Jerry, Sung, Stephen, list,
I didn't discuss discrete versus continuous, just pointed out to Sung
that in quantum mechanics, complex numbers are needed for what many
would call their algebraic properties. I could also have pointed out
that abstract algebra deals with symmetries such as those on which
conservation laws are based.
Best, Ben
On 7/3/2014 12:20 AM, Jerry LR Chandler wrote:
Ben, Stephen, List:
The mathematical thicket is indeed thick, as your historical notes
indicate. Even thicker, in my opinion.
First, Western mathematics developed rather differently than
non-Western mathematics, perhaps because of the Greek origins of
Western thought.
At the root, is the association of Mesopotamian symbol systems which
co-evolved primitive forms of grammar with numbers. The Greeks
separated the usage of sound symbols as either number or noun.
More to the point is the iconic character of geometry - visible lines
showing the relationships in a clear and distinct form, versus the
invisible character of relations created by algebraic notations and
the infinity of geometric forms generated (as in the theories of chaos
and fractals.)
Thus, for CSP, he classified the meanings in the terms, as "thing,
representation, and form". The form was formal - and it could be
either an geometric form or an algebraic form or, for him (but not
modern mathematics) a invisible chemical form.
CSP knew how to index geometric, algebraic and chemical forms as
formula in exact modes of symbolization (within the state of knowledge
as it stood in his day.)
Among the students of CSP writings today, this fact is seldom
acknowledged. The usual linguistic focus of discussions generates
very rich discussions, but, alas, limited understanding of CSP's meanings.
But, I digressed.
The critical question for the non-mathematicians is the usage of
terminology such that discrete mathematics is separated from continuos
mathematics, which CSP crisply separated.
As you correctly point out, this separation is necessary for quantum
mechanics as applied to the individual atoms of chemical elements,
which intermingles discrete numbers, discrete functions and continuous
function.
Indeed, this intermingling of various mathematical forms effectively
prevents the application of exact quantum reasoning to biochemicals
systems... and upward.
At some point, the thickets becomes an impenetrable jungle,
mathematically and logically. Conflating scientific concepts such as
entropy with information, contribute to constructing
logically impenetrable jungles.
Thus, new symbol systems with fresh indexes are introduced to form
the icons associated with the logics of living systems. At the risk
of pushing this metaphor to far, I will add that the logical
disentanglements of jungles into thickets is possible within the
theories of numbers and graphs. This disentanglement requires, among
other things, a careful accounting of the relations among icons,
indices and symbols, along with the corresponding truths derived from
observations.
Cheers
Jerry
On Jul 2, 2014, at 3:57 PM, Benjamin Udell wrote:
Sung, list, I think you're getting into a thicket. Mathematicians
have varied on these questions.
Kronecker (according to Weber 1893) said that God made integers, all
else is the work of people (/Menschenwerk/).
The Nicolas Bourbaki group placed most classical geometry under the
umbrella of abstract algebra, and regarded maths of structures of
order as one of three basic areas (aside from combined areas):
structures of order, structures of group, and structures of space.
Group member Jean Dieudonné once said (I doubt that I can find the
quote) that he thought that perhaps Bourbaki had not paid enough
attention to "combinatorial" areas of mathematics - I guess he meant
the enumerative wing of combinatorics at least, maybe measure theory,
etc., maybe even graph theory (traditionally a wing of combinatorics).
Nature seems to speak in geometries both Euclidean (e.g., complex
numbers) and non-Euclidean.
Complex numbers are historically rooted in algebra (even if one
argues that they're really in geometry), and are important for
representing the amplitudes whose squares are probabilities in
quantum mechanics. Is the use of complex numbers for
quantum-mechanical amplitudes more geometrical than algebraic? Is one
to accept as natural just those algebraic or arithmetical things that
have representations in geometry? Are there any that don't? What
about series, ordered sets, lattices, etc., sets of identificatively
or designatively connective indices ("Toe bone connected to the foot
bone: Foot bone connected to the heel bone....")?
