oct 14 DRAFT SUBJECT TO CORRECTION AND CHANGE

Formal conclusions depend on formal assumptions. This paper sets out
new assumptions for constructing mathematical models from physical models
that differ from those now implicitly used. Using the new assumptions,
derivations of differential equations from coupled physical models change.
Terms that are infinitessimal according to current procedures are finite.
The new conclusions seem justified by logic and evidence.

The idea that mathematics can be an imperfect and obscure representation
of physical circumstances is accepted.

**"As far as the laws of mathematics refer to reality, they are
not certain; as far as they are certain, they do not refer to reality."
Albert Einstein**

Indeed, Einstein and others have expressed surprise that mathematical
equations represent reality at all, and have found that representation
a deep mystery. Many seem to have lost hope of a more "rigorous"
mapping between physical reality and mathematics^{(1)}.
At the same time, the IMPORTANCE of accurate and conceptually clear mapping
has become more and more accepted. According to George Johnson

" **Scientists must constantly remind themselves that the map**

**is not the territory, that the models might not be capturing**

**the essence of the problem, and that the assumptions built**

**into a simulation might be wrong.**^{(2)}
"

How might mathematical mapping more closely approach a cartographic
ideal? The fundamental ideal of cartography seems to be congruence - when
maps are made of territories, the map must represent the details of the
territory that are of interest, in a way that permits checking, back and
forth, between the map and the territory. If there is a character or pattern
on the map, it should be clear what in the territory it refers to, and
it should convey the information about the territory that a user needs
and expects.

Often in cartography, there are maps of maps. The more detailed maps
are unweildy, and so more condensed maps are needed. But for the more condensed
maps to be constructed, they must refer to the more detailed maps - not
directly to the complexities of the territory.

I suggest, with S.J. Kline, that in order to model physical circumstances
into mathematics, we need an intermediate map, capable of representing
details that more abstract mathematical formulations cannot. I suggest
that representations in this intermediate map must be clear before mapping
into more abstract mathematical domains can validly be done. Motivation
for this suggestion comes from neurophysiology, where the standard neural
conduction equation seems to be inconsistent with data. Derivation of a
neural conduction equation according to the mapping scheme below seems
to fit data much better.

The natural world involves details of geometry, dimension, measurement
procedure, and physical law that abstract mathematics does not encompass
or clearly represent. We can map to natural laws because there are many
linear natural law operators^{(3)}, commonly
called dimensional parameters, that have been almost (but not quite) trouble
free. Linear natural law operators (dimensional parameters) encode relations
(including physical dimensions, physical laws, and measurement procedures)
from an EXTRAMATHEMATICAL world into mathematics. We need an intermediate
mathematical mapping domain that can accomodate these linear natural law
operators. Let's call this domain the **concrete domain of the measurable**.
We describe what we want concrete domain of the measurable to do, and the
axioms and rules on which it is based, as follows.

*Because we want map-territory correspondence in the cartographic
representation sense, we want representation in the concrete domain of
the measurable to have
algebra that is consistent with itself, and that fits the constraints of*

*and*

*dimensional consistency.*

*We want to be able to insist on these constencies in every term
of an equation before that equation in the concrete domain of the measurable
can be considered to be defined.*

*We want algebra in the concrete domain of the measurable to be
ARITHMETICALLY CONSISTENT from one unit system to another.*

Our current modeling does not meet these requirements. (Appendix **).
When we construct model equations from coupled physical models according
to the current procedures, we have terms that are not clearly related to
the geometry of the physical model. These terms are not defined in a dimensionally
consistent way. These terms, in the conventionally written finite form,
are also arithmetically inconsistent from one unit system to another.

Hart has shown persuasively that dimensional calculations cannot be
validly done within the dimensionless rules of the real numbers, but must
occur, explicitly or implicitly, with dimensioned scalers^{(4)}.
He has set out a Typed Family of Fields (TFF) formulation to show important,
long neglected aspects of dimensional calculation, particularly those involve
with linear algebra. But before a model based on a physical circumstance
can be mapped into Hart's TFF formulation, it must be mapped in such a
way that geometrical, dimensional, and arithmetical consistency of terms
can be assured. A mapping into a concrete domain of the measurable according
to the following assumption-axioms does this.

