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.

Classifying mathematical models

It is hard to pick where to start, so I’ll default to using popularity as a guide. The second most popular post on the blog (with 20,005 views at the time of writing; if you’re curious, dear reader, the most popular post is on defining empathy, sympathy, and compassion) is this one:

Three types of mathematical models (September 8, 2013)

where I outline my three classifications for the spirit of models: inscillications, heuristics, and abstractions. The first one tries to go ground up, the second to build and mould intuitions, and the third to make formal arguments over models (the post explains it better, of course). I’ve clearly developed a bias for the latter two over the years, but in that 2013 post, I try to be as balanced as I can.

Much more recently, I’ve tried to distinguish the latter two types of models even more clearly, by associating heuristics with idealization and contrasting that to abstraction:

Double-entry bookkeeping and Galileo: abstraction vs idealization (June 15, 2018)

I did this contrast in a historical setting and suggested that even Galileo confused the ideas of abstraction and idealization.

However, it is a bit of a shame that three-fold classification post is so popular — it took me less than 6 months to realize on TheEGG that the classification was incomplete. In fact, Ishanu Chattopadhyay pointed out right away that I was missing an important type of models — abductions:

From heuristics to abductions in mathematical oncology (March 12, 2014)

this category is meant to capture the types of models that are built in machine learning and statistics. But also, I think, operationalizations like our recent paper on the game assay in non-small cell lung cancer or my more general discussion of effective games.

In fact, David Basanta’s tweet summarizing Jacob Scott’s presentation on our non-small cell lung cancer work recently got me back into exploring the difference between abductions and abstractions:

Abstract is not the opposite of empirical: case of the game assay (June 2, 2018)

where I argue that certain empirical measurements work as abstractions. In particular, the physical process that implements the macroscopic observables serves as the abstraction: many microdynamic processes can implement the abstract macroscopic observable. And this implementation can hide unnecessary complexity and thus make the macroscopic theory more tractable and useful than its reductive implementtion.

Unsurprisingly, pointing out this tension between reductive and effective theories is not a new development by me. This was already evident to John Maynard Smith as he worked with George Price on the genuses of evolutionary game theory:

John Maynard Smith on reductive vs effective thinking about evolution (July 17, 2018)

For the most part, the field followed John Maynard Smith’s more reductive view of evolutionary game theory and especially embraced his love of simulations and agent-based models.

Pros and Cons of Computational Models

Another thing that will come as no surprise to you, dear reader, is that in the grand scheme of things, my interest in experiments and effective theories is relatively new. Like so many other evolutionary game theorists following Maynard Smith, I spent a lot of time working on computer simulations.

This has developed into a bit of a love-hate relationship with the approach.

In an early post, I tried to discuss varioues reasons for why one might care about computational modeling:

Three goals for computational models (December 19, 2013)

the 3 that I focus on are: (1) forecasting an external reality, (2) clarifying and formalizing theories (see also: Heuristic models as inspiration-for and falsifiers-of abstractions), and (3) communication and rhetoric.

By the time of writing that post, I was not a huge fan of computational modelling, especially simulations. In the following post I express a particularly negative sentiment (which has softened since) towards simulations through what I called the curse of computing:

Four color problem, odd Goldbach conjecture, and the curse of computing (May 14, 2013)

The central aspect of this curse is the ease of simulation approaches (at least compared to analytics), which often results in people thinking less about the details of their models. You can often see this curse happen with the add-the-kitchen-sink approach to modeling. This bloats models with more and more (often poorly grounded) features. The motivation there seems to be to try to turn a heuristic into an inscillication.

But instead of succeeding, the kitchen-sink approach often seems to leverage an irrelevant rhetorical trick of needless complexity:

Truthiness of irrelevant detail in explanations from neuroscience to mathematical models (January 20, 2015)

Sometimes this complexity can be very dangerous. For example, when mathematical finance hides lies in complexity.

Given all this negative sentiment, I’ve recently tried to offer some balance by considering how Hobbes might react to simulation studies:

Hobbes on knowledge & computer simulations of evolution (August 25, 2018)

where I bring the discussion full circle and present the different ways of knowing inherent in simulations vs experiments. I argue that the two are different but both useful.

And, of course, knowledge is not limited to the above two kinds: “knowledge of” and “knowledge via”. There is also the implicit “knowledge how” of craft. This lets me close with a positive point on computer modelling — and programming more generally:

Techne and Programming as Analytic Philosophy (September 15, 2018)

Whereas moral philosophy aims to turn the implicit knowledge in the craft of the good life into explicit knowledge. Similarly, the computer modeller and programmer have to formalize various implicit and poorly categorized kinds of knowing.

Ethics of Models

Since we’re on the topics of moral philosophy. Let me finish with a discussion of ethics in modelling. My discussion is far too brief given the overwhelming importance of the topic. In fact, even though I give less space to it here, I think ethics in modeling is far more important than the previous two topics.

If we use models for rhetoric — which it feels to me is what most models end up being used for, especially heuristics — then I believe that we have important moral constraints on how we discuss and present our results:

Models, modesty, and moral methodology (April 27, 2018)

Unethical use of models becomes especially bad when it is extracted to mass scales. Thankfully, this doesn’t seem to be much of a problem in fields like mathematical oncology, given how sceptical medics are of math. But I am sure we will have to worry about it… eventually.

For understanding these dangers, I recommend looking at things written by Cathy O’Neil. Here is an analytic index of some of her post:

Weapons of math destruction and the ethics of Big Data (September 5, 2014)

I would also highly recommend reading her recent book: Weapons of Math Destruction. Although the book might be more social than what most modellers are looking for. Which is a shame, since it is an important aspect. To delve a bit deeper on it, I would recommend Abeba Birhane’s resource list on automated systems and bias.

An easier book for modellers might be Feyerabend’s Against Method. If a book is too long then I’ve put together an analytic linkdex on some of Feyerabend’s thoughts:

Philosophy of Science and an analytic index for Feyerabend (September 26, 2014)

These can be especially useful for understanding Box’s famous motto that “all models are wrong”:

Are all models wrong? (November 6, 2013)

And I guess this a fitting — albeit abrupt — end for this post. Especially given my advocacy for falsehood as primary and truth as an end to discussion.

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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.

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