The Reasonable Ineffectiveness of Mathematics in the Biological Sciences


(With apologies to Wigner)

The fact that biology is not rich in theory is well known. Of the theories that do exist (such as Darwin’s theory of evolution), many have never been formulated into mathematical laws. Physics envy is a well-known propensity of those biologists who desire to be working on a more general understanding of whole fields (like cellular biophysics or regulation of gene expression) rather than on uncovering details of particular subsystems.

I used to be certain that the lack of mathematical rigor in biology was the fault of biologists, who generally are unable to grasp the fundamentals of even simple math. But that canard may be less true than it used to be (if it ever really was). With the explosion of systems biology and the development of algorithms for data mining, there are plenty of biologists around who are perfectly familiar with the manipulation of equations and concepts of theoretical formulation.

Certainly for some areas of biological science, good mathematical minds (taking a break from physics) have attempted for quite some time to apply their skill to producing the kind of laws that are (as Wigner put it) “true everywhere on the Earth, was always true, and will always be true.” And yet, success in these attempts has been limited at best.

Evolutionary biology is a good example. We certainly have a wonderful theory, one that has been well formulated, improved upon and expanded over the years to include data and concepts from many disciplines, and that has been supported beyond question as to its truthfulness. And yet, where is the law of evolution? Why can we not formulate a mathematical treatment of the evolutionary process, including inheritance of genotype, selection of the fittest phenotype, and fixation of that genotype in the population?

Of course, there are equations that can approximate some of the steps fairly well, but not the whole process. Is this because biologists are too stupid or lazy to come up with the right answers? Of course not, and even if they were, physicists have fared no better at the task, and not for want of trying.

So, what is the problem? I think the answer might be that biology is, unlike physics, simply not amenable to mathematical description. This is certainly not a new thought; biologists have been saying this for generations, although I have never believed it. The whole point of making a theory is to tame what seems to be a wild, complex mess of unconnected data into a tidy manageable package. But what if tidy manageable packages are something alien to biology? What if there is a fundamental truth behind the enormous difficulty of reducing biology to math?

Of course, mathematics doesn’t describe the real physical world very well, either. What it does is describe models of the real world perfectly, and the best laws for the models are used for real-world calculations. If the model is very good, the laws work well in the real world also.

Perhaps this is the problem. Perhaps there aren’t any really good biological models. In evolution, for example, one can develop a model, but as soon as we have one that can be predicted by an equation, we run into major difficulties. How can we define fitness so that it can be mathematically described? We cannot say that we will determine fitness by the number of offspring produced, because that leads to a circular argument. Fitness is a characteristic that cannot be quantitated a priori . But if we cannot assign a value of fitness to a characteristic, how can we model the process of natural selection?

The fitness issue is only one example of the problems facing the theoretical biologist in the construction and use of models. Take the cell. Or better, any functional cellular component, like ribosomes or chloroplasts. We know a great deal of the detail of the molecular mechanisms of protein synthesis, and we can make impressive videos or elaborate cartoons, but how can such a process be mathematically modeled with any degree of accuracy?

Actually the problem is much worse. There are very few biological entities or concepts that can be defined mathematically with sufficient precision to allow for making models. Even the idea of a species turns out to be very fuzzy. One definition of a species is a group of animals that can produce viable offspring only by mating with each other. The problem is, there are too many exceptions to this rule to make it mathematically precise.  And what about the majority of living species that don’t actually mate (like all the bacteria, and other monists)?

Sharing the same exact DNA sequence doesn’t really work, because individuals within a species don’t share their exact complete sequences, and it is impossible to draw the line between who is in the species and who isn’t on this basis. It seems that the idea of species is useful, but not very exact, and very hard to pin down.

So, it might be necessary, in order to make any mathematically based theoretical progress in biology, to either: 1. Invent some new kind of mathematics or formalism that is conducive to describing biological reality (as was done a number of time for physics); or 2. Give up on a mathematical treatment, and use something completely different. And no, I do not know what that might be, but it might need to stretch what we call science and methodological naturalism a bit. We might even need philosophy to get somewhere!. I can write this scientific heresy here, because this is my blog, and I can write anything I want. If you are reading this, you can also, and I would love to read what you think about this idea.

