Darwin as an early algorithmic biologist

In his autobiography, Darwin remarked on mathematics as an extra sense that helped mathematicians see truths that were inaccessible to him. He wrote:

Darwin Turing HeadbandDuring the three years which I spent at Cambridge… I attempted mathematics… but got on very slowly. The work was repugnant to me, chiefly from my not being able to see any meaning in the early steps in algebra. This impatience was very foolish, and in after years I have deeply regretted that I did not proceed far enough at least to understand something of the great leading principles of mathematics, for [people] thus endowed seem to have an extra sense. But I do not believe that I should ever have succeeded beyond a very low grade. … in my last year I worked with some earnestness for my final degree of B.A., and brushed up … a little Algebra and Euclid, which later gave me much pleasure, as it did at school.

Today, this remark has become a banner to rally mathematicians interested in biology. We use it to convince ourselves that by knowing mathematics, we have something to contribute to biology. In fact, the early mathematical biologist were able to personify the practical power of this extra sense in Gregor Mendel. From R.A. Fisher onward — including today — mathematicians have presented Mendel as one of their own. It is standard to attributed Mendel’s salvation of natural selection to his combinatorial insight into the laws of inheritance — to his alternative to Darwin’s non-mathematical blending inheritance.

But I don’t think we need wait for the rediscovery of Mendel to see fundamental mathematical insights shaping evolution. I think that Darwin did have mathematical vision, but just lacked the algorithmic lenses to focus it. In this post I want to give examples of how some of Darwin’s classic ideas can be read as anticipating important aspects of algorithmic biology. In particular, seeing the importance of asymptotic analysis and the role of algorithms in nature.

To convince you that Darwin embodied some of the leading principles of mathematics, I want to start with R.A. Fisher — an authority for both mathematicians and biologists. In the preface to one of the first works of mathematical biology, Fisher devotes some space to the differences between mathematicians and biologists. For Fisher, those repugnant steps of early algebra are not the foundation of mathematics, but rather a practical technique. The manipulation of mathematical symbols is “comparable to the manipulation of the microscope and its appurtenances of strains and fixatives”. This difference in tools of the trade is real, but superficial.

The substantive difference between mathematicians and biologists is in how their imagination was trained. The biologist’s imagination is trained on the complexity of the actual and the particular: “[biologists] are introduced early to the immense variety of living things; their first dissections, even if only of the frog or dog fish, open up vistas of amazing complexity and interest.” The mathematician’s imagination, instead, is trained on the elegance of the abstract: “The ordinary mathematical procedure in dealing with any actual problem is, after [idealizing] what are believed to be the essential elements of the problem, to consider it as one of a system of possibilities infinitely wider than the actual, the essential relations of which may be apprehended by generalized reasoning, and subsumed in general formulae, which may be applied at will to any particular case considered.” In other words, biologists see the actual particular complexity of the world, while mathematicians idealize and abstract the world.

Darwin could see both the particular and the abstract structure in the world. I think that this ability to not lose the particular while reasoning abstractly is one of the important factors that allowed a break from Aristotelian biology. I do not need to rehearse Darwin’s mastery of the particular — there is no doubt that he was a good naturalist. To highlight Darwin’s mastery of the abstract, I need to show how he build on prior thought.

First, let us turn to Malthus. In his Essay on the Principle of Population, Malthus famously wrote:

Population, when unchecked, increases in a geometrical ratio. Subsistence increases only in an arithmetical ratio. A slight acquaintance with numbers will shew the immensity of the first power in comparison of the second.

This is an asymptotic argument. Although Malthus goes on to estimate the exact geometric ratio involved, he also recognizes that that specific is not essential to his argument. Malthus recognizes that given any geometric factor greater than 1, and any arithmetic factor, eventually the geometric growth would surpass the arithmetic. In modern terminology, Malthus was recognizng and using the fact that 2^{\Theta(n)} \subset \omega(n) and then building a theory on top of this asymptotic separation.

Darwin saw the importance of this observation. He also recognized that the essential aspect of it was the asymptotic separation. It did not matter which particular resources implemented the limiting factor. More importantly, it dod not even matter what specific sub-exponential function they scale with — just that is was sub-exponential. He could abstract this principle as the basis for his struggle for existence. Thus, he could provide an abstract cause for natural selection.

Today, when we make appeals to evolution, we usually place natural selection as primary. For example, when we describe evolutionary medicine, we use terms like “using natural selection to achieve therapeutic goals”. In other words, we use a natural process to achieve our artifice our goals. Similar to how one might use the flow of the river to turn a mill.

But this was not the direction Darwin approached evolution. Instead, he started with domestication before moving on to variation in nature, laying out the struggle for existence, and then finally natural selection in chapter 4. It is only late in chapter 4 that he refers to domestication as `artificial’ selection. Structurally, his argument proceeds from looking at the selection algorithms used by humans and then abstracting it to focus only on the algorithm and not the agent carrying out the algorithm. He then recognizes that the breeder’s role as selector can be replaced by another actor: the struggle for existence. He sees the importance of the algorithm of selection and that it can be implemented in many ways. It is only after we’ve accepted Darwin’s explanation that we proceeded to reaify natural selection and redefine or explain artificial selection in reference to it.

In other words, Darwin’s approach was to project onto nature the human algorithm corresponding to the actions of breeders. Like a computer scientist or mathematician today, he was using his understanding of human procedures to look at nature. And although we now put natural selection conceptually prior to artificial selection, biologists still follow the path from practical to foundational.

Darwin did not express himself with algebra — in the way that Mendel did — and so did not use the manipulations of the mathematician. But this is superficial. More fundamentally, like a good mathematician — he abstracted. In so doing, he showed that he understood something of the great leading principles of mathematics. More specifically, like a computer scientist, Darwin saw how to project algorithms onto Nature. Now his projection provides the light needed to make sense of biology.

