Will Cantor’s Paradise Ever Be of Practical Use?

We really have no way of knowing what early humans thought when they gazed up at the sky. Since everyday practical experience is, by definition, limited to a very small region of space and time, it requires considerable cognitive sophistication to conceive of something – say the night sky – “going on for ever,” let alone to ponder whether that means it is “infinite,” or indeed what “infinite” actually means.

What we do know is that the ancient Greeks made what may have been the first substantial attempt to analyze the notion of infinity, with Zeno of Elea (ca 490-430 BCE) of particular note for his discussion of a number of (seeming) paradoxes that arise from the assumption that space and time are (or are not) infinitely divisible.

Archimedes’ (ca 287-112 BCE) calculations of areas and volumes made implicit use of infinity, and from today’s perspective can be recognized as the forerunner of integral calculus.

Skillful formal – though by modern standards not rigorous – use of the infinitely large and the infinitely small was made by Isaac Newton and Gottfried Leibniz in their development of modern infinitesimal calculus in the seventeenth century, though it was not until the nineteenth century when Bernard Bolzano, Augustin-Louis Cauchy, and Karl Weierstrass finessed the lurking problems of infinity by means of the famous (and for many a first-year mathematics major, infamous) epsilon-delta definitions of limits and continuity.

But none of these developments was about infinity as an entity; the focus rather was on the unending nature of certain processes, starting with counting. It was Georg Cantor (1845 – 1918) who really tackled infinity head on. His proof that the set of real numbers cannot be put into one-one correspondence with the natural numbers, and hence is of a larger order of infinitude, led to a series of papers, published in a remarkable ten-year period between 1874 and 1884, that formed the basis for modern abstract set theory, including the development of a fully formed arithmetical theory of infinite numbers (or “cardinals”).

Reactions to Cantor’s revolutionary new ideas ranged from outraged condemnation to fulsome praise. Henri Poincaré called Cantor’s work a “grave disease” that threatened to infect mathematics, and Leopold Kronecker described Cantor as a “scientific charlatan” and a “corrupter of youth.” Ludwig Wittgenstein, writing long after Cantor's death, complained that mathematics had become “ridden through and through with the pernicious idioms of set theory,” a theory he dismissed as “utter nonsense,” “laughable,” and “wrong.”

At the other end of the spectrum, in 1904, in the UK the Royal Society awarded Cantor its highest award, the Sylvester Medal, and in Germany David Hilbert declared that “No one shall expel us from the Paradise that Cantor has created.”

Having devoted the early part of my professional career to work in (infinitary) set theory, starting with my Ph.D. in “large cardinal theory,” completed in 1971, and moving on to work on alternative universes of sets (a particularly hot topic after Paul Cohen’s introduction of the method of forcing in 1963), in the early 1980s my interests started to shift elsewhere, to questions about information, communication, and human reasoning.

I found myself temporarily back in the world of set theory and the arithmetic of infinite numbers recently, when I was approached by the organizers of the World Science Festival to moderate a panel discussion on the topic of infinity and a more in-depth follow-up the following day.

Both discussions raised the question as to whether study of infinity – in particular the hierarchy of larger infinities that Cantor bequeathed to us – would ever have any practical applications. As panelist Hugh Woodin remarked at one point in the discussion, it is a foolish mathematician who declares that a particular piece of mathematics will not find applications. For instance, G. H. Hardy’s famous statement (in his book A Mathematician’s Apology) that his work in number theory would never find practical application, proved to be spectacularly wrong less than a century later, when number theory became the foundation for internet security.

Hardy’s observation was based on his familiarity of the world he lived in, a world in which the World Wide Web was not even a dream. Today, we cannot know what the world of tomorrow will look like. On the other hand, whatever our children and grandchildren will take for granted, their world will surely be finite, which makes it unlikely that Cantor’s theory – and the almost a century of development in set theory since then – will have practical use.

Or does it? What about calculus? Infinitesimal (!) calculus not only has applications in the modern world, but much of the science, technology, medicine, and even financial structure the underpins our world depends on calculus for its very existence. Applications don’t get more real than that.

True, but the dependence on infinities you find in calculus is essentially asymptotic. What really drives calculus is the unending nature of certain processes on the natural numbers. Talk of “infinitely large” or “infinitely small” is little more than a manner of speaking. Indeed, the epsilon-delta definitions (which do not involve infinities or infinitesimals) are a way to formalize that manner of speaking, effectively eliminating any actual infinite or infinitesimal quantities.

In contrast, much of the work on infinity (more precisely, infinities) carried out in the second half of the twentieth century (when I was working in that area) focused on properties of sets that made their cardinalities super-infinities of different orders: inaccessible cardinals, Ramsey cardinals, measurable cardinals, compact cardinals, supercompact cardinals, Woodin cardinals, and so on. Infinities which dwarfed into invisibility the puny cardinality of the set of natural numbers. Indeed, each one in that sequence dwarfed all its predecessors into invisibility. How could that work find an application?

I’ll lay my cards on the table. I think the chances are that it won’t. But I don’t think it is impossible. Indeed, I began to suspect a possible application in the very domain I worked in after I left set theory.

[This may of course be nothing more than a reflection of having at my disposal a large hammer which made everything look a bit like a nail. But let’s press on.]

The post 9/11 world saw me involved in a series of Defense Department projects the first being improving intelligence analysis (and the others essentially variants of that).

In today’s information rich world, the major nations can be assumed to have access to all the information they need to predict (and hopefully thence prevent) the majority of terrorist attacks. The trouble is, the few data points which must be identified and connected together to determine the likelihood of a future attack are just a tiny few in an overwhelming ocean of data. Even in the era of cloud computing, identifying the key information is analogous to using the naked eye to find a handful of proverbial needles in a non-proverbial field of haystacks.

To all intents and purposes, the available data is infinite. The only hope is to impose some structure on the data that makes it possible for humans and computers to work together on it, narrowing down the focus to the regions more likely to be of significance. Though modern computing systems can sift through massive (finite) amounts of data in a relatively short time, they need to be programmed, and writing those programs (at least, some kinds of them) will require some structure on those large sets of data. A possible place to find the appropriate structure(s) is infinitary set theory. In other words, to develop the appropriate structures, assume the data is infinite. View the infinite as a theoretical simplification of the very large finite. (Economists sometimes make a similar simplifying assumption about economies.)

Do I think this is likely? Frankly, no. But then, neither could Hardy conceive of any practical application of his work in number theory. [Incidentally, like Hardy, I don’t think mathematics needs applications to justify itself. It’s just that the question of application is what this article is about!]

The discussion about large cardinals you will find in those panel discussions at the World Science Festival might seem impossibly abstract and far removed from the everyday world. Indeed, it is. But the questions being discussed all resulted from a process of rigorous, logical investigation that arose directly from late nineteenth century attempts to understand heat flow. History tells us that what begins in the real world, very often ends up being used in the real world.

Prediction is hard, particularly about the future.

Incidentally, how did I end up working on a project for the DoD? They asked me. I might not be the only person to speculate about a possible use of Cantor’s paradise. This is your taxpayer dollars at work.