Hi,
The mathematical theory of categories may provide a rational framework for
characterizing and analyzing the possible connections between physics,
biology, and semiotics. It is for this reason that I begin this post with
some definitions in category theory:
(1) “Comparison and analogy are fundamental aspects of knowledge
acquisition.
We argue that one of the reasons for the usefulness and importance of Category
Theory is that it gives an abstract mathematical setting for analogy and
comparison, allowing an analysis of the process of abstracting and relating
new concepts. This setting is one of the most important routes for the
application of Mathematics to scientific problems. “ (Brown and Porter,
2006).
(2) “We view a category as giving a fairly general abstract context for
comparison. The objects of study are the objects of the category. Two
objects, A and B, can be compared if the set C(A,B) is non-empty and
various arrows A ---> B are ‘ways of comparing them’. The composition
corresponds to: If we can compare A with B and B with C, we should be able
to compare A with C.” (Brown and Porter, 2006).
(3) “. . . a functor . . is a way of comparing categories, . . . “ (Brown
and Porter, 2006).
(4) "“Different branches of mathematics (human knowledge; my addition) can
be formalized into categories. These categories can then be connected
together by functors. And the sense in which these functors provide
powerful communication of ideas is that facts and theorems (regularities;
my addition) proven in one category (discipline; my addition) can be
transferred through a connecting functor to yield proofs of analogous
theorems in another category. A *functor** is like a conductor of
mathematical truth (*my emphasis*)*” (Spivak, 2013).
(A) The scientific controversies surrounding Shannon's entropy H may not
be successfully analyzed without taking into account their semiotic
aspects. Many would agree that, had von Neumann not suggested to Shannon
to name his H function "entropy", a lot of the confusions in the H-S (or
information-entropy) debates might have been avoided.
(B) There is no doubt that a close formal similarity exists between the
mathematical equations of H and S (see Rows 1 and 3, Column 1). But this
is a shallow and superficial reason for giving both functions the same
name, 'entropy', without first checking that both mathematical functions
share some common principles or mechanisms. Since the meaning of 'entropy'
in thermodynamics is relatively well established (e.g., a measure of
disorder, obeying the Second Law), giving this same name to the H function
may lead to unwittingly attributing the same thermodynamic meaning of
entropy to H. In fact many prominent scientists and mathematicians
unfortunately have taken this road, thereby creating confusions among
scholars.
(C) A somewhat similar events have transpired during the past 7 years in
my research. I, with the help of one of my pre-med students at Rutgers,
derived a new equation, called the Planckian distribution equation (PDE)
(see Row 3, Column 2), by replacing the universal constants and temperature
in the Planck blackbody radiation equation (PBRE) (see Row 1, Column 2)
with free parameters, a, b, A and B. For convenience, we define "Planckian
processes" as those physicochemical, biomedical or socioeconomic processes
that generate numerical data that fit PDE, and there are many such
processes found in natural and human sciences [1]. In certain sense, H
function of Shannon is related to the S function of Boltzmann as PDE is
related to PBRE. Therefore, if there are functors connecting PDE and PBRE
(e.g., energy quantization, wave-particle duality) as I strongly believe,
it is likely that there can be at least one functor connecting H and S
which I do not believe is the Second Law as some physicists and
mathematicians claim. As Jon and Stan recently hinted, the functor
connecting H and S may well turn out to be "variety" or "complexity" as
suggested by Wicken [2, p. 186].
(D) In addition to the "mathematical functors" described in (C), there may
be "non-mathematical" or "qualitative" functors connecting H and S on the
one hand and PDE and PBRE on the other, and I humbly suggest that these
"qualitative functors" may be identified with the Peircean sign triad or
semiosis as briefly indicated in Row 6 of the table below.
(E) If the contents of Table 1 turn out to be true in principle, we may be
justified to recognize two kinds of functors in category theory --
"quantiative" and "qualittive" functors", the former belonging to the
domain of mathematics and the latter to that of semiotics.
____________________________________________________________________________________________________
Table 1. A comparison between Shannon's invention of H and my derivation
of PDE from Planck radiation equation.--
____________________________________________________________________________________________________
Shannon's invention of H
Ji's derivation of PDE
____________________________________________________________________________________________________
1. Original Boltzmann equation for entropy S
Planck blackbody radiation equation (PBRE)
equation (1872-5)
(1900)
S = k ln W which generalizes as
U = ((2 pi h c^2)/lambda^5)/(Exp(hc/kT lambda) - 1)
S = - k Sum Pi log Pi
____________________________________________________________________________________________________
2. Insight or microscopic explanation
quantization of action or movement
mechanisms of macroscopic measurements
(as a prelude to organization)
____________________________________________________________________________________________________
3. New equation Shannon equation
Planckian distribution equation (PDE)
invented or invented in 1948
derived in 2008-9
discovered
H = - K Sum Pi log_2 Pi
y = (a/(Ax + B))^5/(Exp(b/(Ax + B)) - 1)
_____________________________________________________________________________________________________
4. Significance H measures the variety or
PDE can be used to measure the degree of
complexity of a message
non-randomness (also called 'order' or
source of a communication
'organization') of a system, called the Planckian
system.
information (I_P) (2015)
______________________________________________________________________________________________________
5. Domain of Any field that can generate
Any field that can generate long-tailed
application probability distributions Pi
histograms that fit PDE
______________________________________________________________________________________________________
6. Semiotic Equation for H was invented but
Equation for PDE was derived from PBRE
problem H had no 'name'. von Neumann
and was found to fit single-molecule enzyme
involving "names" recommended for H the same name
kinetic data, strongly suggesting that the two
and their "objects" 'entropy' used to designate entropy
mechanisms of (i) the energy quantification
or "referents". S in thermodynamics without
and (ii) the particle-wave duality operating atoms
knowing whether or not the
also operate in enzymes. In other words, the new
mechanism or principles
underlying equation discovered here has both a rational name
S can be extended to H.
and reasonable mechanisms suggested by the
In other words, the new equation
name. Hence the semiotic problem faced by
invented by Shannon had no
'genuine' PDE is far less problematic than that faced by H.
name nor 'genuine' object or
referent.
_______________________________________________________________________________________________________
Common principle The Peircean sign triad, or
The Peircean sign triad, or
or mechanism semiosis
semiosis.
connecting different
disciplines or fields
________________________________________________________________________________________________________
If you have any questions, suggestions, or criticisms, please let me know.
All the best.
Sung
Sungchul Ji, Ph.D.
Associate Professor of Pharmacology and Toxicology
Department of Pharmacology and Toxicology
Ernest Mario School of Pharmacy
Rutgers University
Piscataway, N.J. 08855
732-445-4701
www.conformon.net
References:
[1] Ji, s. (2015). Planckian distributions in molecular machines,
living cells, and brains: The Wave-particle duality in biomedical
sciences. Proceedings of the International Conference on Biology and
Biomedical Engineering. Vienna, March 15-17, 2015. Uploaded to
ResearchGate in March, 2015.
[2] Wicken, J. S. (1987). Entropy and Information: Suggestions for
Common Language. Phil. Sci. 54: 176-193.
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