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Oran Magal – Another Look at the ‘Paradox of Analysis’ and the Content of Mathematics
You are cordially invited to attend a McGill Philosophy Workshop presented by Oran Magal on Monday, October 5 at 1:30pm in Leacock 927. Oran Magal Another Look at the 'Paradox of Analysis' and the Content of Mathematics The so-called paradox of analysis is a problem with regard to the notion of conceptual analysis. It can be roughly summarised as follows: if analysis yields nothing new, it is hard to see how it would be genuinely informative. If on the contrary it does yield something new relative to the analysandum, are we justified at all in calling it 'analysis'? In other words, it seems that analysis must be either sterile or trivial. This stands in contrast to our conception of various activities which we take to be analytic; moreover, it puts in question the very idea of analytic truths being anything other than trivial. I will discuss a specific variant of this general problem that arises with regard to mathematics, especially "pure" mathematics. The difficulty is stated beautifully by the mathematician, physicist and philosopher Henri Poincaré (1854-1912), in a passage from his article "On the nature of mathematical reasoning" (orig. 1894): [quote:] The very possibility of the science of mathematics seems an insoluble contradiction. If this science is deductive only in appearance, whence does it derive that perfect rigour no one dreams of doubting? If, on the contrary, all the propositions it enunciates can be deduced from another by the rules of formal logic, why is not mathematics reduced to an immense tautology? The syllogism can teach us nothing essentially new, and, if everything is to spring from the principle of identity, everything should be capable of being reduced to it. Shall we then admit that the enunciations of all those theorems which fill so many volumes are nothing but devious ways of saying A is A? [end quote] This, then, is the perplexity which is the topic of the talk, to put it provocatively: how is it that pure mathematics has any content at all? If it is to be rigorous, it must be wholly deductive; but if so, how is it at all informative? I will lay out some considerations in support of what I take to be a promising approach to this problem: it is to be a holistic approach, but not in the empiricist manner of Quine & Company. It is very much a work in progress, as part of plotting an outline for the dissertation which I should be writing even now, so all comments and suggestions are welcome.