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edited Jun 16, 2020 at 10:30

Quote from Bangs, Crunches, Whimpers, and Shrieks:

Thomson lamps, super $\pi$ machines, and Platonist computers are playthings of philosophers; they are able to survive only in the hothouse atmosphere of philosophy journals. In the end, M—H spacetimes and the supertasks they underwrite may similarly prove to be recreational fictions for general relativists with nothing better to do. But in order to arrive at this latter position requires that one first resolve some of the deepest foundation problems in classical general relativity, including the nature of singularities and the fate of cosmic censorship. It is this connection to real problems in physics that makes them worthy of discussion.

There are also connections to the philosophy of mathematics and to the theory of computability. Because of finitist scruples, some philosophers have doubted that it is meaningful to assign a truth value to a formula of arithmetic of the form $(\exists x_1)(\exists x_2)\dots(\exists x_n)F(x_l,x_2,\dots,x_n)$. It seems to me unattractive to make the truth of mathematical statements depend on the contingencies of spacetime structure. The sorts of arrangements considered above can be used to decide the truth value of assertions of arithmetic with a prenex normal form that is purely existential or purely universal. (Fermat's last theorem, for example, has a purely universal form.) For such an assertion $\gamma_1$ is set to work to check through the (countably infinite) list of $n$-tuples of numbers in search of a falsifier or a verifier according as the assertion to be tested is universal or existential, and $\gamma_1$ reaps from these labors a knowledge of the truth value of the assertion. But as soon as mixed quantifiers are involved, the method fails. However, Hogarth (1994) has shown how more complicated arrangements in general relativistic spacetimes can in principle be used to check the truth value of any arithmetic assertion of arbitrary quantificational complexity. Within such a spacetime it is hard to see how to maintain the attitude that we do not have a clear notion of truth in arithmetic.

Quote from Bangs, Crunches, Whimpers, and Shrieks:

Thomson lamps, super $\pi$ machines, and Platonist computers are playthings of philosophers; they are able to survive only in the hothouse atmosphere of philosophy journals. In the end, M—H spacetimes and the supertasks they underwrite may similarly prove to be recreational fictions for general relativists with nothing better to do. But in order to arrive at this latter position requires that one first resolve some of the deepest foundation problems in classical general relativity, including the nature of singularities and the fate of cosmic censorship. It is this connection to real problems in physics that makes them worthy of discussion.

There are also connections to the philosophy of mathematics and to the theory of computability. Because of finitist scruples, some philosophers have doubted that it is meaningful to assign a truth value to a formula of arithmetic of the form $(\exists x_1)(\exists x_2)\dots(\exists x_n)F(x_l,x_2,\dots,x_n)$. It seems to me unattractive to make the truth of mathematical statements depend on the contingencies of spacetime structure. The sorts of arrangements considered above can be used to decide the truth value of assertions of arithmetic with a prenex normal form that is purely existential or purely universal. (Fermat's last theorem, for example, has a purely universal form.) For such an assertion $\gamma_1$ is set to work to check through the (countably infinite) list of $n$-tuples of numbers in search of a falsifier or a verifier according as the assertion to be tested is universal or existential, and $\gamma_1$ reaps from these labors a knowledge of the truth value of the assertion. But as soon as mixed quantifiers are involved, the method fails. However, Hogarth (1994) has shown how more complicated arrangements in general relativistic spacetimes can in principle be used to check the truth value of any arithmetic assertion of arbitrary quantificational complexity. Within such a spacetime it is hard to see how to maintain the attitude that we do not have a clear notion of truth in arithmetic.

Quote from Bangs, Crunches, Whimpers, and Shrieks:

Thomson lamps, super $\pi$ machines, and Platonist computers are playthings of philosophers; they are able to survive only in the hothouse atmosphere of philosophy journals. In the end, M—H spacetimes and the supertasks they underwrite may similarly prove to be recreational fictions for general relativists with nothing better to do. But in order to arrive at this latter position requires that one first resolve some of the deepest foundation problems in classical general relativity, including the nature of singularities and the fate of cosmic censorship. It is this connection to real problems in physics that makes them worthy of discussion.

There are also connections to the philosophy of mathematics and to the theory of computability. Because of finitist scruples, some philosophers have doubted that it is meaningful to assign a truth value to a formula of arithmetic of the form $(\exists x_1)(\exists x_2)\dots(\exists x_n)F(x_l,x_2,\dots,x_n)$. It seems to me unattractive to make the truth of mathematical statements depend on the contingencies of spacetime structure. The sorts of arrangements considered above can be used to decide the truth value of assertions of arithmetic with a prenex normal form that is purely existential or purely universal. (Fermat's last theorem, for example, has a purely universal form.) For such an assertion $\gamma_1$ is set to work to check through the (countably infinite) list of $n$-tuples of numbers in search of a falsifier or a verifier according as the assertion to be tested is universal or existential, and $\gamma_1$ reaps from these labors a knowledge of the truth value of the assertion. But as soon as mixed quantifiers are involved, the method fails. However, Hogarth (1994) has shown how more complicated arrangements in general relativistic spacetimes can in principle be used to check the truth value of any arithmetic assertion of arbitrary quantificational complexity. Within such a spacetime it is hard to see how to maintain the attitude that we do not have a clear notion of truth in arithmetic.

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created Apr 9, 2018 at 17:59

Martín-Blas Pérez Pinilla

Quote from Bangs, Crunches, Whimpers, and Shrieks:

Thomson lamps, super $\pi$ machines, and Platonist computers are playthings of philosophers; they are able to survive only in the hothouse atmosphere of philosophy journals. In the end, M—H spacetimes and the supertasks they underwrite may similarly prove to be recreational fictions for general relativists with nothing better to do. But in order to arrive at this latter position requires that one first resolve some of the deepest foundation problems in classical general relativity, including the nature of singularities and the fate of cosmic censorship. It is this connection to real problems in physics that makes them worthy of discussion.

There are also connections to the philosophy of mathematics and to the theory of computability. Because of finitist scruples, some philosophers have doubted that it is meaningful to assign a truth value to a formula of arithmetic of the form $(\exists x_1)(\exists x_2)\dots(\exists x_n)F(x_l,x_2,\dots,x_n)$. It seems to me unattractive to make the truth of mathematical statements depend on the contingencies of spacetime structure. The sorts of arrangements considered above can be used to decide the truth value of assertions of arithmetic with a prenex normal form that is purely existential or purely universal. (Fermat's last theorem, for example, has a purely universal form.) For such an assertion $\gamma_1$ is set to work to check through the (countably infinite) list of $n$-tuples of numbers in search of a falsifier or a verifier according as the assertion to be tested is universal or existential, and $\gamma_1$ reaps from these labors a knowledge of the truth value of the assertion. But as soon as mixed quantifiers are involved, the method fails. However, Hogarth (1994) has shown how more complicated arrangements in general relativistic spacetimes can in principle be used to check the truth value of any arithmetic assertion of arbitrary quantificational complexity. Within such a spacetime it is hard to see how to maintain the attitude that we do not have a clear notion of truth in arithmetic.