Mathematics - Incomplete and Contradictory

Newton's (1643-1727) moon and planets are like Heisenberg's electron except that Newton did not realize that location and velocity were relative. Heisenberg (1901-1976) proved that you cannot measure the location AND the velocity of an electron because both are always changing.

Godel (1906-1978) proved that any system is either inconsistent or incomplete. He published his two incompleteness theorems in 1931, following his dissertation in 1929 "On the Completeness of the Calculus of Logic." The grammar of numbers, all the rules for making numbers, will either be incomplete because some rules will be missing, or inconsistent: there will be rules that contradict each other.

Godel demonstrated that certain mathematical theorems can neither be proved nor disproved with accepted mathematical methods. He said, "It will never be possible to acquire the certainty that mathematics does not contain contradictions." There is a class of undecidable problems. AND, there are rules that are true but unprovable. This contradicts the foundation of mathematics that what is true can always be proved.

Godel discovered the impossibility of mathematics as true and provable - there exists a logically necessary impossibility that cannot be avoided. Therefore, any system, like any person, group, culture or organization is either incomplete or inconsistent/contradictory.

Ludwig Wittgenstein (1889-1951) stated in his posthumously published 1965 "On Certainty" that the problem with the terms 'knowledge' and 'knowing' was that they excluded error. It may be that he realized this language problem and thus began to reflect and critique his 1922 Tractatus. His life work took on the character of investigating the contradictions, puzzles and queer notions that he found in linguistic expressions.

One similar problem was discovered by Lacan (1901-1981) in his 1932 thesis "On Paranoiac Psychosis in its Relations to the Personality." Lacan explained that the distinction between the neurotic personality and the psychotic personality. For the neurotic personality, the rules of grammer that produce language are incomplete. Each person has a partially different set of rules for speaking language which results in different uses of language and symptoms. For the neurotic, the rules of language, or what seems intuitive and right, are incomplete. For the psychotic, the rules of language are fixed and contradictory. The fixed contradictory rules produce the symptoms of paranoia and hallucinations!

Lacan's main example is the distinction between the idea of a 'symbolic father,' the person or agency that interferes with the relation of desire between the infant child and the mother-er, and the Names-of-the-Father, the fixed meanings that designate each person, their proper names. There are many symbolic fathers in each person's life - anyone who separates the child from desire, from their mother.  But, there is only one Name-of-the-Father, one's father's name. Lacan asserts that the psychotic has violated the rules of the relation or contract with his father, and has been excluded, separated rigidly from his or her mother. So, the rules of language and meaning are not properly fixed: only some meanings are fixed, and other meanings are variable or loose.

The primary factor affecting the psychotic is the concrete nature of meaning - saying something is like something else as in metaphor or synecdoche is understood by the psychotic as actually existing. "Father" is only understood as a fixed name, the actual person who is one's father, and the 'symbolic father,' or 'cultural father,' is absent.  The fixed nature of meaning overrides the figural and metaphorical understanding of expressions.
Fedric