Friday, October 07, 2016

Boolean algebras, Stone spaces and TGD

The Facebook discussion with Stephen King about Stone spaces led to a highly interesting development of ideas concerning Boolean, algebras, Stone spaces, and p-adic physics. I have discussed these ideas already earlier but the improved understanding of the notion of Stone space helped to make the ideas more concrete. The basic ideas are briefly summarized.

p-adic integers/numbers correspond to the Stone space assignable to Boolean algebra of natural numbers/rationals with p=2 assignable to Boolean logic. Boolean logic generalizes for n-valued logics with prime values of n in special role. The decomposition of set to n subsets defined by an element of n-Boolean algebra is obtained by iterating Boolean decomposition n-2 times. n-valued logics could be interpreted in terms of error correction allowing only bit sequences, which correspond to n<p<2k in k-bit Boolean algebra. Adelic physics would correspond to the inclusion of all p-valued logics in single adelic logic.

The Stone spaces of p-adics, reals, etc.. have huge size and a possible identification (in absence of any other!) is in terms of concept of real number assigning to real/p-adic/etc... number a fiber space consisting of all units obtained as ratios of infinite primes. As real numbers they are just units but has complex number theoretic anatomy and would give rise to what I have assigned the terms algebraic holography and number theoretic Brahman = Atman.

For a details see the articleBoolean algebras, Stone spaces and TGD.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.


At 7:49 PM, Anonymous Anonymous said...

"Abstract. A novel system for representing the rational numbers based on Hensel's p-adic arithmetic is proposed.
The new scheme uses a compact variable-length encoding that may be viewed as a generalization of radix
complement notation. It allows exact arithmetic, and approximate arithmetic under programmer control. It is
superior to existing coding methods because the arithmetic operations take particularly simple, consistent forms.
These attributes make the new number representation attractive for use in computer hardware."


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