Appendix 1: Derivation of a finite increment equation from
a coupled physical model, showing combined effect terms.

Most mathematical models of physical circumstances start
by matching one or more differential equations from a handbook with a physical
case. Thereafter, analysis is carried out according to the rules of abstract
mathematics. In this usual procedure, the differential equation is not
derived from a physical model, but is applied to that model and taken as
an operationally perfect or good enough match.

However, the differential equations themselves are sometimes
inferred directly from physical models in a step by step way from a sketch-model.
This process is called "derivation," and is taught in engineering
schools.

To derive a differential equation
from a physical model:

we
construct a finite scale model that sets out the laws and geometry to be
represented;

then
we derive (one or more) finite increment equation(s) that map that
finite model.

After a finite increment equation
that represents our model has been defined, we may pass that finite equation
to the infinitesimal limit to yield a differential equation.

Equation definition can take special
attention if our finite model includes coupled effects. In such a
case, two equations are implicitly defined in terms of each other.

One of the simplest and most important examples is current
change and voltage change along a length of conductive line (such as a
wire or a length of neural tube.) Current drops **i** are coupled
to voltage and voltage drops according to logic like the following:

**i over the interval is a function of v at x and x+delta x**

** which is a function of i at x and x+delta x**

** which is
a function of v at x and x+delta x**

** which
is a function of i at x and x+delta x**

** and
so on**

Voltage drops are coupled to current and current drops
in the same nested way.

We have to be able to represent
the coupled effects that occur correctly at finite scales, before we can
take the limit of those terms, at successively smaller finite scales, to
the infinitesimal limit. (Current
procedure does not ask for meaningful finite coupled effect terms, and
applies "limiting logic" to these undefined terms, which
are invariably dismissed as 0's or labelled as infinities.)

**Fig. 1** above shows a conducting line that
could be neural conductor.

** v** = voltage **i**
= current

** x** = position along the line alpha=
arbitrary length interval

** R** = resistance/length
**L**=electromagnetic inductance/length

** G**= membrane leakage conductance/length
**C**=capacitance/length

**Fig. 1** above shows an arbitrarily chosen length
alpha, of arbitrary magnitude, which we will call delta **x**.

R, L, G, and C are natural law operators
(Appendix 2.) They represent physical
laws, and are defined as the ratio of one measurable to another under particular
measurement circumstances. The natural law operators, which implicitly
represent much measurement detail, are our interface between the detailed
measurement procedures of physical reality and abstract equation representations
of physical circumstances. The arithmetical
properties of the natural law operators are justified by inductive generalization,
not axiomatic proof. We have no provable
reason to think R, L, G, and C have exactly the arithmetical properties
and restrictions of numbers. In the derivation
below, we'll operate on terms including the symbols R, L, G, and C in the
usual algebraic way, stopping short of algebraically simplifying the terms.
We'll not interpret these terms numerically or physically here in appendix
1, leaving that for appendix 2.

To derive line differential equations in dv/dx and di/dx,
we first need finite difference equations, (delta **v)/(delta x)**
and (delta **i)/(delta x)**. For the finite equations,
we'll be writing out terms that have usually been understood to exist,
but that have been called infinitesimal and neglected. Let's
consider the coupled effects physically.

Voltage drop over delta **x** depends on current.
Current over length delta **x** varies if charge carriers are stored
or discharged in capacitance over the interval delta **x**. The capacitive
effect will depend on dv/dt. Current over delta **x** will also be lost
from membrane leakage over delta **x**. That loss will depend on v.
Voltage drop depends not only on current, through R, and L, but also depends
on interactions between voltage and G and between dv/dt and C.

In an exactly symmetric way

Current drop over delta **x** depends on voltage.
Voltage over length delta **x** varies with inductive reactance,
L di/dt. Voltage over delta **x** will also be lost from interaction
between resistance and current, Ri. Current drop depends not
only on voltage, through G and C, but also depends on interactions between
current and R, and between di/dt and L.

The idea that di/dt depends ONLY on G and C neglects effects
that act over delta **x**. The idea that dv/dt depends ONLY on
R and L neglects crosseffects over delta **x**. (If
the effects are finite over finite lengths, they MUST be represented in
the differential equations that are integrated to represent these finite
lengths.) Appendix 2.

