Mathematics and Biology

Having worked in two traditionally distinct fields, both separately and combining the methods of both, I have more than a few thoughts on what separates these disciplines and how this can be quite detrimental.

As I see it there are two core differences between a mathematical world-view and a biological one: the first is in how knowledge is acquired; and the other is the difference between a dynamical world-view and a static one. What may surprise you is which one is static and which one is dynamic!

Knowledge acquisition

I once asked my, then, colleagues at the École Normale Supérieure about their opinions about Biology. They were all PhDs in biology, with extremely illustrious careers and a place at the top-table in international academia. I marvelled at their ability to figure out a biological system and quickly demonstrate results. Their categorisation of their own field somewhat astonished me.

Biology, as it is practiced, does not require a lot of fore-knowledge. In order to have an impact in any one field of biology you need to learn approximately one textbook worth of knowledge. This is not to disparage the technical complexity of this knowledge, nor the astonishing breadth of all biological knowledge. However, to impact on any single field in biology you need to know only the contents of the appropriate textbook-sized chunk of knowledge. For experts in the field, and my friends are definitively experts, this typically takes 3 months. After 3 months then, they can be effective in the lab and begin to deliver real scientific results.

By comparison, the mathematical sciences (physics, mathematics) have a tower structure. You enter on the ground floor and proceed, acquiring technical knowledge, floor-by-floor. Every piece of learning is built-upon a hierarchy of previous knowledge. It is almost impossible to learn the advanced stuff without a thorough understanding of the basics, and not-so-basics, of the discipline. At the very peak, it is not necessary to know the peak on neighbouring sub-fields, but the trajectory to get to that point only diverges at the last moment. This means it takes years of study to get to a point where a practitioner can deliver meaningful contributions to the wider community.

Mathematics is inherently constructivist and dynamic in nature

I have worked in Pure Mathematics. This is a realm of pure relationships, reminiscent of the Glass Bead Game of Hermann Hesse. You construct a set of rules then follow them to their logical conclusions, constructing theorems and proofs along the way. Judgment of peers in this area is via their ability to demonstrate deep and lateral thinking on themes which others also find of interest. Over time, certain subtopics have shown particular interest due, either to the unexpected richness of their structure, or to some general applicability outside of mathematics. Mathematics is often most successful when it is gaining generalisable insights into the highly specific models often developed in Physics.

Mathematics is inherently about operations or relationships. My favourite work of, almost art-like, pure mathematics is a book called Laws of Form. The author uses the operation of ‘distinction’ (or difference) to build up an entire algebra, equivalent to that which we learned in school. This is a magnificent work, which shows that from a single operation effectively all of mathematics can be constructed. More importantly it shows that mathematics is purely the study of relations and the approach is completely constructionist.

Most of science is reductionist, and mathematics at first glance appears to mirror this. Why study Groups when you can find a more fundamental structure such as a Monad? But this appearance of more and more fundamental units is really just an artefact of how we learn the subject. Practitioners are more interested in finding generalisations; explanations which cover a multitude of cases.

A number of years ago, Robert Rosen wrote a seminal work motivating the use of Category Theory in the study of biology. Category Theory is often thought of the ultimate, theory to rule all theories, in algebraic mathematics. Rosen applied it to study function (of functor) in biology. Unfortunately, he died before producing a follow-up more applied work and true utilisation of Category Theory in Biology has not really taken off. However, his focus on dynamics and searching for the right level of description – as applied to biology – should be inspirational to future generations of mathematical biologists.

The constant focus on difference and generalisation makes mathematics an inherently constructionist and ultimately dynamic world-view.

Biology has a reductionist, static world-view

Despite its aim of understanding living things Biology is largely the study of inanimate systems. Biologists have not yet overcome their urge to kill it and pin it to their specimen boards. The reconstruction of imputed function, in the wild, is retrospective and largely left to the imagination.

The history of biology is one of collecting. Botanists brought home uprooted plant specimens. Lists were compiled. Meta-comparisons of lists were performed. And, incredibly, the theory of evolution was born. I say incredibly, because it is a truly amazing feat of human perceptiveness that a theory of millennia-long levels of influence, changing and transforming species, emerged from such a tradition.

This historic influence has continued right up until the present day. Most students are still confronted, at an early stage in their studies, with lists of plant and animal names to learn. They must also learn phylogenetic classifications, in the true scientific tradition; which means, ignorant of the knowledge that these classifications may have changed even during their own lifetimes. Why is this valuation of rote learning still a feature when the underlying data set is constantly in flux?

