Mathematics and Biology II – Practical considerations

Last month I wrote about the historical world-views of mathematicians and biologists. These articles are part of a planned four part series, in an attempt to first understand and then improve the working relationship between these two key scientific disciplines. This is all a work in progress, so at the end, I will try to take the key learnings from each of the articles and distill them into a single composed article.

This month, I want to discuss the practical considerations why mathematics and biology still don’t work so well together.


Before beginning, I will stipulate that I am a mathematician who does biology, and I am a biologist who does mathematics. This discussion is about the mainstream. And, let’s face it, my own career path does not suggest that these two subjects work too successfully together.

Why don’t biologists do mathematics?

If you ask a biologist why the do what they do, they will typically reply that they love figuring things out, and, are amazed by the natural world. These statements are inherently true. To be a scientist, of any kind, you need to enjoy working on problems until you’ve “figured them out.” I mentioned, in a previous article on failure, that trial-and-error is intrinsic to the scientific process. Additionally, the natural world is, in my opinion, the hardest puzzle of all to try to understand. In the previous article, in this series, I mentioned how every instance in biology is effectively unique making its study considerably more difficult than physics or chemistry.

“What’s your favourite statistical test?”

I was asked this question by some eminent experimental biologists at the École Normale Supérieure (ENS). They were serious. They respected me, knew I was working with top mathematicians, and they wanted my opinion. I am still flabbergasted by it, but it is important to point out that these were not stupid people. Far from it.

I have spent many coffee breaks and other opportunities, over very many years, asking biologists what they think of mathematics. Over time, I’ve honed my ability to ask this question without triggering their inbuilt defensiveness. The answer comes out as almost always the same:

“I chose biology because I wanted to avoid mathematics.”

Most of my friends and colleagues, who studied biology in university, did so because it allowed them to pursue science while maximally avoiding mathematics. Of all of the sciences, this is the one which is the least mathematical, especially in the university syllabi.

At a senior level, these two disciplines have a very short history of working together. Depending on where you pursue your undergraduate studies, in biology, you are still quite likely to encounter: biology professors who are disparaging of mathematics; and, mathematical programmes which deliberately avoid covering the thinking behind the tools being taught.

No discipline can entirely avoid mathematics today. Mathematics departments have massively gained from this situation; increased teaching hours lead to more faculty posts. However, the mathematicians have realised that to provide service teaching to other fields they need to resort to short-cuts. The most common short-cut, in the case of biology, is to teach the rules of statistics as a look-up table. This means, they simplify the practice of statistics to a set of rules. The biology students internalise only the procedure, without taking any of the understanding, and they then operate this procedure for the rest of their careers.

My colleagues at the ENS were far from stupid. They were just used to operating under this paradigm. My response of, “it depends,” came across like a typical nerd response. Imagine asking the most technical person you know how to turn on the radio, and they answer with, “Is it a valve or a transistor radio?” This is how I came across – different worlds!

I know of a number of senior biological scientists who do apply mathematics in their day-to-day research. If you speak French, you can watch this video where Antoine Triller admits that he realised during his career the importance of mathematical approaches and thus incorporated them into the direction of his department. Antoine is the director of the department of biology at the ENS (no, he was not involved in the discussion quoted above) and is known worldwide for his work at the forefront of biological research.

These scientists came to mathematics too late. I described in my world views article how mathematical knowledge must be built in a tower structure. Human life and effort is finite. There is too much expert knowledge required, to correctly apply mathematics, for us to expect a career biologist to just learn the necessary skills later (a more typical approach in biology).

Why don’t mathematicians do biology?

I chose undergraduate mathematics because, for me, it didn’t require much time input. I was sick at the time and found the commitment of lab subjects too time consuming. By contrast, I could roughly follow the mathematical lectures then turn up on the day of the exam and work out the answer from first principles. I am not going to pretend that I recommend this approach, but I am far from unique among mathematically talented students.

Mathematical students tend to be pretty sharp and they like to see the answers to questions quickly. When they can’t see the answer, they get curious, and often work night and day until they get it. They are obsessive and their major limitations are those imposed by their own lack of knowledge and their own physiological inability to work non-stop on a problem.

None of this conforms to the rhythmic nature of lab research. As an undergraduate, physics and chemistry labs are tailored to fit into periods suitable for a single class. I can’t comment on biology, as I never took it, but it appears to follow this same rhythm. As a (post-)graduate student and later, these experiments may become more drawn out but they typically follow some kind of regular sampling interval (most likely with declining frequency later in the sample period).

