Save for a typo now corrected, this paper has been uncriticised.
Surviving a posting at Los Alamos is not peer review. I have not
submitted a paper for review since this LANL filing. For about the last
four months, I've tried to justify the work, beyond any possible doubt,
based on experimental data alone. In this case "beyond doubt" is a very
high standard. People are reluctant to believe that the ARITHMETICAL
rules we now use in physical modeling can be incomplete, and that the
incompleteness could have persisted over centuries. For smaller stakes,
I'd find the current data more than sufficient. Much data acts to strongly
support the Showalter-Kline model, and many experts are in sympathy with
this. But the supporting results are not in the form of simple,
logically coercive tests. I've come to feel that an experiment needs
to be done.
(According to the Showalter-Kline model, conduction velocity in
channel - closed neural dendrites will be independent of frequency
above a threshold - according to the present theory, that velocity
will vary as the square root of frequency.)
That's the difference between a brain capable of switched resonance and
passive memory, and one that is not. To motivate the experiment, and
disciplined investigation of the Showalter-Kline mathematics, the idea
that the arithmetic of physical representation can be investigated anew
has to be THINKABLE.
Over years, Professor Stephen J. Kline and I have encountered enormous
resistance to the idea that our culture's arithmetic of measured quantities
could be incomplete. The key question, on which the gift of review depends,
is
A. Could the ARITHMETICAL rules we now use in physical modeling be
incomplete?
People in the *Historia Matematica* group may be the best qualified
people in the world to address this question, and especially to address
the "thinkability" of this question in the context of the history of
mathematics and mathematical physics.
I'm making the appendices of "A Modified Equation for Neural
Conduction and Resonance" available in separate form. These appendices
set out the math itself:
Appendix 1: Derivation of a finite increment equation from a coupled
physical model, showing combined effect terms.
http://www.wisc.edu/rshowalt/lanap1_2
Over much time and many contacts, I've never had the derivation
of these combined effect terms questioned by a professional,
BUT NO PROFESSIONAL I'VE TALKED TO HAS BEEN CLEAR ABOUT WHAT
THESE COMBINED EFFECT TERMS MEAN, NUMERICALLY OR DIMENSIONALLY,
AT FINITE SCALE. Limiting arguments, that generate infinitesimals
and infinities, describe limits of well defined series of finite
terms. If the finite terms are not workably defined, the
limiting argument cannot be right.
To answer this question about the numerical and dimensional meaning of
combined effect terms, one needs clear self-consistency tests set out in
Appendix 2: Representing Physical Models as Abstract Equations:
Procedures Inferred from experimental mathematics.
http://www.wisc.edu/rshowalt/lanap2
Algebraic simplification of combined effect terms must be done at
unit scale (aka point form).
The derivation procedures shown often have negligible quantitative
consequences, but can have enormous implications in certain parametric
ranges.
( See http://www.wisc.edu/rshowalt/ranges/ )
The effective inductance of the neural conduction equation can be > 10^15:1
larger than now calculated with the additional rule of measurement domain
arithmetic applied correctly. The scientific and medical implications
of this are important. Two issues of human importance concern epilepsy
and ventricular fibrillation, one a major cause of human suffering, the
other the leading direct cause of death in the industrial world. In my
experience, many neuroscientists are prepared to consider this "if only
the mathematicians permit it." To get this matter of math and physiology
considered and tested, question A above needs to be "thinkable."
Background discussions:
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 cited 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. I'd be proud to have any
of you read it and comment on points made in it. At the start of it,
I respond to the following welcome question:
budrap - Jun 11, 1998 EST (#584) ... - When you think about calculus
... Calculus seems to get the right answers without providing a very
useful/informative map of the territory.
..... "is there a Copernican/Keplerian analog that might allow us
to more clearly comprehend a part of physics that calculus now masks
in infinitesimals?"
The discussion involves a good deal about the history of the problem and
its significance that may be of interest to contributors in *Historia
Matematica*.
