U. of Wisconsin, Madison Stanford
University

**BACKGROUND TO THIS DISCLOSURE**:

**We show, in symbols and by numerical example,
that our culture's conventional limiting arguments, which are essential
parts of our mathematical modeling procedures, are now degenerate because
they apply to terms that are not defined. Terms
are not defined because they include groups (of dimensional parameters ^{(1)}and
spatial increments) that are not defined. Terms
that include these groups must be properly interpreted and defined in terms
of dimensional theory before the conventional limiting process can be validly
applied. When this is done,
these terms, which are now dismissed as infinitesimals or labeled as infinities,
are finite. **

* When
we derive DIFFERENTIAL EQUATIONS from PHYSICAL models, we start by setting
up FINITE INCREMENT EQUATIONS that represent the model completely (or,
if the equations involve some recursive relation, we represent the model
to some specified degree of completeness.) *

** Before
we can use these FINITE INCREMENT EQUATIONS we must know what they mean,
term for term. We must be able
to represent EVERY TERM by a specific measurement procedure. (EVERY
TERM must make sense in terms of measurement theory.)
**

** ONLY
THEN can we take the limit of the FINITE
INCREMENT EQUATION and reduce it to a VALID
DIFFERENTIAL EQUATION.**

**We have come to feel that our core simulation problems
involve the DERIVATION of differential equations.**

**3. Complete physical model specification
requires a full set of operational measurement procedures that define the
variables and dimensional parameters in the model.**

**The
differences between the new procedure and the old one come because of the
discovery that the dimensional parameters are not just numbers, but are
subject to arithmetic restrictions (or, and this is the same thing, must
be interpreted in terms of measurement theory ^{5}.)
The requirements of measurement theory make the mathematics that applies
directly to concrete circumstances different from abstract mathematics.**

According
to the first, more literal method Maxwell cites, we have "difficulty"
interpreting some (cross effect) terms that interested Maxwell particularly.
Indeed, with no more information than Maxwell had, we
cannot interpret them at all. We are stopped.

THEREFORE
we make an assumption, based on faith and experience. We
make that assumption along with Maxwell, giants before him (Newton, LaPlace,
LaGuerre, and Fourier) and workers since. **As
a culture, we decide to act AS IF our physical quantity representing symbols
may be abstracted into simple numbers in our intermediate calculations.
This ASSUMPTION has produced equations that fit experiment innumerable
times. But it remains a pragmatic
ASSUMPTION with
no logically rigorous basis at all**.

The assumption CANNOT be justified by claiming universal success for our culture's analysis. Our culture's analysis has generated more successes than a person could reasonably understand and count, but our culture's analysis has also failed (often inexplicably) more often than any person could review or count. The reporting career of George Johnson, and the text of REALITY BYTES, provide many examples. Computational fluid mechanics, most fields of engineering, and most fields of physical and biological science offer many examples where analysis fails or cannot be brought to bear, for reasons that seem unclear to all concerned.

The
strangeness of the assumption that the symbols that we use to represent
measurable things are "just numbers" may help explain why some
smart, careful students, who wish to carefully and redundantly trace decisive
stages of logic as they imagine them and learn them, can distrust mathematical
modeling procedures, can refuse to learn and use mathematical modeling,
and can come to fear mathematics. These students
can see how measurements (or measurable quantities) can be set out as numbers.
They can see, or think they can see, how quantities can interact according
to patterns, and how these patterns can be symbolized by terms in equations.
But at the start of their reasonings and visualizations,
quantities are more than stark numbers - they exist in contextual detail,
that detail can be expanded to measurement procedures and apparatus, and
the terms correspond to pictures. In the course of calculation,
in an untraceable jump made without __any __detailed explanation
or justification, dimensional quantities previously linked to measurement
are stripped into simple, decontextualized numbers. These
decontextualized numbers appear in terms that cannot be pictured by any
physically connected step-by-step process, although image analogies corresponding
to these terms can sometimes be constructed. These issues
bothered Maxwell, and have bothered us, for conceptual reasons and for
computational reasons as well. In Maxwell's electromagnetic studies, and
in neurophysiology today, the standard mathematical modelling assumptions
yield very wrong answers.

To review:

On Maxwell's first assumption, that
deals with quantities that are traceably connected a physical model in
measurable physical context, we as a culture have terms that are difficult
(impossible) to interpret.

On Maxwell's second assumption, that strips the measurable
quantities to "mere numerical quantities," these terms fit readily
into our culture's calculus apparatus, and can quite often be "shown"
by a limiting argument to be infinitesimal or infinite.

Maxwell
was unhappy with his second assumption, and searched for ways to make his
first assumption computationally workable until his death.
Maxwell was NOT convinced that crosseffect terms that mathematicians were
calling infinitesimal really were infinitesimal. But
he did not have a mathematically coherent reason to doubt the standard
mathematics. Maxwell had not seen that there were some
symbols, that he was using just as if they were simple dimensional numbers,
that were not just dimensional numbers.

Here
is a crossterm, one of many considered later. This crossterm
is encountered in the modeling of a conductive line (a wire or a neuron.)
**R** is resistance per unit
length. **C** is capacitance per unit length. **R**
and **C** are defined as ratios of measurables, in (fairly complex)
measurement procedures invisibly encoded when the letters **R** and
**C** are written. **R** and **C** are dimensional
parameters. **x** is a length along the line.
**v **is voltage. **t** is time. James
Clerk Maxwell called crossterms like this "difficult
to interpret" in 1879.

Crossterms like this make no
sense in terms of any measurement
procedure. We as a culture now dispose of them by a
limiting argument that assumes they are validly written, when they are
not. As a culture, we now say that

And so terms like this are discarded as "infinitesimal."
We discard these terms without clearly defining what they mean first.

With
knowledge that the dimensional parameters have arithmetical restrictions,
we proceed differently. It happens
(and this is new knowledge) that mixed products of **R**, **C**,
and length are only defined in point form (or when interpreted by an operational
measurement procedure, which produces the same thing).
The arithmetic of the limiting argument above,
that looks so compelling, is undefined and actively misleading. The
correct interpretation of the crossproduct, writing out **R **and
**C** so that their numerical and unit parts are visible, proceeds as
follows. We put the length in point form, which is **(1
meter) _{p}** for a meter length unit.

We then do arithmetic to evaluate the product of

(The idea that (1 meter)

We've
been applying limiting arguments to undefined terms, and have been getting
wrong answers. We have to set out terms like this in
a way that makes measurable sense before applying limiting arguments to
them. It turns out that, for these terms, we "get
to point scale" by substituting point forms of our spatial variables
directly. (Or by doing an operational measurement procedure
that amounts to this.) At point scale, our arithmetic
is defined. This would seem natural and logical to any
fresh student, and no more arbitrary than many other things she has to
learn. But people are not accustomed to it. We
are asking that people consider a new rule that restricts their arithmetic.
Discussion of this can quickly come to resemble an "argument about
religion." Perhaps this isn't surprising.

People
often learn their mathematics as a matter of faith, based on patterns of
seeing that may be incomplete or wrong, but that may seem compelling to
the people who accept them. Mistakes based on "mathematically perfect"
derivations can be hard to see, and can last a long time.
For example, an important mistake in fluid mechanics lasted 150 years,
defeated some famous analysts, and still persists in some elementary physics
texts. The inviscid equations of Leonard Euler were used
to predict flows. Those equations "show" that
wings have neither drag nor lift. The analysis was corrected
by Ludwig Prandl in 1904, and his change made modern fluid mechanics possible.
The change took new insights (the idea of the boundary layer) and new techniques
(asymptotic perturbation theory). The Euler analysis
made a deeply buried (and plausible) assumption (no viscous effects, even
at the walls) that wrecked predictions. It was a long
time, even after Prandl pointed out the right answer, before everyone could
see the buried mistake, see Prandl's better analysis, and learn to use
Prandl's breakthrough. Even though everybody knew that
the Euler equations weren't working, there was still resistance. A
later modification of flow theory, by Kline^{(8)}
, took fifteen years and the work of several coworkers and many graduate
students to become established. Again, most of the fluid
mechanikers who resisted the new way really knew the old way was wrong,
at some level. Even so, there was much resistance.
Established ways of thought and reflex patterns of work are hard to change.

The
resistance to special treatment of the dimensional parameters has been
far beyond anything in our experience. We have difficulty
because people do not know that the old analysis has a problem.
Many regard the possibility of the problem we speak of as unbelievable.
We question limiting arguments based on ill defined terms, and limiting
arguments have a special and high status in mathematics for historical
reasons. We are asking for modification of patterns that are
**very **deeply habituated. If one looks
at a trained analyst at work, in either the "pure" or engineering
sciences, one will see limiting arguments and the arithmetical usages around
them applied with unconscious-reflexive facility, as spontaneously as breathing.
These processes are "done from the wrist" as if they were spinal
reflexes. One would not be looking at a trained analyst
otherwise. Analysts judge each other by this sort of
reflexive facility.