**ASSUMPTIONS FIT TO CONCRETE, MEASURABLE MODELING THAT CAN INTERFACE
WITH ABSTRACT MATHEMATICAL DOMAINS.**

The following definitions and axioms appear to make step-by-step sense
of modeling from the physical-measurable to the abstract mathematical.
The requirements of the concrete and the abstract are not the same.

**definition: CONCRETE: considered in direct and rule-bound connection
to a particular object or a particular class of objects under specifically
described circumstances: specifically connected to embodiment. **

**definition: ABSTRACT: considered apart from any application to a
particular object or specific instance: separated from embodiment.**

There must logically be two kinds of **domains**.

**First, concrete domains of the measurable must exist
that accommodate algebra, but that also include full details of system
specification and operational measurement definition that permit physical
laws related to specific, concrete circumstances to be expressed in terms
that include linear natural law operators, relevant dimensions,
and system variables.**

**Second, an abstract domain must exist, where entirely abstracted
symbols (real numbers or dimensioned scalars) are manipulated. **

We must be able to start from a *concrete domain of the measurable*,
map results into the *abstract domain*, do abstract mathematical operations
in the *abstract domain*, and then relate the results of these abstract
operations back into a *concrete domain of the measurable*. We must
be able to take these steps without losing information of concrete importance.**
**

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

Resistance per unit length is an example of a linear natural law operator.
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 linear natural law operator in Ohm's law for a wire. Other
physical laws have other linear natural law operators. __ The linear
natural law operators (dimensional parameters) encode experimental information
into a compact, linear form.__

Here are some linear natural law operators (dimensional parameters) in common use:

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

These are not axiomatic constructs, but they are essential in the modeling
of physical laws.

Terms that express physical laws have a particular structure that relates
a * linear natural law operator*, a space-time geometry, and
a system variable. Newton's law of inertia, Ohm's law, applied to a lump
resistor, and Ohm's law applied along a wire, offer examples of that structure.

These equations make sense in reference to specific physical situations
with specific geometries, and must be expressed in consistent unit system.
The linear natural law operators **m**, **R _{p}** and

**{f/a}** ; **{v _{1-2}/i}** ; and

for specific values of these dimensional variables that apply GENERALLY
within a domain of definition. It is the generality of the linear natural
law operators which makes them more than algebraic entities. It is this
generality that makes it possible for us to describe physical laws algebraically.
On the right hand side of () and () there is a natural law operator defined
at a point or between fixed references, times an independent variable.
The product determines a resultant variable on the left side. On the right
hand side of () there is a natural law operator defined per unit length,
associated with a specific length, times an independent variable. The product
determines a resultant variable on the left side.

**Here are some simple axioms, that permit us to define physical model
equations completely in a concrete domain of the measurable. After
this complete definition is done, we may readily map these equations into
the domain of the algebra or a domain of dimensioned scalars.
With these axioms, we can define the terms Maxwell found "difficult
to interpret."**

**AXIOMS INVOLVING LINEAR NATURAL LAW OPERATORS (DIMENSIONAL PARAMETERS)
and the GEOMETRY of CONCRETE EQUATIONS.**

**A1) Axiom: The axiom of geometrical consistency.** When an equation
refers to a particular geometry, then EVERY TERM must refer to that particular
geometry. If the equation refers to an increment of volume (length, area,
time), EVERY TERM must contain an increment of volume (length, area, time)
and a linear natural law operator expressed per unit volume (length, area,
time).

