Truth as Eigenform

definition_of_truth.md

Definition of Truth

Jason the Goodman <[email protected]>:

How about defining "truth" with "stability" of the coupling loop between "the object" and the cognitive system coupling with the "object"? If the process stabilizes and an eigenvalue of the loop emerges, we say "a truth" is found. This would refocus our attention from the "object" itself to the nature of the cognitive system, which may include animals and robots in addition to humans. My tentative way to upgrade from first-order thinking to the second-order thinking. Then, instead of searching for "truth", we search for the "Lyapunov potential function" for the situation if we could find one...

Louis H Kauffman <[email protected]>

Eigenform is important way to formalize a kind of stability. Truth is a special kind of eigenform, not just any eigenform. Truth means the truth of a PROPOSITION about something. So we need to have a language involved and the notion that the propositions are talking about some domain where it is possible to check whether the proposition’s statement is correct.

Jason the Goodman <[email protected]>:

would you define your "special kind" here - how specific it has to be? And, what kind of eigenform is not "a truth" for a dog/cat or for a robot? Could you please show me.

Louis H Kauffman <[email protected]>:

Given a mapping F:D —> D where D is any mathematical domain (e.g. an arbitrary set), an eigenform is a fixed point of F, possibly in some larger domain D, extending D. In this way, every F has a corresponding eigenform E with F(E) = E.

The way this is proved is to construct E = F(F(F(F…))), the infinite concatenation of F upon itself. Sometimes this formal eigenform is related to the original domain D and sometimes it absconds (or transcends) well beyond that domain. There are many examples. I will give a few for reference.

1. D = R, the real numbers. F(x) = 1 + 1/x. Here E = 1 + 1 /(1 + 1/(1+ 1/ …)) is the formal eigenform, but there is a real number x such that x = 1 + 1/x and we can take x = (1 + sqrt(5))/2. Furthermore, the finite iterations 1, F(1), F(F(!)), … are:

• 1, 1+ 1/1 = 2/1, 1 + 1/(1 + 1/1) = 3/2,
• 1 + 1/(1 + 1/(1 + 1/1)) = 5/3, …
• and these have (1 + sqrt(5))/2 as their limit.

So in this case the formal eigenform can be identified with an element of the original domain D.

2. D = R, the real numbers. F(x) = -1/x. Now there is no real number x such that x= -1/x. And the formal eigenform E = -1/-1/-1/-1/-1/-1/… does not correspond to any element of R, but it can be matched up with the complex number i (in the domain of complex number that extend R) with i^2 = -1. In this way the complex number i can be interpreted in terms of an “oscillating” formal number that is an eigenform.

3. Let F(x) = <x> where this means “put brackets around x”. Then E = <<<<<…>>>>>. This is a quintessential formal eigenform.

4. Let F(x) = “I am the one who says x.”. Here F(x) is a sentence and for example F(red) = “I am the one who says red.” This may be true or false about me. But F(I) = “I am the one who says I.”. That is true and could be regarded as a definition of “I”. So in this sense "F(I) = I.” This example has to do with truth.

5. Let F(x) = This sentence has x words. Then F(5) is true and we regard F(5) not as equal to 5 but as correctly referring to itself and so true.

6. More generally, I shall say that a statement P is a “propositional eigenform” if it refers to itself, true or not. For example. This statement is false. This statement is fifty words long. This statement is your cerebral cortex. This statement is the last thing you will read before enlightenment. This statement is written in Russian on a spaceship headed to interstellar space. This statement is Gogol’s nose. This statement is self-referential.

7. Of the propositional eigenforms in 6. only the last one is true by my reckoning. But now lets see. Let P be any statement about some state of affairs that could be either true of false. Consider the following statement: S(P) = This statement is true if and only if P is true. Then S(P) is a propositional eigenform and S(P) is true exactly when statement P is true! We have shown that propositional truth can be matched with the existence of a true propositional eigenform.