Dear FISers:
First, a brief review of previous posts.
Molecular bionetworks are composed from the chemical elements into
informed / organized associations. Such networks are not composed de
novo from individual elements, rather each bionetwork is a further
step in the emergence of the history of the species, the history of
the individual, the emergence of time itself. Concepts of scale, of
matter, of mathematics and of philosophy are entangled in
descriptions of biological behavior of metabolic networks.
In the opening remarks, I wrote:
Historically, the investigation of empirical basis of molecular
bionetworks (metabolism) started shortly after Pastuer�s pioneering
experiments on the causes of fermentation (1870�s). At that time,
the fermentation of grape juice was difficult to control but of great
economic importance. (Bad wine sells cheaply!) Quantitative
analysis of yeast fermentations showed that each molecule of glucose
generated two molecules of the 2-carbon alcohol and two molecules of
carbon dioxide. The yeast cell informed the specific destruction of
the sugar. (If the 6-carbon sugar was merely burned, then it
produced exactly six molecules of carbon dioxide.) Thus, the
thermodynamic question of why the yeast cell did not completely burn
up the 6- carbon sugar entirely into carbon dioxide arose.
3. From an informational perspective, what does this incomplete
fermentation process of glucose suggest about the role of
thermodynamics in living processes?
In the post of Nov 28, 2005, a list of bionetworks was suggested:
When we speak of bionetworks we wish to include the following classes
of networks, starting from simple and growing increasingly perplex:
1. the networks of chemical communications of simple cells that
generate cellular functions and reproduction
2. the networks of chemical communications among the cells of a
multicellular organism
3. the networks of electrical communication in organisms with central
nervous systems
4. the networks of internal mental communication in higher organisms
such that consciousness is generated.
5. the networks of external communications among members of social
and cultural communities.
6. the networks of eco-system "communications" that sustain the
dynamics of ecosystems as organizations of species and geological
circumstances.
(This categorization of bionetworks is somewhat arbitrary but is
adequate for this general discussion. The important point, with one
exception, is that the concept of bionetworks is given clarity and
distinctiveness by separation of Aristotelian categories. )
In this post, the concept of number in the context of bionetwork
theory will be discussed in light of comments from Stan, Pedro, Bob
and others.
The thoughtful reader will recall that Ted G. stimulated parts of
discussion with a list of his questions about the challenges facing a
new theory of information. The present commentary looks at these
issues from an historical perspective of number.
First, one notes that the concept of number is integral to what
Wittgenstein referred to as "ordinary everyday language." The
mathematical edifices constructed over the centuries within the
mathematical community are closely related to specialized
definitions. Such specialized definitions serve to force a
particular choice of logical relations on the mind. The choices of
such definitions are, of course, informed choices, informational
choices. The choices are guided by consistency with other
definitions AND with consistency with calculations. The number of
such mathematical definitions is very large, certainly in the
thousands and more probably in the tens of thousands. One notes that
each mathematical definition is a particular choice from among many
choices. Thus, one can imagine, roughly speaking, the calculation of
the information content of mathematical language with reference to
Wittgenstein's ""ordinary everyday language" as the reference. For
example, if the mathematical term, point, was one of exactly 16
different definitions of the word "point" in everyday language, then
four bits of information would be contained in the mathematical term
"point". (Of course, this is a vast over-simplification of the
informational theory of language. For example, the Mac dictionary
gives 18 separate definitions of point as a noun and 6 separate
definitions as either a transitive or intransitive verb. Of the 18
denominations of the term "point", about one-fourth have mathematical
connotations.) From a mathematical perspective, any such calculation
of the information content of a mathematical term, would be an object
of scorn or ridicule. The counter argument is simple. Mathematics
depends on using language in a coherent way such that the principles
for making generic calculations can be made. The number of possible
ordinary language definitions of a mathematical term contribute
little if anything to the objective of making calculations!
In terms of third order cybernetics, the concept of a number as a
point on a line is a quantity, is a position and is a possession of
the line. Thus, a number as a point of reference on the real number
line, is, in the sense of Aristotelian categories, a member of three
denominations of categories.
