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Georg Cantor showed that some infinities are bigger than others. Did he assault mathematical wisdom or corroborate it?
In 1883, the brilliant German mathematician Georg Cantor produced the first rigorous, systematic, mathematical theory of the infinite. It was a work of genius, quite unlike anything that had gone before. And it had some remarkable consequences. Cantor showed that some infinities are bigger than others; that we can devise precise mathematical tools for measuring these different infinite sizes; and that we can perform calculations with them. This was seen an assault not only on intuition, but also on received mathematical wisdom. In due course, I shall sketch some of the main features of Cantor’s work, including his most important result, commonly known as ‘Cantor’s theorem’. But first I want to give a brief historical glimpse of why this work was perceived as being so iconoclastic. Ultimately, my aim is to show that this perception was in fact wrong. My contention will be that Cantor’s work, far from being an assault on received mathematical wisdom, actually served to corroborate it.
The standard conception of the infinite is that which is endless, unlimited, unsurveyable, immeasurable. Ever since people have been able to reflect, they have treated the infinite with a curious combination of perplexity, suspicion, fascination and respect. On the one hand, they have wondered whether we can even make sense of the infinite: mustn’t it, by its very nature, elude our finite grasp? On the other hand, they have been reluctant, indeed unable, to ignore it altogether.
In the fourth century BCE, Aristotle responded to this dilemma by drawing a distinction. He believed that there is one kind of infinity that really can’t be made sense of, and another that is a familiar and fundamental feature of reality. To the former he gave the label ‘actual’. To the latter he gave the label ‘potential’. An ‘actual’ infinity is one that is located at some point in time. A ‘potential’ infinity is one that is spread over time. Thus an infinitely big physical object, if there were such a thing, would be an example of an actual infinity. Its infinite bulk would be there all at once. An endlessly ticking clock, on the other hand, would be an example of a potential infinity. Its infinite ticking would be forever incomplete: however long the clock had been ticking, there would always be more ticks to come. Aristotle thought that there was something deeply problematic, if not incoherent, about an actual infinity. But he thought that potential infinities were there to be acknowledged in any process that will never end, such as the process of counting, or the process of dividing an object into smaller and smaller parts, or the passage of time itself.
Aristotle’s distinction proved to be enormously influential. Its importance to subsequent discussion of the infinite is hard to exaggerate. For more than 2,000 years, it more or less had the status of orthodoxy. But later thinkers, unlike Aristotle himself, construed the references to time in the actual/potential distinction as a metaphor for something more abstract. Having a location ‘in time’, or being there ‘all at once’, came to assume broader meanings than they had done in Aristotle. Eventually, exception to an actual infinity became exception to the very idea that the infinite could be a legitimate object of mathematical study in its own right. Cue Cantor…