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.

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Overcoming folk-physics: the case of projectile motion for Aristotle, John Philoponus, Ibn-Sina & Galileo

A few years ago, I wrote about the importance of pairing tools and problems in science. Not selecting the best tool for the job, but adjusting both your problem and your method to form the best pair. There, I made the distinction between endogenous and exogenous questions. A question is endogenous to a field if it is motivated by the existing tools developed for the field or slight extensions of them. A question is exogenous if motivated by frameworks or concerns external to the field. Usually, such an external motivating framework is accepted uncritically with the most common culprits being the unarticulated ‘intuitive’ and ‘natural’ folk theories forced on us by our everyday experiences.

Sometimes a great amount of scientific or technological progress can be had from overcoming our reliance on a folk-theory. A classic examples of this would be the development of inertia and momentum in physics. In this post, I want to sketch a geneology of this transition to make the notion of endogenous vs exogenous questions a bit more precise.

How was the folk-physics of projectile motion abandoned?

In the process, I’ll get to touch briefly on two more recent threads on TheEGG: The elimination of the ontological division between artificial and natural motion (that was essential groundwork for Darwin’s later elimination of the division between artificial and natural processes) and the extraction and formalization of the tacit knowledge underlying a craft.
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Looking for species in cancer but finding strategies and players

Sometime before 6 August 2014, David Basanta and Tamir Epstein were discussing the increasing focus of mathematical oncology on tumour heterogeneity. An obstacle for this focus is a good definitions of heterogeneity. One path around this obstacle is to take definitions from other fields like ecology — maybe species diversity. But this path is not straightforward: we usually — with some notable and interesting examples — view cancer cells as primarily asexual and the species concept is for sexual organisms. Hence, the specific question that concerned David and Tamir: is there a concept of species that applies to cancer?

I want to consider a couple of candidate answers to this question. None of these answers will be a satisfactory definition for species in cancer. But I think the exercise is useful for understanding evolutionary game theory. With the first attempt to define species, we’ll end up using the game assay to operationalize strategies. With the second attempt, we’ll use the struggle for existence to define players. Both will be sketches that I will need to completely more carefully if there is interest.

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John Maynard Smith on reductive vs effective thinking about evolution

“The logic of animal conflict” — a 1973 paper by Maynard Smith and Price — is usually taken as the starting for evolutionary game theory. And as far as I am an evolutionary game theorists, it influences my thinking. Most recently, this thinking has led me to the conclusion that there are two difference conceptions of evolutionary games possible: reductive vs. effective. However, I don’t think that this would have come as much of a surprise to Maynard Smith and Price. In fact, the two men embodied the two different ways of thinking that underlay my two interpretations.

I was recently reminded of this when Aakash Pandey shared a Web of Stories interview with John Maynard Smith. This is a 4 minute snippet of a long interview with Maynard Smith. In the snippet, he starts with a discussion of the Price equation (or Price’s theorem, if you want to have that debate) but quickly digresses to a discussion of the two kinds of mathematical theories that can be made in science. He identifies himself with the reductive view and Price with the effective. I recommend watching the whole video, although I’ll quote relavent passages below.

In this post, I’ll present Maynard Smith’s distinction on the two types of thinking in evolutionary models. But I will do this in my own terminology to stress the connections to my recent work on evolutionary games. However, I don’t think this distinction is limited to evolutionary game theory. As Maynard Smith suggests in the video, it extends to all of evolutionary biology and maybe scientific modelling more generally.

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Personal case study on the usefulness of philosophy to biology

At the start of this month, one of my favourite blogs — Dynamic Ecology — pointed me to a great interview with Michela Massimi. She has recently won the Royal Society’s Wilkins-Bernal-Medawar Medal for the philosophy of science, and to celebrate Philip Ball interviewed her for Quanta. I recommend reading the whole interview, but for this post, I will focus on just one aspect.

Ball asked Massimi how she defends philosophy of science against dismissive comments by scientists like Feynman or Hawking. In response, she made the very important point that for the philosophy of science to be useful, it doesn’t need to be useful to science:

Dismissive claims by famous physicists that philosophy is either a useless intellectual exercise, or not on a par with physics because of being incapable of progress, seem to start from the false assumption that philosophy has to be of use for scientists or is of no use at all.

But all that matters is that it be of some use. We would not assess the intellectual value of Roman history in terms of how useful it might be to the Romans themselves. The same for archaeology and anthropology. Why should philosophy of science be any different?

Instead, philosophy is useful for humankind more generally. This is certainly true.

But even for a scientist who is only worrying about getting that next grant, or publishing that next flashy paper. For a scientist who is completely detached from the interests of humanity. Even for this scientist, I don’t think we have to concede the point on the usefulness of philosophy of science. Because philosophy, and philosophy of science in particular, doesn’t need to be useful to science. But it often is.

Here I want to give a personal example that I first shared in the comments on Dynamic Ecology.
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Token vs type fitness and abstraction in evolutionary biology

There are only twenty-six letters in the English alphabet, and yet there are more than twenty-six letters in this sentence. How do we make sense of this?

Ever since I first started collaborating with David Basanta and Jacob Scott back in 2012/13, a certain tension about evolutionary games has been gnawing at me. A feeling that a couple of different concepts are being swept up under the rug of a single name.[1] This feeling became stronger during my time at Moffitt, especially as I pushed for operationalizing evolutionary games. The measured games that I was imagining were simply not the same sort of thing as the games implemented in agent-based models. Finally this past November, as we were actually measuring the games that cancer plays, a way to make the tension clear finally crystallized for me: the difference between reductive and effective games could be linked to two different conceptions of fitness.