Best, Ben
On 7/2/2014 1:02 PM, Sungchul Ji wrote:
Hi,
The following response from Ed Dellian seems to raise an interesting
possibility for evaluating the UAM thesis. According to Ed, I should not
include geometry as a part of "mathematics" in the expression
"Unreasonable Arbitrariness of Mathematics (UAM)". I tend to agree with
him, since my evidence supporting UAM comes mainly from algebra, not
geometry. If this turns out to be true on further inquiry, then the UAM
thesis may have to be modified as
"The Unreasonable Arbitrariness of Algebra (UAA)" (070214-1)
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
---------------------------- Original Message ----------------------------
Subject: AW: "Unreasonable Arbitrariness of Mathematics":: Evidence
From: "Ed Dellian"<[email protected]>
Date: Wed, July 2, 2014 5:08 am
To: "'Sungchul Ji'"<[email protected]>
"'Malcolm Dean'"<[email protected]>
--------------------------------------------------------------------------
Dear Sung,
you're absolutely right on UAM with respect to arithmetic and algebra.
This branch of mathematics is arbitrary as it is a product of human logic
and reason only, based on logic, and therefore - as all of logic -
ultimately based on the principle of non-contradiction. In arithmetic and
algebra, this principle appears as the tautology " A = A " (equivalently,
"A is not "non-A"), on which all "equations" are built. As a consequence,
different mathematical equations can fit to describe experimental data,
none of which equations tells anything about the real meaning of the data.
It was Leibniz who held the principle A = A “to be the foundation of
"all of mathematics" (see Leibniz's second letter to Caroline Princess of
Wales, paragraph 1, in: The Leibniz-Clarke exchange of letters, 1715/6).
But Leibniz was wrong when he included Euclidean geometry. This geometry
is not an „arbitrary“ human product but is rooted in Nature herself:
it is the “language of Nature†(Galileo, 1623). Therefore, Galileo and
Newton chose Euclidean geometry as the mathematical tool for their natural
philosophy and the theory of motion. Note that Galileo’s Discorsi of
1638 (I’ve edited a new German translation recently) and Newton’s
Principia (which I edited in German in 1988) as well are throughout based
on synthetic geometry, not on the calculus differentialis! This
“synthetic†geometry (contrary to arithmetic and algebra, and also
contrary to Cartesian analytical geometry) knows a non-tautological
principle A : B = C = constant (the principle of proportionality of
heterogeneous entities A * B) that allows for a rational mathematical
relation between natural entities of a different kind – a relation that
would be impossible in arithmetic and algebra. (Note that algebra
doesn’t know the basic “algorithm†of proportion theory A/B = C =
constant, nor does it know natural constants which actually always are
geometric proportionality constants). And, this is the reason why
synthetic geometry passes the analytic power of tautological arithmetic
and algebra, and even allows for discovering the unknown on the basis of
the known. Whenever natural experience brings lawful relations between
heterogeneous natural entities to light, as for instance the relation
between “energy†E and “momentum†p in applications of the Maxwell
theory, the correct resulting mathematical description will always be a
geometric proportionality that mirrors the naturally existing structure,
not something that has been taken from the human brain and arbitrarily
imposed on nature. This was the case when John Henry Poynting (1884)
discovered the relation E/p = c = constant. The same thing happened when
Planck found a relation between “energy†E and the frequency f of
light, a relation that urged him to form the relation E/f = h = constant.
This is a geometric proportionality again according to the pattern A/B = C
= constant. Geometric proportion theory also provides the basic structure
of the double helix which is a quaternate proportion according to A : B =
C : D. Therefore natural science should eventually accept the truth that
“The language of nature is not algebra†(cf. my 2012 essay on my
websitewww.neutonus-reformatus.com <http://www.neutonus-reformatus.com/>
).
All the best,
Ed.
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