Effective games from spatial structure

For the last week, I’ve been at the Institute Mittag-Leffler of the Royal Swedish Academy of Sciences for their program on mathematical biology. The institute is a series of apartments and a grand mathematical library located in the suburbs of Stockholm. And the program is a mostly unstructured atmosphere — with only about 4 hours of seminars over the whole week — aimed to bring like-minded researchers together. It has been a great opportunity to reconnect with old colleagues and meet some new ones.

During my time here, I’ve been thinking a lot about effective games and the effects of spatial structure. Discussions with Philip Gerlee were particularly helpful to reinvigorate my interest in this. As part of my reflection, I revisited the Ohtsuki-Nowak (2006) transform and wanted to use this post to share a cute observation about how space can create an effective game where there is no reductive game.

Suppose you were using our recent game assay to measure an effective game, and you got the above left graph for the fitness functions of your two types. On the x-axis, you have seeding proportion of type C and on the y-axis you have fitness. In cyan you have the measured fitness function for type C and in magenta, you have the fitness function for type D. The particular fitnesses scale of the y-axis is not super important, not even the x-intercept — I’ve chosen them purely for convenience. The only important aspect is that the cyan and magenta lines are parallel, with a positive slope, and the magenta above the cyan.

This is not a crazy result to get, compare it to the fitness functions for the Alectinib + CAF condition measured in Kaznatcheev et al. (2018) which is shown at right. There, cyan is parental and magenta is resistant. The two lines of best fit aren’t parallel, but they aren’t that far off.

How would you interpret this sort of graph? Is there a game-like interaction happening there?

Of course, this is a trick question that I give away by the title and set-up. The answer will depend on if you’re asking about effective or reductive games, and what you know about the population structure. And this is the cute observation that I want to highlight.

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Plato and the working mathematician on Truth and discourse

Plato’s writing and philosophy are widely studied in colleges, and often turned to as founding texts of western philosophy. But if we went out looking for people that embraced the philosophy — if we went out looking for actual Platonist — then I think we would come up empty-handed. Or maybe not?

A tempting counter-example is the mathematician.

It certainly seems that to do mathematics, it helps to imagine the objects that you’re studying as inherently real but in a realm that is separate from your desk, chair and laptop. I am certainly susceptible to this thinking. Some mathematicians might even claim that they are mathematical platonists. But there is sometimes reasons to doubt the seriousness of this claim. As Reuben Hersh wrote in Some Proposals for Reviving the Philosophy of Mathematics:

the typical “working mathematician” is a Platonist on weekdays and a formalist on Sundays. That is, when he is doing mathematics, he is convinced that he is dealing with an objective reality whose properties he is attempting to determine. But then, when challenged to give a philosophical account of this reality, he finds it easiest to pretend that he does not believe in it after all.

What explains this discrepency? Is mathematical platonism — or a general vague idealism about mathematical objects — compatible with the actual philosophy attributed to Plato? This is the jist of a question that Conifold asked on the Philosophy StackExchange almost 4 years ago.

In this post, I want to revisit and share my answer. This well let us contrast mathematical platonism with a standard reading of Plato’s thought. After, I’ll take some helpful lessons from postmodernism and consider an alternative reading of Plato. Hopefully this PoMo Plato can suggest some fun thoughts on the old debate on discovery vs invention in mathematics, and better flesh out my Kantian position on the Church-Turing thesis.

Blogging, open science and the public intellectual

For the last half-year I’ve been keeping TheEGG to a strict weekly schedule. I’ve been making sure that at least one post comes out during every calendar week. At times this has been taxing. And of course this causes both reflection on why I blog and an urge to dip into old unfinished posts. This week I deliver both. Below is a linkdex of 7 posts from 2016 and earlier (with a few recent comments added here and there) commenting on how scientists and public intellectuals (whatever that phrase might mean) should approach blogging.

If you, dear reader, are a fellow science blogger then you might have seen these articles before. But I hope you might find it useful to revisit and reflect on some of them. I certainly found it insightful. And if you have any important updates to add to these links then these updates are certainly encouraged.

