Sung wrote:
"Mathematics is a species of the sign according to Peirce. (070114-1)
All signs are arbitrary according to Saussure. Therefore,
mathematics is arbitrary."
This is invalid for two reasons:
First, it is logically invalid. The error is called 'the fallacy of four
terms'. A valid syllogism must have exactly three unambiguous categorical
terms. But the term 'mathematics' as outlined above is ambiguous.
Mathematics is both a process and a producer of models. The specific model
chosen may be 'arbitrary' in the sense that it is symbolic, but the process
used within that mathematical reasoning is not arbitrary. There are two
meanings of mathematics in the syllogism above.
Second, it is only symbolic signs that are arbitrary, in the sense that
their models are assigned by an external agent; iconic and indexical
semiosic relations are not arbitrary. Saussure's semiology applies only to
symbols, therefore, there are also TWO meanings of the term 'sign' in the
above syllogism.
Edwina
----- Original Message -----
From: "Sungchul Ji" <[email protected]>
To: "Malcolm Dean" <[email protected]>
Sent: Tuesday, July 01, 2014 7:37 PM
Subject: [PEIRCE-L] Re: "Unreasonable Arbitrariness of Mathematics"::
Evidence
Malcolm,
Did I attach the evidence for UAM (unreasonable aribtrariness of
mathematics) to this email ? If not, it is now attached below.
As you can see in the table attached, each of the two sets of experimental
data, i.e., the single-molecule enzyme kinetic data and the decision-time
histogram, can be fit into two very different mathematical equations --
the Planck distribution and a double-exponential function for the former
and the Planck distribution and the ex-Gaussian distribution for the
latter. These recent findings of mine suggested to me that mathematical
models may be arbitrary. In fact you do not need any experimental data to
come to such a conclusion, since the same conclusion can be reached at by
simply combining the definition of the sign given by Peirce and the
linguistic principle of the arbitrariness of signs apparently first
articulated by Ferdinand de Saussure:
"Mathematics is a species of the sign according to Peirce. (070114-1)
All signs are arbitrary according to Saussure. Therefore,
mathematics is arbitrary."
Let me refer to Statement (070114-1) as the UAM (Unreasonable
Arbitrariness of Mathematics) thesis. If we apply the UAM thesis to
quantum mechanics (as a body of experimental data on the microscopic
world), it may be predicted that
"There should exist at least two mathematical models of (070114-2)
quantum phenomena that are compatible with all known
experimental data."
I suggest that Prediction (070114-2) based on the UAM thesis has been
validated by the fact that physicists, after over one century of quantum
mechanical research, are still passionately debating as to which of the
two theories are true - i) the non-deterministic, probability-wave theory
of Bohr, Bon, Heisenberg, etc, or ii) the deterministic, pilot-wave theory
of de Broglie, Schroedinger, Einstein, Bohm, and their more recent
followers, including my fellow professor at Rutgers, S. Goldstein, and his
research group.
Since Statement (071014-2) seems to be well supported by the modern
history of physics, it may be further asserted that
"The century-old debate between Einstein and Bohr on (070114-3)
the mathematical nature of quantum mechanics may be
ascribed to the principle of the arbitrariness of
mathematics specifically and the principle of the
arbitrariness of signs more generally. There may be
neither winner nor loser on this debate, since both
schools of thought can be judged valid within their
own theoretical framework."
If the above argument proves to be valid, the century-old debate in
physics may end up being finally solved by the knowledge gained outside of
physics, namely in biology, linguyistics and semiotics. It is possible
that the problem under debate is/was too complex to be resoved within
physics, just as some of the current probelms fased in biology may turn
oout to be too difficult to be solved within biology or even within
natural sciences assisted by mathemtics and high-power computer science.
With 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
Thanks.
I think the enthusiasm with which mathematicians present their "accuracy"
is produced by very narrow observations of the phenomena which happen to
match their expectations. The world is fuzzy and messy, except in rare
cases. The mathematics students I have observed have little
self-reflection. They do not see that their mathematical experiences are
special mental abilities, similar to religious states, in that
mathematics
is a form of generalization. I once asked Sean Carroll about the ontology
of his particle physics. "They are just numbers, statistics," he replied.
My departed friends, Michel and Francoise Gauquelin, once set out to
finally disprove Astrology. They gathered large statistical databases
(30k)
and compared planetary positions at birth with adult professions. And lo!
Given a complicated statistical method, they were able to demonstrate a
significant correlation. American skeptics decided that since this was
the
most promising evidence for planetary effects in human affairs, a simple
replication would provide a knock-out blow to the superstition. And lo!
With a U.S. sample size of 15k, they replicated the findings of the
Gauquelins. Did they honestly report the results? Of course not. The
statistical method was now "problematic" and "unclear." The results were
"dubious."
Personally, I no longer care whether planetary effects are acknowledged,
or
whether psi is shown to exist. I'm satisfied that a bunch of skeptics
proved that mathematicians can lie just like the rest of us.
M.
On Sun, Jun 29, 2014 at 4:20 AM, Sungchul Ji <[email protected]> wrote:
Hi,
In 1960, Eugene Wigner made the following statement in a paper with the
now-iconic title, â?oUnreasonable Effectiveness of Mathematicsâ?. In
it, he
cites a question presumably raised by one of his students at Princeton
(hence I will refer to it as the â?oWerner questionâ? for convenience):
". . . someone came to me and expressed his bewilderment [The
remark to be quoted was made by F. Werner when he was a student in
Princeton.] . . . with the fact that we make a rather narrow selection
when choosing the data on which we test our theories.
â?~How do we know that, if we made a theory which (062914-1)
focuses its attention on phenomena we disregard
and disregards some of the phenomena now commanding
our attention, that we could not build another theory
which has little in common with the present one but
which, nevertheless, explains just as many phenomena
as the present theory?â?T
It has to be admitted that we have no definite evidence
that there is no such theory."
Reading between the lines, it seems reasonable to infer that Wigner was
fully aware of the incompleteness of his famous thesis that
â?oMathematics is unreasonably effective.â? (062914-2)
which we may refer to as the â?oWigner thesisâ?.
In other words,
â?oThe Wigner thesis cannot provide any definitive (062914-3)
answer to the Werner question, neither positive
nor negative, and hence is incomplete.â?
The reason that the Wigner thesis is incomplete is because of
â?oThe Unreasonable Arbitrariness of Mathematics (UAM)â?
(062914-4)
The purpose of this email is to bring to your attention some recent
developments in theoretical biology (see Table 1 attached) that support
the UAM thesis: i.e.,
â?oTwo or more mathematical models can simulate a given (062914-5)
set of experimental data, just as two or more signs
can represent a given object in semiotics.â?
I think Statement (062914-5) is consistent with the facts (i) that
mathematical models are examples of Peircean signs (or, more
specifically,
the â?~argument-symbolic-legisignâ?T) and (ii) that all signs are
arbitrary ,
according to Saussureâ?Ts thesis of the â?oArbitrariness of Signsâ?
[2]. I am
assuming here that
â?oSaussureâ?Ts dyadic theory of signs can be viewed as (062914-6)
a reduced version of Peirceâ?Ts triadic theory of signs
-- reduced in the sense that the role of the semiotic
agent was not EXPLICITLY taken into account.â?
With 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
[1] Wigner, E. (1960). The Unreasonable Effectiveness of Mathematics
in the Natural Sciences. Communications in Pure and Applied
Mathematics 13 (I). Available at
https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html.
[2] See, for example,
http://en.wikipedia.org/wiki/Sign_(linguistics).
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