This showed a new door for me: philosophers of biology have already done extensive conceptual analysis of different versions of fitness. Unfortunately, due to various time pressures, I could only peak through the keyhole before rushing out my first draft on the two conceptions of evolutionary games. In particular, I didn’t connect directly to the philosophy literature and just named the underlying views of fitness after the names I’ve been giving to the games: reductive fitness and effective fitness.

Now, after a third of a year busy teaching and revising other work, I finally had a chance to open that door and read some of the philosophy literature. This has provided me with a better vocabulary and clearer categorization of fitness concepts. Instead of defining reductive vs effective fitness, the distinction I was looking for is between token fitness and type fitness. And in this post, I want to discuss that distinction. I will synthesize some of the existing work in a way that is relevant to separating reductive vs. effective games. In the process, I will highlight some missing points in the current debates. I suspect this points have been overlooked because most of the philosophers of biology are focused more on macroscopic organisms instead of the microscopic systems that motivated me.[2]

Say what you will of birds and ornithology, but I am finding reading philosophy of biology to be extremely useful for doing ‘actual’ biology. I hope that you will, too.

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Cataloging a sparse year of blogging: IMO workshop and preprints

Happy 2018!

With 2017 finally behind us, TheEGG enters its 8th calendar year. This past year has been a slow one for the blog, with only 10 new articles and two posts cataloguing 2016 (on cancer and on more theoretical aspects of evolution and general modelling). Half the months were barren: I posted nothing in March, April, May, July, August, September; and only October and November saw more than one post. But those two months of activity were good. We saw the list of TheEGG authors joined by David Robert Grimes, Vincent Cannataro, and Matthew Wicker; plus the return of Robert Vander Velde.

If you’re keeping score at home, this means that I only wrote six new articles last year.

As in the past, I want to start the new year by summarizing the old.

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Ontology of player & evolutionary game in reductive vs effective theory

In my views of game theory, I largely follow Ariel Rubinstein: game theory is a set of fables. A collection of heuristic models that helps us structure how we make sense of and communicate about the world. Evolutionary game theory was born of classic game theory theory through a series of analogies. These analogies are either generalizations or restrictions of the theory depending on if you’re thinking about the stories or the mathematics. Given this heuristic genealogy of the field — and my enjoyment of heuristic models — I usually do not worry too much about what exactly certain ontic terms like strategy, player, or game really mean or refer to. I am usually happy to leave these terms ambiguous so that they can motivate different readers to have different interpretations and subsequently push for different models of different experiments. I think it is essential for heuristic theories to foster this diverse creativity. Anything goes.

However, not everyone agrees with Ariel Rubinstein and me; some people think that EGT isn’t “just” heuristics. In fact, more recently, I have also shifted some of my uses of EGT from heuristics to abductions. When this happens, it is no longer acceptable for researchers to be willy-nilly with fundamental objects of the theory: strategies, players, and games.

The biggest culprit is the player. In particular, a lot of confusion stems from saying that “cells are players”. In this post, I’d like to explore two of the possible positions on what constitutes players and evolutionary games.

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Cataloging a year of cancer blogging: double goods, measuring games & resistance

Happy year of the Rooster and 2017,

This month marks the start of the 7th calendar year of updates on TheEGG. Time to celebrate and summarize the posts of the year past. In 2016 there was the same number of posts as 2015, but instead of being clustered in a period of <7 months, they were more uniformly distributed across the calendar. Every month had at least one new post, although not necessarily written by me (in the case of the single post by Abel Molina in October). There were 29 entries, one linkdex cataloging 2015, and two updates on EGT reading group 51 – 55 & 56 – 60.

In September, as part of my relocation from Tampa to Oxford, I attended the 4th Heidelberg Laureate Forum. I wrote two pieces for their blog: Alan Turing and science through the algorithmic lens and a spotlight on Jan Poleszczuk: from HLF2013 to mathematical oncology. You can read those (and more posts coming this year) on their blog. I won’t go into more detail here.

As before, this post is meant to serve as an organizing reference and a way to uncover common themes on TheEGG. A list of TL;DRs from 2016. The year was split up into four major categories: cancer, complexity & evolution, other models, and philosophy. The cancer posts make up almost half the articles from last year, and are further subdivided into three subsections: double goods game, experimental game theory, and therapy resistance. I want to focus on these cancer posts for this linkdex, and the other three categories in the next installment.

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Multiple realizability of replicator dynamics

Abstraction is my favorite part of mathematics. I find a certain beauty in seeing structures without their implementations, or structures that are preserved across various implementations. And although it seems possible to reason through analogy without (explicit) abstraction, I would not enjoy being restricted in such a way. In biology and medicine, however, I often find that one can get caught up in the concrete and particular. This makes it harder to remember that certain macro-dynamical properties can be abstracted and made independent of particular micro-dynamical implementations. In this post, I want to focus on a particular pet-peeve of mine: accounts of the replicator equation.

I will start with a brief philosophical detour through multiple realizability, and discuss the popular analogy of temperature. Then I will move on to the phenomenological definition of the replicator equation, and a few realizations. A particular target will be the statement I’ve been hearing too often recently: replicator dynamics are only true for a very large but fixed-size well-mixed population.

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