Glossary

Parenthetical references indicate the essay in which the term first appears.

Antikythera Mechanism — An ancient Greek mechanical device, dating to roughly 100 BCE, used to predict astronomical phenomena. Often regarded as the world’s first known analog computer. (The Mechanized Hum)

Arithmetization of Syntax — Gödel’s method of representing symbols, formulas, and proofs as natural numbers. This allows statements about the structure of a formal system to be expressed within arithmetic itself. (Paradox and Self-Reference)

Behavioral Zombie — A hypothetical being that behaves exactly like a conscious person while lacking subjective experience. Often used in discussions of consciousness, computation, and physicalism. (Mind and Machine)

Church–Turing Thesis — The claim that any function that can be effectively computed by an algorithm can be computed by a Turing machine. It is not a mathematical theorem but a widely accepted principle connecting informal notions of computation with formal models. (Mind and Machine)

Computation — The execution of a rule-governed procedure that transforms inputs into outputs according to a formally specified process. (Mind and Machine)

Consistency — A property of a formal system in which no contradiction can be derived. If a system is consistent, it cannot prove both a statement and its negation. (Paradox and Self-Reference)

Diagonal Argument — A method introduced by Georg Cantor for demonstrating that certain sets are larger than others. Gödel later adapted a diagonal-style construction to produce a sentence that refers to its own provability. (Paradox and Self-Reference)

Formal System — A system consisting of symbols, rules for forming expressions, axioms, and rules of inference. Mathematics can be expressed within formal systems, which allow proofs to be treated as purely syntactic procedures. (Paradox and Self-Reference)

Gödel’s incompleteness Theorems

Gödel Numbering — A method introduced by Kurt Gödel in which every symbol, formula, and proof in a formal system is assigned a unique natural number. This allows statements about syntax to be translated into statements about numbers. (Paradox and Self-Reference)

Gödel Sentence — A sentence constructed within a formal system that effectively asserts its own unprovability. If the system is consistent, the sentence is true but cannot be proven within that system. (Paradox and Self-Reference)

Gödel, Kurt — A logician and mathematician whose incompleteness theorems transformed the foundations of mathematics and profoundly influenced later discussions of computation, truth, and mind. (Paradox and Self-Reference)

Goodstein’s Theorem — A theorem concerning sequences of natural numbers that is true but unprovable within Peano Arithmetic. It is often cited as a natural example of incompleteness. BETTER (Mind and Machine)

Hilbert’s Program — David Hilbert’s attempt to place mathematics on secure foundations by formalizing it and proving its consistency using finitary methods. (Paradox and Self-Reference)

Hypercomputation — Hypothetical forms of computation that exceed the capabilities of Turing machines. (Mind and Machine)

Imitation Game — See Turing test

Incompleteness — The property of a formal system in which some true statements cannot be proven within the system. See Gödel’s Incompleteness Theorems (Paradox and Self-Reference)

Liar Paradox — A paradox arising from a self-referential statement such as “This sentence is false.” Any attempt to assign it a truth value leads to contradiction. (Paradox and Self-Reference)

Lucas, J. R. — British philosopher best known for applying Gödel’s incompleteness theorem to arguments against mechanism in “Minds, Machines and Gödel” (1961). (Mind and Machine)

Materialism

Mathematical Understanding — The capacity to grasp mathematical meaning and truth, as distinguished from the formal manipulation of symbols. (Mind and Machine)

Mechanism — The philosophical view that mental processes are entirely the result of physical or computational processes. BETTER (Mind and Machine)

Mechanist Thesis — The claim that human reasoning, understanding, and possibly consciousness can be fully captured by computation. BETTER (Mind and Machine)

Meta-Language — A language used to describe, analyze, or reason about another language or formal system. (Paradox and Self-Reference)

Metamathematics — The study of mathematics itself using mathematical methods, including the investigation of formal systems, proofs, consistency, and completeness. (Paradox and Self-Reference)

Paris–Harrington Theorem — A combinatorial statement that is true but unprovable in Peano Arithmetic, often cited as a natural example of incompleteness. BETTER (Mind and Machine)

Penrose, Roger — Mathematical physicist and Nobel laureate known for developing and extending the Gödelian critique of mechanism in works such as The Emperor’s New Mind and Shadows of the Mind. BETTER (Mind and Machine)

Provability — The property of a statement that can be derived from the axioms and rules of a formal system. (Paradox and Self-Reference)

Reductionism — The view that complex phenomena can ultimately be explained entirely in terms of simpler physical components and their interactions. BETTER (The Mechanized Hum)

Russell’s Paradox — A contradiction discovered by Bertrand Russell in naive set theory involving the set of all sets that do not contain themselves. (Paradox and Self-Reference)

Self-Reference — The property of a statement, system, or process that refers to itself. (Paradox and Self-Reference)

Semantics — The meaning or interpretation of expressions within a language or formal system. BETTER (Paradox and Self-Reference)

Soundness — A property of a formal system in which every provable statement is true in the intended interpretation. (Mind and Machine)

Strong AI — The view that appropriately structured computation is sufficient for genuine intelligence and consciousness. (Mind and Machine)

Syntax — The formal structure of expressions in a language or formal system. Syntax concerns rules for manipulating symbols without regard to meaning. (Paradox and Self-Reference)

Turing Machine — An abstract model of computation introduced by Alan Turing consisting of a tape, a set of states, and a set of transition rules. (Mind and Machine)

Turing Test — A behavioral criterion proposed by Alan Turing in which a machine is judged intelligent if its conversational performance is indistinguishable from that of a human. BETTER (Mind and Machine)

ω-Consistency — A stronger form of consistency used in Gödel’s original proof of the First Incompleteness Theorem. Rosser later showed that ordinary consistency is sufficient. BETTER (Paradox and Self-Reference)