Here are a list of papers on parallel machines, philosophy, and the foundations of mathematics. Time of last update followed by date of first copy, is in parentheses.
- Parallel Machines (1997, 1987) Because it is time-dependent, parallel computation is fundamentally different from sequential computation. Parallel programs are non-deterministic and are not effective procedures. Given the brain operates in parallel, this casts doubt on AI's attempt to make sequential computers intelligent. Very slightly modified from version published in Minds and Machines 7: 543-551, 1997.
- Logical Philosophy (2001, 1979) Part I (Logic): Meaning, Concepts, Reference, Assertions, and Truth. Part II (Philosophy): Philosophy, Epistemology, Determinism vs Free Will, Ethics, and the Meaning of Life.
- A Comprehensive Solution to the Paradoxes (1 Sept 2000, 1994) A philosophical explanation why logic and mathematics should restrict themselves to arithmetic comprehension.
- A Philosophical Introduction to A Foundation of Elementary Arithmetic (1 Jan 2001, Aug 1999) What is a foundation of elementary arithmetic?
- A Foundation of Elementary Arithmetic (28 Apr 2002, 1987) A derivation of the propositions of elementary arithmetic from first principles.
- Dedekind's Proof (10 Dec 2001, April 2000) Dedekind's method to prove there are an infinite number of things, sometimes derided, is not so bad as all that.
- Proving Peano's Axioms (10 Dec 2001, Sept 2000) A continuation of Dedekind's Proof, this proves the Peano Axioms from the axioms of A Foundation of Elementary Arithmetic.
- Systems for a Foundation of Arithmetic (16 Mar 2002, Dec 2001) A more elaborate and extensive working of many of the themes in "Dedekind's Proof" and "Peano's Axioms," it contains some new results as well. The file is in .pdf format.
- Consistency and Existence (1 Oct 2000, Apr 2000) Existence implies consistency, but not the contrary.
- The Existence of Numbers (Or: What is the Status of Arithmetic?) (3 June 2002) A discussion of arithmetical ontology and its implications for the status of arithmetic.
- Against Angels and the Cantorian-Fregean Theory of Number (16 Mar 2002) Originally a posting to sci.logic and sci.math, it disputes the notion that one-to-one correlation is the basis of number.
- "True" Arithmetic Can Prove Its Own Consistency (6 April 2004, May 2002) A strong system of arithmetic (though weaker than Peano Arithmetic), corresponding to arithmetic without the assumption of closure, can prove its own consistency, and indeed the consistency of stronger systems. The file is in .pdf format and is version 5, where consistency is now not only proven in the fashion of Godel numbering, but in the stronger sense of non-coded sequences.
- Proving Quadratic Reciprocity (12 November 2003, June 2003) F, the system of arithmetic considered in Consistency, is used to prove more theorems, including Quadratic Reciprocity. This confirms the power of F, which is essentially second-order arithmetic without the ad infinitum assumption that there is always a next number. The file is in .pdf format.
- Equivalence of F with a Sub-Theory of Peano Arithmetic (19 Feb 2005) In a short, technical note, the system of arithmetic, F, introduced in Systems for a Foundation of Arithmetic and used in "True" Arithmetic Can Prove Its Own Consistency and Proving Quadratic Reciprocity, is demonstrated to be equivalent to a sub-theory of Peano Arithmetic (which does not, most notably, include the Successor Axiom). The file is in .pdf format.
- Three Theorems of Godel (10 Nov 2005) Three theorems of Godel - the Completeness and the First and Second Incompleteness Theorems - are looked at through the prism of the Successor Axiom. The file is in .pdf format.
- Arithmetic without the Successor Axiom (10 Feb 2006) A book-length self-contained exposition of Arithmetic without the Successor Axiom. It shows how to develop arithmetic without the Successor Axiom up through Quadratic Reciprocity and goes on to show how the system can prove its own consistency. The file is in .pdf format.
- Who Needs (to Assume) Hume's Principle? (Aug 2008, 23 July 2006) Neo-logicism uses definitions and Hume's Principle to derive arithmetic in second-order logic. This papers investigates how much arithmetic can be derived using definitions alone, without any additional principle such as Hume's. The file is in .pdf format.
- Comments on Naming and Necessity (3 March 2007) Comments offered on Saul Kripke's Naming and Necessity. The file is in .pdf format.
- The Solution to the Liar's Paradox (13 March 2007) The solution to the Liar's Paradox is explained. The file is in .pdf format.
- Proving Bertrand's Postulate (31 Dec 2007) Second-order arithmetic minus the Successor Axiom is used to prove Bertrand's Postulate, by representing larger numbers via second-order objects. The file is in .pdf format.
- General Arithmetic (24 February 2008) An investigation of arithmetic with just induction and one other axiom, the functionality of successorship. The file is in .pdf format.
- A Theory of Meaning (30 November 2008) A short exposition of a theory of meaning. The file is in .pdf format.
- Descriptive Ethics (26 August 2009) Remarks on descriptive ethics. The file is in .pdf format.
- Depression in a One-Good Barter Economy (21 February 2010) How a one-good economy without money can fall into economic depression. The file is in .pdf format.
- A Note on the Berry Paradox (5 March 2011) A short explanation of the solution to the Berry Paradox. The file is in .pdf format.
- A Natural First-Order System of Arithmetic Which Proves Its Own Consistency (8 Nov 2011) A natural and strong first-order system can prove its own consistency. The file is in .pdf format.
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