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13 Responses to The Reasonable Ineffectiveness of Mathematics in the Biological Sciences

  1. Ethan Ortega says:

    Fascinating. I’m an avid lover of mathematics and am fascinated by its progression (or discovery) throughout human history as we sought to learn more of nature. It seems to often go hand in hand with physics and chemistry but I have rarely heard it used in biology. I like to think that God has immeasurably more mathematics for us to discover, yet regarding His living creation, I’m never one to assume that the mysteries will ever be fully solved in this life.

  2. You are right, Ethan. There is some math in biology, but not much. I agree with you that God needs to show us a lot more about how to use math to make sense of our biological discoveries. I also agree that we will never finish learning about the mysteries of our world. I find that to be a good thing.

  3. Robert Byers says:

    it might also be that evolutionary biology is not true. I think math is just a special case of the true measurement of the universe created by God. So is biology not organized like physics/ I don’t see why not. The same creator. It probably is just so much more complex then physics which is probably overrated for complexity.
    if biological processes were logical and based on principals then its a option math could explain them. So its a great clue , probably, that evolutionism is just not true. A hint.

  4. I dont agree Robert, that the difficulty of explaining evolution mathematically says anything about its truthfulness. As I said, this difficulty applies to all of biology, not just evolution. Biological principles are definitely not logical, and while they are based on principles, there are exceptions everywhere. I am sure there must be some systematic way to understand biology that might be non mathematical. I mentioned philosophy, but it could also be theology.

  5. SheilaDeeth says:

    Makes me think of wave-particles, Heisenberg uncertainty, and, on a less esoteric note, the mathematics of climate. The line between man’s attempt to understand and man’s attempt to control is always a fine one – often mathematical.

  6. Sy

    Your piece interfaces rather interestingly with what Joshua Swamidass has been saying (on various platforms) recently. He’s staunchly defending methodological naturalism (by insisting that science needs to be humble about how much it can do – what can’t be covered by MN isn’t science), but is also defending maths in biology (by focusing almost exclusively, it seems to me, on neutral theory and its maths). But to him the boundaries of science need to be clearly delineated (by the profession) so that niche construction, quantum theory and natural selection are “in”, but irreducible complexity, providence or astrology are “out”.

    But if you’re right – and I think you have to be – that biology is no more amenable to reductive maths than, say, psychology or history; and if one might need to stretch methodological naturalism out of shape to accommodate it, then you’re really saying that biology is only partly scientific.

    I’m not here talking about the spiritual questions like the soul or anything, but simply about shrews eating worms, tadpoles becoming frogs and so in.

    So is it philosophy? Surely not, because it’s based on empirical observation (as was astrology, of course!). Is it an art? That doesn’t seem a good description either. The only answer I can think of is, I think, rather the same as yours – that the secure boundaries that make “science” appear so different from, and more reliable than, other human activities will gradually disappear, so that science will be seen to be messily human (and perhaps return to seeking the divine!).

    • I totally agree with you Jon, but dont tell anyone! I think we see that withdrawal from “orthodox” (or simply pure MN) science most clearly in biology, but as Sheila intimates in her comment, it might also apply to some of the “newer” physics as well. We can even look at math itself, and be consumed with wonder that some simple equations lead to the beauty of fractal sets, with their infinite complexity.

      Another clue that we might be right is the strident defense of MN and the “scientific method” and the holy, lofty position of science above all other human endeavors by so many extremely passionate advocates, both atheists and theists. In my experience, it appears to me to be some sort of rule that the more passionate, even fanatical defenses are raised in support of an idea, the more likely it is about to fade away.

  7. Good point Sheila. I love the fact that physicists used new mathematical constructs (like operators) to deal with the new phenomena they were seeing. I wish I, (or any biologist) were able to do that.

  8. Charles says:

    Great article!

    It reminds me of the excitement I felt when I first figured out the “mathematical” reason for why codons had to be at least three bases long in order to code for the number of amino acids there are. 2^4 just didn’t cut it. As you said, it definitely doesn’t seem that something as diverse and complicated as biology will ever be boiled down to just a few equations.

  9. Thank you Charles. I hope all is well, I am taking a bit of a vacation for the summer, but I hope to see you back here in September.

  10. Pingback: Black Gunk | The Book of Works

  11. Reblogged this on Richard's Watch and commented:
    Professor Seymour Garte’s post makes a very interesting contribution to our thread on maths, science and scripture, especially bearing in mind the Book of Genesis.

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