What conclusion can be drawn from the preceding two paragraphs?
I invite the readers to open these paragraphs in differernt domains
of discourse.
The paragraphs are written for the purpose of exposing the fact that
the usage of number as atomic number is different from any of the
above connotations. Atomic numbers are used to construct molecular
words as a collection of objects attached to one another. In other
words, the conceptual framework from which bionetworks are
constructed is a conceptual framework that uses the concept of number
in a way that differs from usage of number in number theory, in
geometry, and in Shannon information theory.
This introduces a dilemma. Atomic numbers are used as exact numbers
with respect to the list of pure forms of matter (such as gold, lead,
silver, copper, carbon, ....) and as logical objects; as abstract
logical objects with particular logical properties. In order to
distinguish between the two usages, the two definitions, I will use
the term, proto-numbers when I am thinking about the logical
objects. The particular logical properties I refer to are the
specific attachments of electrical particles to one another to
generate stable collations of particles. Continuing the
composition, stable collations of particles can form generative
bionetworks. The inferential logic of such collections is not the
logic of Frege, of Russell, of Quine, of Kripke or of McGinn. So, I
will use the term "proto numbers" to refer the logical relations that
generate the "ordinary everyday language" of biology and medicine.
As an aside, I note that the discovery of "proto numbers" at the turn
of the 20th Century led to the development of quantum mechanics and
subatomic physics. The term "proto" is used in the sense that proto
numbers are the historical precursors of the integers. The logic of
proto numbers bears resemblance to the early Greek concept of
"polygonal numbers."
With these introductory remarks, I turn to older material.
First, the issue of optical isomers, fermentation of glucose, and
thermodynamics as mentioned in the third question. Thermodynamics,
as a part of the traditional engineering sciences, was derived from
natural language by twisting he definitions of natural language into
mathematical forms that correspond with real numbers. The theory is
thermodynamics includes variables for volume, temperature, pressure,
energy forms, number and a non-variable term, the gas constant.
Proto numbers as such do not enter into the small set of
thermodynamic equations. From an Aristotelian perspective, the genus
of thermodynamic systems excludes living systems. (And ecosystems!)
The spectacular success of thermodynamic sciences resides in the
description of various heat engines where the critical variables are
changes in volume, in pressure and and temperature. In the
application of thermodynamics to real systems, all particles must
enter in as "number". The logical properties of proto numbers enters
into thermodynamic equations via the notion of equilibrium constants
as pure numbers, not as collations of electrical particles.
So, returning to the fermentation of wine, the narrative of "why"
glucose in not fermented completely to CO2 (carbon dioxide)? I
suggest that it is primary a mis-understanding about the languages of
systems. The language of the question is constructed from two
different number systems, one describing glucose in terms of proto
numbers, the other describing the quasi - equilibrium under the local
circumstances in the fermentation vat, in thermodynamic terms. When
we consume the wine, our bodies complete the "fermentation" of
glucose to carbon dioxide, with possible pleasant effects. I am
personally very thankful for the proto numbers; if only real numbers
existed, we could have wine fermentation! :-) The optical isomers,
are, of course, an exclusive property of informed proto numbers and
are not expressed in thermodynamic language.
I now turn to Stan's Post of Nov. 30. (Vol. 488, #1.)
Stan - the focus on Pattee's arguments about initial conditions in
not particularly meaningful for bionetwork theory. (If you have
counter-examples to this statement, I would be pleased to hear them.)
What does it mean to initialize a bionetwork? Is any other
initialization outside of evolution possible? (Sporulation is a
special case that requires a special discussion.) Such expressions
as "constructed initial and boundary conditions", which are based in
real numbers and thermodynamics, do not describe the conditions
internal to a cell. Such properties are intrinsic to the concept of
a bionetwork for a living system. In particular, Stan, given that
the logic of proto numbers is fundamentally different from the logic
of the real numbers, I do not see how these concepts aare relevant
to bionetworks. Perhaps you have some ideas on how you might
transform this hypothetical bio-logic into the logic of the proto
numbers?