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About Artem Kaznatcheev
From the Department of Computer Science at Oxford University and Department of Translational Hematology & Oncology Research at Cleveland Clinic, I marvel at the world through algorithmic lenses. My mind is drawn to evolutionary dynamics, theoretical computer science, mathematical oncology, computational learning theory, and philosophy of science. Previously I was at the Department of Integrated Mathematical Oncology at Moffitt Cancer Center, and the School of Computer Science and Department of Psychology at McGill University. In a past life, I worried about quantum queries at the Institute for Quantum Computing and Department of Combinatorics & Optimization at University of Waterloo and as a visitor to the Centre for Quantum Technologies at National University of Singapore. Meander with me on Google+ and Twitter.

3 Responses to Darwin as an early algorithmic biologist

  1. Rob Noble says:

    You (and Fisher) frame this as mathematician versus biologist, but couldn’t it more generally be theoretician versus empiricist? Theoreticians can idealize and abstract the world without necessarily needing non-elementary mathematics. And it’s unclear to me why Darwin’s thought process should be considered an instance of algorithmic biology in particular, as opposed to just theoretical biology (I would consider algorithmic biology to be a subset of the intersection of theoretical biology and mathematical biology).

    • Rob Noble says:

      Possible definitions:
      * A mathematical biologist uses mathematical tools;
      * A computational biologist uses computational tools;
      * A theoretical biologist uses abstraction and theoretical analysis;
      * An algorithmic biologist uses methods from computer science.
      If you consider abstraction to be necessary for algorithmic biology then algorithmic biology is a subset of theoretical biology. In any case, just because someone uses abstractions doesn’t necessarily make them an algorithmic biologist (nor a mathematician); it makes them a theoretical biologist.

    • Thanks for the comment!

      I did not mean to propose a delimitation of what makes a biologist or what makes a mathematician. But on rereading the post, I can see how my references to Fisher might imply that I was aiming to do so. I got a lot of push-back about this on Reddit, too. They really didn’t like this post (at least compared to most of my other ones). So let me clarify.

      I turned to Fisher’s distinction not to define what mathematicians or biologists are, but to zoom in on two aspects that they differ on: manipulation and abstraction. I sided with Fisher in dismissing the first as superficial and the second as important. I then wanted to argue that Darwin had some very interesting abstractions. Since abstraction is one (among many) of the “great leading principles of mathematics”, I thus aimed to contradict Darwin’s humble quote.

      With that out of the way. I want to zoom in on what I think you’re saying: abstraction is a feature that all theorists have, and it should be attributed to theorists and not mathematicians. This is a point we can definitely debate on.

      Let’s start in Fisher’s time. From the preface of his book, I don’t think he meant (just) empirical biologists when he was writing. I also don’t think there was a huge empirical/theoretical biology division at the time, but I could be wrong. I think that Fisher was talking about theoretical biologists (who in that time would also be familiar with basic experimental methods from their schooling) and he did not see them as doing abstraction or idealization in the same way as mathematicians do.

      I haven’t read enough work from Fisher’s time to know if he was right then. But I think he is certainly right now. Most of the work I read by theoretical biologists (and I only really read work by theoretical biologist, not experimentalists since that goes way over my head) does not do abstraction in any way that feels like what I mean by mathematical abstraction (as I tried to describe in this post). Rather, it proceeds by analogy and usually by analogy between particular cases. This is an extremely useful process, and in some ways it resembles abstraction, but I don’t think it is the same.

      That being said, there are certainly some people who would label themselves as theoretical biologist who use abstraction. And some people who would label themselves as mathematicians who use analogy. But one activity is more central to one field and the other to the other.

      Now, you could argue that Darwin was — under my arbitrary terminology — drawing an analogy and not an abstraction. This might be the case. The reason I considered it an abstraction is because he seemed to engage with the multiple realizability or an algorithm. And the recognition of something as an algorithm (i.e. a substrate-independent process with feedback and regulation) at a time when people did not clearly recognize algorithms as such. To me the move between the struggle for existence in a population doing ‘natural selection’ to itself versus the choices of a farmer doing artificial selection to her crops/animals is a big leap as an analogy. But it is a much more reasonable step if one first abstracts the idea of an algorithm as something that can be implemented by the activity of various agents. And then focuses on the properties that agent must have to be a ‘correct’ implementation.

      Of course, one might be able to go through all the above mental gymnastics for any analogy.So maybe my distinction is idle and in that case the only saving feature of my post might be drawing attention to those mental gymnastics?

      I will have to reflect on this more and write a better follow up post.

      Finally, to your definitions of fields. If we are going to define fields, I prefer sociological definitions from which we try to extract some family resembles in conceptual approaches. I have limited exposure to mathematical biology and limited experience doing what I have decided to call ‘algorithmic’ biology. But for me, the biggest mental distinction I have from mathematical biologists is again by analogy. In this case to the difference between pure and applied math. Both use ‘mathematical tools’ but not in ways that are all that compatible. As such for me an algorithmic biologists would be someone who tries to be a theoretical computer scientists (branch of pure mathematicians; i.e. relationship between cstheory and cs is not like relationship between theoretical biology and biology) while thinking about biology.

      This is probably another thing I need to reflect on and write about explicitly. It won’t help anyone else, but it’ll let me sort out my own perceptions a bit.

      In practice, since ‘algorithmic biology’ is a made up word, I do very little of it. Most of my papers are probably mathematical biology or theoretical biology. As for labels applied to individual people, I think that is just social convention — since such labels are primarily for the benefit of others (although they do often restrict what we think of ourselves). And I think most would label Darwin as a naturalist, and not any of the other terms we discussed. Although I really enjoyed your Venn diagram.

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