When1: 1979
When2: 2007
Who: Douglas Hofstadter [Hofstadter, Douglas]
What: computer scientist
Where: USA
works\ Godel, Escher, Bach: an Eternal Golden Braid [1979]; Mind's I [1982: with Daniel Dennett, editors]; I Am a Strange Loop [2007]
Detail: Mental events are recursive self-representational loops {strange loop}. The physical basis of loops is the molecular-behavioral loop. Consciousness is higher-order thoughts or reports accompanying unconscious mental states, so brain can monitor itself {higher-order thought, Hofstadter}. This control system allows recursion through self-representations. Mental states have different levels.
Brain has complex patterns, some of which are self-referentional. Lower animals, mammals, primates, children, adults, brain-damaged adults, and senescent people have no, some, half, medium, or high self-reference. Also, self-reference can have one, some, many, or infinite numbers of levels. People can nest things to infinite self-reference. Brain complex patterns are entirely physical at microscopic levels but have descriptions, and causes and effects, that use intentions at higher levels. Self-reference threatens paradox, runaway feedback, inconsistency, and incompleteness.
Strange loops feed back, cross levels, and go back to previous loop stages.
By the theory of types, a set cannot contain itself and a proposition cannot refer to itself.
A true proposition has a proof, which makes it true. A false proposition has no proof, showing it is false. False propositions lead to contradictions.
A different integer can represent each symbol. The sequence of primes can represent each position in a string. The prime raised to the integer represents the symbol at the position. For example, if symbol = is at position 1 and integer 5 represents symbol =, 2^5 = 32 represents the string "=". For more than one position, multiply the primes raised to powers. For example, if integer 2 represents symbol 4, the string "= 4" can be 2^5 * 3^2 = 288. In reverse, knowing the number 288 (Gödel number) and factoring into primes gives the symbol string. Formulas and Gödel numbers have one-to-one mapping and so are analogous. Their meanings are the same, but the concepts differ. Natural numbers can represent any pattern, have unlimited expressivity, and are like universal language.
Formal systems cannot prove that a Gödel number is the number of a true formula.
Symbol strings can represent propositions and inference rules, or not. Proofs derive propositions from previous propositions using rules of inference, and arithmetic calculations on proposition and rule Gödel numbers are equivalent to proofs. Proofs have Gödel numbers.
Some proposition Gödel numbers are in a system, as valid formulas, and the rest are out. Valid formulas come recursively from earlier valid formulas and must get larger. Some proposition Gödel numbers are valid formulas and provable. Provable formulas come recursively from earlier valid formulas and can be smaller or larger.
By describing Gödel numbers using their computation methods, formulas can contain their Gödel numbers. Proposition subjects and verb phrases have smaller Gödel numbers than whole propositions. Propositions can have verb phrases as subjects. Propositions about themselves are not provable.
In formal systems, proofs always find true propositions (consistency). If propositions about themselves were provable, formal systems find that the statement "propositions about themselves are not provable" is false. This is inconsistent.
Formal systems can prove all true propositions (completeness). If propositions about themselves are not provable, formal systems cannot find the true statement "propositions about themselves are not provable". This is incomplete.
I is a symbol that perception sometimes triggers in brains. I becomes larger over development, with more perceptions, results of actions, memories, beliefs, goals, feelings, and imaginings. Brain has structures larger than molecules and neurons and even neuron assemblies and brain regions. Such structures correspond with objects and events in the physical world and so are analogies.
Brains have many symbols and can make symbol patterns. Some brains can make symbol patterns that refer to symbol patterns. Symbol patterns can communicate.
Universal Turing machines can read and write descriptions of themselves (and so any machine).
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Date Modified: 2022.0224