We
would all be lost without our reflexes, that may derive from culture but
that become a part of us. Once a person is committed
to a reflex, it is awkward to consider it, or to doubt it, or to try to
change it. In our experience, if one asks a trained analyst
to change her limiting procedures, or the arithmetical usages around them,
one encounters some of the same difficulties one might have if one asked
that analyst to stop breathing. For one thing, the analyst
will perceive the request as an attack on her competence.
For another, the analyst will perceive the request as a crazy, literally
unthinkable request. The fact that the old processes
have been involved in much successful work will be taken as reason to trust
the old procedures beyond doubting. The notion that our
limiting processes is associated with flawed arithmetic in some cases will
be dismissed. If one persists, and one is taken more
seriously, one is likely to encounter fear responses.
If one is taken very seriously, one may encounter panic responses combined
with disorientation responses, closely followed by various defensive and
hostile responses. We have seen these responses from
individuals committed to the current analytical usages. We have seen these
responses from groups committed to these current analytical usages. These
are not responses that fit people for detached judgement.

We
are dealing with a matter that is a large scale issue of life and death.
It is an issue of great scientific interest.
Our conclusions need to be checked. For mathematicians,
physicists, and many engineers these logical issues are not easily addressed
because they are linked to contextual, reflexive, institutional issues
that are intense but that are not simple. For such people,
we are saying something that is not only "unbelievable," but
also "unseeable" in the paradigm conflict sense Kuhn and others
have described.

People
who are not so reflexively engaged, and who are not so indoctrinated, are
needed to look at the work, and check it step-by-step. People
not blinded by their reflexes and their indoctrinations will be better
able to judge the argument and evidence here than the "experts"
so blinded. People interested in truth, who see the stakes,
but do not feel they have a lot to lose if the argument goes one way rather
than another will be better able to judge the argument and evidence here
than the "experts" who are also parties-in-interest. We
care about our results too much to be fully trustable judges of them. We
want our results to be right. Some expert physicists and mathematicians
will care about our results too much to be fully trustable judges of them,
because if we are right, we will invalidate much of the work they have
done, that they rely on, and that they revere. Such experts
will want our results to be wrong.

No
one is to blame for these entirely human motivations. Even
so, we need umpires, who can look at the work from a certain distance,
and care for the truth. The medical implications of this work
are so large that right answers are what should matter here.

We
have been in correspondence with George Johnson since December of last
year. We've decided that the work requires checking
from a broadly based audience that includes mathematicians and physicists,
**but does not include them exclusively**. We
have concluded that an extensive disclosure-submission in a New York Times
forum under George Johnson is a proper way to address that audience. We
have found that we cannot reasonably rely on peer review, unassisted, in
its usual form for this particular case. Peer review
is valuable but not perfect. We strongly support peer
review as the standard pattern of scientific and professional engineering
evaluation. One of us (Kline) has more than 170 peer
reviewed articles, and has reviewed at least as many. (Kline,
no stranger to peer review, is one of the leaders in fluid mechanics, especially
computational fluid mechanics, this century.) But in
our particular and unusual case we are, reluctantly but definitely, attacking
the "invisible colleges" of physics and mathematics on decisive
ground. We are doing so as engineers, that is, as outsiders.
Stakes are high and, in our experience, emotions related
to the work also run high. Peer review was never meant for this. We
need, in addition to checking from mathematicians and physicists, checking
by quantitatively competent people who are NOT mathematicians and are NOT
physicists. There are many such people in the United
States. The mathematics that we disclose is not intrinsically
difficult once it is pointed out. It may be checked in
a matter-of-fact fashion by professionals of all kinds. But
the new mathematics may be **most **difficult for mathematicians and
physicists, because it requires that they change deeply established patterns
of thought and reflex.

** Practical
implications of our work include a reinterpretation of important aspects
of neurophysiology, some plain matters of life and death, some as interesting
as memory. We say that the effective inductance of small
neural lines is now understated by factors of 10 ^{10}-10^{19}
(that is, effective neural inductance is understated by 10,000,000,000:1
to 10,000,000,000,000,000,000:1 . )
We believe that serious mistakes in neural science and medicine
have been made, and are being made because of this mistake, and some other
mistakes that follow from the same mathematical misunderstanding.
For scientific, medical, and moral reasons, it is important that we be
checked in this. **

**THE DIMENSIONAL PARAMETERS:**

Readers
of** George Johnson **and **REALITY BYTES **may not be surprised
when we say that there is an error at the interface between the measurable
world and our culture's mathematical modeling. They may be surprised
when they see how basic the error is, and how old it is. We were surprised
that the error has existed since Isaac Newton's time.
Since that time it has been assumed that the "mapping patterns"
built into analysis somehow exactly fit the "territories" of
the measurable world. The reason for the fit has been
thought vague, or magical.

The
fit has been better grounded than many have thought. Where
the map-territory fit exists, the fit is there, in the cases we've examined,
because we as a culture have always had sharp, mechanistic "mapping
tools" that we have reason to trust from measurement experience. However,
we as a culture have not recognized these ubiquitous "mapping tools"
as the logically special entities that they are. We as
a culture have sometimes used these tools with facility. As a culture,
we have never used them with understanding. Nor have
we as a culture always used these mapping tools perfectly.

The
linkage between the measurable world and our culture's more-or-less abstracted
equation representations of that world occurs via **dimensional parameters**,
common number-like entities, used for centuries, that encode and abstract
experimental information in our culture's physical laws.
These dimensional parameters ARE the link between
the physical laws that can be measured, and (more or less) abstracted equations.
The dimensional parameters have seemed to be so trouble-free that no one
has looked very hard at them. We found that we had to
do so.

Here are some directly measurable dimensional parameters:

Mass, density, viscosity, bulk modulus,
thermal conductivity, thermal diffusivity, resistance (lumped), resistance
(per unit length), inductance (lumped), inductance (per unit length), membrane
current leakage (per length), capacitance (lumped), capacitance (per unit
length), magnetic susceptibility, emittance, ionization potential, reluctance,
resistivity, coefficient of restitution, and many more.

These dimensional parameters are not abstractions: each
of them corresponds to the detailed context of a measurement procedure.
(Usually, you'd need a sketch, and some instructions, to describe that
measurement procedure.) The dimensional parameters are
each expressible in units, and the unit system can be reduced to units
of length, mass, and time. But the units notated in a
dimensional parameter are **only** defined in the context of a SPECIFIC,
and sometimes very detailed, measurement procedure.

** In
addition to dimensional parametric functions, there are also compound dimensional
parameters, made up of the products or ratios of dimensional parameters
and other dimensional variables taken together.** **RC** in equations
2-3 above is a compound dimensional parameter.

A
famous class of the compound dimensional parameters is the dimensionless
numbers, such as the Reynolds number used in fluid mechanics. These
"dimensionless numbers" exist in a strongly specified dimensional
context: they exist in consistent dimensional systems (systems that may
be reduced to length, mass, and time and a tightly specified list of measurement
procedures.) The dimensionless parameters are defined
in terms of context-complete, context-specific measurement procedures applied
to a particular circumstance. For instance, a particular
Reynolds number will apply to a particular airplane model at a particular
angle of attack at a particular velocity for a particular fluid, and to
other geometrically similar systems where ratios of inertial to viscous
forces are the same. A Reynolds number or other dimensionless
number has an exponent of 0 for all of the dimensions in the consistent
dimensional system that it is defined in. The numerical value
of the Reynolds number or other dimensionless number is exactly the same
for any other consistent dimensional system subject to EXACTLY the same
measurement procedures for EXACTLY the same physical circumstances.
The "dimensionless" numbers exist in a dimensional world that
must be specified in much detail. They are not "just
numbers" in the sense of "just abstract numbers." Again
for emphasis: the dimensionless numbers and dimensional parameters relate
to concrete, measurable things. They are not abstract.
They are connected to context and specific embodiment.
All the dimensional parameters can be defined according to the following
pattern.

**DEFINITION: A
dimensional parameter is a "dimensional transform ratio number"
that relates two measurable functions numerically and dimensionally. The
dimensional parameter is defined by measurements (or "hypothetical
measurements") of two related measurable functions A and B. The dimensional
parameter is the algebraically simplified expression of {A/B} as defined
in A = {{A/B}} B. The dimensional parameter is a transform
relation from one dimensional system to another. The dimensional
parameter is also a numerical constant of proportionality between A and
B (a parameter of the system.) The dimensional parameter
is valid within specific domains of definition of A and B that are operationally
defined by measurement procedures.**

Example: Resistance per unit length
of a wire, **R**, is the ratio of voltage drop per length of that wire
to current flow over that length of wire. A resistance
per unit length determined for a specific wire for ONE specific length
increment and ONE specific current works for an INFINITE SET of other length
increments and currents on __that__ wire (holding temperature the same,
and assuming wire homogeneity.) **R** is the dimensional
parameter in Ohm's law for a wire. Other physical laws
have other dimensional parameters.

