Quick introduction: the algorithmic lens

Computers are a ubiquitous tool in modern research. We use them for everything from running simulation experiments and controlling physical experiments to analyzing and visualizing data. For almost any field ‘X’ there is probably a subfield of ‘computational X’ that uses and refines these computational tools to further research in X. This is very important work and I think it should be an integral part of all modern research.

But this is not the algorithmic lens.

In this post, I will try to give a very brief description (or maybe just a set of pointers) for the algorithmic lens. And of what we should imagine when we see an ‘algorithmic X’ subfield of some field X.

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Danger of motivatiogenesis in interdisciplinary work

Randall Munroe has a nice old xkcd on citogenesis: the way factoids get created from bad checking of sources. You can see the comic at right. But let me summarize the process without direct reference to Wikipedia:

1. Somebody makes up a factoid and writes it somewhere without citation.
2. Another person then uses the factoid in passing in a more authoritative work, maybe sighting the point in 1 or not.
3. Further work inherits the citation from 2, without verifying its source, further enhancing the legitimacy of the factoid.
4. The cycle repeats.

Soon, everybody knows this factoid and yet there is no ground truth to back it up. I’m sure we can all think of some popular examples. Social media certainly seems to make this sort of loop easier.

We see this occasionally in science, too. Back in 2012, Daniel Lemire provided a nice example of this with algorithms research. But usually with science factoids, it eventually gets debuked with new experiments or proofs. Mostly because it can be professionally rewarding to show that a commonly assumed factoid is actually false.

But there is a similar effect in science that seems to me even more common, and much harder to correct: motivatiogenesis.

Motivatiogenesis can be especially easy to fall into with interdisiplinary work. Especially if we don’t challenge ourselves to produce work that is an advance in both (and not just one) of the fields we’re bridging.

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Cataloging a year of metamodeling blogging

Last Saturday, with just minutes to spare in the first calendar week of 2019, I shared a linkdex the ten (primarily) non-philosophical posts of 2018. It was focused on mathematical oncology and fitness landscapes. Now, as the second week runs into its final hour, it is time to start into the more philosophical content.

Here are 18 posts from 2018 on metamodeling.

With a nice number like 18, I feel obliged to divide them into three categories of six articles each. These three categories: (1) abstraction and reductive vs. effective theorie; (2) metamodeling and philosophy of mathematical biology; and the (3) historical context for metamodeling.

You might expect the third category to be an after-though. But it actually includes some of the most read posts of 2018. So do skim the whole list, dear reader.

Next week, I’ll discuss my remaining ten posts of 2018. The posts focused on the interface of science and society.
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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.

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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’.
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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.

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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.

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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.

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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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Separating theory from nonsense via communication norms, not Truth

Earlier this week on twitter, Brian Skinner wrote an interesting thread on how to distinguish good theory from crackpottery. He started with a trait that both theorists and crackpots share: we have an “irrational self-confidence” — a belief that just by thinking we “can arrive at previously-unrealized truths about the world”. From this starting point, the two diverge in their use of evidence. A crackpot relies primarily on positive evidence: he thinks hard about a problem, arrives at a theory that feels right, and then publicizes the result.

A theorist, on the other prong, incorporates negative evidence: she ponders hard about a problem, arrives at a theory that feels right and then proceeds to try to disprove that theory. She reads the existing literature and looks at the competing theories, takes time to understand them and compare them against her own. If any disagree with hers then she figures out why those theories are wrong. She pushes her theory to the extremes, looks at its limiting cases and checks them for agreement with existing knowledge. Only after her theory comes out unscathed from all these challenges does she publicize it.

For Skinner, this second prong is the definition of scholarship. In practice, coming up with a correct theory is mostly a painful process of discarding many of your own wrong attempts. A good theorist is a thorough, methodical and skeptical of their own ideas.

The terminology of crackpottery vs scholarship is probably overly harsh, as Skinner acknowledges. And in practice, somebody might be a good theorist in one domain but a crackpot elsewhere. As Malkym Lesdrae points out, there are many accomplished accademics who are also crackpot theorists: “Most often it’s about things outside their field of specialty”. Thus, this ideal self-skepticism might be domain specific.

It is also a destructive ideal.

In other words, I disagreed with Skinner on the best way to separate good theory from nonsense. Mostly on the framing. Skinner crystalized our disagreement in a tweet: whereas he views self-skepticism as I an obligation to the Truth, I view a similar sort of self-reflective behavior as a social obligation. I am committed to this latter view because I want to make sense of things like heuristic models, where truth is secondary to other modelling concerns. Where truth is not the most useful yardstick for checking the usefulness of model. Where you hear Box’s slogan: “all models are wrong, but some are useful.

Given the brief summary of Skinner’s view above — and please, Brian, correct me in the comments if I misrepresented your position — I want to use the rest of this post to sketch what I mean by self-reflective behavior as a social obligation.
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