Let's derive voltage and current equations that include
crosseffects. We'll write our voltage and current functions as **v(x,t)**
and **i(x,t)**. We assume homogeneity and symmetry for our conductor.
We assume that, for the small lengths of interest, the average voltage
(average current) across the interval delta **x** is the average
value of voltage (current) at the endpoints of the interval delta **x**.

Writing down voltage change as a function of the natural
law operators and variables that directly affect voltage, and centering
our interval at x, so that our interval goes from **x-(delta x)/2**
to **x+(delta x)/2**, we have:

The current change equation is isomorphic:

Note that :

Equation (1) includes **i(x+(delta x)/2,t)** and its time derivative.

**i(x+(delta x)/2,t)** is __defined__ by equation (2).

Equation (2) includes **v(x+(delta x)/2,t)** and
its time derivative.

**v(x+(delta x)/2,t)** is __defined__ by equation
(1).

Each of these equations requires
the other for full specification: each contains the other.

If the cross-substitutions implicit in these equations
are explicitly made, each of the resulting equations will also contain
the other. So will the next generation of substituted equations, and the
next, and so on. This is an endless regress. Each substitution introduces
new functions with the argument **(x+(delta x)/2)**, and so there
is a continuing need for more substitutions. To achieve closure,
one needs a truncating approximation.

Expression of current, voltage, and their time derivatives at x, the midpoint
of the interval, truncates the series.

A key point of this paper is that we have not been sure of how the arithmetic (and the dimensional or scale limitations) of these symbols have worked. If we ask

"If N effects represented by distributed parameters interact on or in ONE piece of space, should the numerical value of that piece of space multiply into the expression representing the interaction ONCE or N times?"

the conventional answer is "N times." But we are not actually sure of this, and the answer that consistency requires is "once." (See Appendix 2.) Let's proceed with these substitutions, associating symbols without interpreting them numerically or physically.

For example

is

which expands algebraically to

These terms would be simpler if voltage averages and derivative
averages were taken at the interval midpoint, **x**, as follows:

How may terms like those of (6) be interpreted, physically
and numerically, at finite scale? In these expressions, two
natural law operators are EACH associated with the SAME interval of length.
Do the lengths multiply? On what authority do we say that the
lengths multiply? Are there restrictions on the scale at which the
multiplication can be done? If the lengths do multiply, what
does this represent physically? Does the multiplication make
numerical sense, and is that multiplication consistent with tests the expression
must pass? It turns out that if we apply
standard arithmetical rules to these crossterms, we are led to mathematical
and physical inconsistencies. (Appendix 2)
We have no axiomatic reason to be surprised by this.

The equation below shows voltage change over an interval
of length delta **x**, divided by the length delta **x**
to produce a gradient form analogous to a derivative. Terms
derived from three stages of cross substitution are shown. Symbols
are grouped together and algebraically simplified up to the point where
the meaning of further algebraic simplification of relations in the dimensional
parameters **R**, **L**, **G**, **C**, and delta **x**
becomes unclear. Expresssions in curly brackets are NOT YET
DEFINED.

The current gradient equation over the same interval is
isomorphic to **7** with swapping of **v** for **i**, **R**
for **G**, and **L** for **C**.

Whenever coupled physical effects
act over an interval of space, combined effect terms are to be expected.
Rules for their interpretation must be found. Those
rules are beyond the authority of the axioms of pure mathematics, but consistent
rules for interpreting these expressions can be inferred from mathematical
experiments. (Appendix 2)

Combined effect terms such as those shown here are seldom derived, because they are thought to always vanish in the limit. That assumption embodies an assumption about scale or numerical limits on the arithmetic involved.

However, expressions such as those in the curly brackets
of (7), interpreted in a consistent way, are finite, and yield finite terms
in differential equations. Often such combined effect terms are negligible,
but sometimes they are important.

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

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

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

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

Appendix 2: REPRESENTING PHYSICAL
MODELS AS ABSTRACT EQUATIONS: PROCEDURES INFERRED FROM EXPERIMENTAL MATHEMATICS

Procedures for representing physical models in equation
form cannot be determined from our axioms because our axioms are limited
to abstract domains. But representation procedures can be examined by means
of experimental mathematics. Valid representation procedures must be consistent
with computational consistency tests. Current techniques for calculating
the interaction of several natural laws over a spatial increment fail tests
that valid representation requires, and are ruled out. A consistent technique
is proposed. According to the proposed technique, terms in some equations
that have been thought to be infinitesimal are finite. Implications in
neural medicine and other fields that deal with the brain appear to be
large (see appendix 4).