**A2) Axiom: The axiom of dimensional consistency.** Every variable
and natural law operator in the equation must be defined in a consistent
system of units that may, with sufficiently careful and complete specification,
be reduced to units of length, mass, and time.

a21 Every variable in the equation must be set out in that same system
of units.

a22 Every linear natural law operator in the equation must be in that
same system of units. This means that every linear natural law operator
must be expressible as a numerical value times a unit group (reducible
to L,M,T or units traceable to L, M, T) where ALL lengths (explicit or
implicit) are the same unit of length, ALL masses (explicit or implicit)
are the same unit of mass, and ALL times (explicit or implicit) are the
same unit of time.

**A3) Axiom: CONSTRUCTED LINEAR NATURAL LAW OPERATORS**: When two
linear natural law operators (dimensional parameters) occur in product
or ratio form as a result of a valid algebraic construction in the concrete
measurable domain where they are defined, these dimensional parameters
can be algebraically simplified together. Numerical parts are multiplied
(divided) together. Unit exponents are added (subtracted.) The result is
a CONSTRUCTED LINEAR NATURAL LAW OPERATOR. This CONSTRUCTED LINEAR NATURAL
LAW OPERATOR is a LINEAR NATURAL LAW OPERATOR, and may be used in later
calculations in the same ways as any other.

**A1**, **A2**, and **A3** express long-accepted results used,
explicitly or implicitly, since before Maxwell's time, and still used.
But if A1-A3 are applied to models of coupled physical circumstances, without
more, we come to terms that Maxwell called "difficult to interpret."
The following frequently recurring fact causes a recurrent kind of problem.

**Fact: If we construct terms of a finite increment model that correspond
to N physical effects in interaction together, each per unit length (area,
volume, time), we get a term which has N length (area, volume, time) increments
associated together. **

**Problem: Since the equation is set out to correspond to only ONE
length (area, volume, time), what do we do with the N-1 other lengths (areas,
volumes, times) ? **

**THE STANDARD ANSWER NOW IS TO APPLY A LIMITING ARGUMENT TO THESE
**

**N-1 LENGTHS (AREAS, VOLUMES), which invariably sets the crossterms
to 0.** When this is done, a procedure from the domain of the algebra
is applied to another domain. A limiting argument is applied to physically
senseless finite increment terms of the form

**compound linear natural law operator**

times

**increment expression that refers to nothing identifiable **

times

**increment expression that refers to the model geometry**

times

**independent variable**.

Such a finite increment term is geometrically undefined,
dimensionally inconsistent with other terms in the equation in which it
occurs, and arithmetically inconsistent from one system of units to another.

Appendix 1 shows, in detail, the
ill definedness, dimensional inconsistency, and arithemetically inconsistency
with unit system change that these conventional terms have.

**KEY ASSUMPTION:** We can avoid these difficulties and make sense
of these terms if we assume that the extra increments can be, and must
be

interpreted as dimensions (length, area, volume, time)

or

interpeted as dimensions times the unity operator

[ (1 unit length); (1 unit area); 1 (unit volume) ]

or

interpreted in an algebraic simplification of a compound natural law
operator including increments, with every natural law operator implictly
converted to a scale where the increment scale is unit scale. After algebraic
simplification, the compound natural law operator formed is reconverted
to the unit system being used.

These three forms of the KEY ASSUMPTION are operationally the same.

If we make this KEY ASSUMPTION, then A4 permits us to represent all
the terms that occur in our physical modeling in a way that matches the
physical model they correspond to geometrically and dimensionally. The
terms formed are arithmetically consistent from one unit system to another.

**A4) Axiom: CONSTRUCTED NATURAL LAW OPERATORS INCLUDING INCREMENTS:
Algebraic formulation **

Suppose a term is written as a result of a valid algebraic model construction,
and two or more natural law operators and one or more spatial increments
occur in the term. Let the axiom of geometrical consistency require that
these natural law operators and one or more of these spatial increments
be grouped together. THEN the natural law operators and spatial increments
that are grouped together can be algebraically simplified together, multiplying
(dividing) numerical parts and adding (subtracting) unit exponents to form
a CONSTRUCTED LINEAR NATURAL LAW OPERATOR. **This CONSTRUCTED LINEAR NATURAL
LAW OPERATOR is a LINEAR NATURAL LAW OPERATOR, and may be used in later
calculations in the same ways as any other.**

A4 corresponds to a PHYSICAL implication:

**A5) Axiom: CONSTRUCTED NATURAL LAW OPERATORS INCLUDING INCREMENTS:
Physical formulation**

There can be PHYSICAL interactions between several kinds of physical
laws, that occur over a length, or over an area, or over a volume, or over
a period of time. These effects may be expressed by CONSTRUCTED NATURAL
LAW OPERATORS (DIMENSIONAL PARAMETERS) that include both NATURAL LAW OPERATORS
and spatial increments.