__ The
dimensional parameters encode experimental information into a compact,
linear form. This form may be denoted by a symbol exactly
like the symbols used to denote simple dimensional numbers, and that is
now done. Such symbols, representing encoding of context-bound
measurement information, are now used exactly as dimensional numbers
inside our culture's physical equations. __
The distinction between the dimensional parameters and ordinary dimensional
numbers exists functionally and has done so from the first equation definitions
of physical law. However, to our knowledge that distinction has not been
clearly defined before. So far as we can tell, the distinction
has not been thought important.

** If
one asks "how do our culture's equations
connect to the measurable world,"**
a major question in

__It is HERE
that the measurable interfaces our culture's symbolic mathematical representations.__

This paper sets out the material at a level that concerned James Clerk Maxwell. It is, we believe, at a level that would most concern a new student, or someone, not a mathematician, looking at the issues involved.

**I. 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. M.R.
Showalter**

**Checking so far:**

This posting is not our first effort to get this mathematics checked. We don't believe that George Johnson would permit this posting if he were not satisfied that we'd worked to get checking elsewhere. Nor is our math unchecked. Nor have uncorrected mistakes in the mathematics been found by anyone. (People have expressed aversion to our work, but that is a different thing. ) Nonetheless, the mathematics, which is at the core of physical modeling, has not been accepted or seriously discussed, by mathematicians or physicists. We do not think it right to review our efforts to get this math checked in detail here. We can say that, for six years, we've worked hard to get this math, and work leading up to it, critically reviewed and discussed. We've attempted to get checking of the work in every effective way we've been able to think of. For the last year particularly, after the arithmetical limitations of the dimensional parameters were clearly identified, we've tried to get the work checked. We've not been called wrong for explicit, traceable reasons that could stand up to examination (except for a much appreciated question about scale choice, which has led us to use MKS-Georgi units.) Nonetheless, the work has been dismissed, and seems to have been treated as unthinkable. Mathematicians and physicists of distinct good will have had this reaction, and sometimes their reactions have involved visible signs of personal stress. Our work has been "undiscussable." Late last year, one of us (MRS) sent a series of e-mail messages, describing the math in some detail, and soliciting checking of it, to every member of the Department of Mathematics at the University of Wisconsin, Madison and to most of the neuroscientists at U.W. These email messages, and a background essay on checking efforts sent to George Johnson in January 1997, are available to interested parties (FTP:angus.macc.wisc.edu/pub2/showalt). There were no responses to my requests from mathematicians or physicists solicited, nor did substantial help from neuroscientists occur. People watching, including a mathematically gifted Dean, felt that there was some obligation for the mathematicians to respond if they had found a mistake. Those who examine the transmissions may agree. Nonetheless, the mathematics, which is at the core of physical modeling, and which has immediate and large medical and scientific implications, was not discussed.

" ** We
need to establish one mathematical point. Once
established, it need not take up much paper, either. Suppose that a short
piece of less than a page, including a few definitions fore and aft and
the following language, were published in NATURE.**

(the modified differential equation
derivation procedure, with intensification for currently undefined terms,
was here.)

or

we may consider each symbol as denoting only the numerical value of the corresponding quantity, the concrete unit to which it occurs being tacitly understood.

How crossterms happen in the derivation of coupled equations from models:

We find it convenient to write these as an input specification list like this:

with the variables listed before the semicolon, and the dimensional parameters afterwards. For this line conduction model, we assume that

and

Writing down voltage change as a function of the dimensional
parameters and variables that directly affect voltage, we have.

Writing down current change as a function of the dimensional
parameters and variables that directly affect current, we have.

We may equally well rewrite (4a) and (4b) going from points
**x- delta x** to **x+delta x**, so that the interval is centered
at **x**.

Note that equation (5a) includes **i(x+delta x/2)** and its time derivative.
d**i(x+delta x/2)/dt** is defined by equation (5b). Equation (5b) includes
**v(x+delta x/2)** and its derivative d**v(x+delta
x/2)/dt** is defined by equation (5a).
Each of these equations requires
the other for full specification: * each contains the other.
*If the cross-substitutions specified implicitly
above are explicitly made, each of the resulting equations will also each
contain the other. So will the generation of equations
following, and the next, and so on. This is an endless regress.
Each substitution introduces new functions
with the argument

**! ! ! ! ! IMPORTANT
! ! ! ! !**

** We
cannot assume that terms in this kind of coupled expansion are unimportant
until we know what the symbols in them mean, and what the arithmetical
rules that apply to those symbols are.**

** ****We
MUST assume that these terms have a dimensional interpretation consistent
with the other terms in the same equation.
We must assume that all the terms, properly determined, have the same dimensional
unit exponents. We must
assume that all the terms, properly interpreted, are meaningful (have specific
numerical and dimensional interpretations) at all spatial scales in the
equation's domain of definition.**

**! ! ! ! ! ! ! ! ! ! !
! ! !**

We can proceed with substitutions like the following,
associating symbols without interpreting them numerically or physically.
For example

is

which expands algebraically to

These terms would be simpler if voltages and derivatives
of voltages were taken at the interval midpoint, **x**. But even so
simplified, *it is terms of this kind that are
"difficult to interpret" in Maxwell's sense*. (Maxwell
was an industrious analyst living in an analytically competitive world,
and when he wrote "*difficult to interpret*" we believe
that he must have meant *"operationally impossible to interpret"*.)

Consider crossterms like these:

If one wishes to speak of expressions like those
of (9), what do they mean for finite delta when the symbols are considered
to stand for fully physical things, or complete models of physical things,
subject to the detailed physical rules that stand behind the model?
How do you interpret them with a sketch interpreted by measurement procedure?
** In discussions with mathematicians, engineers, and scientists, the
first author (MRS) was not (for three years) able to find anyone
who was confident of the meaning of these kinds of expressions at finite
scales (or, as a matter of logic, when length was reduced to an arbitrarily
small value in a limiting argument.)** The
equations below shows voltage change over an interval of length delta

The equation for **i(x+delta x/2,t)-i(x-delta x/2,t)**,
is isomorphic to 10 (or 11 below), with swapping of the variables **v**-**i**
and the dimensional parameters **R**-**G**, and **L**-**C**.

Let's
rewrite the neural conduction equation (10), dividing each term by the
length increment, and assuming that length increment is so small that we
can approximate the gradient of voltage per unit length as a derivative.
In (11) we substitute the word "length"
for delta **x**. This substitution is a useful notational
step, perhaps most useful because it helps us to think without our reflexes
engaged about what the notion of length (or length^{2}, length^{3},
etc.) may mean in the terms of this equation.

10 may be "simplified" according to common arguments
to (**12**)

In current usage, Maxwell's second
assumption-method seems to "define" all these "difficult"
terms, below the first line in **10**, **11** or **12** (and then
seems to define them out of existence.) The definition is never tested,
because these terms are dismissed by a limiting argument that all trained
analysts now trust but do not examine. (Group consensus and truth
are not the same.) All the terms in the curly brackets are thought to be
"infinitesimal in the limit as **x** goes to 0" and are discarded
(usually they are never written.) These terms are
then dismissed based on a limiting argument that ASSUMES that these terms
are DEFINED at finite scales according to Maxwell's second method.
*They are not consistently defined according
to Maxwell's second method, as has been assumed: the assumption that they
are well definined leads to contradictions.*

** When
these cross terms are consistently defined, they become easy to interpret.
**To show this, we'll start by showing how the assumptions of Maxwell's
second method fail logically and numerically.

Difficulties
with dismissal of these terms on the basis of meaning are set out elsewhere^{(9)}.
However, even setting these aside, the dismissal fails several closure
tests.

(Closure tests are standard tests all through measurement theory. One
goes around a sequence that should be a cycle. If one
is not exactly where one began after a "cycle" that "cycle"
is defective. It has failed a closure test. Surveying
offers a common example. If a surveyer measures around
a mountain (or a lot) and comes back to a particular stake, the measured
position of that stake "around the loop" must match, within measurement
error, the position measured for that stake before going around the loop.
Kirchoff's laws in electricity are closure definitions useful as laws.)

Here
is a cycle that should close, but that does not. At a
finite scale, before taking the limit, the terms below the first line of
equations like **10 **or **11** are supposed to represent finite
quantities. Yet when we as a culture take the limit as
delta **x** goes to zero, these crossterms are infinitesimal (0), and
are discarded in the conventionally derived differential equation. Now,
let's take that differential equation, which has the "infinitesimal"
terms stripped away. Let's integrate it back up to specific
scale **x**. **We get an equation that lacks
the crossterms that we know existed at scale x in the first place**.
We should have closure, and do not. When one or more of the dimensional
parameters

In
addition, the expressions in curly brackets in (**10** and **12**)
fail as representations of a physical circumstance. First,
the terms fail the test of homogeneity.
Different descriptions of the same thing,
that should have the same total size, have different sizes.
For instance, consider a curly bracketed expression in **(delta x) ^{2}**.
If delta

These
expressions are also meaningless because they are constructed on the basis
of a type invalid arithmetic. The loop test below showed
us that so clearly that we SAW the inherent
error in our culture's arithmetical assumptions, an error that we had been
looking at, or looking near, for a long time before.
The loop test then served as a "testbed" that told us what the
arithmetical restrictions of the dimensional parameters were.

