**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. **

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 delat**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, 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.

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 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. 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 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) 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 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 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. 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.

**AGAIN, 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, 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, the operationally valid experimental rules for representation can be found. 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.

**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.)

__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 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 conductance
equations that apply to a line conductor, such as a wire, the Heaviside
equations, offer 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:

The terms on the right hand side contain products of natural law operators
that 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 other 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) externally to the 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 terms such
as those shown in 7b must be 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. 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 7a-b
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 the numerics of some of
these terms, but that cannot change 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 as follows: would be algebraically simplified
as follows:

numerical
part: Numerical parts of the natural law operators making up
the constructed natural
law operator would be multiplied (divided). (Numerical
parts of any increments in the constructed natural law operator
are not part of the multiplication - the value is therefore numerically
independent of the increment scale at which it is evaluated.)

dimensional
part: dimensional exponents of all natural law operators and
any associated increments
in the constructed natural law operator would be
added (subtracted).

This rule produces constructed natural law operators that are increment
scale insensitive. **This rule is exactly the procedure used
for centuries for compound natural law operators without increments. **

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

The rule may be rephrased, and was expressed as follows in the main
paper:

* "the
expressions within the curly brackets are physical interpretations of natural
laws that happen
to have been "effectively measured" at scale x. To compute a
natural law coefficient that
fits the term, we convert x to *

* (1
length unit) *

* and
algebraically simplify on that basis." *

The S-K equation follows from application of this rule to constructed
natural law operators that include spatial increments.

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 interact
arithmetically. That seems reasonable, and arguments for the arithmetic
seem strong. Still, this arithmetic is an unprovable assumption. We have
gone beyond the range where axioms determine results.

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.

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

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.

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

**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.