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.
-----------------------------
PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON PEIRCE-L
to this message. PEIRCE-L posts should go to [email protected] . To
UNSUBSCRIBE, send a message not to PEIRCE-L but to [email protected] with the
line "UNSubscribe PEIRCE-L" in the BODY of the message. More at
http://www.cspeirce.com/peirce-l/peirce-l.htm .