We
should all know that a "number" or "expression" that
can be manipulated by "proper arithmetic" and permissible unit
changes so that it has any value at all is meaningless.
Let's look at a simple loop test, analogous to many closure tests in physical
logic. In Fig. 2, an algebraically unsimplified dimensional
group that includes products or ratios of dimensional numbers, such as

is set out in meter length units at **A**. This quantity
is algebraically simplified directly in meter units to produce "**expression
1**," dealing with the dimensional parameters and increments as
"just numbers." The same physical quantity is also
translated from **A** into a "per meter" basis at **C**.
The translated quantity at **C** is then algebraically
simplified to **D** in the same conventional fashion. The
expression at **D**, expressed in meter length units, is converted to
a "per meter" basis to produce "**expression 2**."
__Expression 1 and Expression 2 must be the same,
or the calculation is not consistent with itself.__
Quantities like those in
the curly brackets of **10**-**11** consistently fail the loop test.
By choice of unit changes, and enough changes, such "quantities"
can be made to have ** any numerical value at all**.
Expressions such as these are

Here is a an example of how our symbolic-numerical conventions "go around the loop."

The loop test fails!

**The loop test fails because a
standard procedure is flawed. **

**Before algebraic simplification,
going from one unit system to another adjusts not just the numerical value
of dimensional properties in the different unit systems, but numerical
values corresponding to the spatial variable (length), as well. **

**After algebraic simplification,
one has a compound dimensional property - adjusting it to a new unit system
corresponds to adjusting numerical values that correspond to the unit change
for the dimensional properties only, with no corresponding adjustment
for the absorbed spatial variable.**

The result is an irreversible, numerically
absurd, but now standard mathematical operation. The difficulty has been
deeply buried and hidden in our culture's notation.

** Note
that the loop test of Fig. 2 only fails for terms that are now routinely
discarded as "infinitesimal" or "infinite" without
detailed examination**.
Multiplication and division of groups of dimensional numbers (and dimensional
parameters) without spatial or temporal increments passes the loop test
without difficulty, with numerical values handled arithmetically, and dimensional
exponents added and subtracted in the usual way. Dimensional
parameters associated with increments according to the exact pattern that
defined the dimensional parameters in the first place also work.

Problems
now arise when we as a culture associate several dimensional parameters
together with several increments, where the increments correspond to different
physical effects evaluated over the same interval. These
problems arise in notations that we as a culture have never understood
at finite scales - we as a culture have no reason for confidence for the
notational procedures we've been using.
The procedure we as a culture have used for evaluating such circumstances,
as we've notated them, involves a kind of multiple counting that yields
perverse results, as the loop test shows. We
want to incorporate a rule that avoids mistakes like this.
The rule needed restricts multiplication
and division of dimensional parameters and increments to intensive (point)
form, except for circumstances where the dimensional parameters are associated
with increments in the exact way that defined them. It
requires us to clarify our concept of increments (of length, area, volume,
or time) defined **at a point**.

**! ! ! IMPORTANT
! ! !**

** Most
of the time, engineers and scientists have been dealing with differential
equations (defined at points) rather than finite increment equations (defined
over intervals). When differential
equations have been manipulated, the difficulties shown by the loop test
have not occurred because every term in the differential equations
have already been in point form. **

** However,
DERIVATIONS of differential equations from coupled physical circumstances
HAVE involved finite interval models, and need to be reexamined to see
if terms have been lost. "Rigorous proofs"
of some of our trusted differential equations can no longer be relied on.**

** !
! ! ! ! ! ! ! !**

Notated
as we as a culture have notated them, and interpreted as we as a culture
have interpreted them, entities that represent coupled relations are physically
unclear and numerically undefined. Although we as a culture
may notate the interaction of resistance, capacitance, and resistance together
in interaction with the measurable **di/dt** over the interval **x**
as

where delta **x** is a number times a length unit, the loop test shows
that this notation, literally interpreted, does not correspond to any consistent
numerical value when unit systems are changed, and then changed back. (14)
shows a common and long trusted notation that does not reflect limitations
on the arithmetic permissible with the dimensional parameters.

Let's
rewrite (**14**) setting out a notation that makes explicit problems
we need to solve concerning our notation of "length" in this
expression:

What well defined notion can we as a culture have
of "length" in the limit as scale shrinks to a point?
Surely, no number of length units can suffice as such a well defined notion
of length "at a point", because whatever number of length units
we choose, we can always pick a smaller number still.
We as a culture can't be correct if we mean that (but in our culture's
limiting arguments, that assume an invalid arithmetic, we do mean that.)
*We as a culture need a notion of "length" that makes focused
sense at a point, because differential equations are defined at points.
*

**If** we assume, incorrectly, that the symbols in

and similar expressions are indistinguishable from the abstract numbers
we are used to manipulating in our differential equations, **then**
we are faced with a contradiction.

On
the other hand, if we remember that these symbols represent concrete, measurable
circumstances, then we face no contradiction. We do face
a conceptual challenge. We
must face the need to reassess some of our assumptions.
We need to recall that more complicated systems often have rules that less
complicated systems do not have. For
this reason, we are not OBLIGATED to assume that the rules of arithmetic
that apply to these more complicated entities, in their more complicated
concrete context, are exactly the same as the arithmetical rules that apply
to the simpler system of abstract numbers.
We can determine what the arithmetical rules in the concrete domain are,
rather than assume these rules on the basis of some plausible analogy to
abstract mathematics. For a long time, this conceptual
challenge was too much for us. It was not obvious to
us that we could question arithmetical usages in the concrete domain.
The loop test above showed us that we had to. The dimensional
parameters are subject to a type restriction that does not exist in the
abstract domain.

Once
we realize that we may determine the arithmetical rules of concrete domain
mathematics, we face the question: how do we get these symbols and symbol
groups to make fully consistent sense in terms of measurement? We
need to consider what consistent sense in terms of measurement means.

Here is James Clerk Maxwell, starting from the first page
of the first volume of his **A TREATISE ON ELECTRICITY & MAGNETISM:**
(Dover)

** "PRELIMINARY.**
**ON THE MEASUREMENT OF QUANTITIES**

**"1.] Every
expression of a Quantity consists of two factors or components.
One of these is the name of a certain known quantity of the same kind as
the quantity expressed, which is taken as a standard of reference.
The other component is the number of times the standard is to be taken
in order to make up the required quantity. The standard quantity is technically
called the Unit, and the number is called the Numerical Value of the quantity.**

** "There
must be as many different units as there are different kinds of quantities
to be measured, but in all dynamical sciences it is possible to define
these units in terms of the three fundamental units of LENGTH, TIME, and
MASS. **

**. . .
"2.]
In framing a mathematical system we suppose the fundamental units of length,
mass, and time to be given, and deduce all the derivative units from these
by the simplest obtainable definitions. . . . The formulae at which we
arrive must be such that a person of any nation, by substituting for the
different symbols the numerical values of the quantities as measured in
his own national units, would arrive at a true result.**
. . .

** " This
is most conveniently done by ascertaining the dimensions (Maxwell
means the exponent here) of every unit in terms of the three fundamental
units. . . . . .
For instance, the scientific unit of volume is always the cube whose side
is the unit of length. If the unit of length varies, the unit of volume
will vary as its third power . . . . . **

**" A
knowledge of the dimensions (Maxwell means exponents) of units furnishes
a test which ought to be applied to the equations resulting from any lengthened
investigation. The (unit
exponents) of every term of such an equation, with respect to each of the
three fundamental units, must be the same. If not, the
equation is absurd, and contains some error, as its interpretation would
be different according to the arbitrary system of units which we adopt.
"**

P.W. Bridgman made Maxwell's position more realistic and
definite by clarifying the notion of measurement.
Measurement involves operational procedures in a specific, step-by-step
context. The operational procedures are implicit, and
logically necessary, but are not expressed in the abstract algebraic symbols
themselves, which show only units raised to exponents.

An important point, for Maxwell, for Bridgman, and for
anybody else who wants valid equations, is as follows:

**ALL the operational procedures must refer only to steps
where, for specific units of LENGTH, MASS, and TIME, (call them L, M, and
T) the measured quantity (or partial operational procedure involved in
the derivation of the quantity) could be reduced to **

**L ^{n1} M^{n2} T^{n3}**

**where n1, n2, and n3 are exponents that may be integers
or fractions. If and only if every measurable in THE ENTIRE
SYSTEM OF UNITS is in this form, then the choice of particular length scale,
mass unit, and time unit is arbitrary. **

The
requirement that the ENTIRE SYSTEM be consistent is a severe one in the
case of units that combine mechanical and electrical (and even more, mechanical,
electrical, and magnetic) quantities. There is ONE such system now in common
use. The CONSISTENT SYSTEM now in use is the MKS-Giorgi
unit system^{(10)}. (An
infinite number of other unit systems would be possible, compared to the
MKS-Giorgi unit system, but it would be a lot of work to get even one other
one, because of the coupling between the definitions of length, mass, time,
the electrical units and the magnetic units, and because of the hard word
required to reexpress a large number of dimensional parameters in the new
system.)

