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guslA couple weeks ago I made a big edit on the Wikipedia article titled "Mathematical_models_in_physics", deleting some stuff that seemed to imply that "mathematics is not always faithful to physics" because of the Banach-Tarski Paradox.
I posted my justification for this here
Also here:
"An example of how geometry does not accurately represent the universe comes in the Banach-Tarski paradox which have consequences such as that a marble can be cut up into finitely many pieces and reassembled into a planet, or a telephone could be cut up and reassembled as a water lily. These transformations are not possible with real objects made of atoms, but are possible with their geometric shapes."
The Banach-Tarski paradox involves cutting an object into parts that are not Lebesgue-measurable, so it shouldn't be at all surprising that they can be recombined into objects of different sizes.
Physical objects are always Lebesgue-measurable, so the above "example" is irrelevant to the application of geometry in physics. The reason these transformations are not possible is because it's impossible to divide physical objects into non-Lebesgue measurable parts.
So you could say that the mathematical theory has "extra stuff" which does not have a physical interpretation. This is very similar to how, in recursion theory, there exists a whole theory of oracles and hypercomputation, which is irrelevant to real computers. The only part of recursion theory that can be interpreted "physically" is the part that talks about finite machines running in finite time.
The paragraph seems to be implying that geometry gives an inaccurate representation of the universe. That seems to mean that geometry return falsities. While different theories of space need different geometries, and perhaps no known geometry is "the true geometry" of space, the Banach-Tarski paradox is not an example of geometry's limitations. It is merely a mathematical truth that cannot be translated into a statement about physics (so it can neither lie nor tell the truth).
- Gustavo Lacerda
I posted my justification for this here
Also here:
"An example of how geometry does not accurately represent the universe comes in the Banach-Tarski paradox which have consequences such as that a marble can be cut up into finitely many pieces and reassembled into a planet, or a telephone could be cut up and reassembled as a water lily. These transformations are not possible with real objects made of atoms, but are possible with their geometric shapes."
The Banach-Tarski paradox involves cutting an object into parts that are not Lebesgue-measurable, so it shouldn't be at all surprising that they can be recombined into objects of different sizes.
Physical objects are always Lebesgue-measurable, so the above "example" is irrelevant to the application of geometry in physics. The reason these transformations are not possible is because it's impossible to divide physical objects into non-Lebesgue measurable parts.
So you could say that the mathematical theory has "extra stuff" which does not have a physical interpretation. This is very similar to how, in recursion theory, there exists a whole theory of oracles and hypercomputation, which is irrelevant to real computers. The only part of recursion theory that can be interpreted "physically" is the part that talks about finite machines running in finite time.
The paragraph seems to be implying that geometry gives an inaccurate representation of the universe. That seems to mean that geometry return falsities. While different theories of space need different geometries, and perhaps no known geometry is "the true geometry" of space, the Banach-Tarski paradox is not an example of geometry's limitations. It is merely a mathematical truth that cannot be translated into a statement about physics (so it can neither lie nor tell the truth).
- Gustavo Lacerda
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(no subject)
Date: 2005-08-01 12:24 pm (UTC)(no subject)
Date: 2005-08-01 12:26 pm (UTC)(no subject)
Date: 2005-08-01 12:29 pm (UTC)(no subject)
Date: 2005-08-01 12:50 pm (UTC)(no subject)
Date: 2005-08-01 01:04 pm (UTC)BTW: I don't see your edit in the history page, unless you're Darastrix. And even then, that edit was from 15 July.
(no subject)
Date: 2005-08-01 01:07 pm (UTC)(no subject)
Date: 2005-08-01 01:13 pm (UTC)(no subject)
Date: 2005-08-03 11:03 pm (UTC)"The axiom of choice seems quite plausible since we are allowing "arbitrary" mappings s. In categories in which only geometrically "reasonable" mappings are allowed, the axiom of choice is usually not true; but this only points out that such categories are distinct from the category of constant sets and arbitrary maps, which itself exists as an especially simple extreme case with which to contrast the others (Cantor's abstraction). Some of the opposition to the axiom of choice stems from its consequence, the Banach-Tarski Paradox... The abstractness of the sets, correlated with the arbitrary nature of the mappings, makes such paradoxes possible, but of course such paradoxes are not possible in the real world where things have variation and cohesion and mappings are not arbitrary. Nonetheless, since Cantor mathematicians have found the constant noncohesive sets useful as an extreme case with which continuously variable sets can be contrasted."
F.W. Lawvere and R. Rosebrugh, Sets for Mathematics, CUP, 2003, p. 86-7.
Lawvere himself was originally motivated by a desire to see continuum mechanics placed on an axiomatic foundation and eventually developed the smooth topos to serve as the realm for it. The category of smooth manifolds is not a topos, so it needed to be expanded, but it was key to him that this expansion not introduce "arbitrariness" into the possibilities for mappings as it would if it were blown out into the whole topos of sets. That's really the depth of my understanding of that at this point. Not exactly in the mathematical mainstream. ;)