According to A5), and axioms A1-A4), some terms now called infinitesimal,
and some terms now called infinite, are finite terms. We can show how this
comes about mechanically, on the basis of the assumptions A1-A4. We can
then consider the results, in terms of cartographic reasonableness, and
in comparison to data.

After a construction in the concrete domain of the measureable is done,
mapping from this more detailed map to a simpler, more abstract domain
(one of Hart's Typed Family of Fields, commonly thought of as the domain
of the algebra, is straightforward. It may be done by taking a limit of
the model equation. For differential equations, this is the same as transfering
the equation, symbol for symbol, into the more abstract domain.

Let's apply these axioms and derive a neural conduction equation, starting
within a concrete domain of the measurable that corresponds to the physical
conduction line that we are modeling.

To derive a differential equation from a physical model in classical
physics (still the dominant physics in practical work) we argue as follows.

1. We construct a model (including a sketch, and any necessary information)
at a finite scale that represents the physical laws and geometry in question.

2. We derive finite increment representative equation(s) in a concrete
measurable domain. We set up a representation that maps the model completely
(or, if the equations involve some recursive relation, we represent the
model to some specified degree of completeness.)

3. We define each term in each equation so that it fits our model geometrically
and dimensionally.

4. We then map our equations, fully defined in 3, into the domain of
the algebra.

Let's proceed to infer a finite increment equation for electrical transmission along a line, a case that is inherently coupled.

**Fig. 1** shows a neural conductor (axon or dendrite considered
as a transmission line). A tubular membrane is filled with and surrounded
by an ionic fluid. The fluid inside the tube carries current (and signal)
and has resistance **R** and electromagnetic inductance **L** per
unit length. The outer fluid is grounded. The membrane separating these
conducting fluids has capacitance and leakage conductance per unit area.
We speak of the following variables and parameters:

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

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

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

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

**Fig. 1** shows an arbitrarily chosen length , which we will call
**x** because that is commonly done. is picked from other indistinguishable
lengths. (For consistency, the length of is a number times the unit of
measure used in the calculation in the **x** direction.) We are giving
length a two-symbol name that includes a separate number, . Other entities
that have numerical values in our calculations are denoted as single symbols,
and do not have separate numerical symbols (such as ) associated with them.

We need finite difference equations, **v(x,x,t)** and **i(x,x,t)**.
We face the following coupled circumstance.

Because voltage drop over a length depends on current, and current over
length partly depends on charge carriers stored in capacitance or lost
through membrane leakage over , we know that **v/x** for **x=** __must
partly depend on effects including C and G__ over length

and

Because current drop over length depends on voltage change over , and
voltage change over partly depends on **R** and **L**, we know that
**i/x** for **x=** __must partly depend on effects of R and
L__ over length

From such interactions, it follows that:

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

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

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

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

**and so on**

The logic for **v** is symmetric.

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

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

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

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

**and so on**

Our **i** equation implicitly contains our v equation (and its derivatives.)
Our **v** equation implicitly contains our **i** equation (and its
derivatives.) When we write out our **v** and **i** equations more
explicitly, we still have implicit couplings between **i** and **v**
functions. In turn, we can write these implicit equations out more explicitly.
We still have couplings. As we write our relations out in more and more
explicit form, we get more and more terms, in an endless regress.

An expansion according to this logic is set out in Appendix 1. The expressions
are set out in a CONCRETE DOMAIN OF THE MEASURABLE according to algebra,
without yet interpreting the terms geometrically and dimensionally.