Some seem to feel that mathematics is axiomatic construction
and nothing else, but that sometimes, nevertheless, that axiomatic construction
can be mapped to some useful work. The jump from the abstract to the concrete
is held to occur by some discontinuous and unexplained process. A smoother,
better explained transition between the abstract and concrete seems desirable.
Mathematics already interfaces with experimental usages, and has long been
pushed toward experimental approaches by the computer^{(1)}.

G.C. Chaitin has shown that many things in pure math are
"true for no (axiomatically provable) reason at all^{(2)}."
Chaitin suggests that where existing axioms don't apply, new organizing
assumptions may be considered, and may be useful. K. Godel advocated experimental
approaches in mathematics on similar grounds^{(3)}.
Even the interior of mathematics has experimental aspects. Some degree
of experimental math seems justified and useful even in number theory.

The interface between abstract mathematics and the representation
of physical circumstances can be investigated experimentally, as well.

There may be many reasons to investigate this interface
between abstraction and concrete representation. My main one is concern
about the correct form of the neural transmission equation. Medically important
differences in neural line inductance, that can be 10^{18}:1 or
larger, hinge on a question that is beyond the jurisdiction of the axioms
of pure mathematics. That question can be clarified, and perhaps entirely
resolved, with experimental mathematics.

Conclusions based on mathematical experiments always lack
the certainty of an axiomatic basis. Even so, some much-tested conclusions
may be useful, and using them as new assumptions can permit useful logical
work that would not be possible otherwise. Experiment-based inferences
(assumptions) are now widely used in cryptography and other computer-based
fields.

Results of mathematical experiments cannot prove with
axiomatic certainty, but can disprove. When mathematical experiments show
counterexamples to an assumption, that assumption has been ruled out.

Even within pure math, where axioms reign, there are good
reasons to use experimental approaches to test and organize ideas that
we may wish to use, where our axioms cannot be brought to bear. This supplements
axiomatic usages without violating them.

In mathematical representation of PHYSICAL circumstances,
set out in terms of experimentally derived physical laws, we are using
mathematical techniques beyond where the axioms of pure math apply. If
we are to proceed at all, we must use experimental mathematics.

Here is the logic that experimental work has:

E1.
In experimental work, candidate assumptions are somehow recognized or guessed.

(No testing
can happen before we focus on something to test.)

E2.
Candidate assumptions are tested against evidence. So long as an assumption

survives all
tests, it is used (with some wariness) as a provisional assumption.

E3.
Assumptions that evidence contradicts are rejected, or the assumptions
are modified

so that they
do fit evidence.

If we use these experimental approaches
we may sometimes usefully organize, extend, and focus our knowledge beyond
the realm of our axioms. If we do not use these
approaches, we cannot go beyond our axioms at all.

When we mathematically represent a physical circumstance,
we are beyond our axioms. Let's call that representation process "p-m
representation" for "representation from physical model to mathematical
model."

(We'll assume that a workable p-m representation can be
reversed in a m-p representation so that we can start with a physical model,
convert it into a statement in abstract mathematics, operate on the abstract
mathematical statement, and then relate that statement in abstract mathematics
back to the physical model without misinterpreting or losing information
of interest to us.)

We have NO axioms for p-m representation
or m-p representation. We must determine the representation procedures
of valid p-m representation and m-p representation on EXPERIMENTAL grounds.

Here is the p-m representation problem in more detail.
When we derive an equation representing a physical model, reasoning from
a sketch and other physical information, we write down symbols and terms
representing physical effects. We may write down several stages of symbolic
representation before we settle on our "finished" abstract equation.
As we write our symbols, we implicitly face the following question:

Question:
WHEN can we logically forget that the symbols we write represent a physical
model? WHEN can we treat the
equation we've derived from a physical model as a context-free abstract
entity, subject only to
the exact rules of pure mathematics?

We can never do so on the basis
of rigorous, certain, clearly applicable axioms. There are no such axioms.
We cannot avoid making an implicit assumption that says

"THIS
equation can be treated as a valid abstract equation, without further concern
about its

context or origin, because it seems right
to do so, or because it is traditional to do so.

