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From: Brent Meeker <meekerdb.domain.name.hidden>

Date: Wed, 20 Sep 2000 12:39:22 -0700

Logic and number theory and in fact any mathematical theory are independent of

physical reality since they are simply sets of axioms and their consequences according to some rule of inference.

Euclidean geometry didn't change in 1823; only the idea that it necessarily

applied to the physical world changed. There are many physical entities that

number theory does not apply to. It's just that the the non-applicability is

so obvious that one doesn't usually think of number theory applying. For an

example that sometimes trips up 4th graders; consider that there are two school

clubs, one with 15 members and one with 20 members. They have picnic together.

How many are at the picnic?

Similarly the application of logic is empirical. Ordinary predicate calculus

has a limited range of applicability - you get into trouble if you quantify

over predicates. Logic ordinarily deals with propositions which have

true-false values and connectives (and, or,...). But it could be extended to

include things like 'red implies not-green'. All of this is independent of the

physical world. It's just that we like to adapt our language so as to make it

easy to talk accurately about the world.

Brent Meeker

*> Conventional wisdom of course says that logic and hence number theory
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*> are independent of physical reality. Suppose you asked in 1822,
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*> whether the rules concerning plane geometry would be different in
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*> other universes. Then the answer would have been that a system of
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*> axioms and hence geometry are independent of physical reality.
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*> However in 1823 Bolyai and Lobachevsky independently realized that
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*> entirely self-consistent "non-Euclidean geometries" could be created
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*> in which the parallel postulate did not hold. Today, the discussion
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*> would be quite different. For geometry, the cosmology of universes
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*> with a range of curvatures are now considered reasonable.
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*>
*

*> In his book, The Structure and Interpretation of Quantum Mechanics,
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*> Hughes mentions works which assert that the rules of logic may be
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*> empirical. I find your question stimulating, in that an implication
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*> of logic being empirical is the charming speculation as to whether
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*> there could be universes which operate with different versions of the
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*> distributive law.
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*>
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*> References:
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*> quant- ph/0001074
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*> An Epistemological Derivation of Quantum Logic
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*> John Foy
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*> math. HO/9911150
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*> Machines, Logic and Quantum Physics
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*> David Deutsch and Artur Ekert
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*>
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*> John
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*>
*

Regards

Received on Wed Sep 20 2000 - 13:20:09 PDT

Date: Wed, 20 Sep 2000 12:39:22 -0700

Logic and number theory and in fact any mathematical theory are independent of

physical reality since they are simply sets of axioms and their consequences according to some rule of inference.

Euclidean geometry didn't change in 1823; only the idea that it necessarily

applied to the physical world changed. There are many physical entities that

number theory does not apply to. It's just that the the non-applicability is

so obvious that one doesn't usually think of number theory applying. For an

example that sometimes trips up 4th graders; consider that there are two school

clubs, one with 15 members and one with 20 members. They have picnic together.

How many are at the picnic?

Similarly the application of logic is empirical. Ordinary predicate calculus

has a limited range of applicability - you get into trouble if you quantify

over predicates. Logic ordinarily deals with propositions which have

true-false values and connectives (and, or,...). But it could be extended to

include things like 'red implies not-green'. All of this is independent of the

physical world. It's just that we like to adapt our language so as to make it

easy to talk accurately about the world.

Brent Meeker

Regards

Received on Wed Sep 20 2000 - 13:20:09 PDT

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