Our crossterms in () are not yet clearly related to the model that generated
them.

1. For a single term, we could make different interpretations of the
symbols in the term that apply to different geometries.

2. For a single term, we could make different interpretations of the
symbols that would correspond to different units.

Consider one term, grouped in different algebraically permitted ways

These different groupings, that are algebraically the same, have different
physical interpretations.

In 7b we interpret every crossterm of 7a in this same way, so that every
crossterm consists of an independent variable, an interval **x**, and
a compound linear natural law operator (dimensional parameter) shown prior
to algebraic simplification.

the compound natural law operators (dimensional parameters) simplify
to

In 7c, the natural law operators and constructed natural law operators
are broken down into their numerical and dimensional parts. Every natural
law operator and constructed natural law operator has a dimensionality
of v^{n} t^{m} Q^{o} x^{p} with dimensionality
expressed in ratio form (negative dimensionality in the denominator, positive
in the numerator.) Dimensions to the 0 power are not shown. Every term
has net dimensions (of voltage) as it must.

So far, all we have done is INTERPRET what the terms in 7a MUST mean
to satisfy dimensional consistency and geometrical consistency in the CONCRETE
DOMAIN OF MEASUREMENT in which the conduction model from which they were
derived exists.

To map the model into the domain of the algebra, we must take a limit. Here is the basic logic to used to map from the concrete domain into a differential equation in the domain of the algebra.

Our abstract equations may be written more simply, by making our dimensional
notation implicit.

This differential equation, in the domain of the algebra, is subject to exactly the same rules as other differential equations.

However, it contains terms, corresponding to the crosseffects in 7a,
that would have been called "infinitesimal" according to the
current practice, which applies a limiting procedure to geometrically and
dimensionally undefined terms. Whether these finite terms are indistinguishable
from 0, or very large, depends on the numerical size of the natural law
operators (dimensional parameters) R, L, G, and C.

Comparison of a wire case and a neural dendrite case shows the contrast
in values of these crossterms that can occur.

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

**R = .14 x 10 ^{-4} ohm/cm C
= 3.14 x 10^{-9} farads/cm**

**G = 3.14 x 10 ^{-10} mho/cm L
= 5 x 10^{-9} henries/cm**

Here is (**1**) with the numerical value of terms set out below the
symbolic expression of the terms:

Here are the corresponding natural law operator (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.

**R = 1.4 x 10 ^{10} ohm/cm L = 5 x 10^{-9} henries/cm**

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

This **R** is 10^{11} larger than in the previous case of
the wire.

The two cases modeled by the same equation are radically different.
Cross product terms that are trivial in the wire case are dominant in the
neural dendrite case. In the wire case, calling these crossterms "infinitesimal"
is a near-perfect approximation. In the dendrite case, calling these crossterms
"infinitesimal" is gross error.

I have shown a different approach to deriving differential equations
from physical models, based on different assumptions and procedures. We
may ask three kinds of questions about these changed derivations.

First, we may provisionally grant the assumptions, and then ask how

the derivational consequences work out.

The derivation of the S-K transmission equation shows those consequences.
Under the new procedure, when there are coupled circumstances, coupled
terms now called infinitessimal will be finite. The size of these finite
terms will depend on the size of the natural law operators in the model
under consideration.

Secondly, we may ask if the assumptions of the new procedure are required
for symbolic consistency in the derivations. A major test is consistency
from one unit system to another, an issue that applies to measurable circumstances,
but that is not a concern in the abstract domain of the algebra.

The loop test shows this.

Thirdly, we may ask if the new assumptions yield derivations that

FIT DATA that is not consistent with the standard derivations.

I have a linear algebra derivation of the conduction equation

that fits. Spice simulation
of the transmission equation

Neural data fits.

Soon, I hope, QM data will fit.

