Celebrating the International Day of Mathematics – on the mathematical imagination and the infinite

The International Day of Mathematics, also known as Pi Day, is celebrated every year on 14 March. It is a day for professionals and enthusiasts alike to highlight the elegance and endless possibilities that mathematics offers to the world. From the intricate patterns of nature to the complex algorithms at the forefront of today’s most exciting technologies, this day serves as a reminder of how deeply entwined mathematics is woven into the fabric of our daily lives.

The philosophy of mathematics is an interesting branch of philosophy that studies the assumptions, foundations, and implications of mathematics. One aspect of this is the concept of infinity – its various paradoxes present unique challenges in the philosophy of mathematics, leading to deep inquiries about the nature of numbers, sets, and the universe.

Today, on the International Day of Mathematics, Asst Prof Daniel Waxman from the NUS Department of Philosophy shares with NUS News his thoughts on mathematical imagination and the infinite, and how the imagination plays a significant role in our understanding and justification of certain mathematical concepts.

Paradox at the heart of mathematics

In 1901, the world of mathematics was left reeling by Bertrand Russell's discovery of a paradox at the heart of set theory. Consider – Russell's argument went – the set R of all sets that do not contain themselves. Is R a member of itself?

There is no way to answer this question without contradicting oneself.

The reasoning is as follows: if R is a member of itself, then, by its definition as the set of all sets that do not contain themselves, it is not a member of itself. But if R is not a member of itself, then by analogous reasoning it is a member of itself!

It follows that certain highly appealing mathematical principles governing the existence and behavior of sets – now known as naive set theory – are inconsistent: they allow us to prove claims of the form P and not-P. Inconsistency is an extremely undesirable property for a mathematical theory to possess. If a theory is inconsistent, then using standard logical rules, it can be shown to prove any statement whatsoever, even absurd claims like 0=1.

The inconsistency of naive set theory spurred the development of alternatives. More than a hundred years later, we have a relatively good idea how to formulate versions of set theory that do not rely on the problematic principles used by Russell to derive a contradiction.

Justifying consistency

So the question arises: why do we think that these theories are any better than naive set theory? How can we know that they are in fact consistent? The space of possible answers here is constrained by another landmark result in mathematical logic in the 20th Century: Gödel's Second Incompleteness Theorem.

This result tells us, roughly, that any consistent theory of a certain minimal strength is unable to prove its own consistency. So, any knowledge that our mathematical theories are consistent must either come from a proof within some other mathematical theory, or via non-deductive means altogether.

Using our imagination

In some recent work, I have been exploring one way of implementing the second option: in particular, the possibility that we might get justification in consistency using our imaginative faculties.

In one respect, the idea is simple. We have a faculty of sensory imagination. Imagine a cat leaping at a bird, or Bruce Springsteen playing guitar, or driving on a wet road at night. If you do, you will have experienced an imaginative episode that involves entertaining "mental images" – pictures in the mind's eye, or sounds in the mind's ear. Philosophers have argued for centuries about what imaginative episodes of this kind can justify.

My own view is that, if we are able to clear-headedly imagine a scenario, we can conclude – at least in the absence of any countervailing evidence to the contrary – that whatever holds in the scenario is at least possible. So – if we are able to imagine scenarios in which our mathematical theories hold – then there is at least the prospect of justifying their consistency.

Imagining the infinite

There is, however, a major stumbling block. Virtually any mathematical theory studied today whose consistency is seriously in question is infinite. Think for instance of arithmetic, the theory of the natural numbers. To imagine a scenario in which all of the basic axioms of arithmetic are true, it would have to be populated with infinitely many objects: 0,1,2,3... and so on, indefinitely. But how could we ever possibly imagine an infinite scenario? Surely, given the limitations of our finite human minds, this is out of the question?

In fact, I think it's entirely plausible that we can imagine infinite scenarios. Drawing on some recent work in the cognitive science of perception and imagination, I argue that we ought to regard our imaginative capacities as being inherently dynamic.

Here is a simple example: imagine a perfectly rigid playing-card painted entirely red on one side and entirely green on the other. We can all imagine such an object, I take it. But notice that no single static mental image can display all of the information about the imagined scenario! Any perspective from which we "view" the card will at best capture part of one side.

The lesson is that we ought to think of the content of our imagination as extending beyond a single static image, to include the ways in which scenarios are dynamically developed or elaborated in imagination. But once the dynamic character of imagination is recognised, there is nothing stopping us from considering infinite scenarios!

Try to imagine a number-line, for example, of the kind used to teach school-children the whole numbers. Now consider what happens as you "pan your mental camera right": if you are anything like me, you will extend the line so that, no matter how far you go, you imagine more strokes/numbers than before. While it's true that at any given time you will be "viewing" a finite image, the crucial point is that you will have an additional disposition to extend that finite image even further, going on in the same way, indefinitely, no matter how far you proceed.

So it's reasonable to describe the imagined number line as infinite: not because at any point we have a static image containing infinitely many objects, but because no matter how far rightward our mental camera pans, at no point will the rightward scanning ever lead to the numbers terminating.

Of course, even if everything I have said is right, we have achieved at best some kind of justification in the consistency of arithmetic – an infinite theory, to be sure, but an extremely small and tame infinity by the standards of mathematicians! If I am right, then humans may well have the capacity to – strictly and literally – imagine the infinite. But much interesting work remains in assessing just how far our imaginative capacities are able to take us, and how the foundations of mathematics might be justified.

Daniel Waxman is an Assistant Professor in Philosophy at NUS. He has a PhD in Philosophy from New York University and an MMathPhil from the University of Oxford. Before Singapore, he taught in New York, Oxford, and Hong Kong. In the first semester of AY24/25, he will be teaching a course on the ‘Introduction to Philosophy of Mathematics’.