We have made the jump from concrete representation
to valid abstraction HERE."

This assumption may happen to be
right in the case at hand. But the assumption about p-m representation
is not provably true from the axioms and procedures of pure mathematics.
People go ahead and make these sorts of assumptions
as they work. They cannot avoid doing so. Right or wrong, they are making
"experimentally based" assumptions in their representation-derivations.
People have made these implicit assumptions without recognizing the essentially
experimental nature of their proceedings. It is better that this experimental
nature be recognized, so that consistency checks can be applied to the
unprovable steps. Any inconsistencies involved with these implicit steps
may then be identified.

For any particular case of p-m representation, decisions
are being made in a context of EXPERIMENTAL MATH at the interface between
abstract math and physical circumstances. If a counterexample or inconsistency
pertaining to a p-m representation usage is found, that is an extra-axiomatic
circumstance. The extra-axiomatic usages that are failing as p-m representative
tools should be modified so that they pass the consistency tests right
p-m representation takes. Such modifications may disturb habits, but they
need not, and commonly cannot, disturb the axioms of pure mathematics.

The Kelvin-Rall neural transmission equation derivation
is based on an implicit, unprovable assumption about p-m representation:.

USUAL P-M REPRESENTATION
ASSUMPTION: Abstract mathematical usages
and p-m representative usages are the SAME. When we are representing
a physical circumstance with mathematical symbols, those symbols
are NUMBERS, and nothing more, the instant they are written down.
All our rules of abstract mathematics apply immediately to our symbolic
constructions.

On the basis of this USUAL P-M REPRESENTATION ASSUMPTION,
all of the crossterms in equations 7, 8,
and 9 are ill defined. Here is equation 7, derived in detail in Appendix
1. At a finite scale delta **x** each of these crossterms
(terms below the first line) must correspond to finite physical effects.
We have NO axiomatic guidance for computing these compound expressions.

We are referring to products of p-m representation procedures,
not to axiom-based entities. We must judge the procedures
we use to compute these compound expressions by experimental standards.
Do these representations map the territories we expect when we check them?
We may if necessary modify those procedures for consistency
without violation of any axiom.

We must know what these representations mean numerically.
If our computation is valid, the magnitude of a term at a set value of
delta **x** and a set value of independent variable must be unique.
After all, our limiting argument is an argument that deals with a decreasing
sequence of finite terms. Before we can
validly take the limit of equation 7, set
out in the main paper, and derived in more detail in Appendix
1), and derive a differential equation from it, we must know the magnitude
of the crossterms for any
finite delta **x** we choose. If
we proceed according to the USUAL P-M REPRESENTATION ASSUMPTION, we find
that our crossterms are not well defined. Equation
7b is the finite increment form of equation 7, which is in (delta v)/(delta
x) form.

The indeterminacy of these crossterms according to the
USUAL P-M REPRESENTATION ASSUMPTION can be shown in the following ways.
The difficulties set out below also apply to other crossterms
that represent the combination of physical laws over an increment of length,
area, volume, or time.

Numerical indeterminacy under "permitted" algebraic
manipulations:

We have been taught to assume that the crosseffect-containing
terms such as the curly bracketed terms in (7) consist of symbols that
are "just numbers." We should be able to algebraically simplify
each of these crossterms in many different sequences that involve dimensional
unit changes, so long as the end of each of the sequences is in the same
dimensional units. The numerical values of all such paths should be the
same. They are not. See Fig. 1 below. An algebraically
unsimplified dimensional group that includes products or ratios of dimensional
numbers, such as one of those in curly brackets in **(7)**, is set out
in cm length units at **A**. This quantity is algebraically simplified
directly in cm units to produce "**expression 1**." The same
physical quantity is may be translated from **A** into a "per meter"
basis at **C**. The translated quantity at **C** can then be algebraically
simplified to **D**. The expression at **D**, expressed in meter
length units, is converted to a "per cm" basis to produce "**expression
2**." __Expression 1 and Expression 2 must be the same, but they
are not. The calculation is not consistent with itself.__

By repeating different "valid" computational loops in this way,
any of the crossterms in curly brackets in (7) or (7b) can be changed
to any value at all, large or small. This is not the valid
arithmetical behavior that we conventionally and thoughtlessly expect!
The loop test of Fig. 1 above shows that these crossterms
are meaningless as usually calculated, and the reason is as follows:

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, as well.