A mathematically-inclined student can rarely speed-up a lab experiment through insight or getting to the answer faster. And this peeves them.

I see the world in black or white

Biology, in appearance, is messy and imprecise. It is the opposite of the perfect world, which exists in the imaginations of many mathematicians. You constantly have to fudge, or use your judgment, on measurements.

We sometimes try to classify the messiness as different forms of noise. But this is only part of the story. Understanding the precise type of noise is vital if you want to understand its impact on a biological system. I am using mathematical terms here, to explain this issue, but this is not how it is typically introduced in a biology class.

Mathematically-inclined students shy away from studies in which their judgment, in the absence of bottom-up rules, should be used. For this reason, they much prefer studies in Computer Science. Since childhood, I could explain to you the functioning of your computer software from the level of PN-junctions upwards. Again, I am not alone in this. Telling me to not ask any questions and just use Method X tends to turn me off. But this rote-learning of technique is common in biological studies.

My favourite technique is…

Mathematicians are far from perfect. Most of us have our preferred tools. And we tend to use them for every problem. This is not such a good idea when studying an applied problem. And, it is a particularly bad idea when studying biological problems.

One of my favourite books is The Mathematics of Behavior. I love this book because in every chapter it moves on to a new mathematical technique and how it is used in behavioural science (psychology, biology, etc.). What you quickly grasp, in reading this book, is that to most effectively study behaviour you need to choose the right mathematics for the particular problem. It is even evident, in the book, that over time the best method has changed for any given application.

If you have spent 10 years advancing, and perfecting, your ability in a given technique it is very difficult to openly admit that – given advances in data capture techniques, or in our understanding of the underlying biology – a different technique would now be more appropriate. It is even harder to be this honest at earlier stages in the analysis, when you just want to have an impact before you walk away.

John D. Cook’s blog is a rare example of somebody applying the breadth of mathematics, with enormous insight, to biological problems.

Structural problems

Finally, there are the strange few, like me, who stray into biology from mathematics or, like one of my PhD supervisors, who stray from biology into mathematics. If I claim (in my previous article) that there is huge value in combining mathematics with biology we must do really well, right?

Actually no. I refer to the problem here as one of structural problems. Or, in start-up speak, the interests are not aligned.

A mathematician who spends his time working on biology gets no respect from fellow mathematicians. The techniques she uses are rarely cutting-edge enough. Mathematics is a discipline and it judges its own by the standards of people who spend all of their time studying pure mathematics. Applied mathematics has existed for a very long time (here is a gratuitous link to one of my favourite books, as an example). But applied mathematics is dominated by physics, with long established techniques and methodologies.

Mathematical biology is in its infancy by comparison. And, like all fields in their infancy, the productivity as measured by papers churned out is considerably lower. In applied mathematics it is possible to reliably produce a paper for a top-flight journal every six months. In mathematical biology the publication rate is less than one per year, and typically in much lower-ranked journals.

This is basically an issue of career. There is nothing intrinsically wrong with the publication rate in mathematical biology. But it will lead to a very painful road to academic tenure (and to grant funding).

Right now, the optimal, from an academic career point-of-view, is a biologist who dabbles in mathematics. My former field, of computational neuroscience, was dominated by experimental neuroscientists who made simplistic models of their latest experiments in order to publish them in the top journals. The mathematical models are trivial, and are typically produced by masters students in their labs, but they give the aura of mathematisation without taxing the brain of the reader.

This phenomenon is wrong, on so many levels, that I may devote a full article to it sometime. But, for now, let’s just say that the mathematics are not really adding anything. It’s a waste of opportunity. However, it is a well known route for people from the experimental side to boost their publication rankings enormously. Similarly, it is well known from the mathematical side that a postdoc spent doing experiments, while it may be tedious, demonstrates the ability to direct experiments (it shouldn’t but it does) and can greatly aid with job applications later.

So, it is possible to lead a double-life and combine both mathematics and biology. But the rewards are greatly enhanced for those who only simulate the process. Or, better yet, for the experimentally-inclined who supervise others who do the mathematics.

People are not stupid. They see this and go elsewhere.

Next time

Until now, I have omitted the elephant in the room: Bioinformatics. Next month, I will discuss the impact of bioinformatics and statistics on biology and also why I don’t see it as a model for the future mathematisation of biology.


My thought and arguments are formed by my time in academia. But I am also relatively versed in the application of mathematics to biology in industry. I may introduce an article to this series similarly comparing why this career path also has problems.

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