For
the finite increment equations that we derive to make sense, they must
make dimensional sense. EVERY term must be a valid term.
From the loop test, we know that every one of the terms
including curly bracketed crossterms below is arithmetically undefined
when the symbols in it are interpreted by abstract arithmetic. What
interpretation __can__ we apply to the cross effect terms? Our
interpretation must be well defined and consistent with measure theory.

Let's refer to equation (17). Let's
ask a standard question: "how do
we, in a meaningful, fully defined way, take the limit of BOTH SIDES of
this equation?"

On the
left side of equation (17), the limit as delta x approaches 0 is dv/dx.
(That limit is the definition of dv/dx). dv/dx is defined at a point -
we may say that it is in point form.

On the right side of equation (17),
let's proceed term for term, from left to right

There is nothing to do for the -Ri
term - it is already in point form

There is nothing to do for the -
L di/dt term - it is already in point form

(R and L are numerically and dimensionally the same, applied
to any length of the homogeneous wire, or applied to any point on that
wire. *Other dimensional parameters also stay the same,
numerically and dimensionally, for any interval.*)

We now have 11 terms, each with
a collection of symbols within curly brackets that is UNDEFINED in the
limit, without some additional convention or notation.

Within these curly brackets, in addition to products of
the dimensional parameters R, L, G, and C, are associated (multiplied?)
spatial variables. The association of the dimensional
parameters and the spatial variables DOES NOT CORRESPOND EXACTLY TO THE
CONDITIONS FOR WHICH THE DIMENSIONAL PARAMETERS ARE DEFINED. THE PRODUCT
(OR RATIO) OF THE DIMENSIONAL PARAMETERS AND SPATIAL (OR TEMPORAL) VARIABLES
IS NOT DEFINED FOR GENERAL VALUES OF THE SPATIAL OR TEMPORAL VARIABLES.
The loop test shows this. Some other cyclic closure tests
also show this.

So
our culture's standard limiting argument for these curly bracketed expressions,
taking delta x smaller and smaller, till these curly bracketed expressions
can be dismissed, isn't valid because that
limiting argument uses arithmetic that isn't valid.

We want to proceed in the closest
**POSSIBLE** analogy to our standard limiting argument.

If
the curly bracketed terms are to make sense in differential and finite
difference form (and we assume they MUST make sense) then the
curly bracketed expressions must all simplify to COMPOUND DIMENSIONAL PARAMETERS
(that do not change in numerical value or dimensionality from point scale
to any finite scale for a homogeneous system.)

So
we need to be able to interpret the curly brackets in terms of a Bridgman
operational procedure. We can do this, but our operational
definition requires a length scale of 1 length unit.
We are constrained to the 1 length unit scale by requirements we've touched
before, with respect to consistent unit systems.

To interpret any of the curly bracketed terms in equation (8) according
to a Bridgman operational effective measurement procedure, we can proceed
as follows. Since the curly brackets involve undefined arithmetic, we can't
interpret them directly. For any of the
curly bracketed expressions, we can, however, proceed as follows:

1. We can take each curly bracketed
term out of the equation, to evaluate it by measurement theory

2. We can set the length scales
at 1 length unit, and after doing that, algebraically simplify the term
into a compound dimensional parameter. (This is the Bridgman operational,
dimensionally correct procedure.)

3. We can then reinsert the curly bracketed term, which is now a (compound) dimensional parameter, where it was before. This dimensional parameter will be numerically and dimensionally the same for WHATEVER length scale it is evaluated for, including point scale.

If we choose point scale, we've
come as close as we can, considering the arithmetical limitations we've
been dealing with, to taking a valid limit of that term.

We can proceed, term for term, in
this way. Doing so, we come as close as possible to taking a limit, term
for term. In the end, every one of our terms consists
of a dimensional parameter (that may be a properly simplified compound
dimensional parameter) times a measurable quantity. Every
one of these dimensional parameters, and every one of these measurables,
exists in the same compatible system of units.

There
is another interpretation that gets us to the same result in an operationally
simpler way. We don't have to "conceptually remove
our groups for a measurement procedure" and we don't have to "take
a limit." Suppose,
rather than work through "an operational measurement procedure"
for each term, as described above, and then putting our term at point scale,
we'd substituted a "point form of length" with a value of (1
meter)_{p} for every instance of length. Suppose we then algebraically
simplify, term for term, using usual arithmetic for dimensional numbers.
Our end result would be at point scale, and would be dimensionally and
numerically the same as before. **WE'D HAVE CONCRETE
EQUATIONS THAT COULD BE MAPPED INTO DIFFERENTIAL EQUATIONS IN THE ABSTRACT
DOMAIN OF THE CALCULUS ON A SYMBOL-FOR-SYMBOL BASIS.**

The
logic is as follows. We are trying to express "length"
at a point (or over a differentially small length. We
are trying to express length in (point) form. **R**
and **C** are already in intensive (point) form, both numerically and
dimensionally. That is, **R** and **C, **both in
units that are per unit length**,** are in a form that is **numerically
independent of length**. This **R **and **C **works
for any interval, no matter how short, and works at point scale, without
numerical change. **di/dt** is also in intensive form
(defined at a point). **We
as a culture need an intensive form for length.** Here
is what is needed, expressed in words. We need to interpret
"length" as "the property of length in per unit length units."

Let's
think of what we as a culture already do when we reason from measurement.
Our measurement procedures define things in terms of spatial variables
(length, area, volume, time) and other dimensions (voltage, charge, and
many others). The measurements are inherently finite
in nature. Still, we all speak of properties in INTENSIVE
FORM, defined for points. An intensive property or intensive
variable has a well-defined value for each point in the system it refers
to. For instance, we speak of "resistance per unit
length defined at a point" even though a point has 0 length. The
numerical-scaling argument we use to arrive at intensive properties and
variables is simple and nearly reflexive. To **intensify**
our properties and variables, we say that **the
property (or variable) at a point is the property (or variable) that we
get from a logic of interpolation from a finite scale to finer and finer
scales. The interpolation assumes homogeneity of the model down to a vanishing
spatial (and/or temporal) scale." **For
example, consider the notion of resistance per unit length. Let's idealize
the wire as a line. The resistance **R** expresses voltage gradient
per unit length, per unit current. For any interval that includes length,
the basic notion of resistance can be directly defined "per unit length".
Resistance per unit length at a point has the same numerical value and
the same units as resistance per unit length for some finite increment.
We do not use a limiting process.

For example, we find the value of internal energy per unit mass
at a point for a homogeneous system by dividing the internal energy of
the system by the mass of the system. We do not use a
limiting process. Other properties can be defined in
similar ways "per unit area" or "per unit length" over
finite areas, or finite volumes. But
the notion of "length (or area, or volume) at
a point" is an abstraction. This extremely useful abstraction
is much older^{(11) }than some of our
rigorous calculus formality^{(12)}.
In thermodynamics and elsewhere, we as a culture
don't intensify our extensive variables by a calculus argument of any kind.
**We just assume that the property we're considering is homogeneous. Then
we write our intensive variables directly.**

The abstract notion of length or area or volume "at a point" is already embedded in many of the intensive properties in common use. Using meter and second units, the intensive forms (point forms) of length, area, volume and time are:

**Length at a point: { 1 length/length (length unit)}**
**in meter units:
(1 meter) _{p}**

**An instant in time: { 1 ( time)/( time))
(time unit) } in second time units: ( 1 second )**

The dimensions of length, area, and volume are length
to the first, second and third power respectively. The coefficients are
the identity operator, 1 because, for even the smallest imaginable numerical
values of length, **l**, area, **a**; or volume **v**

** l**/**l** = 1
**a**/**a** = 1 **v**/**v** =
1 and **t**/**t** = 1

We can rewrite (**10**) as

Substituting the intensive form of length into (**10**) or (**12**)
in place of delta **x**, we may algebraically simplify the bracketed
expressions in the equation(s). This separates **R**, **L**, **G**
and **C **into numerical parts (**R**_{n}, **L**_{n},
**C**_{n}, and **G**_{n}) that are algebraically
simplified together, and unit groups that are algebraically simplified
together. We'll choose MKS-Georgi units, with v as volts, Q
as coulombs, and t as time in seconds. We get:

We could have continued the expansion process that produced
(**10**) and (**11**) and gotten more terms if we had wished to do
so.

