6 found
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  1. Arithmetic Without the Successor Axiom.Andrew Boucher - manuscript
    Second-order Peano Arithmetic minus the Successor Axiom is developed from first principles through Quadratic Reciprocity and a proof of self-consistency. This paper combines 4 other papers of the author in a self-contained exposition.
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  2.  68
    The Existence of Numbers (Or: What is the Status of Arithmetic?).Andrew Boucher - manuscript
    I begin with a personal confession. Philosophical discussions of existence have always bored me. When they occur, my eyes glaze over and my attention falters. Basically ontological questions often seem best decided by banging on the table--rocks exist, fairies do not. Argument can appear long-winded and miss the point. Sometimes a quick distinction resolves any apparent difficulty. Does a falling tree in an earless forest make noise, ie does the noise exist? Well, if noise means that an ear must be (...)
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  3. Comments on Naming and Necessity.Andrew Boucher - manuscript
    I recently had the occasion to reread Naming and Necessity by Saul Kripke. NaN struck me this time, as it always has, as breathtakingly clear and lucid. It also struck me this time, as it always has, as wrong-headed in several major ways, both in its methodology and its content. Herein is a brief explanation why.
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  4.  20
    Dedekind's Proof.Andrew Boucher - manuscript
    In "The Nature and Meaning of Numbers," Dedekind produces an original, quite remarkable proof for the holy grail in the foundations of elementary arithmetic, that there are an infinite number of things. It goes like this. [p, 64 in the Dover edition.] Consider the set S of things which can be objects of my thought. Define the function phi(s), which maps an element s of S to the thought that s can be an object of my thought. Then phi is (...)
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  5.  36
    The Solution to the Sorites Paradox.Andrew Boucher - manuscript
    The solution to the Sorites Paradox is discussed.
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  6. Who Needs (to Assume) Hume's Principle?Andrew Boucher - manuscript
    Neo-logicism uses definitions and Hume's Principle to derive arithmetic in second-order logic. This paper investigates how much arithmetic can be derived using definitions alone, without any additional principle such as Hume's.
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