Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

154 johannes hafner and paolo mancosuwill be relevant to our discussion. In order to introduce them we need toclarify three notions: argument pattern, generating set for , and conclusionset for .Argument pattern: Kitcher derives this notion from specific examples takenfrom the natural sciences, e.g. Newtonian mechanics and Darwin’s theory ofevolution. We will illustrate it after giving the definitions with an examplefrom mathematics. Let us begin with the notion of schematic sentence. Thisisanexpression obtained by replacing some or all of the non-logical expressions ina sentence by dummy letters.Asetoffilling instructions tells us how the dummy letters in a schematicsentencearetobereplaced.A schematic argument is a sequence of schematic sentences.A classification for a schematic argument is a set of sentences which tellsus exactly what role each sentence in a schematic argument is playing, e.g.whether it is a premise, which sentences are inferred from which and accordingto what rules, etc.A general argument pattern 〈s, f , c〉 is a triple consisting of a schematic arguments, asetf of sets of filling instructions, and a classification c for s.An example from mathematicsConsider the problem of determining the equation of the line tangent to theparabola y = 2x 2 + 3x + 1 at point (1,6). We can solve the problem usingderivatives as follows.1. [2x 2 + 3x + 1] ′ = 4x + 32. [4x + 3] x=1 = 73. Thus the tangent line to 2x 2 + 3x + 1at(1, 6) is(x − 1)7 = (y − 6).A schematic argument for determining the tangent line to a differentiable curvef (x) at a point (x 0 , y 0 ) can be obtained from the above as follows.1S. [f (x)] ′ = g(x)2S. [g(x)] x=x0 = c3S. Thus the tangent line to f (x) at (x 0 , y 0 ) is (x − x 0 )c = (y − y 0 ).Filling instructions:Replace f (x) by a description of the function under consideration.Replace g(x) by a description of the derivative of f (x).Replace c by the value of g(x) at x 0 .Classification:1S and2S are premises. 3S follows from 1S and2S bycalculus.