The **di/dx** equation analogous to (**19**) is

Each
term consists of one (compound) dimensional parameter times a measurable.
These differential equations, when integrated to length delta **x**,
reconstruct the values that apply to that length **x+delta x**,
with no lost terms. **Every term in these differential equations passes
the loop test. ** We may map these differential equations
symbol-for-symbol into corresponding partial differential equations.
We may map these differential (or corresponding partial differential) equations
symbol-for-symbol into the domain of the algebra. These
equations are different equations from the Kelvin-Rall equations now used
in neurophysiology. They
contain the Kelvin-Rall terms, but have other terms as well.

An
important difference is the effective inductance term.
**For unmyelinated axons and dendrites in the neural
range of sizes, the numerical magnitude of R ^{2}C/4 is between
10^{12} and 10^{21} times larger than L, depending on dendrite
diameter and other variables. **
This term, which is much too small to measure in large
scale electrical engineering, is a dominant and practically important term
at neural scales in neural tissue.

Many
terms now thought to be "infinities" are also finite terms when
they are correctly interpreted in intensive form.

**Physical
domains that include dimensional parameters that represent measurable circumstances,
differ from the domain of the algebra.** Unless we
know this, we can discard important terms, and delude ourselves, or form
false infinities, and delude ourselves.

**Here
again is the procedure by which we can map a defined circumstance from
the real, dimensional world into a concrete model including equations;**
map that concrete model into abstract equations;
and, after abstract manipulations **map the results of our abstract analysis
back into a concrete model that may be tested against results in the world.
**

**PROCEDURAL RELATIONS
BETWEEN CONCRETE PHYSICAL MODELS AND ABSTRACT MATHEMATICAL REPRESENTATIONS
OF THEM**

** A
concrete physical model directly connected to the details of measurement
must be a finite scale model. The finite scale requirement
occurs for two related reasons. First, geometrical details
of specification involve lines and space-filling geometries that become
degenerate at the point scale of a differential equation.
Second, ALL our measurement procedures are finite scale procedures.
No real measurements occur instantaneously, or at a true geometrical point.
For these reasons, equations representing a concrete physical model in
step-by-step detail must be finite increment equations. The
measurements must be expressed in a system of consistent units ^{4}.**

**A (necessarily finite) concrete
physical model must be** c**ompletely specified before concrete finite
increment equations** **can be derived from that model. **

**1. Complete physical model specification
requires an input** **specification list setting out the variables of
the model,** **and setting out the physical laws at play in the model
with** **their corresponding dimensional parameters (3). **

**2. Complete physical model specification
requires a sketch** **showing the relationships of geometry, variables,
and** **physical law being modeled**

**3. Complete physical model specification
requires a full set of operational measurement procedures that define the
variables and dimensional parameters in the model.**

**From this physical model specification,
one or more concrete** **finite increment equations can be derived setting
out the** **quantitative relations of the model. **

** ALL
the terms in these concrete finite increment model equations MUST make
sense by the standards of measurement theory ^{5} before the concrete
equations can be validly transformed via a limiting argument into the abstract
domain (5) in which differential equations exist.**

** This
means that ALL terms must, when properly interpreted, be defined according
to a step-by-step measurement or calculational procedure consistent with
the unit system being used. Such a step-by-step procedure is an operational
measurement procedure. When this is done for every term, the model equation
is in consistent units and is in "measure theory consistent form."
**

**Iff all the terms in the concrete
finite increment model equations are reduced into measure-theory consistent
form, then the model equations can be mapped, using a conventional limiting
process, into differential equations that exist in the abstract domain
in which differential equations exist.**

** **

**NEW INFORMATION.
Dimensional parameters are the entities that express
concrete physical modeling laws in measurable form. The
dimensional parameters are not just numbers.
**

**Arithmetic with a dimensional
parameter in the measurable domain is only defined when it corresponds
EXACTLY with the definition of that dimensional parameter,
or when the dimensional parameter is interpreted
as part of a group operational measurement procedure performed at unit
scale for the spatial variables in the group.
**

**(An alternative statement is
that, for equations in gradient form that are otherwise ready to be reduced
to derivatives, groups of dimensional parameters and spatial variables
must be algebraically simplified at POINT SCALE.
The point forms ^{6} of the spatial
variables, in MKS units, are**

**length at a point: (1 meter) _{p}**

**volume at a point: (1 meter ^{3})_{p}
a
point in time: (1 second)_{p}**

**WORK IN THE ABSTRACT DOMAIN:**

**Calculation according to the rules of analysis
can then be validly done, on these valid differential equations.**

**REMAPPING OF ABSTRACT RESULTS
INTO A CONCRETE CONTEXT:**

**If a result of analysis is to be applied to the
measurable** **model from which it traces, groups in terms that act
as if they** **are dimensional parameters in the domain of calculus
can be** **interpreted as constructively derived dimensional parameters
in** **the measurable model system. Once this interpretation is made,**
**the calculus model may be mapped back into the measurable model**
**on a term for term basis, for use or checking.**

We have faced the following question repeatedly:

"**If this new procedure, with its new interpretation
of crossterms, is right, wouldn't that invalidate many results that everyone
has good reason to trust**?"

The answer is no.

We've
been dealing with the conduction line equations. We've
derived a new conduction equation, with new terms. In
the cases where the old derivation works well empirically, the new procedure
works just as well. The new terms are too small to matter.
In the case of neural function, where the old derivation fails on many
counts, the new derivation has the properties needed to describe behavior^{(13)}.
**The same equation, with the same terms, has enormously
different properties depending on what the numerical values of the dimensional
parameters in it happen to be. ** Here is a partial
expansion of the **dv/dx** line conduction equation:

**R** is resistance per length, **L** is inductance
per length, **G** is membrane conductance per length, and **C **is
capacitance per length.

For
many values of **R**-**L**-**G**-**C**, all the terms in (**1**)
that include products of **R**, **L**, **G**, and **C **will
be very small. However, some of these crossproduct terms
can be enormous for different values of **R**, **L**, **G**, and
**C**. Comparison of a wire case and a neural dendrite
case shows the contrast in values that can occur. **R**
is resistance per length, **L** is inductance per length, **G** is
membrane conductance per length, and **C** is capacitance per length.

For
many values of **R**-**L**-**G**-**C**, all the terms in (**1**)
that include products of **R**, **L**, **G**, and **C**will
be very small. However, some of these crossproduct terms
can be enormous for different values of **R**, **L**, **G**, and
**C**. Comparison of a wire case and a neural dendrite
case shows the contrast in values that can occur.

For a 1 mm copper wire with ordinary insulation and placement,
typical values of the dimensional parameters would be:

**R = 1.4 x 10 ^{-3} ohm/meter C = 3.14 x 10^{-7}
farads/meter**

**G = 3.14 x 10 ^{-8} mho/meter L = 5 x 10^{-7}
henries/meter**

Here is (**19**) with the numerical value of terms
set out below the symbolic expression of the terms:

Here are the corresponding dimensional parameter values
for a 1 micron diameter neural dendrite, assuming accepted values of axolemma
conductivity, capacitance per membrane area, and inductance per unit length
(volume conductivity 1.1 x 10^{4} ohm-meter g = .318 mho/meter^{2}
c=10^{-2}farads/meter^{2}.)

**R = 1.4 x 10 ^{12} ohm/meter L = 5 x 10^{-7}
henries/meter**

**C = 3.14 x 10 ^{-8} farads/meter G = 3.14 x
10^{-6} mho/meter**

Note that **R** is 10^{15} larger than in the
previous case of the wire.

**For a neuron situated as real neurons are, surrounded by a glial
cell and a fluid cleft, C would be about two orders of magnitude lower.
Even taking this into account, for the same equation,
cross product terms that are trivial in the wire case are dominant in the
neural dendrite case. **

For
the wire case, the numerical values of the primary terms (the numerical
values of the dimensional parameters **R**, **L**, **G**, and
**C**) are compared with the numerical values of the numerically most
important cross product terms in the voltage gradient transmission equation
below.

For this copper wire case, **none** of the cross product
terms are large enough to attend to in a practical modeling equation. The
sensible formula to use for the wire values of **R**, **L,** **G**,
and **C** is the same one the limiting procedure would produce:

In the neural dendrite case, some of the crossterms that
were trivial in the wire case become DOMINANT terms. For these neural value
of **R**, **L**, **G**, and **C**, two cross product terms
that were too small to attend to in the wire case have become dominant
terms. Magnetic inductance, **L**, has become too small to include,
because for these values of **R**, **L**, **G**, and **C**
the **R ^{2}C cross product term is 3 x 10^{22} times bigger
than L, and an important modeling term.**

These
finite crossterms make the difference between the brain as a high **Q**
system with sharply switched resonant communication and information storage,
and the brain as an overdamped system without resonance, that appears incapable
of any significant information processing at all because phase distortion
in Kelvin-Rall is prohibitively large. This makes the
difference between a role for inductance in ventricular fibrillation and
no role for inductance in ventricular fibrillation. This
makes the difference between epilepsy as a resonant phenomenon, and no
role for resonant coupling and resonant neural destruction in resonance.
This makes the difference between no plausible memory models that can handle
complex information, and switched resonance memory models that appear to
be able to handle the information brains do handle. That
is to say, the difference between the limiting procedure for deriving differential
equations, and the intensification procedure, involves matters of real
neuroscientific interest, including some plain matters of life and death.

