Tuesday, June 12, 2007

Quid es Veritas?

On CBR, a person I dearly respect but disagree with often had this to say yesterday:

What I'm saying mostly is that there is an objective reality, completely independent of our beliefs.

A god (or gods) either exists or it doesn't. Magic and the supernatural are either real or they aren't.

There's no objective evidence that any method other than the scientific one produces results. Science works whether you believe in it or not.

What scientists and theologians are trying to do is basically the same thing: explain the nature and reality of the universe and the things in it.

To which I must reply with this:

Binary logic is not isomorphic. The Ancient Greeks certainly thought so, which is part of the reason why they hated irrational numbers, to the point of (if I recall correctly) drowning guys who suggested that the square root of two was an actual number. In the system of binary logic, everything neatly divides into "true" and "false" and "All truth values are decidable" is considered axiomatic.

We ran into a few problems with that in the twentieth century, namely with two chaps named Wittgenstein and Gödel. Wittgenstein convincingly argued that there are no absolute rules governing any one bit of language, and thus there is no way to determine if something is true in all circumstances. For example, the word "Fire" doesn't obey absolute rules.

As a noun, "There's a fire over there."
As a command or warning, "Fire!"
As a question, "Fire?"
As the answer to a question, "Fire."

So, Wittgenstein concluded, there is no way within the bounds of human language to determine absolute truth, only ways of determining truth within specific language games that had rules of construction and truth-bearing. Symbolic logic is a language game, as is science, as is theology. Not only communication but means of thinking are human language, including math. This is where Gödel comes in.

Kurt Gödel was only 24 when he received his doctorate from the University of Vienna and only 25 when his Incompleteness Theorems were published in 1931 (yeah, I know, I feel inadequate too). The implications of the proofs are astounding.

  1. Someone introduces Gödel to a UTM, a machine that is
    supposed to be a Universal Truth Machine, capable of correctly
    answering any question at all.

  2. Gödel asks for the program and the circuit design of the
    UTM. The program may be complicated, but it can only be finitely
    long. Call the program P(UTM) for Program of the Universal Truth

  3. Smiling a little, Gödel writes out the following sentence:
    "The machine constructed on the basis of the program P(UTM) will never
    say that this sentence is true." Call this sentence G for Gödel.
    Note that G is equivalent to: "UTM will never say G is true."

  4. Now Gödel laughs his high laugh and asks UTM whether G is
    true or not.

  5. If UTM says G is true, then "UTM will never say G is true" is
    false. If "UTM will never say G is true" is false, then G is false
    (since G = "UTM will never say G is true"). So if UTM says G is true,
    then G is in fact false, and UTM has made a false statement. So UTM
    will never say that G is true, since UTM makes only true statements.

  6. We have established that UTM will never say G is true. So "UTM
    will never say G is true" is in fact a true statement. So G is true
    (since G = "UTM will never say G is true").

  7. "I know a truth that UTM can never utter," Gödel says. "I
    know that G is true. UTM is not truly universal."

What Gödel succeeded in proving was that in any given mathematical or logical system, by using the axioms of said system you would inevitably run into questions that cannot be decided one way or the other. You can always, of course, jump to other systems to decide them, but all this does is create an even larger system with the same problem. Any given logical or mathematical system is necessarily incomplete.

So, the assumption "All truth values are decidable" inevitably leads to the conclusion "Not all truth values are decidable." Crazy, isn't it? And if you bring in other truths from other systems to help, you just delay the problem into something else. It's turtles all the way up.

This means that we live in an environment of "truths" that are constantly overlapping and interacting with one another, and indeed competing, but that there is no absolute truth that transcends the whole of the universe anywhere, only very carefully defined rule sets that if operated correctly lead to truths that exist absolutely only within the confines of their specific rule sets. Binary logic is a language game/rule set that leads to specific results in specific circumstances but by necessity it can never truly encapsulate the whole of the universe. Hence, not everything operates precisely on a true/false paradigm, as the idealist/rationalist/positivist tradition argues.

This also means that theology and science are different language games/rule sets, and thus such contradictory statements as "There is no God/are no gods" and "There is a God/are gods" can both be simultaneously true.

There's a fellow in the Bible I believe who asked a very interesting question: Quid es veritas? What is truth?

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