHEMATICS sets out questions members
of *Historia Matematica* are especially qualified to answer.
PROPOSED ROUGH HISTORY sets out what I believe may have happened
historically, as a result of the undetected error I discuss, and how
that error has been patched, and has caused trouble.
BIBLIOGRAPHICAL BACKGROUND shows references to the Showalter-Kline
work, what it involves, and who has reacted to it.
Members of *Historia Matematica* may be the best qualified group
in the world to consider the question:
"Is this THINKABLE?"
I'm committed to trying to find the right answer, not merely my own answer.
I'd be grateful for any responses any of you might give me, either on
*Historia Matematica* or privately. If I've made a mistake, I'll let
interested parties know that promptly. If a mistake is not found, I'll
have taken a step tending to increase the crediblity of the work.
That credibility is important for two reasons:
1) It will help get the work reviewed, as mathematics and mathematical
physics.
2) It will help me motivate experiments in neurophysiology that will
show whether my neural conduction model is right or wrong.
Respectfully,
M. Robert Showalter