In the course of that forum discussion, I made available on the web
the same papers submitted here, in a previous form. More than 15 people
pulled down the papers. I don't know their names, but many of these
people came from distinguished universities and institutions. I heard
of no objections to the work.
None of this says the work is right, but it tends to reinforce the
"thinkability" of the work.
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, once Question A becomes "thinkable."
model that follows from careful consideration of Question A, in light of
experimental math in some ways similar to the experimental math that
G. C. Chaitin uses. I discus how the model might be proved or disproved,
established or rejected.
********************
Last year, before my friend S.J. Kline's death, we submitted a
number of papers to NATURE. They were too long, and we knew it. Our
motivation was not to have the papers published as submitted. Instead, we
hoped that NATURE might help us get checking on our material, which we
knew people didn't like to think about.
The editors of NATURE did not help us with the checking we'd asked
for. However, to our honor and surprise, they seem to have given serious
consideration to publishing the work. In response to the draft submissions
referenced below, the editors of NATURE wrote a gracious, supportive, and
seemingly reluctant rejection letter. That careful letter, I believe, was
intended to help us. I have made the letter available on my website in two
forms:
text at http://www.wisc.edu/rshowalt/natletshrt/
full facsimile at http://www.wisc.edu/rshowalt/naturlet/
NATURE's letter included this:
"Although it is sadly the case that some studies simply do not
lend themselves to the NATURE format, this need not mean that our
readers are left in the dark about the latest developments. As
you know, we frequently discuss such work in the context of our
News and Views section, and if you were to send us preprints of
your present papers when they are finally accepted elsewhere for
publication, we could explore the possibility of doing likewise
with your work."
Those familiar with NATURE rejections expect them to be terse, and this
was not. NATURE's letter indicates that the editors of an outstanding
journal that specializes in neurobiology found the material significant
and plausible. That doesn't make us right. It does tend to make more
thinkable the question on which all our work depends:
A. Could the ARITHMETICAL rules we now use in physical modeling be
incomplete?
The papers Professor Kline and I submitted to NATURE are referenced here
to show the potential importance in neuroscience and neural medicine that
the S-K equation has. These papers will be rewritten and submitted to a
neurophysiological journal when the central mathematical foundation of the
S-K derivation is peer reviewed. That depends, more than anything else,
on the thinkability of question A, our core question.
HYPOTHESIS: DENDRITES, DENDRITIC SPINES, AND STEREOCILIA HAVE RESONANT
MODES UNDER S-K THEORY by M.R. Showalter at
http://www.wisc.edu/rshowalt/hypothesis/
REASONS TO DOUBT THE CURRENT NEURAL CONDUCTION MODEL by M.R. Showalter
at
http://www.wisc.edu/rshowalt/doubt/
A NEW PASSIVE NEURAL EQUATION. Part a: derivation by M.R. Showalter
at
http://www.wisc.edu/rshowalt/deriva/
A PASSIVE NEURAL EQUATION: Part b: neural conduction properties by
M.R. Showalter at
http://www.wisc.edu/rshowalt/derivb/
Here are the physical derivation (math) papers we submitted to NATURE.
MODELING OF PHYSICAL SYSTEMS ACCORDING TO MAXWELL'S FIRST METHOD by
M.R.Showalter and S.J.Kline at
http://www.wisc.edu/rshowalt/maxmeth/
EQUATIONS FROM COUPLED FINITE INCREMENT PHYSICAL MODELS MUST BE
SIMPLIFIED IN INTENSIVE FORM by M.R.Showalter and S.J. Kline at
http://www.wisc.edu/rshowalt/pointfrm/
If equations derived according to Maxwell's 1st method are right,
inferences from experiments are only valid over a RESTRICTED range
by M.R. Showalter and S.J. Kline at
http://www.wisc.edu/rshowalt/ranges/
These pieces are still right, but I'm more clear now than Steve and
I were on the nonaxiomatic nature of the world of measurement.
