Re: Skolem's Paradox and why is math the way it is?
From: J.E. (troubled6man_at_yahoo.com)
Date: 09/21/04
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Date: 21 Sep 2004 06:40:07 -0700
> A different point of view may help.
Thank you for trying to help. If you do want to help, please respond
to my comments towards your reply. The comments get better farther
down, I only included the earlier comments for "completeness" since I
noticed that you were fond of that concept. Sorry, no more jokes, I
promise.
> 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?
Electrons are infinitely small as far as we can tell. I don't care
about large lumpy projectiles, but the electromagnetic FIELD of an
electron IS pretty darn amooth, is it smooth enough to meet your
criteria? How about the electrical potential generated by it?
> Of course not.
Oops, sorry, no wait, you're saying that electrons are composed of
parts? What evidence do you have?
> 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.
I'm interested in fundamental physics, not approximations to lumpy
bumpy imperfect composite objects like protons or basketballs.
Electrons are PERFECTLY identical, each and EVERY one. That's pretty
darn perfect.
> 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.
I actually disagree on this count, and this makes me unpopular in
physics circles, because my standards require more work, ugh. Dirac
theory is more accurate theory than Pauli theory, which is more
accurate than Heisenberg theory. But the complex number in Heisenberg
(an alleged non-relativistic) theory, remain as unexpected holdovers
from the relativistic theory. I believe that if you do correct
quantum AND relativistic physics and THEN do the approximations that
SOME things are much smaller than others or that some things are
mostly constant over a region or that some things mostly cancel each
other out over a region, then one gets the correct approximation and
that SOME relativistic and quantum effects will go away, but that
OTHERS stay. I think intrinsic spin is a relativistic quantum effect,
and in the non-quantum non-relativistic approximations it still holds
two electrons symmetrically in helium and it still gives rise to
magnetism in paramagnets and ferromagnets. If you started with
Newton, then magnetism would be a bit hard to explain. A big lumpy
bumpy hunk of metal might appear to be moving non-relativistically,
but maybe just maybe on a very small fine scale, there are parts of
the whole moving very fast indeed, that's really the issue, so one
needs to be accurate from the beginning and then EACH and EVERY
approximation needs to be justified to see if is valid for the
measurements you are going to take and the range over which you will
average and so on.
> 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.
...
> 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.
Projectively defining closure does NOT help physics. There is no
UNIQUE square root of any number, it is supposed to be the inverse of
a non-injective function. And introducing an UNINTERPRETED i, makes
things MUCH worse not BETTER. There are many physical things that
square to negative one, and how to do it depends on how you are
interpreting BOTH the original number and the square root function.
It sounds like you expect me to find it NECESSARY to push coordinates
around instead of physical objects. It's just a model! If the
negative one represents fliping points on a line across on origin,
then the square root of that operation is to rotate the line pi/2
radians about in a plane containing the line. And clifford numbers,
not uninterpreted imaginary numbers provide a better framework for
that.
The models you describe have ONE function, to EASE the proving of
MATHEMATICAL results. Mathematician bring in an uninterpreted "i"
(projective definitions instead of contructive definitions) so that
they can prove theorems that hold
for ANYTHING that meets their definitions, but then it becomes VERY
HARD to later make the equations be again about real physical things
once you step into such a mathematical trap. I've seen students never
come back, they spend weeks in some platonic math fantasy camp and
come back acting like an uninterpreted "i" exists and start putting it
everywhere it's CONVENIENT for their calculations so that they don't
have to THINK about what is really going on, and then they present
results, and I ask them to explain it and they can NOT explain it
because they walked through a uninterpreted hinterland to get their
results. If we forced people to use interpreted results the whole
time, then they could trace their calculations to meanings.
Who CARES if complex numbers "solve" polynomial equations? If the
polynomial is 5th degree or higher then it isn't going to be solved
anyway, and no polynomial is very physical anyway since they blow up
far from the origin. Closure doesn't give you clifford numbers, but
THINKING about the INTERPRETATION of a model of geometry DOES. And
complex numbers aren't GOOD enough to do that, we have to make MATRIX
REPRESENTATIONS of clifford numbers if we insist on using complex
numbers. So we end up with clifford numbers anyway, whereas if we had
USED clifford numbers EARLIER then we wouldn't have NEEDED complex
numbers. So somplex numbers are a WASTE and just CONFUSING and
unnecissarily COMPLICATE things.
> 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.
I lost you here, the rationals have the property that no matter how
close any two distinct point are on a line, one can find an unbounded
quantity of point between them.
> 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.
Heisenberg SHOULD have used the OLD mathematics of CLIFFORD algebras,
as should have Einstein since that works for all THREE systems.
Newton is excused becaused #1 he invented the calculus and it was
sufficient for the task and #2 clifford algebra wasn't around yet.
But Einstein and Heisenberg were LET DOWN by their "mathematical
friends" who ignored clifford algebra "because" it was "algebraically"
isomorphic to "a matrix algebra". The INTERPRETATION is different
than a matrix algebra and a different INTERPRETATION leads to
DIFFERENT class of isomorphism (for different structure). And using
fields (in the physics sense) of clifford algebra gives DIFFERNT
results (like no big bang) than using manifolds, so don't give me some
lecture about it not mattering.
> 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.
Assuming and approximating are totally different horses. And I am
concerned with FUNDAMENTAL physics, NOT computationally efficient
approximations of lumpy atomic matter interacting in a messy
complicated world. My dream is of a model with ONE object with enough
degrees of freedom to model the real world that moves from one state
to another based on it's current state, no reference to any other
model or outside potential or anything. I have to figure out what
degrees of freedom to have, and how many and how it evolves. It's a
tall order, but I was hoping for inspiration from cleaning house and
focusing on physics education. That way if I fail at the dream
personally, then at least I made existing physics easier to teach to
students.
- Next message: flip: "Re: Nora Baron"
- Previous message: Georg: "Re: Math books to master math of string theory"
- In reply to: Martin Schlansky: "Re: Skolem's Paradox and why is math the way it is?"
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- Reply: Shmuel (Seymour J.) Metz: "Re: Skolem's Paradox and why is math the way it is?"
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