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Mark Foote

Post title:  Things as They Are & Logic

(Nov 24 2015 at 03:02 PM)



The book I've been studying for the last while says this:

"In 1908 Brouwer, in a paper entitled 'The untrustworthiness of the principles of logic', challenged the belief that the rules of the classical logic, which have come down to us essentially from Aristotle (384-322 B.C.), have an absolute validity, independent of the subject matter to which they are applied. Quoting from Weyl 1946, 'According to his view and reading of history, classical logic was abstracted from the mathematics of finite sets and their subsets. ...Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all of mathematics, and finally applied it, without justification, to the mathematics of infinite sets.'"

("Introduction to Metamathematics", Kleene, pg 46-47)

The book continues with a cautionary tale about a mathematician named Hilbert:

"... Hilbert conceded that the propositions of classical mathematics which involve the completed infinite go beyond intuitive evidence. But he refused to follow Brouwer in giving up classical mathematics on this account.

To salvage classical mathematics in the face of the intuitionistic criticism, he proposed a program which we can state preliminarily as follows: Classical mathematics shall be formulated as a formal axiomatic theory, and this theory shall be proved to be consistent, i.e. free from contradiction."

(Ibid, pg 53)

In 1931, Godel published two theorems that demonstrate that if a formal axiomatic theory is consistent, then it is necessarily incomplete (some things that are known to be true are not provable within the theory), and if a formal axiomatic theory is complete, then it is inconsistent (a contradiction can be derived within the theory).

Godel's theorems put an end to "Hilbert's Program", as it was called:

"There has been some debate over the impact of Godel's incompleteness theorems on Hilbert's Program, and whether it was the first or the second incompleteness theorem that delivered the coup de grace."

(from "Hilbert's Program", Stanford Encyclopedia of Philosophy)

We know that as much as anything can be established in mathematics, Godel's theorems have been established. We know this because mathematicians have now found ways to use computers to check every aspect of proof in some theorems; the first theorems that were tested in this manner were Godel's, and the computer verified their logical validity.

For me, it all boils down to keeping a watchful eye whenever someone talks in terms of a "completed infinity" instead of in terms of particulars and relationships. Logic does not give consistent results when applied to completed infinities.

Here are some of the categories which appear to me to be "completed infinities", to which logic has been applied (and inconsistent results, obtained): God, the Cosmos, Emptiness, the Void, Nirvana, never, always, past, future, dharma, dharmakaya; all Buddhas past, present, and future; the self.

When I let go of my notions of "completed infinity", I am left with things just as they are; turns out, that's the domain where the logic we're all familiar with applies.

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