The Creativity and Structure of Pure Mathematics
American mathematician Dr. Holly Krieger is a lecturer at Britain’s University of Cambridge and Director of Studies and Fellow in Maths at Murray Edwards College, one of the few women-only colleges of Cambridge. She’ll be speaking at John Adams Institute on October 10 about the beauty of symmetry, one of the most visibly aesthetic principles of math. We spoke with Holly to learn more about the creativity and structure of pure mathematics.
By Katherine Oktober Matthews
Many people avoid math the same way they avoid the dentist. Tell us how you see it – what attracts you?
I am attracted to a certain formalism and logical structure. The abstraction level I find very appealing, but also, that it’s possible to really understand. One moment, you have no clue what some piece of mathematics is saying or how it’s working, and suddenly there’s like this neurological shift and it clicks into place. Then, not only do you understand it, but it’s almost incomprehensible that you didn’t understand it five seconds ago. And I find that really satisfying.
More broadly, how do you see math’s role in our lives?
Math plays a couple of different roles. The first is direct and practical, and the same reason why history and social sciences are part of a liberal arts education, which is that knowing some mathematics is necessary to have a meaningful voice in a democratic society. Quantitative information informs policy decisions at every level of government, and when an elected official shows you a graph and claims that this graph explains why they need your vote/money/time, it’s important that you’re able to independently estimate the quality of the information they’re giving you.
The second is a broader role, which I’ve heard referred to as “teaching people to think”. Deductive reasoning and abstract thinking – by which I mean considering objects via their relationships instead of their more basic properties – can be taught in various subjects, but mathematics is the only one which distills these skills and teaches them separately from opinions and empirical observations, which are subject to error and not easily generalized to other settings.
You’re in a field of mathematics called arithmetic dynamics. Can you tell us a bit more about that?
Let me tell you about those two words separately first. Arithmetic is the study of the whole numbers, and particularly the prime numbers, which are the building blocks of all whole numbers by using multiplication. Dynamics is the study of how systems defined by simple rules behave in the long term. Arithmetic dynamics is the combination of the two – but let me illustrate by giving you an example of the type of question one could ask in this field. The Fibonacci sequence is a well-known dynamical sequence: the simple rule defining it is to start with 1 and 1, and to get the next number, take the sum of the previous two: 1,1,2,3,5,8, and so on. An arithmetic question one could ask about this dynamical sequence is: are infinitely many Fibonacci numbers prime? We actually don’t know the answer to this question, though it is centuries old.
What makes this arithmetic question interesting?
I have no clue if there would ultimately be some applicability to it, maybe or maybe not. To me, the interestingness of the question is the basicness of it. You could explain the question to a very young child – there’s no technical definitions that need to be expressed, you’re just talking about adding things, and then to talk about primes, you need to understand multiplication. Even if we don’t know of any application to any particular branch of research, it’s such a natural question to ask because in many ways the primes are the building blocks of arithmetic. So, when I say it’s “interesting,” it’s not just because it’s an old question, which it is, but also because it seems so basic yet we can’t answer it.
Which gives you more pleasure in math: a problem that you can solve, or a problem that you can’t?
This is such a unique and wonderful question. I wish I could give the romantic answer that an unsolved problem is a sort of blissful intellectual agony to me…but I’m far too practical for that to be true. I like to solve problems! My mathematical personality is more conqueror than explorer, though I could see that changing as I age and broaden my knowledge.
How do modern mathematicians solve problems?
There is quite a lot of sitting and thinking with a notepad or blackboard, for me at least – I think most deeply about mathematics when I’m writing a lot. But for me it tends to be a balance of collaboration and solitary computation; often I’ll discuss potential techniques or interesting questions with my colleagues, then we’ll sit down separately and try to tease out whether any of our grandiose schemes will actually work. It’s very common that something which seems only a small conceptual hurdle in conversation turns out to be a significant technical difficulty once you try to set down rigorous mathematics.
Mathematics is not referred to as an experimental science, but I often find that it is. If I have an idea or question, I’ll nearly always try a simple or easy test case before putting in serious time trying to prove it. That feels a lot like an experiment to me, though fortunately the type that can be done outside an expensive or remote laboratory.
So, despite all our technological advancement, the go-tos are still blackboards and notepads. Are computers capable of this kind of problem solving or is exploring the unknown of mathematics fundamentally a human task?
That’s a question that’s under a lot of debate now. I’m an optimist about progress, so my personal opinion is that yes, there will be a point at which my job can be done by a machine; but that’s not to say that that point will be in my lifetime. If you asked an artist the same question, I think you’d get a lot of objections, but I’d also assert the same answer: at some point in the future, machines will be capable of creating meaningful art. So, I don’t think it’s a fundamentally human pursuit, but that’s because I don’t think creativity is a fundamentally human pursuit.
Considering that computers compute so much better than us, where is it that they fall short in math?
So far, in the realm of mathematical proofs, very short. In particular, they have no idea how to ask an interesting question – at least, what humans would consider an interesting mathematical question.
Do you see advanced mathematical problem solving as creative or logical?
It’s both. The creativity is the top-level part of the work. It’s intuition and association and the ability to make connections based on prior experience. It’s about cooking up interesting questions, and noticing patterns, or realizing that something about this feels like something else I’ve thought about, and asking, “Is there some relationship between them?” It’s not until you sit down and crank out the very rigid, logical structure – which can result in beautiful proofs – that the straight forward logic comes in.
You’re coming to speak at JAI about the mathematics of symmetry – tell us a bit about that. How should we see symmetry as a mathematical concept?
Symmetry is a particularly great topic for understanding that notion of how mathematics is about abstract reasoning that I mentioned earlier. The mathematical notion of symmetry is invariance under change: can I apply a transformation to this object without changing it? This is an example of moving from the description of the object (round) to a description of its relationship to other things (in this case, itself).
A mathematician might be even more interested in the question: is its collection of symmetries a complete description of a circle? In other words, is it the only curve which can be rotated about any angle and remain the same? That’s a fun question to think about a little bit if you haven’t contemplated it before.
Symmetry is also a concept that is used in deep and abstract mathematics that is nonetheless interesting and accessible to non-mathematicians, as we see it every day in biology, art, and architecture. For me, that makes it particularly interesting to discuss with people who may have neutral or negative perceptions of the appeal of pure mathematics.