After
algebraic simplification, 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.

THIS IS AN EXTRA-AXIOMATIC CONCERN:
WE HAVE A REPRESENTATION PROCEDURE THAT NEEDS MODIFICATION, BUT NO CHALLENGE
TO VALID AXIOMS.

Contradiction between differential equations and the models
they came from.

Suppose we assume that the symbols in the crossterms are
all "just numbers." When we take the limit as delta **x**
goes to zero on that assumption, these crossterms are all infinitesimal.
So the differential equation we derive on this basis lacks
these crossterms.

We take our differential equation, and integrate it back
up to a specific scale delta **x**. We get an equation
that lacks the crossterms that we know existed at scale delta **x**
in the first place. The values
at the same point, derived by two "correct calculations" are
inconsistent, and can be very different.

THIS IS AN EXTRA-AXIOMATIC CONCERN:
WE HAVE A REPRESENTATION PROCEDURE THAT NEEDS MODIFICATION, BUT NO CHALLENGE
TO VALID AXIOMS.

Crossterms also fail a standard test map-representations
should pass - the test that the whole should equal the sum of its parts:

In physical representations, wholes should equal the sums
of which they consist. Consider any of the terms below the
first line of 7 or 7b. Suppose any term, evaluated at interval delta **x,**
is instead set out as the sum of a number of intervals adding up to interval
delta **x**. If delta **x** is divided into n pieces, and
those n subintervals are computed and summed, that sum will be is only
1/nth (or 1/n^{2}) the value for the same expression computed over
interval delta **x**, taken in one step. We can make the value
of the term on the interval delta **x** vary widely, depending
on how many subintervals we choose to divide delta **x** into.
This cannot represent PHYSICAL behavior. These terms are supposed
to represent physical behavior.

THIS IS AN EXTRA-AXIOMATIC CONCERN:
WE HAVE A REPRESENTATION PROCEDURE THAT NEEDS MODIFICATION, BUT NO CHALLENGE
TO VALID AXIOMS.

The USUAL P-M REPRESENTATION ASSUMPTION
is that the symbols we write down are "just numbers" the instant
we write them down. In the case of these crossterms that represent multiple
physical effects over the same spatial increment, the usual assumption
fails. So we need to look more closely at the details of what we are representing,
the symbols we use to do that representing, and the procedures that apply
to those symbols. We need representative
procedures, that interface with our physical model on the one side, and
interface with abstract mathematical usages on the other side, that avoid
the representative contradictions shown above.

When we look at how physical models are represented by mathematics, we have NO axioms to rely on, and we have NO valid intuition to guide us. We must rely on the ordinary patterns of experimental investigation.

According to E-3 below, we are seeking a modification
of current p-m representation procedure that maps these crossterms validly
into axiomatic, abstract mathematics *without changing other p-m representation
procedures, now currently established, that we have no reason to doubt.
*

*We are violating no valid axiomatic
principles when we use experimental approaches to find a p-m representation
that passes all operational tests needed to validly map to abstract equations.
* If a valid p-m representation procedure is found, that
empowers axiomatic mathematics, and in no way diminishes it.

If we use the patterns of experimental logic and investigation
with the same care that other people have applied to many other technical
problems, operationally valid experimental rules for representation can
be found, tested, and verified.

Here again
are the experimental patterns:

E1. In experimental work, candidate assumptions are somehow recognized or guessed. (No testing can happen before we focus on something to test.)

E2. Candidate assumptions are tested against evidence. So long as an assumption survives all tests, it is used (with some wariness) as a provisional assumption.

E3. Assumptions that evidence contradicts
are rejected, or the assumptions are modified so that they do fit evidence.

This is not axiomatics, but we are beyond the axioms of pure mathematics,
and have no other axioms. Experimental logic and investigation are all
we have.

Operational definition of representative entities and inference
of arithmetical rules that apply to them in p-m representation.

