Realizing Reason by Macbeth Danielle

Author:Macbeth, Danielle
Language: eng
Format: epub
Publisher: OUP Oxford
Published: 2014-11-15T00:00:00+00:00

5.5 Conclusion

“Mathematics,” Grabiner (1974, 364) remarks, “is that area of human activity which has at once the least destructive and still the most fundamental revolutions.” Nowhere is this more evident than in the nineteenth-century revolution in mathematical practice, the revolution that was, as Stein describes it, a second birth of the subject. Mathematics was not merely to be done in a new and better way, as Descartes had shown with his method that it could be. It was to be raised anew on purely conceptual foundations by means of reason alone.45 What Kant had declared impossible was now enacted. Indeed, it almost seems in retrospect, that Kant had only to make explicit that early modern mathematics, like its ancient counterpart, constitutively involves something we can call pure intuition for mathematicians to rise up and prove him wrong.

The revolution, I have suggested, unfolded in two distinguishable waves. First, from reasoning in the symbolic language of arithmetic and algebra, mathematicians began self-consciously to reason on expressions in that language—something eighteenth-century mathematicians such as Euler had in fact already been doing, albeit quite unselfconsciously and unsystematically. Indeed, one can find this sort of reasoning here and there already in Descartes’ Geometry. The symbolism is not only a medium within which to discover new results as Euclid’s diagrams are; its equations can also exhibit discernible patterns that can be exploited in the course of mathematical inquiry. At the second stage of the revolution, somewhat ironically, all appeals to symbolisms were to be eschewed. The symbolic language that in the seventeenth and eighteenth centuries had opened up a whole new world of mathematics (and physics) was now to be regarded as providing a merely contingent and external perspective on what now appeared to be the real subject matter of mathematics. What had seemed in the eighteenth century to be the thing itself, namely, the formula, had been revealed to be only its outer clothing. What matters is not the formula through which we first gain cognitive access to a function but the function and, in particular, the intrinsic properties in virtue of which it is the function it is.

Kant’s critical philosophy, although not a prerequisite of the revolution, nevertheless profoundly influenced the articulation both of the nature and fatal flaws of the old regime and of the fundamental character and constitution of the new. Not Kantian pure intuition but Kantian ideas, that is, concepts of reason conceived as intelligible unities, are now to be seen as what mathematics is really about. Finally, after over twenty-five hundred years of growth and development, mathematics has, so it seems, become a self-standing discipline, the work of pure reason wholly unfettered by the contingencies of our form of sensibility. But how can this be? Surely mathematics needs to answer to something independent of its own activity, and if it does not answer to anything independent of its own activity then it really is, as since the nineteenth century it has so often been taken to be, nothing more than a mere game.

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