sourceType:: book author:: Neal Stephenson sourcePublication:: Anathem ref:: noteTitle:: Anathem; Neal Stephenson. (book)
Anathem; Neal Stephenson. (book)
acknowledgements section has references to sources that informed the story: http://nealstephenson.com/acknowledgments.html
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https://www.daviddeutsch.org.uk/books/the-fabric-of-reality/
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Our Mathematical Universe http://arxiv.org/pdf/0704.0646
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Penrose posits, in The Emperor’s New Mind (ISBN 978-0192861986) and Shadows of the Mind (ISBN 978-0195106466)
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Kurt Gödel is the one thinker who knit together all of the other influences mentioned in these Acknowledgments. I need to mention a few things about Gödel now, gleaned from several different sources, and it seems most expedient to list all of the sources first. So, here they are:
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Palle Yourgrau. Gödel meets Einstein: Time Travel in the Gödel Universe (1999) ISBN 978-0812694086
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Palle Yourgrau. A World Without Time: The Forgotten Legacy Of Godel And Einstein (2004) ISBN 978-0465092949
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Rebecca Goldstein. Incompleteness: The Proof and Paradox of Kurt Godel (2005) ISBN 978-0393051698
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Hao Wang. A Logical Journey: from Gödel to Philosophy (1997) ISBN 978-0262231893
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Kurt Gödel. Collected Works. Edited by Solomon Feferman, John W. Dawson Jr. et al. (1990) ISBN 0195147219
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Freeman Dyson. Private communication, 2007
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Verena Huber-Dyson. Private communication, 2006
As explained by Hao Wang, the general model that Gödel used to think about mathematical Platonism ran something like this:
- Entities that are the subject matter of mathematics exist independently of human perceptions, definitions and constructions.
- The human mind is capable of perceiving such entities.
Item 1 above seems uncontroversial to many and is believed, at least to some extent, by nearly all mathematicians as well as many who adopt a “common sense” approach to such questions; as an example, anyone who believes that 3 was a prime number a billion years ago, agrees at least to some extent with Item 1.
Anyone who espouses (1), however, must supply an account of how it is that the human mind is capable of obtaining knowledge about mathematical entities, which, according to (1), are non-spatiotemporal and do not stand in a normal causal relationship to the entities that make up the physical universe. Various arguments have been put forward to explain this seeming paradox; for a useful summary, see Mark Balaguer's entry on Platonism in Metaphysics in the Stanford Encyclopedia of Philosophy and for a more thoroughgoing treatment read his Platonism and Anti-Platonism in Mathematics (ISBN 978-0195143980)
Gödel’s approach to (2) is as follows:
2a. “Something besides the [physical] sensations actually is immediately given.” Gödel refers to these givens as “data of the second kind.”
2b. “It by no means follows, however, that the data of this second kind, because they cannot be associated with actions of certain things upon our sense organisms, are something purely subjective. Rather they, too, may represent an aspect of objective reality, but, as opposed to the sensations, their presence in us may be due to another kind of relationship between ourselves and reality.”
2c. “I conjecture that some physical organ is necessary to make the handling of abstract impressions (as opposed to sense impressions) possible…Such a sensory organ must be closely related to the neural center for language.”
The first two quotes above are from Gödel’s 1964 paper What is Cantor’s continuum problem? and the third is from Hao Wang’s A Logical Journey. These three postulates will hereinafter be denoted as the second kind of data; the other kind of relationship; and Gödel’s Organ.