The jump between a physical system
model, defined in terms of drawings, measurement procedures and other detail,
and the abstract mathematical representation of it is taken for granted,
but not usually set out clearly. S.J. Kline and I have
tried to understand at a defined, procedural level how measurable circumstances
are mapped to mathematical equations. Kline had written a respected
book tightly connected with the subject^{(4)}.
A first task was to identify the natural law operators,
sometimes called dimensional parameters, in procedural detail.

The natural law operators are the entities that interface
between our experimental measurements and the formalities of abstract,
symbolic mathematics. Here are some directly measurable
natural law operators (often referred to as properties):

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, . . . .

There are many, many more.

**All are defined according to the same pattern:**

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

Example: 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.) (Unrelated measurables
could also be expressed as ratios, but such ratios would describe only
one point, not an infinite set of points.)

__The natural law operators
are not axiomatic constructs. They are context-based linear constructs
that encode experimental information__.

We are concerned with the arithmetical properties of the
natural law operators because of the inconsistencies related to crossproducts
including spatial entities that have been discussed above.

Let's review the arithmetical properties relating to the
natural law operators that we have no reason to doubt, and much reason
to be sure of:

Natural law operators work just
like dimensional numbers when they are used in exact correspondence with
the equation that defines them.

For example, resistance per unit length is the numerical
and dimensional transform that expresses Ohm's law, and acts "just
like a number" in expressions of Ohm's law.:

Natural law operators may be combined to form compound
natural law operators.

**DEFINITION: A compound natural law operator
is a "dimensional transform ratio number" that relates two measurable
functions numerically and dimensionally. The compound natural law operator
is a transform relation from one dimensional system to another. The
compound natural law operator is also a numerical constant of proportionality
between one measurable value and another. The compound natural law
operator is the product or ratio of two natural law operators, sometimes
in association with a spatial increment. The compound natural law operator
is valid within specific domains of definition of the natural law operators
that define it.**

Natural law operators act "just like numbers"
when they multiply or divide to form a compound natural law operator that
does not include an increment of space (length, area, volume, or time.)
The Heaviside equations, the conductance equations that apply
to a line conductor, such as a wire, are examples. Here is the Heaviside
equation for voltage, and the constructed natural law operators that apply
to it, operationally defined. The products LC, RC, and LG are compound
natural law operators that relate the derivatives and variables shown.
They are calculated, numerically and dimensionally, just like other products
of dimensional numbers:

LC, RC, LG, and RG act as compound natural law operators
as follows:

Mathematical and engineering practice has long depended
on our ability to multiply and divide natural law operators in this (scale
independent) way. There is NO axiomatic
reason why we can treat natural law operators as ordinary dimensional numbers
when we calculate compound natural law operators that do not include spatial
increments. But we have solid experimental support for the
fact that we can do so. That evidence goes back to celestial mechanics
calculations now nearly three hundred years old, and has been essential
all through the history of mathematical physics.

We have practically no experience
with compound natural law operators that contain spatial increments, however.
J.C. Maxwell and others worked with such constructs,
and were often frustrated in calculational sequences. Indeed,
for reasons reviewed above, we have solid calculational experimental support
for the fact that we CANNOT treat compound natural law operators including
spatial increments, such as those in the curly brackets below, as "just
numbers."

However one may wish to describe or think about our difficulties
with these constructs, what is numerically essential is that we infer a
rule that is a valid p-m representation. In
physical representations, wholes should equal the sums of which they consist.
This is an essential test in cartography,
the literal mapping of physical spaces that is the type case of our representations.
If the sum of a term
over an interval is to be independent of the number of (evenly divided)
subintervals into which that interval is divided, that term must be proportional
to the following relation:

Every term on the right side of 7b is already linearly related
to length (m=1) by the delta x outside the curly bracketed compound natural
law operator expressions. The compound natural law
operator terms cannot have any length dependence at all. Otherwise,
the terms cannot describe physical behavior. The argument for other
compound natural law operator terms (with area or volume increments) will
be the same.

For numerical consistency, compound natural law operator
expressions in terms such as those shown in 7b must be valid numerical
coefficients numerically independent of increment scale, just as other
natural law operators are independent of increment scale.