(This is worth checking, worth fighting
about, worth getting right.)

Current
neurological modeling of dendrites uses the same equation that models the
wire, radically understating effective inductance. Sensible
modeling formula for the dendrite involves the following quite different
choice of terms. (Note that this equation is a selection
of important terms among others; it is an approximation that approximates
reality well enough in a particular regime of **R-L-G-C**.)

The effective inductance is 3 x 10^{22} times larger than would
be predicted in the Kelvin-Rall equation. The effective inductance term
goes as the inverse cube of line diameter; for a .1 micron diameter dendritic
spine, effective inductance would be 1000 times larger still.
**Lines that have been thought devoid of inductance, and incapable of
inductive effects such as resonance, have very large effective inductance.**

Note also
that the damping effect normally produced by **R** is now mostly produced
by an **R ^{2}G/4** term. Changing

The
notion that solutions are parameter dependent is well established in some
fields, for example viscous fluid flow. That notion applies
to these new terms. In the neural transmission case and
elsewhere, crossterms now dismissed as infinitesimals are finite, and some
are large. Electrical measurements testing the conduction
equations have been carefully and accurately done, over a limited range
of parametric values. They have been "experimentally perfect"
over that range. Nevertheless, too large an extrapolation
beyond that tested range of parametric values can be treacherous.
We say that this kind of extrapolation has been treacherous in neurophysiology.
**The dimensional parameters that we as a culture must all use in our
physical representations operate with type-restricted arithmetic rules.
Terms that we as a culture have thought to be zeros are
finite. We as a culture must learn to take this into
account when we extrapolate an equation that may be "a perfect match
to experiments" in one range of parameters into some very different
range of parameters.** This provides some reason for
caution when physical situations with enormously different values of physical
parameters are modeled with the same equations. "Perfect
fit" to experiments in one range of parameters does not insure even
a good fit in some other parametric range.

M. Robert Showalter Madison,
Wi. USA

Stephen Jay Kline Stanford,
Ca. USA

**APPENDIX 1**

Here
is the letter from **NATURE** in response to a very long and unconventional
submission that we sent them. We sent that submission,
to the most elite academic journal we knew, hoping to get their help in
securing checking of the intensification process. **NATURE**
did not give us the help we asked for. Perhaps they were
right not to do so. They did send the following letter, which was a kindness.

........................................................................

*****************************************

Here is the text:

Letter from Karl Ziemelis,

Physical Science Editor

NATURE dated 11 April 1997

Dr. M.R. Showalter

Department of Curriculum and Instruction

School of Education

7205B Old Sauk Road

Madison, Wi 53717

Dear Dr. Showalter,

Thank
you for your letter of 20 February and for your seven linked submissions.
I apologize for the unusual length of time that it has
taken us to get back to you, but please understand that the sheer volume
of interrelated material that you submitted took us rather longer than
we had hoped to read and digest. This delay is all the
more regrettable as the work is not in a form that we can offer to consider
for publication.

As you already clearly realize, the space available in the journal clearly poses a fundamental problem. In my letter of 31 October 1994, I had hoped to explain what we look for in a NATURE paper - in essence, we do not see ourselves as an outlet for exhaustive presentations of scientific problems (regardless of their potential importance), but as a forum for presenting brief and focused reports of outstanding scientific and/or technological merit. An additional, but equally important, characteristic of NATURE papers is that they should be self-contained: sub-dividing an extensive body of work into numerous (but intimately linked) "Nature-length" contributions is simply not a realistic option.

You are clearly appreciative of this fact, in that your stated intention is not so much to dominate our pages with your submissions, but to seek peer review on the work in its entirety, and defer until later the decision about what form any published form might take. This is not, however, a service that we could realistically offer - quite aside from the fact that it would be placing an unrealistic burden on our referees, we aim to send out for peer review only those papers that we judge to be appropriate (at least in principle) for publication in NATURE, in accordance with the criteria outlined above.

This
is not to deny that within your seven manuscripts there may be the essence
of a NATURE paper; but given the time constraints under which we work,
the onus must be on the authors, rather than on the referees and editors,
to construct such a paper from the vast amount of material supplied.
But I have to say that the need for extensive cross-referencing apparent
in the present manuscripts suggests to us that the likelihood that such
a paper would be readily forthcoming is not too high.
It is therefore our feeling that your interests would be better served
by pursuing publication of the work in a more specialized journal having
the space to accommodate the lengthy exposition that your work so clearly
requires.

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.

Once again,
I am very sorry that our decision must be negative on this occasion, but
I hope and trust that you will rapidly receive a more favorable response
elsewhere.

Yours sincerely,

Karl Ziemelis

Physical Science Editor

**This is a rejection from NATURE,
not an acceptance. The favorable language in it stands
for much less than a conventional peer review. Even so,
we believe the letter does tend to support the view that our work is plausible
enough, and important enough, to be worth checking. **

**Some background on Stephen J. Kline and M. Robert Showalter**

We
have each been thinking about the interface between measurement and modeling
for much of our professional lives, and have been thinking about this interface,
together, for more than a decade. Showalter has been
more focused on the math-physics interface set out here, and issues related
to it. Steve Kline is the more physical of the two of
us, the more keyed to images. Showalter is the more concerned
with formality and logic. We share an interest in the
history of ideas of modeling and analysis, and a sense of modeling and
analysis as a body of human patterns and assumptions that exist, and may
be improved, in a historical context. The point of departure of our work
together is Steve's **SIMILITUDE AND APPROXIMATION THEORY** (McGraw-Hill
1964, Springer-Verlag 1986). Kline has invested several
thousand hours in our work together, much of it supervising and disciplining
Showalter, and making sure that the work stayed physical, and stayed connected
with known problems. Showalter has put in much more time
than Kline.

**Stephen Jay Kline** is Clarence
J. and Patricia R. Woodard Professor, Emeritus, of Mechanical Engineering
at Stanford University, and also Professor of Values, Technology, Science,
and Society at Stanford. He is author of more than 170
papers and two books, and editor of six other volumes. Kline
was the founding Chairman of the Thermosciences Division in Stanford's
Department of Mechanical Engineering. He is also one of the four founding
faculty members in Stanford's program in Science, Technology, and Society.
The largest number of his technical publications are in fluid mechanics,
and report results of experiments, new forms of analysis, computation,
and the description of new instrumental methods. In 1996
the Japanese Society of Mechanical Engineers (JSME) decided to ask the
outstanding contributors to various fields how they had pursued their work.
They designated Kline as the most important contributor to fluid mechanics
in the 20th century. While some other worker might have
been reasonably chosen, Kline's distinction in experimental and computational
fluid mechanics is undoubted.

**Stephen Jay Kline** has received the following **honors**
and **awards**:

**ASME Melville Medal for best paper of the year 1959.**

**ASME Centennial Award for career contributions to fluid mechanics**

**George Stephenson Prize of the British Institution of Mechanical
Engineers**

**Golden Eagle award from U.S. CINE and Bucraino Medal (Italian) for
the film "Flow Visualization" (30 minute educational film).**

**Gold Medal of the Chinese Aero/Astro Society**

**Member, National Academy of Engineering**

**Honorary Member, ASME (the highest honor ASME gives)**

**Fellow, American Association for the Advancement of Science**

Books by **Stephen J. Kline **

**SIMILITUDE AND APPROXIMATION THEORY **(McGraw-Hill, 1964; Springer-Verlag,
1986)

**CONCEPTUAL FOUNDATIONS FOR MULTIDISCIPLINARY THINKING **(Stanford
University Press, 1995)

for more on S.J.Kline, see http://www-tsd.stanford.edu/tsd/ressjk.html
klinelink

**M. Robert Showalter **is a graduate
of Cornell University. Rather than undertake an academic
career, Showalter set out to make "analytical invention" a possibility
and to make "analytical engineering" more efficient.
He was interested in questions like "**How do you define and design
an optimal structure in a fully specified, complicated, fully commercial
circumstance?**" For instance, suppose an airplane
design needs a wing, to mount an engine and connect to a specific fuselage.
How do you arrive at a FULLY optimized design for that wing, in a case
of real technical complexity, with "optimal" a defensible word
in terms of all the technical considerations that pertain at all the levels
that matter in the case (aerodynamics, structure, fabrication, maintenance,
cost)? How do you even approach such a job?
To make sense of such a job would require superposition, coupling together,
and repeated solution of sometimes complicated systems of equations.
It would require new techniques of specification, so that system problems
could be packaged in ways that could be isolated for analysis, and then
reinserted into context. It would require sharp identification
of "linchpin" problems that, if solved, would enlarge the realm
of the possible. It would require solution of some "linchpin"
problems concerning mathematical technique. Showalter
found himself forced to attend to such issues of mathematical technique.
For his adult life, he has believed, with Kline, that the "miracle
of modeling and analysis" is not how good mathematical modeling sometimes
is, but how seldom useful, and how mysteriously treacherous, analytical
modeling is when it is applied to problems complicated enough to be of
commercial interest. Although it may be "a poor
craftsman who blames his tools" Showalter has long had a focused interest
in finding and fixing the defects in our culture's analytical tools.

