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"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).
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content assessment scale. It is of interest to the following WikiProjects: | |||||||
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"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).