For background, we also submitted an annotation of an excellent survey
article, showing how the new theory fit what was known.
A verbatim copy of COMPUTATION AND THE SINGLE NEURON by Christof Koch
taken from NATURE, 16 January, 1997 annotated and with two appendices
by M.Robert Showalter at
http://www.wisc.edu/rshowalt/kochanno/
The NATURE submissions fit data rather well, but I'd used an inconsistent
system of units (MKS units are needed for consistent crossterms.) That
would have produced an error, but the error is balanced by the effects of
glial clefts that surround neural lines, using an analysis that depends
on S-K. Results fit data well again:
"The Glial membrane-fluid cleft-neural membrane arrangement cuts effective
neural capacitance, greatly increasing signal conduction velocity and
greatly reducing the energy requirement per action potential", by M.R.
Showalter at
http://www.wisc.edu/rshowalt/cleft/
******************************************************************
One can also show the inadequacy of our arithmetic in physical description
by means of linear algebra. This has been done in
"A REDERIVATION OF THE ELECTRICAL TRANSMISSION LINE EQUATIONS
USING NETWORK THEORY SHOWS NEW TERMS THAT MATTER IN NEURAL
TRANSMISSION."
http://www.wisc.edu/rshowalt/kirch1/
which starts:
"Some of the logic of neurophysiology depends on the mathematical
form of the passive electrical transmission equations that apply
to nerves. The electrical transmission line is modeled here by
network theory. The network model is inconsistent with the accepted
transmission equations,
dv/dx = -Ri -L di/dt and
di/dx = -Gv -C dv/dt.
The network model shows an effective inductance that depends on R
and C."
The presently accepted equations for dv/dx and di/dx are based on a
limiting argument that depends on currently accepted arithmetic in the
measurement domain. These equations have no dependence on R and C.
Linear algebra programs of very high quality and reliability generate
dependancies on R and C.
This inconsistency is of practical importance. The inconsistency, which
is not now understood, can generate an initially small but explosive
error. We must understand what happens here if we are to validly trust
our numerical integrations, Numerical integrations are essential parts
of high-stakes technical work in the fusion and nuclear fields, and
elsewhere.
To expect ordinary people to evaluate this issue, and to expect people
to take the issues involved seriously enough so that publication is
reasonable, question A must again be thinkable:
A. Could the ARITHMETICAL rules we now use in physical modeling be
incomplete?
So long as question A is unthinkable, the issue is unthinkable. The
*Historia Matematica* group is uniquely qualified to think about
question A.
Given the medical stakes, and some other stakes as well, I feel that,
one way or another, I have to do two things:
1. I must convince the math profession that arithmetical procedures
at the foundation of our derivations of differential equations from
coupled PHYSICAL models has to be determined by experimental math,
because the dimensional numbers that apply measurement are extra-
axiomatic. S-K derivations follow from experimental consistency.
Question A must be "thinkable" before these arguments can be
seriously evaluated.
and
2. I must convince the neuroscientists of the S-K neural transmission
model. Some very good ones, in my experience, stand quite ready to
be convinced. The issues is derivation of the neural equation, and
that depends on a longstanding matter of unidentified arithmetic. If
there is a scale constraint on algebraic simplification of certain
terms in physical modeling, switched neural resonance makes sense.
Otherwise, it does not. Surrounding neural details will be useful,
and I've got a lot of them. I can show that neural lines "look"
strongly inductive, and that "granting the math" a great deal of
known detail follows. But, given the stakes and the doubts at play
in this case, persuasive sequences have to be SIMPLE. The core
demonstration will have to be a direct experimental measurement.
That measurement can be done, but hasn't been. To motivate that
measurement, Question A must be "thinkable."
Here is Question A again:
A. Could the ARITHMETICAL rules we now use in physical modeling be
incomplete?
The Historia Matematica group is uniquely qualified to think about
question A. I believe that, if question A can be seriously considered,
all the rest follows easily, and much of medical use can be done.
Thank you for your attention,
M. Robert Showalter