Re: Skolem's Paradox and why is math the way it is?
From: Martin Schlansky (mschlans_at_optonline.net)
Date: 09/21/04
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Date: Tue, 21 Sep 2004 06:54:13 GMT
A different point of view may help.
Mathematical systems, like continuum mathematics, are logically constructed
entities that, although originally inspired by real world phenomena, have
evolved into systems that are based on sets of elements that are "ideal" and
not real. So it is amazing to find that all mappings of the continuous
real line of points(e.g.,n-dimensional space, curves and surfaces) are
omnipresent in all of classical physics; These mappings represent objects
such as "infinitesimal points", and analytic curves and surfaces that are
"infinitely smooth." There are no such types of "ideal" objects in the real
world. Can you find a piece of matter that is infinitesimally small or a
projectile that follows a path that is perfectly smooth? Of course not. Yet
we use continuum mathematics to state and define the laws of physics and to
derive and define solutions. The correct answer is that continuum
mathematics when correctly applied is only a very fine approximation to real
world events. Comprehensive , well defined physical theories are evaluated
on the degree to which they correctly predict outcomes and events, as
measured by a community of learned individuals and "to within certain
constraints and assumptions" which are often implicit. When applied outside
these conditions all physical theories break down; If these constraints are
violated the mathematical system will continue to give results but these
will not match the physical outcomes and events. This is why physical
theories are constantly being reworked. When you move to the world of atoms
the laws of classical physics break down and must be supplanted by the laws
of quantum physics. When you move to objects that move at very high speeds
and great distances you must adopt the physics of Relativity. Note that even
though Relativity encompasses and supercedes Newtonian physics you wouldn't
want to apply the equations of Relativity to every day(normal speeds and
distances) phenomena because the mathematically correct solutions would be
overly complex , unintuitive and unnecessary; rather it would be better to
use the approximations predicted by Newtonian mechanics as is done.
Why does continuum mathematics represent ideal points, curves and surfaces?.
The answer is that the mathematical system we call algebra must contain a
property we call algebraic closure. An algebraic system defines a set of
elements(eg,numbers) and operations(add,subtract,multiply,divide) that are
allowed to operate on these elements. In every case we want to treat the
algebra as a tool so that no matter how complex our models and calculations
are we get intermediate and resulting values that lie within the defining
set of numbers. When this is not the case we must expand the set of numbers
so that closure is achieved. If we start with the natural
numbers(1,2,3,...), we are blocked by our inabilty to get a the result for
2-5, so we extend the natural numbers to become integers
(... -3,-2,-1,0,1,2,3, ...). Then 2-5 = -3 is in the set. If we now have the
integers we are blocked by our inability to get a result for 3/2, so we
extend the integers to become rational numbers (... -15/4,1/2, 2/3, 3/2,
21/13 ...). If we now have the rational numbers we are blocked by our
inabilty to get a result for square root of 2, so we extend the rationals
to become real numbers. If we have the real numbers we are blocked by our
inabilty to get a result for square root of -1, so we extend the real
numbers to become the complex numbers(i=sqrt(-i), 2+5i, -2/3+7i). Bear in
mind that we must have closure because very often our mathematical models,
formulas and calculations involve complex algebraic manipulations on many
variables. We do not know whether a particular variable will be an integer
or a rational number or a real or complex number. That is, we must produce a
correct answer whatever the values of the variables may be. It is the
necessity of closure that has logically induced ideal objects like the
infinitesimal points on the real line. No matter how close any 2 points are
on the real line you can find an infinity of points between them. This is
obviously an ideal but logical construct, necessary if we want to maintain
closure and uniqueness of elements.
Indeed new mathematical systems are constantly being synthesized and
discovered. This is what modern mathematics is all about. Newton used the
new mathematical system called the calculus to state his laws of motion,
Einstein used new mathematical objects called tensors and Riemannian
geometry to state the laws of General Relativity, Heisenberg used a new
mathematical system called matrix algebra to state one version of the laws
of quantum mechanics.
So even though real number analogs may not exist in the real world
physicists construct theories that are approximations to the real world. For
example like the real numbers you "assume" that a fluid is a continuum of
points contained within a small volume of space. This model gives rise to
the partial differential equations of fluid mechanics which at the macro
level predicts outcomes for fluid type phenomena ,to a degree of
correctness, which is highly satisfactory.
In any case you may be interested in books that explain the nature of
physical theories. I'm sure there are several. The one I know of is
"Foundations of Physics" by Robert Bruce Linsey and Henry Margenau,
publisher Dover Books.
"Shmuel (Seymour J.) Metz" <spamtrap@library.lspace.org.invalid> wrote in
message news:414e25b3$17$fuzhry+tra$mr2ice@news.patriot.net...
> In <39d6e584.0409182125.2d1a5ab1@posting.google.com>, on 09/18/2004
> at 10:25 PM, troubled6man@yahoo.com (J.E.) said:
>
>>I have real concerns (as a scientist) about how to make representations
>>of mathematical objects,
>
> Why? You claim to be a physicist; your job is to make models of the
> real world. Models of Mathematical objects can be safely left to those
> that work in Logic, Mathematic and Metamathematics.
>
>>and I don't see how maintaining the fiction that
>>real numbers that we can't describe somehow "exist"
>
> I can't see how claiming that a "collapse of the wave function", that
> we can neither model not observe, exists. The existence of
> nonconstructible Mathematical objects however, is necessary for the
> tools that working physicists use.
>
>>The axioms of ZF assert that the empty set exists. The axioms of ZF
>>give conditions under which a new set exists constructed from finite
>>statements and already existing sets. All sets that are asserted to
>>exist by ZF can be "constructed" in this sense.
>
> No. The axioms of ZF are sufficient to prove the existence of
> uncountable sets.
>
> --
> Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>
>
> Unsolicited bulk E-mail subject to legal action. I reserve the
> right to publicly post or ridicule any abusive E-mail. Reply to
> domain Patriot dot net user shmuel+news to contact me. Do not
> reply to spamtrap@library.lspace.org
>
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