A philosophy of mathematics

The reason there is only one possibility, the reason that the allusions are mere clothing, is that unlike in an experiment, in a proof or calculation the process and result are related internally: “ suppose I do this [ weigh the balls ] again’ – here the ‘this’ doesn’t include this result, otherwise it is not an experiment, but a calculation – there is an internal relation. The conditions of the experiment don’t include the result “ [ LFM, p.97]. wiitgenstein argues “ there is a difference between a process having a result and being its own result “ [AWL, p.r 88]. A difference in result between our experiment can be due to many causes: the mechanism needs oiling, dust has gotten in the scales, one of the balls has become chipped, and so forth. A difference in result in calculation can be due only to this: one of us has made a mistake. The roles an experiment and a proof play are thus entirely different. If we are certain the scales are accurate, we can use them to test the weight of the balls, if we are certain of the weight of the balls, we can use them to test  the accuracy of the scale. Proofs and calculation, however, are like the standard meter bar in paris [ see PI, so], the bar can be used to test whether a stick is a meter long, but the stick cannot be used to test wheter the standard meter bar is a meter long. The proposition of mathematics have acquired a special role.

            If the roles of mathematical and empirical proposition were conflated, however, then mathematics would lose its peculiar inexorability and its objectivity, falling into the anarchist camp. “ the question arises ,” wittgestein says in an extended argument, “ what we take as criterion of going according to the rule” [RFM, VI – 16 ]. He then describes three failed alternatives: “ is it for example a feeling of satisfaction that accompanies the act of going according to the rule?? Or an intuition (intimation) that tells me I have gone right? Or is it certain practical consequences of proceeding that determine whether I have really followed the rule?” the rejected alternatives are carefully chosen examples of psychologism, identifying mathematical correctness with a  psychological criterion. The second is an example of how wittgenstein sometimes characterized intuitionism . the third exemplifies a crude kind of pragmatism ( a kind that both wittgenstein and william james have been accused of holding but which, in fact, both explicitly repudited ). Wittgenstein’s argument of rejecting these alternatives is similar to Frege’s arguments againts psycologism.