SSwrites:
philosophically MATTER is delayed energy, stickiness
and delay, obstruction to the flow of energy.
JLRC:
I think you are reversing the concepts of source and consequence.
Only because of the concreteness of matter do we have some
understanding of energy as a mathematical abstraction.
In particular, the proto numbers of the individual species of matter
are the sources of our concept of real numbers in the sense of self -
reference. The concept of energy is a generic concept, not a concept
of species, a continuous variable, not a discrete variable.
Bionetworks are composed of particular molecular species,
mathematical graphs so that one must find a logical path from such a
view of matter to a particular species.
SS:
In contrast, NUMBER
concerns the size of distinct collections of matter -- most primitively,
just: one, some, many.
JLRC:
By starting with the abstraction of size, you miss the point.
Proto numbers stand in exact correspondence with the integers of
everyday ordinary language.
SS:
Finally, IDENTITY is based in the uniqueness of
configurations of matter.
JLRC:
In the first part of your answer, you deploy real numbers to invoke
energy as a source.
In the comment on identity, you invoke the proto number concept of
identity as a species.
(In the real numbers, identity is associated with the operations of
addition and multiplication.)
SS:
But suppose we were to take seriously a switch from integers to real
numbers in describing elements. Ae you willing to assert that an
element
would show itself as, e,g.,13.00000000000...000...infinity?
JLRC:
As noted above, the proto numbers are precursors of real numbers.
As such, they correspond exactly with the natural language integers,
1,2,3,... N, where N is a 'small' number, the list of chemical
elements is finite..
Proto numbers are monomials.
When you write, 13.00000000000...000...infinity?, you presuppose that
the proto numbers are polynomials and that the concept of extension
as used in the real numbers applies to proto numbers. Both
presuppositions fail in proto numbers.
SS:
Even if you
would opt for that, where IS any individual of this element at this
moment?
JLRC:
Again, you invoke a property of real numbers and the physical
tradition when you ask about where IS an element at this moment.
For bionetworks, the self-referential property of the proto number
system renders such language as artificial.
The self-referential character of proto numbers / DNA / bionetworks
is sufficient to conduct biological function.
If you wish to invoke the concept of a common spatial axis for
locating matter, then the location will be expressed in terms of real
numbers (using such concepts as center of gravity and linear vector
spaces), not proto numbers and illations based on informed
relationships.
Stan: It seems to me that you ought to try your hand at explaining
the optical isomer problem and the wine fermentation problem in terms
of your entropic digestion of nature.
Turning to Pedro's post of December 2 (vol. 446, #2), Pedro writes:
PM:
and a dismissal of the importance of quantum mechanics, contemplated
as "merely describing the motions of electrical particles in space
and time."
JLRC:
The phrase "a dismissal of the importance of quantum mechanics" is
your phrase, not mine.
I only sought to place the role of quantum mechanics (QM) in
perspective relative to the theory of bionetworks. QM is an
extremely important theory for relating physical concepts of motion
in space. The fact that is restricted to a simple conceptualization
of time severely restricts its utility for biological purposes, such
as the reproduction of bionetworks.
The proto numbers are the source of quantum mechanics. It is the
concepts of number and electrical particle that the concepts of force
and motion are applied to. After all, what are the variables of the
Schodinger equation for molecules?
Before quantum mechanics can be a description of bionetworks, one must:
First, identify all (the thousands, the tens of thousands, more,
depending on the biological species) the members of the bionetwork.
Secondly, one must identify exactly which proto numbers are present
in each identity.
Thirdly, one must identify exactly the ratio of the proto numbers
with respect to one another in each identity.
Fourthly, one must compose the proto numbers into a unique pattern of
relations such that the abstract identity of the pattern is in
correspondence with properties of the identity.
After these four steps are completed, one can then apply the concepts
of motion in space and temporal changes in local position to the
pattern of proto numbers, that is, attempt to write a version of a
Schodinger - like equation for each member and for the system
as a whole.
In other words, it is necessary to make calculations with proto
numbers before calculations with real numbers in quantum mechanical
relations can be considered.
