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Quantitative Epistemology the Structure of Logical Knowledge Michael Barany Submitted 9 April 2009 1 Logical Knowledge The Pythagorean theorem states that in any right triangle the legs of length a and b and hypotenuse of length c are related by the equation a2 b 2 c 2 1 Equation 1 represents an element of mathematical knowledge More generally the Pythagorean theorem is an element of logical knowledge But the equation itself does not make this so The Pythagorean theorem is an element of logical knowledge because of the way it is established and the way it is used It is established through a step by step logical derivation from a collection of foundational principles It is used in combination with other established elements of knowledge in a particular way to establish further elements of logical knowledge Logical knowledge is recognizable less by the contents of its assertions than by the logical way in which they are connected How then is the Pythagorean theorem connected to other statements In Euclid s Elements it appears as the 47th proposition of Book I after a long sequence of definitions postulates common notions and other propositions Euclid does not hold a monopoly on proofs of the theorem however Loomis 1968 compiles over 250 pages of proofs and demonstrations of the theorem When taught to school children the theorem appears after explanations of such things as right triangles side lengths and square numbers In more 1 advanced mathematical settings it is often stated without proof and is used as a foundational idea with which to establish other statements In this essay we shall seek an account of logical knowledge which applies to all these different settings for the Pythagorean theorem and indeed to all works which might be called logical 1 In so doing we shall treat logical knowledge as a system and ask what features of the system are properties of the system itself as opposed to the isolated meanings and referents of the statements therein Roughly

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