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
>> website www.neutonus-reformatus.com <http://www.neutonus-reformatus.com/>
>> ).
>>
>> All the best,
>>
>> Ed.
>>
>
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