Models as maps and maps as interfaces

One of my favorite conceptual metaphors from David Basanta is Mathematical Models as Maps. From this perspective, we as scientists are exploring an unknown realm of our particular domain of study. And we want to share with others what we’ve learned, maybe so that they can follow us. So we build a model — we draw a map. At first, we might not know how to identify prominent landmarks, or orient ourselves in our fields. The initial maps are vague sketches that are not useful to anybody but ourselves. Eventually, though, we identify landmarks — key experiments and procedures — and create more useful maps that others can start to use. We publish good, re-usable models.

In this post, I want to discuss the Models as Map metaphors. In particular, I want to trace through how it can take us from a naive realist, to critical realist, to interface theory view of models.

Bourbaki vs the Russian method as a lens on heuristic models

There are many approaches to teaching higher maths, but two popular ones, that are often held in contrast to each other, are the Bourbaki and Russian methods. The Bourbaki method is named after a fictional mathematician — a nom-de-plume used by a group of mostly French mathematicians in the middle of the 20th century — Nicholas Bourbaki, who is responsible for an extremely abstract and axiomatic treatment of much of modern mathematics in his encyclopedic work Éléments de mathématique. As a pedagogical method, it is very formalist and consists of building up clear and most general possible definitions for the student. Discussions of specific, concrete, and intuitive mathematical objects is avoided, or reserved for homework exercises, Instead, a focus on very general axioms that can apply to many specific structures of interest is favored.

The Russian method, in contrast, stresses specific examples and applications. The instructor gives specific, concrete, and intuitive mathematical objects and structures — say the integers — as a pedagogical examples of the abstract concept at hand — maybe rings, in this case. The student is given other specific instances of these general abstract objects as assignments — maybe some matrices, if we are looking at rings — and through exposure to many specific examples is expected to extract the formal axiomatic structure with which Bourbaki would have started. For the Russian, this overarching formalism becomes largely an afterthought; an exercise left to the reader.

As with many comparisons in education, neither method is strictly “better”. Nor should the names be taken as representative of the people that advocate for or are exposed to each method. For example, I am Russian but I feel like I learnt the majority of my maths following the Bourbaki method and was very satisfied with it. In fact, I am not sure where the ‘Russian’ in the name comes from, although I suspect it is due to V.I. Arnol’d‘s — a famous Russian mathematician from the second half of the 20th century — polemical attack on Bourbaki. Although I do not endorse Arnol’d attack, I do share his fondness for Poincaré and importance of intuition in mathematics. As you can guess from the title, in this article I will be stressing the Russian method as important to the philosophy of science and metamodeling.

I won’t be talking about science education, but about science itself. As I’ve stressed before, I think it a fool’s errand to provide a definition or categorization of the scientific method; it is particularly self-defeating here. But for the following, I will take the perspective that the scientific community, especially the theoretical branches that I work in, is engaged in the act of educating itself about the structure of reality. Reading a paper is like a lesson, I get to learn from what others have discovered. Doing research is like a worksheet: I try my hand at some concrete problems and learn something. Writing a paper is formalizing what I learned into a lesson for others. And, of course, as we try to teach, we end up learning more, so the act of writing often transforms what we learned in our ‘worksheet’.

The Noble Eightfold Path to Mathematical Biology

Twitter is not a place for nuance. It is a place for short, pithy statements. But if you follow the right people, those short statements can be very insightful. In these rare case, a tweet can be like a kōan: a starting place for thought and meditation. Today I want to reflect on such a thoughtful tweet from Rob Noble outlining his template for doing good work in mathematical biology. This reflection is inspired by the discussions we have on my recent post on mathtimidation by analytic solution vs curse of computing by simulation.

So, with slight modification and expansion from Rob’s original — and in keeping with the opening theme — let me present The Noble Eightfold Path to Mathematical Bilogy:

1. Right Intention: Identify a problem or mysterious effect in biology;
2. Right View: Study the existing mathematical and mental models for this or similar problems;
3. Right Effort: Create model based on the biology;
4. Right Conduct: Check that the output of the model matches data;
5. Right Speech: Humbly write up;
6. Right Mindfulness: Analyse why model works;
7. Right Livelihood: Based on 6, create simplest, most general useful model;
8. Right Samadhi: Rewrite focussing on 6 & 7.

The hardest, most valuable work begins at step 6.

The only problem is that people often stop at step 5, and sometimes skip step 2 and even step 3.

This suggests that the model is more prescriptive than descriptive. And aspiration for good scholarship in mathematical biology.

