stoutfellow

In the course of the nineteenth century, mathematics was utterly transformed, in its subject matter, in its methodology, and in its philosophy. In this post, I'll discuss the nature of that transformation in general terms; in the posts to come, I'll describe how that transformation played out in various branches of mathematics.

The transformation had two major aspects, one, as it were, reaching downward and the other upward, and they fed off each other. On the one hand, each of the major branches of mathematics - algebra, analysis, and geometry - rested, to different degrees, on shaky foundations. It became necessary for those foundations to be reconstructed more rigorously, whether by the clarification of critical definitions or by the establishment of firm (in the case of geometry, firmer) axioms. Rigorization, then, was one part of the change in mathematics.

The other aspect was a sharp increase in abstraction. Up to this point, mathematics was generally considered as, in some fashion, describing the real world - perhaps an idealized form of that world, perhaps one only approximable by human tools, but nonetheless real. The possibility of exploring mathematics that did not directly deal with the world emerged strongly in this period.

On the one hand, the push for rigorization encouraged greater abstraction; once axioms have been laid out, it becomes conceivable to change those axioms. This step is not an inevitable one - it took millennia before anyone seriously tried modifying Euclid, for example - but once it was taken, even in one place, the possibilities suddenly became visible to many, and new kinds of mathematics mushroomed.

On the other hand, once mathematics became detached from the outer world, rigorization became increasingly necessary. If your thinking deals with the world as it is, a failure of rigor is, perhaps, not as serious, because there is a "reality check" available. More generally, intuition about the real world can, at least, point to what might be true, and guide the theorem-prover; once that is lost, the only lifeline available is close attention to rigor.

There's more to be said about the nature and consequences of this mathematical revolution, but the next few Rambles will focus on specific examples before we return to these issues.

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