Even so, for DIMENSIONAL consistency, the dimensional
exponents of the increments in the compound natural law terms must be ADDED
in the usual way. We know that in a valid equation, every term
must have the same net dimensions. (Suppose not: with an algebraic rearrangement,
one side of the equation would have different dimensions from the other.)
In appendix 1, equation 7 is derived by valid
dimensional number algebra - every term is dimensionally correct.
In every term where an increment occurs, its dimensionality is added in
computation of the dimensionality of the term. We have found
reason to change (restrict) the numerical arithmetic procedures used
to simplify (define) some of these terms, but the changes must preserve the
calculation of dimensionality, which is correct.

We infer the following P-M REPRESENTATION RULE:

ASSUMPTION: When the symbols that represent natural laws
are combined to form a new natural law, there are special rules for putting
them together. Only AFTER combination according to these rules can a symbolic
construction be formed that can be dealt with according to ordinary rules
of algebra.

Specifically: Constructed natural law operators in combined
effect terms will include constructed natural law operators comprising
several natural law operators and (perhaps) increments of space or time
variables. Constructed natural law operators are computed (would be algebraically
simplified) as follows:

numerical part: Numerical parts of the natural law operators making up the constructed natural law operator are multiplied (divided). Numerical parts of any increments in the constructed natural law operator are not part of the multiplication or division (i.e. are set at a numerical value of 1.0) (The numerical value of the constructed natural law operator is therefore numerically independent of the increment scale at which it is evaluated.) .........This requirement is satisfied if we restrict the algebraic simplification to an increment scale with a numerical coefficient of unity in the dimensional system in which algebraic simplification is done.)

dimensional
part: dimensional exponents of all natural law operators and
any associated increments in
the constructed natural law operator would be added (subtracted). This
requirement is also satisfied
if we restrict the algebraic simplification to an increment
scale with a numerical coefficient of unity in the dimensional
system in which algebraic simplification is done.)

This rule produces constructed natural law operators that
are increment scale insensitive. Once the constructed natural
law operators are algebraically simplified (that is, defined in an arithmetically
workable way) these operators can apply to any scale.

For compound natural law operators without increments, this rule reduces
to the procedure we've used for centuries. This rule
differs for compound natural law operators that have included increments,
and avoids the self-contradictory behavior these entities have had.

According to this rule, crossterms in equations derived
from coupled physical circumstances are numerically determinant under
permitted algebraic manipulation. There is no longer any contradiction
between between differential equations and the models they came from. Wholes
equal sums of parts.

The rule may be rephrased, in a way some may find easy
to understand, and was expressed as follows in the main paper:

When we derive a finite increment
equation from a coupled finite increment physical model, that equation
will include terms that represent crosseffects including several physical
law operators and several increments in interaction together.

We have no axiomatic basis for deciding what the proper scale or unit system for algebraic simplification of these terms should be.

We know that choice of simplification scale and unit system matters numerically. Therefore, consistency requires us to specify the scale-unit system conditions for valid algebraic simplification.

Self-consistent results are obtained if we insist that algebraic simplification be done at a physical scale (or length, area, volume, etc) with a numerical value of 1.0 in the unit system in which algebraic simplification is done. This physical scale can be as large or small as we choose, since we can also choose any consistent unit system for expressing our measurements. After algebraic simplification (at a numerical scale of unity) we can convert our calculation to whatever consistent unit system we choose.

For example, the expressions within the curly brackets of equations (7), (8) and (9) are physical interpretations of natural laws that happen to have been "effectively measured" at scale delta x. To compute a natural law coefficient that corresponds to the expression in the curly brackets, and that is valid at any scale, including differential scale, we convert to a consistent measurement unit system where length delta x is 1 length unit. (Or we evaluate a "delta x" of 1 length unit in the measurement unit system we are using.)

With our unit system (or measurement) chosen so that the numerical value of delta x=1, we algebraically simplify the expressions in the curly brackets That done, we convert back to the unit system of our overall calculation, if we have departed from it. We have an equation that is arithmetically isomorphic to ordinary algebra, that will not generate false infinitesimals or infinities. The equation we had before was not arithmetically isomorphic to ordinary algebra, and our false infinities and infinitesimals trace from that.

We may also say::

"If N effects represented by distributed parameters interact on or in ONE piece of space, the numerical value of that piece of space should multiply into the expression representing the interaction ONCE, not N times".

The old, conventional answer is "N times." The difference between "once" and "n times" is usually insignificant, but in the case of neural transmission makes a very large difference.

We may also infer a consistent notation for evaluation of equations like (7), (8), and (9), in another way, at a differential (point) scale.