Some terms in finite increment equations that are derived from abstract
algebra without further interpretation do not refer to model geometry,
are dimensionally inconsistent with the equation in which they occur, and
are arithmetically inconsistent from one unit system to another. Now, these
terms are discarded as infinitessimal without considering these definitional
difficulties. The mapping into the concrete domain of the measurable avoids
these difficulties according to procedures that make these terms finite.
Let's take a closer look at what these logical difficulties are. Let's
consider the terms in equation 7a that are conventionally dismissed by
a limiting argument.

These terms were all constructed to refer to a particular model with
**= x**.

On the right hand side of the first line there are two terms, that are
not dismissed by limiting arguments, that refer clearly to model geometry.
These terms ( ) are dimensionally consistent with each other and with the
left hand side of the equation. These terms are arithmetically consistent
from one unit system to another

In the terms in (), an increment, **x**, refers directly to length
in Fig **. A natural law operator (R or L) defined per unit length, acts
over length **x**, and transforms a variable (**i** or **di/dt**)
numerically and dimensionally. The dimensionality of these terms is voltage,
with all the other dimensions taken to the exponent 0, as it must be.

These terms pass a conservation test that representations of physical
circumstance must pass. If **x** is divided into **N** pieces, and
the expressions evaluated over those **N **subintervals are computed
and summed, that sum is the same as it would be for **x** taken in one
step.

These terms are also arithmetically consistent from one unit system
to another. Let's look at a simple loop test, analogous to many closure
tests in physical logic. In **Fig. ~1**, an algebraically unsimplified
term such as one of those in ( ) is set out in cm length units at **A**.
This quantity is algebraically simplified directly in cm units to produce
"**expression 1**." The same term 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 cm" 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
( ) pass this essential loop test. These terms are arithmetically consistent
from one unit system to another.

Below the first line of ( ) are terms in finite increment equation ( ) that are typically dismissed by a limiting argument. At finite scale, these terms fail the tests the terms in ( ) pass. These terms do not refer to the model geometry that they were constructed to represent. These terms are dimensionally inconsistent with the equation in which they occur. These terms

are arithmetically inconsistent from one unit system to another.

In the terms in ( ), there are one or more increments, labelled **x**,
that cannot refer directly to length in Fig **. It is not clear how the
**x** symbols in these terms refer to the physical model these terms
were constructed to represent.

These () terms are dimensionally inconsistent with the equation in which
they occur. To see this, it suffices to see that these terms all **fail**
a conservation test that representations of physical circumstance must
pass, and that the terms in () do pass. For the terms in (), if **x**
is divided into **N** pieces, and the expressions evaluated over those
**N **subintervals are computed and summed, that sum is NOT the same
as it would be for **x** taken in one step. The whole does not equal
the sum of its parts. With conventional patterns of interpretation, we
cannot say that these terms are equal to a compound dimensional parameters
expressed "per unit length" times a length increment corresponding
to model geometry. These terms are somehow "per unit length^{2}",
of "per unit length^{3}," and are therefore dimensionally
inconsistent with the terms in ( ), which are "per unit length."

These ( ) terms are also numerically meaningless at finite scale, as
shown below. 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 the same
loop test referred to before in **Fig. 666**. An algebraically unsimplified
term in () is set out in cm length units at **A**. This quantity is
algebraically simplified directly in cm 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 cm" basis to produce "**expression
2**." __Expression 1 and Expression 2 must be the same, or the
calculation is not consistent with itself.__ Terms like those in the
curly brackets consistently fail the loop test.

The loop test fails!

The loop test fails because a standard procedure is flawed when applied
to terms like those in ().

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 notation.

Note that the loop test of **Fig. 666** only fails for terms that
are now routinely discarded as "infinitesimal" or "infinite"
without detailed examination.

Derivation of equations such as () according to the rules of a concrete
domain of the measurable, as described here, avoids these logical difficulties.
The occurrence of these logical difficulties should cast doubt on the notion
that derivations such as the derivation of the Kelvin-Rall transmission
equation are "rigorous" and scientifically true derivations.

2. Johnson, G. (1997) __Proteins Outthink Computers
in Giving Shape to Life__**NEW SCIENTIST** March 27, 1997.