David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
We present mathematics from a foundational perspective as a hierarchy in which each tier consists of a language, a logic, and a mathematical theory. Each tier in the hierarchy subsumes all preceding tiers in the sense that its language, logic, and mathematical theory generalize all preceding languages, logics, and mathematical theories. Starting from the root tier, the mathematical theories in this hierarchy are: combinatory logic restricted to the identity I, combinatory logic, ZFC set theory, constructive type theory, and category theory. The languages of the first four tiers correspond to the languages of the Chomsky hierarchy: in combinatory logic Ix = x gives rise to a regular language; the language generated by S, K in combinatory logic is context-free; first-order logic is context-sensitive; and the typed lambda calculus of type theory is recursively enumerable. The logic of each tier can be characterized in terms of the cardinality of the set of its truth values: combinatory logic restricted to I has 0 truth values, while combinatory logic has 1, first-order logic 2, constructive type theory 3, and categeory theory omega_0. We conjecture that the cardinality of objects whose existence can be established in each tier is bounded; for example, combinatory logic is bounded in this sense by omega_0 and ZFC set theory by the least inaccessible cardinal. We also show that classical recursion theory presents a framework for generating the above hierarchy in terms of the initial functions zero, projection, and successor followed by composition and m-recursion, starting with the zero function I in combinatory logic This paper begins with a theory of glossogenesis, i.e. a theory of the origin of language, since this theory shows that natural language has deep connections to category theory and since it was through these connections that the last tier and ultimately the whole hierarchy were discovered. The discussion covers implications of the hierarchy for mathematics, physics, cosmology, theology, linguistics, extraterrestrial communication, and artificial intelligence.
|Keywords||No keywords specified (fix it)|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library||
References found in this work BETA
No references found.
Citations of this work BETA
No citations found.
Similar books and articles
Andreas Knobel (1993). Constructive Set Theoretic Models of Typed Combinatory Logic. Journal of Symbolic Logic 58 (1):99-118.
Kenny Easwaran (2010). Logic and Probability. Journal of the Indian Council of Philosophical Research 27 (2):229-253.
P. T. Johnstone (1987). Notes on Logic and Set Theory. Cambridge University Press.
Maarten Wicher Visser Bunder (1969). Set Theory Based on Combinatory Logic. Groningen, V. R. B. --Offsetdrukkerij (Kleine Der a 3-4).
J. Roger Hindley (1986). Introduction to Combinators and [Lambda]-Calculus. Cambridge University Press.
Katalin Bombó (2005). The Church-Rosser Property in Symmetric Combinatory Logic. Journal of Symbolic Logic 70 (2):536 - 556.
Hao Wang (1981/1993). Popular Lectures on Mathematical Logic. Dover Publications.
Ignacio Jané (1993). A Critical Appraisal of Second-Order Logic. History and Philosophy of Logic 14 (1):67-86.
Angelo Margaris (1967/1990). First Order Mathematical Logic. Dover Publications.
Added to index2009-01-28
Total downloads10 ( #165,643 of 1,410,540 )
Recent downloads (6 months)2 ( #108,810 of 1,410,540 )
How can I increase my downloads?