In the rest of the post, I want to reflect on if it is the right aspiration. And also add some detail to the steps.

Minimal models for explaining unbounded increase in fitness

On a prior version of my paper on computational complexity as an ultimate constraint, Hemachander Subramanian made a good comment and question:

Nice analysis Artem! If we think of the fitness as a function of genes, interactions between two genes, and interactions between three genes and so on, your analysis using epistasis takes into account only the interactions (second order and more). The presence or absence of the genes themselves (first order) can change the landscape itself, though. Evolution might be able to play the game of standing still as the landscape around it changes until a species is “stabilized” by finding itself in a peak. The question is would traversing these time-dependent landscapes for optima is still uncomputable?

And although I responded to his comment in the bioRxiv Disqus thread, it seems that comments are version locked and so you cannot see Hema’s comment anymore on the newest version. As such, I wanted to share my response on the blog and expand a bit on it.

Mostly this will be an incomplete argument for why biologists should care about worst-case analysis. I’ll have to expand on it more in the future.

The wei wu wei of evolutionary oncology

The world was disordered, rains would come and the rivers would flood. No one knew when. When it rained, plants would grow, but no one knew which were fit to eat and which were poisonous. Sickness was rife. Life was precarious.

The philosopher-king Yu dredged the rivers, cleaned them so they would flow into the sea. Only then were the people of the Middle Kingdom able to grow the five grains to obtain food.

Generations later, Bai Gui — the prime minister of Wei — boasted to Mengzi: “my management of the water is superior to that of Yu.”

Mengzi responded: “You are wrong. Yu’s method was based on the way of the water. It is why Yu used the four seas as receptacles. But you are using the neighbouring states as receptacles. When water goes contrary to its course, we call if overflowing. Overflowing means flooding water, something that a humane man detests… As for Yu moving the waters, he moved them without interference.”

Although Yu made changes to the environment by digging channels, he did so after understanding how the water flowed and moved naturally. He did so with knowledge of the Way. Yu’s management of water was superior to Bai Gui’s because Yu’s approach was in accordance with the Way. This is what evolutionary oncology seeks to achieve with cancer treatment. By understanding how the dynamics of somatic evolution drive tumour growth, we hope to change the selective pressures in accordance with this knowledge to manage or cure the disease.

Mathtimidation by analytic solution vs curse of computing by simulation

Recently, I was chatting with Patrick Ellsworth about the merits of simulation vs analytic solutions in evolutionary game theory. As you might expect from my old posts on the curse of computing, and my enjoyment of classifying games into dynamic regimes, I started with my typical argument against simulations. However, as I searched for a positive argument for analytic solutions of games, I realized that I didn’t have a good one. Instead, I arrived at another negative argument — this time against analytic solutions of heuristic models.

Hopefully this curmudgeoning comes as no surprise by now.

But it did leave me in a rather confused state.

Given that TheEGG is meant as a place to share such confusions, I want to use this post to set the stage for the simulation vs analytic debate in EGT and then rehearse my arguments. I hope that, dear reader, you will then help resolve the confusion.

First, for context, I’ll share my own journey from simulations to analytic approaches. You can see a visual sketch of it above. Second, I’ll present an argument against simulations — at least as I framed that argument around the time I arrived at Moffitt. Third, I’ll present the new argument against analytic approaches. At the end — as is often the case — there will be no resolution.

Methods and morals for mathematical modeling

About a year ago, Vincent Cannataro emailed me asking about any resources that I might have on the philosophy and etiquette of mathematical modeling and inference. As regular readers of TheEGG know, this topic fascinates me. But as I was writing a reply to Vincent, I realized that I don’t have a single post that could serve as an entry point to my musings on the topic. Instead, I ended up sending him an annotated list of eleven links and a couple of book recommendations. As I scrambled for a post for this week, I realized that such an analytic linkdex should exist on TheEGG. So, in case others have interests similar to Vincent and me, I thought that it might be good to put together in one place some of the resources about metamodeling and related philosophy available on this blog.

This is not an exhaustive list, but it might still be relatively exhausting to read.

I’ve expanded slightly past the original 11 links (to 14) to highlight some more recent posts. The free association of the posts is structured slightly, with three sections: (1) classifying mathematical models, (2) pros and cons of computational models, and (3) ethics of models.