As a child in the 80’s and 90’s I was amazed by the idea of genes, and more importantly of junk DNA. I didn’t get it. I could not understand why scientists, my father included, could think that the vast majority of the coding that makes us human is junk. It felt like a case for the Emperor’s New Clothes, to me, at the time.

Biology, in the 90’s, was settled. We could read the code; hence the enormous efforts which went into the Human Genome Project. And we would shortly be able to understand it. Then everything would be ok.

The picture which emerged over the following decade was quite different. It is no longer surprising to me that biologists originally thought that so much code was junk, since it didn’t seem to code for anything. It took the emergence of epigenetics to move the debate along. But even then, the instinct remained and most biologists moved from a static code vision to a much more complex static code vision.

In my view, biologists are prone to a mental blindspot when it comes to their experimental setups. The scientific method forces us to design controllable experiments. The goal is to design a test which is not subject to outside influences. Hence, for example, we often remove cells from the host body before running tests on them in a dish. Within the accepted bounds of their domain, this leads to very powerful results. But this removes most of the richness of biological behaviour and is the modern day equivalent of pinning butterflies to cork boards. Reconstruction of the emergent real-world behaviour is almost impossible from such restricted settings.

To this day, the cutting-edge of experimental neuroscience is writing papers about the surprising result that, as we gain access to recordings from more and more neurons, we see very simple patterns of neuronal behaviour in our experiments. The technical term is a low-dimensional projection. However, the experimental setups which remove extraneous influences are designed to be extremely low-dimensional in their degrees-of-freedom. So is there anything noteworthy, about the low-dimensionality of the neuronal activity, apart from our surprise at seeing it?

Biology is enormously complex. I personally consider it to be more complex than astrophysics. It is the study of systems, which have been shaped by evolution, to find specific solutions to their ecological niche. This means that every instance is always a singular instance. The fact that the founders of this discipline have found ways of transcending this issue and clumping things that look like one more of those together is extremely laudable.

I am not here to insult my colleagues, who are immensely skilled and knowledgeable. They often do know the limitations of their experimental setups. But they allow the mental skipping over of these limitations to infuse their ability to think about the larger system.

Reductionism is leading to a static view of biology, which is incapable of forming a synthesis with real-world behaviour and requirements and killing the applicability of biological studies.

Implications

The different models of knowledge acquisition in biology as compared with mathematical sciences means that biologists are typically much more comfortable in working with ambiguity and lack of knowledge. They have to, because their systems are so unbelievably detailed that nobody can every know everything about the system. They have an approach to work which means working strictly within the bounds of their field of knowledge and working together with others at the interface points.

Mathematically inclined people are terrified by ambiguity. In extreme cases they use stochastic processes, but even these are not inclined towards ambiguity – they are extremely well defined. The majority of mathematically trained people avoid biology at all costs, and restrict their applied work to engineering-led systems.

This is a shame. Mathematical people have a lot to contribute to biology and biology is a very satisfying system on which to apply mathematical tools. The mathematical study of artificial neural networks has contributed enormously to our understanding of information processing in the brain. The methods developed, such as the replica method, are deeply technically chanllenging as a mathematician to study. Meanwhile, the short-loop feedback from biologists and their experimental results is extremely satisfying as a work modality.

Biology too could learn a lot from adopting a more mathematical mindset. I have worked extensively with biologists and their fear of mathematical thinking immobilises them. They don’t even realise that much of their thinking around ‘models‘ exactly mirrors that of mathematicians and their models. With just a tiny amount of mathematical thinking, as opposed to stamp collecting, the self-consistency of biology could be much greater. And our ability to better predict based on our ground-level understanding massively enhanced.

Epilogue

I moved out of pure mathematics because, despite the inherent beauty of the work I wanted something which would show more immediate, tangible, impact on the world.

Nowadays, my most defined skillset in mathematics is in complex and adaptive systems and stochastic processes. This spans everything from traditional differential equations, often coupled to learning processes, to Bayesian inference models. Having an understanding of so many mathematical processes allows me a level of insight into many of the most important questions of our day. I spend a considerable amount of my time reflecting on how and where learning occurs in systems in which the ongoing dynamics are often influenced by the learning process, and vice-versa. I work in both AI and biology, so I get to see how these ideas play out in explaining the brain and immune systems, but also in machine simulacra of these systems. This back and forth is at the core of my working life.

If you are a mathematician looking for a more grounded working environment, I strongly encourage you to consider biology. And, if you are a biologist who is currently afraid of mathematical thinking, talk to me; I am developing tools which will allow you to use your inherent thinking patterns but enable you to move through implications and outcomes much faster, ultimately increasing your productivity and biological output.