Quantum mechanics is a well grounded physical theory that was
constructed for the explicit purpose of calculating the motion of
electrical particles in space and with time. If we do not keep this
fact in perspective, we lose our perspective of the interplay among
philosophies of concept, number, size, theory and fact. Stan's
narrative is a crisp example of the knotty interplay among language,
number and philosophy. From a practical perspective, the
Aristotelian categories (with a concept of matter substituted for the
concept of substance) provide a useful framework for separating the
wheat from the chaff.
PM continues in the post of Dec 8, 2005, (vol. 488, #3)
In my view, what changed dramatically was not only the relationship
with numbers but, above all, the way to "validate" knowledge: from
"disputatio" to "experimentatio" (rhetorics becoming subsidiary to
experimental procedures--the big point of the scientific revolution).
This point is not trivial concerning our nascent field, as quite many
parties disregard any empirical grounding (abandoning the quest for
"facts" to ingrain their theoretical-doctrinary views). Given that,
potentially, fis also implies a new vision on info and knowledge, and
the contemporary role of sciences at the social realm, this may lead
to fine-tune our views on what thought-collectives actually do... in
any case, the "mystification of numbers" looks a very good point for
future discussions.
JLRC:
I concur with your over-all view.
The notion of the "mystification of numbers" is a much deeper notion
that expresses sentiments about the philosophy of mathematics in
relation to the proto numbers.
The logic of the proto numbers is a simple logic using integers and
ratios of integers. This logic differs from the usual abstraction in
that it is grounded in the chemical elements, in calculations that
are extendable to molecules of unbounded size, not the abstractions
related to continuity.
With the works of Galileo, Descarte, and Newton, mathematics started
down a road to the detailed analysis of continuity. Beginning with
G. Cantor, the importance of calculation began to be minimized. In
pure mathematics today, the concepts of abstraction, of abstraction
of abstractions, of abstraction of abstractions of
abstractions, ... continue on indefinitely, whether calculations
are possible or not. Bionetworks theory is based on exact chemical
calculations. it is the introduction of continuity into such a
system that is difficult.
In a certain sense, pure mathematics has returned to the philosophy
of validation via "disputatio"; of course, applied mathematics must
remain in contact with the reality of validation by
"experimentatio." For example, take Conway's construction of
"surreal" numbers. Are surreal numbers really numbers?
The methods of commutative proof within the proto number system is
based on validation by "experimentatio."
(In order to attempt to avoid being misunderstood, I feel obligated
to add to that I enjoy pure mathematics immensely and have been
blessed with the opportunity to study it intensely for several
years. It is my belief that modern mathematics is created within
natural language within our minds. If we wish use the wonderful
logical insights and delightful coherences among modern mathematic
structures, such as category theory, we must seek the correspondence
relations and document the commutative diagrams with facts.)
PM:
Thus, to continue with Jerry's taxonomy, the problem is that the self-
production network is a mere label for an enormous landscape of
molecular interactions and processes of quite a heterogeneous nature.
Let me take home the term "molecular landscape".
JLRC:
I concur. The heterogeneous nature is essential. At present, we have
no way to express precisely what we wish to include under the concept
of "heterogeneous nature."
I have carefully avoided, in so far as possible, the topics of time,
of dynamics, of reproduction.
A bionetwork has been, generally speaking, discussed as a static
structure.
The inadequacy of this approach is readily apparent.
But, I would also note that this is exactly what Shannon does when he
invokes the concept of encoding a message. An encoded message is a
static object. To introduce a change in an encoded message is to
introduce an error into the message.
The critical distinction, from a conceptual veiw point, is that the
bionetwork messages contain the source of motion for organism. A
network of Shannon messages is completely inert. No necessary
relation is essential among Shannon messages. By contrast, the
collations of proto numbers composed into bionetwork messages include
the messages necessary for reproduction, contingent on the necessary
environment.
PM continues (vol 488, #5) in a quote from Paul Davies:
"Quantum Mechanics thus provides an automatic discretization of
genetic information [without the need for complex intermediate
chemistry]... but how complexity emerges in quantum systems is a
subject still in its infancy..."