In
the course of this work, Showalter learned technical fear, a sense of the
limitations of his own mind compared to the complexities of the world,
a strong sense of the necessity for specification and clear definition,
immunity to the notion that any person or group had to be right about anything,
and a belief that the most sophisticated and usable patterns for technical
description and technical thinking had been evolved, by tens of thousands
of smart and motivated people over many years, in patent practice.
He came to believe, and still believes, that specification for optimization,
and specification for clear theorizing under defined circumstances, is
best done building on these patent usages. These patent
usages involve pictures, step-by-step specifications related to the pictures,
and definitions, called claims in patent practice, that define the subject
being treated in sharply delineated natural language. Mathematics
that fits in a physical context, Showalter believes, should be clearly
connected to that context as set out in a picture-specification-definition
format. He believes that the conveniences of abstraction
must link clearly to this sort of specification and concreteness if a mathematical
result is to be applied, and applied consistently, and applied by human
beings.

Showalter
spent some time, with the support of investors, as an "analytical
inventor," working to optimally redesign internal combustion engines.
Kline worked with him on this project for several years, about half time.
Kline and Showalter became friends as well as coworkers during this period.
During this same period, a former Vice President of Engineering of Ford
Motor Company worked about half time on Showalter's project. Showalter
has 23 U.S. patents. He achieved some "choreographed"
control of mixing and combustion related flows, and got much reduced emissions,
and significantly improved fuel economy, by doing so. The
work was intended to meet the most rigorous EPA emission standard, including
the .4 gram/mile NOx standard. That standard was rescinded
and never enforced. After the emissions work, in an effort
to salvage his enterprise, Showalter worked out ways that appeared to have
the practical potential to reduce engine friction about tenfold.
However, he could not achieve commercial ring oil control on the friction
reduction package, and that oil control was key to all the rest.
It was not a hit or miss problem: he had to model coupled elastic and hydrodynamic
equations on the ring design to keep track of ring oil flows.
The coupling terms that had to be accounted for were all mismodeled as
"infinitesimal" by limiting arguments that, Showalter now knows,
did not accomodate the arithmetical limitations of the dimensional parameters.
Perturbation approaches could not do this job. Showalter's
project failed, and he failed physically for a time.

The
project may not have failed, and would not have failed as it did, if Showalter
had not had epileptic difficulties that incapacitated him for several years.
One inconvenience during this period was loss of the ability to read. Some
other skills were lost, too. Showalter found the work
required to regain these skills hard but interesting.
Showalter earned a Professional Engineer's designation toward the end of
this period. During this period, Showalter gained a steady
interest in brain function, in health and disease, and became convinced
that current brain models, such as they were, were in gross error.

In
1988, Showalter enrolled in the Department of Education, at U.W. Madison,
because of an interest in how people learn to read, and an interest in
other issues in learning as a developmental process.
He was interested, from the first, in neural sciences as well as education,
and in the interfaces between these fields. Although some neural scientists
took an interest in him, he could not, as a practical matter, enroll in
the neural science program because he believed that the Kelvin-Rall transmission
equation was grossly wrong, and Kelvin-Rall is a well-enforced article
of faith in the neural sciences. By about 1990, Showalter,
with Kline's help, had identified the correct neural transmission equation,
with effective inductances 10^{10}-10^{19} greater than
that of Kelvin-Rall, and fit it to David Regan's data, and to other data.
Showalter became convinced that nothing realistic or useful in brain modelling
could be done, by himself or by anyone else, until the error in the neural
transmission equation was found and fixed. Therefore,
as a modeler, he had nothing he could do but keep at the problem, with
Kline's help, until it cracked.

Showalter
has long believed that many questions about development and learning that
are important in education hinged on the same neural transmission issues
that mattered in neural science and medicine.

Especially
since 1992, Showalter has been working nearly full time, with Kline's help,
to find the defect at the interface between measurement and analysis that
he and Kline knew must be there. Together, we have found
that defect, that traces to the arithmetical restrictions on the dimensional
parameters. We report the fix to this modeling difficulty
in this transmission.

In
Showalter's view, the most serious mathematical impediment to analytical
engineering, analytical invention, and analytical modeling applied to science
that has existed in the past is this interface defect that has caused misspecification
of differential equations. That defect, pending checking,
has now been found and fixed. He hopes to put the new
tool to use in neural modelling, the study of efficient and comfortable
teaching, and elsewhere.

1. **DEFINITION:** A **dimensional parameter **is a "dimensional
**transform ratio number" that relates two
measurable functions** **numerically and
dimensionally. The dimensional parameter is** **defined
by measurements (or "hypothetical measurements") of two**
**related measurable functions A and B. The dimensional
parameter is** **the algebraically simplified
expression of {A/B} as defined in A = {{A/B}} B. The dimensional parameter
is a transform** **relation from one dimensional
system to another. The dimensional** **parameter
is also a numerical constant of proportionality between** **A
and B (a parameter of the system.) The dimensional parameter is**
**valid within specific domains of definition of
A and B that are** **operationally defined
by measurement procedures. **

**Here are some dimensional parameters: mass,
density, viscosity, bulk modulus, thermal conductivity, thermal diffusivity,
resistance (lumped), resistance (per unit length), inductance (lumped),
inductance (per unit length), membrane current leakage (per length), capacitance
(lumped), capacitance (per unit length), magnetic susceptibility, emittance,
ionization potential, reluctance, resistivity, coefficient of restitution,
and many more.**

**See section devoted to the dimensional parameters
in the main text.**

**2. Showalter, M.R. REASONS TO DOUBT
THE CURRENT NEURAL CONDUCTION MODELsubmission E.**

**3. Feynaman, R.P., Leighton, R.B, &
Sands, M. THE FEYNMAN LECTURES ON PHYSICSV. 2 Table 18-1 Addison-Wesley
1964**

**4. An equation is in consistent units
if it can be expressed, term for term, in any unit system consisting of
the same system of operational measurement procedures and different units
of length, mass, and time raised to exponents n, m, p**

**L ^{n} M^{m} t^{p}**

**5. The analytical foundations of measurement
theory related to the requirement here were set out by J.C. Maxwell in
sections 1] - 6] of A TREATISE ON ELECTRICITY & MAGNETISM v.1 Dover.
p 1-6. Maxwell's algebraic examples must be interpreted in light of Bridgman's
emphasis that the units of length, mass, and time in the algebra exist
in the tightly specified context of detailed operational measurement procedures.
These procedures are typically complicated enough to require specification
by a sketch and a short verbal instruction. ALL members of a set of consistent
unit systems have EXACTLY the same operational procedures, and apply different
units of length, mass, and time to these consistent operational procedures.**

**6. the p subscript is for labeling,
and does not effect the arithmetical function of the numerical and dimensional
parts of these point forms.**

**7. Maxwell, J.C. A TREATISE ON ELECTRICITY
& MAGNETISM V. 2 Dover Press, modification from 3'd edition, 1891,
twelve years after Maxwell's death.**

**8. See Kline's Stanford Web page, and
its discussion of the work on streak formation, and the change in paradigm
that it involved.** **http://www-tsd.stanford.edu/tsd/ressjk/html.**

**9. Submission A. Showalter, M.R., Kline,
S.J. Modelling of physical systems according to Maxwell's First Method.**

**10. (For our purposes, we can call this
the MKS-volt-coulomb system.) If you look in Rojansky's ELECTROMAGNETIC
FIELDS AND WAVES (Dover) you'll see tables inside the front and back covers
that give a nice sense of how constrained the choice of dimensional system
is when mechanical, electrical, and magnetic units are combined. **

**11. In PRINCIPIA MATHEMATICA
(1687) Book 2, following prop XL, Isaac Newton discusses the propagation
of sound. He employs two numbers that moderns would call "dimensional
parameters" in his treatment. The first is mass of air per unit volume
at a point. The second is compressibility of air per unit volume at a point.
These dimensional entities are only experimentally definable in finite
terms, but they are set out in intensive (point) form. Numerically and
dimensionally, the intensive and extensive form of these numbers is the
same. **

**12. Compare Newton in the 1680's versus
the work of Weierstrass and his school in the 1870's, set out in H. Poincare
L'oeuvre mathematique de Weierstrass" Acta Mathematica,
XXII, 1989-1899, pp 1-18.**

**13. Showalter, M.R. submissions E, and
H. **