The notion of a dimensional parameter at a point, or of spatial increments at a point, has long been ill-defined. What does "resistance per unit length" mean at a point? Doesn't that require a notion of "length at a point?" What might "length at a point" be? In the evaluation and interpretation of compound natural law operators including spatial increments, the algebraic simplification is an "effective measurement." We need notations for the spatial increments at a point, that make measurement sense, and that yield results that work consistently when they are integrated. The following convention passes consistency tests.

When we derive a differential equation
(defined at a point) from a coupled finite increment physical model, we
must put ALL the variables and increments in our model equation into POINT
FORM prior to algebraic simplification. The point forms of spatial quantities
and time (in cm and second units) are:

Differential equations so derived, and integrated to a finite scale, correspond
to equations evaluated at that finite scale by the rule above. Again, false
infinitesimals and false infinities are avoided.

The S-K equation follows from application of this rule to
constructed natural law operators that include spatial increments, and
the results are the same ones that follow from algebraic simplification
of crossterms at a unity spatial scale, followed by passing to the infinitesimal
limit.

We can represent combined physical effects that act over
spatial increments as finite terms in differential equations.

Summary: Experimental Math at the
edge of axiomatics:

This appendix has treated calculations
at the INTERFACE between abstract mathematics and the measurable world.

In mathematical representation of PHYSICAL circumstances, set out in terms
of experimentally derived physical laws, we are using mathematical techniques
beyond where the axioms of pure math apply. If we are
to proceed at all, we must use experimental mathematics. This
paper has done so.

The results
are not so sure as axiomatic results can be, and the negative results are
more sure than the positive ones.

We can rule out current interpretations of
crossterms that call them infinitesimal in the limit. That is a strong
result.

We can suggest a P-M REPRESENTATION RULE that
is a simple change to a currently accepted rule. The
P-M REPRESENTATION RULE is consistent with all physical and mathematical
issues that have been considered. The P-M REPRESENTATION RULE
is a suggestion, that we can hold to be probable, and that we can compare
to further calculations and to physical data. The rule assumes that
the natural law operators that multiply numerically or divide numerically
in compound natural law operators with increments interact arithmetically
in the same way that natural law operators in compound natural law operators
without increments do, but that spatial increments must be evaluated
at a numerical value of unity.
That seems reasonable, and arguments for the arithmetical restriction seem
strong. Still, this arithmetical procedure is an unprovable
assumption applied to extra-axiomatic circumstances. We have
gone beyond the range where axioms determine results. There
is no trick that can conjure up axioms for us here: we must work on an
experimental basis, as we have done here.

However, the results so far are
useful. The Kelvin-Rall neural conduction equation, which lacks
inductance, is strongly ruled out. The Showalter-Kline
neural conduction equation follows from a consistent, reasonable procedure
that can be tested further. It
is reasonable that we should be left with a conclusion of experimental
math that must be subject to further experimental verification or disproof.

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Dedication: Professor Stephen J.
Kline, of Stanford University, author of SIMILITUDE AND APPROXIMATION THEORY^{4}
and one of the great mathematical and experimental fluid mechanicians of
this century, was my partner in the work leading up to this paper. We
worked together on this for almost ten years, up to his death in November
of 1997. Steve's contributions were many and indispensible. Steve thought
hard about the problems of physical representation, and was completely
clear about the need to find and fix an error at the interface between
the representation of coupled physical models at the level of a sketch,
and representation by a differential equation. The
notion that measurables, and constructions of measurables, were ENTIRELY
outside the axioms was hard for both of us. Steve kept thinking
about it, and kept me thinking about it, till his life ended.

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

NOTES:

1. G. C. Chaitin "Randomness in arithmetic and the decline and fall of reductionism in pure mathematics" p. 25 in G.C. Chaitin THE LIMITS OF MATHEMATICS Springer-Verlag, Singapore 1998.

2. G.C. Chaitin "An Invitation to Algorithmic Information Theory" in Chaitin, op. cit. p. 80

3. K. Godel, COLLECTED WORKS, V.3 manuscript "*1951" cited in Chaitin, op. cit. p.85

4. S.J. Kline SIMILITUDE AND APPROXIMATION THEORY McGraw-Hill, 1967, Springer-Verlag, 1984.

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