This is an intriguing quote. Is it more than an effort seeking to
finesse a deep and intractable mathematical dilemma? As Koichiro has
pointed out, along with many others, physicist have a deep problem in
coping with the existence of the integers, exact numbers. However,
the units of measurement of a quanta are ergs - seconds. Thus, how
can one have a duration without continuity? The source of this
dilemma, at its roots, is the nature of a geometric line, not a
material dilemma, rather a temporal dilemma, a geometric dilemma.
Davies imaginative usage of language is absolutely charming. The
question is: Does the structural framework of his logic allows him
any other alternative than to claim a biological attribute for a
mathematical necessity? How should the concept of time enter into
bionetworks? As an absolute concept? As a geometric concept? As a
product relation? As a tensor? As a contingency with the
surroundings of the organism? Or, is network time more like a
Kirchoff's network of electrical flows over a collation of proto
numbers?
(For those interested in a long philosophical discussion of these
concepts, Alicia Juarrero's book, Dynamics in Action, Intentional
Behavior as a Complex System, MIT Press, (1999) is highly
recommended. For a contrary view, Susan Oyama, Evolution's Eye.)
Of course, since quantum mechanics itself is based on proto numbers,
his entire line of reasoning about genetic discretization is suspect.
Bob Ulanowicz (vol. 488, #5) writes:
The problem, as I see it, is that science labors under a metaphysic that
has outlived its usefulness. It is a ragged remnant of what two hundred
years ago passed for Enlightened science, but most colleagues cling
to the
shreds with a ferocious tenacity!
JLRC:
As I look back over the past twenty years of WESS, Bob, I think I see
a few signs of progress in the community at large. But, at the core,
the utility of certain ideas for engineering, for warfare, for public
persuasion, for economic advantage, continue to carry the public
conceptualization of science.
One deep issue is the issue of how language is used. If the
scientific community does not demand a consistent usage of language,
then the intrinsic flexibility of technical sentences can be given
multiple interpretations. It is for this reason that I greatly
admire the mathematics community. Mathematicians, in general, have
extremely high standards when it comes logical integrity and
consistency. Chemists hold to a similar standard of proof for
chemical structures. The tensions between rigor and imagination are
severe.
Igor writes (vol. 488, 6)
A cell mechanism A adapts (measures) to the cell environment, which
it is part of. During this adaptation it increases its complexity and
changes (by its very presence) the cell environment, which thus also
increases its complexity. Given that there are many such mechanisms
(enzyme catalyzed chemical reactions, organelles, membranes, etc),
each mechanism increases its complexity by reflecting on the presence
of the totality of the other mechanisms.
JLRC:
Reading your words from a 'big picture' frame of mind, I concur.
I would prefer to express the concepts in terms of the flow of
electrical particles...
The term "mechanism" can be used in either a mathematical sense, a
chemical sense, or a general sense. In what sense is it being used
in this paragraph?
More specifically, in the narrative you provide, how does one create
relations between electrical particles and the dynamics of
bionetworks? Although only a small number of different electrical
particles exist, how does one create the range of organisms in an
ecosystem?
How can we make such concepts EXACT such that they become a
meaningful method of scientific prediction? This is the challenge
for bionetworks at all levels of perplexity. How does individuality
arise from such a narrative?
What would be a theory of information or communication such that a
generative narrative would be the natural extension of the dynamical
system describing the collation of electric particles?
I hope that I have been able to express the logical issues
surrounding bionetworks in a compelling way and introduce the
mathematical concept of proto numbers in a digestible manner. In
closing, I note that the conceptualization of proto numbers was
discussed in papers presented in Baden-Baden and Liege last summer.
Well, once again I have been unable to respond to all issues.
Hopefully the issue of "causa aequat effectum" for bionetworks can be
addressed before we move on to the next topic.
Happy Holidays to All!
Cheers
Jerry
Jerry LR Chandler
Research Professor
Krasnow Institute for Advanced Study
George Mason University
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Received on Sat Dec 24 00:45:15 2005
