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added reference from von Neumann
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John Sidles
John Sidles
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Sometimes mathematics is not only about the methods of the proof, it is about the statement of the proof. E.g., it is hard to imagine an theorem-searching algorithm ever finding a proof of the results in Shannon's 1948 Mathematical Theory of Communication, without that algorithm first "imagining" (by some unspecified process) that there could BE a theory of communication.

Even so celebrated a mathematician as J. L. Doob at first had trouble grasping that Shannon's reasoning was mathematical in nature, writing in his AMS review (MR0026286):

[Shannon's] discussion is suggestive throughout, rather than mathematical, and it is not always clear that the author's mathematical intentions are honorable.
The decision of which mathematical intentions are to be accepted as "honorable" (in Doob's phrase) is perhaps very difficult to formalize.

[added reference]

One finds this same idea expressed in von Neumann's 1948 essay The Mathematician:

Some of the best inspirations of modern mathematics (I believe, the best ones) clearly originated in the natural sciences. ... As any mathematical discipline travels far from its empirical source, or still more, if it is a second or third generation only indirectly inspired by ideas coming from "reality", it is beset by very grave dangers. It becomes more and more purely aestheticizing, more and more l`art pour le art. ... Whenever this stage is reached, the only remedy seems to me to be the rejuvenating return to the source: the reinjection of more or less directly empirical ideas.
One encounters this theme of inspiration from reality over-and-over in von Neumann's own work. How could a computer conceive theorems in game theory ... without having empirically played games? How could a computer conceive the theory of shock waves ... without having empirically encountered the intimate union of dynamics and thermodynamics that makes shock wave theory possible? How could a computer conceive theorems relating to computational complexity ... without having empirically grappled with complex computations?

The point is straight from Wittgenstein and E. O. Wilson: in order to conceive mathematical theorems that are interesting to humans, a computer would have to live a life similar to an ordinary human life, as a source of inspiration.

Sometimes mathematics is not only about the methods of the proof, it is about the statement of the proof. E.g., it is hard to imagine an theorem-searching algorithm ever finding a proof of the results in Shannon's 1948 Mathematical Theory of Communication, without that algorithm first "imagining" (by some unspecified process) that there could BE a theory of communication.

Even so celebrated a mathematician as J. L. Doob at first had trouble grasping that Shannon's reasoning was mathematical in nature, writing in his AMS review (MR0026286):

[Shannon's] discussion is suggestive throughout, rather than mathematical, and it is not always clear that the author's mathematical intentions are honorable.
The decision of which mathematical intentions are to be accepted as "honorable" (in Doob's phrase) is perhaps very difficult to formalize.

Sometimes mathematics is not only about the methods of the proof, it is about the statement of the proof. E.g., it is hard to imagine an theorem-searching algorithm ever finding a proof of the results in Shannon's 1948 Mathematical Theory of Communication, without that algorithm first "imagining" (by some unspecified process) that there could BE a theory of communication.

Even so celebrated a mathematician as J. L. Doob at first had trouble grasping that Shannon's reasoning was mathematical in nature, writing in his AMS review (MR0026286):

[Shannon's] discussion is suggestive throughout, rather than mathematical, and it is not always clear that the author's mathematical intentions are honorable.
The decision of which mathematical intentions are to be accepted as "honorable" (in Doob's phrase) is perhaps very difficult to formalize.

[added reference]

One finds this same idea expressed in von Neumann's 1948 essay The Mathematician:

Some of the best inspirations of modern mathematics (I believe, the best ones) clearly originated in the natural sciences. ... As any mathematical discipline travels far from its empirical source, or still more, if it is a second or third generation only indirectly inspired by ideas coming from "reality", it is beset by very grave dangers. It becomes more and more purely aestheticizing, more and more l`art pour le art. ... Whenever this stage is reached, the only remedy seems to me to be the rejuvenating return to the source: the reinjection of more or less directly empirical ideas.
One encounters this theme of inspiration from reality over-and-over in von Neumann's own work. How could a computer conceive theorems in game theory ... without having empirically played games? How could a computer conceive the theory of shock waves ... without having empirically encountered the intimate union of dynamics and thermodynamics that makes shock wave theory possible? How could a computer conceive theorems relating to computational complexity ... without having empirically grappled with complex computations?

The point is straight from Wittgenstein and E. O. Wilson: in order to conceive mathematical theorems that are interesting to humans, a computer would have to live a life similar to an ordinary human life, as a source of inspiration.

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John Sidles
John Sidles
  • 1.4k
  • 18
  • 39

Sometimes mathematics is not only about the methods of the proof, it is about the statement of the proof. E.g., it is hard to imagine an theorem-searching algorithm ever finding a proof of the results in Shannon's 1948 Mathematical Theory of Communication, without that algorithm first "imagining" (by some unspecified process) that there could BE a theory of communication.

Even so celebrated a mathematician as J. L. Doob at first had trouble grasping that Shannon's reasoning was mathematical in nature, writing in his AMS review (MR0026286):

[Shannon's] discussion is suggestive throughout, rather than mathematical, and it is not always clear that the author's mathematical intentions are honorable.
The decision of which mathematical intentions are to be accepted as "honorable" (in Doob's phrase) is perhaps very difficult to formalize.