## Antoni Gaudi and learning algorithms from Nature

Happy holidays.

A few days ago, I was exploring Barcelona. This means that I saw a lot of architecture by Antoni Gaudi. His works have a very distinct style; their fluid lines, bright colours, myriad materials, and interface of design and function make for very naturesque buildings. They are unique and stand in sharp contrast to the other — often Gothic revival and Catalan Modernisme — architecture around them. The contrast is conscious; when starting out, Gaudi learned the patterns of the neo-Gothic architecture then in vogue and later commented on it:

Gothic art is imperfect, only half resolved; it is a style created by the compasses, a formulaic industrial repetition. Its stability depends on constant propping up by the buttresses: it is a defective body held up on crutches. … The proof that Gothic works are of deficient plasticity is that they produce their greatest emotional effect when they are mutilated, covered in ivy and lit by the moon.

His buildings, however, do not need to be overgrown by ivy, for Gaudi already incorporates nature in their design. I felt this connection most viscerally when touring the attic of Casa Mila. The building was commissioned as an apartment for local bourgeois to live comfortably on the ground floor off the rents they collected from the upper floors. And although some of the building is still inhabited by businesses and private residence, large parts of it have been converted into a museum. The most famous part among tourists is probably the uneven organic roof with its intricate smoke stacks, ventilation shafts, and archways for framing other prominent parts of Barcelona.

This uneven roof is supported by an attic that houses an exhibit on Gaudi’s method. Here, I could see Gaudi’s inspiration. On display was a snake’s skeleton and around me were the uneven arches of the attic — the similarity was palpable (see below). The questions for me were: was Gaudi inspired by nature or did he learn from it? Is there even much of a difference between ‘inspired’ and ‘learned’? And can this inform thought on the correspondence between nature and algorithms more generally?

I spend a lot of time writing about how we can use algorithmic thinking to understand aspects of biology. It is much less common for me to write about how we can use biology or nature to understand and inspire algorithms. In fact, I feel surprisingly strong skepticism towards the whole field of natural algorithms, even when I do write about it. I suspect that this stems from my belief that we cannot learn algorithms from nature. A belief that was shaken, but not overturned, when I saw the snake’s skeleton in Gaudi’s attic. In this post, I will try to substantiate the statement that we cannot learn algorithms from nature. My hope is that someone, or maybe just the act of writing, will convince me otherwise. I’ll sketch my own position on algorithms & nature, and strip the opposing we-learn-algorithms-from-nature position of some of its authority by pulling on a historic thread that traces this belief from Plato through Galileo to now. I’ll close with a discussion of some practical consequences of this metaphysical disagreement and try to make sense of Gaudi’s work from my perspective.

## 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.

## A year in books: Neanderthals to the National Cancer Act to now

A tradition I started a couple of years ago is to read at least one non-fiction book per month and then to share my thoughts on the reading at the start of the following year. Last year, my dozen books were mostly on philosophy, psychology, and political economy. My brief comments on them ended up running a long 3.2 thousand words. This time the list had expanded to around 19 books. So I will divide the summaries into thematic sets. For the first theme, I will start with a subject that is new for my idle reading: cancer.

As a new researcher in mathematical oncology — and even though I am located in a cancer hospital — my experience with cancer has been mostly confined to the remote distance of replicator dynamics. So above all else these three books — Nelson’s (2013) Anarchy in the Organism, Mukherjee’s (2010) The Emperor of All Maladies, and Leaf’s (2014) The Truth in Small Doses — have provided me with insights into the personal experiences of the patient and doctor.

I hope that based on these reviews and the ones to follow, you can suggest more books for me to read in 2016. Better yet, maybe my comments will help you choose your next book. Much of what I read in 2015 came from suggestions made by my friends and readers, as well as articles, blogs, and reviews I’ve stumbled across.[1] In fact, each of these cancer books was picked for me by someone else.

If you’ve been to a restaurant with me then you know that I hate choosing between close-to-equivalent options. To avoid such discomfort, I outsourced the choosing of my February book to G+ and Nelson’s Anarchy in the Organism beat out Problems of the Self by a narrow margin to claim a spot on the reading list. As I was finishing up Nelson’s book — which I will review last in this post — David Basanta dropped off The Emperor of All Maladies on my desk. So I continued my reading on cancer. Finally, Leaf’s book came towards the end of the year based on a recommendation from Jacob Scott. It helped reinvigorate me after a summer away from the Moffitt Cancer Center.

## Cataloging a year of blogging

Happy Old New Year.

January 2016 is the the start of the 6th calendar year and the 41st month with updates to TheEGG. The reason for the large discrepancy between these two numbers is occasional months without activity. The past year was exceptional in this regard with the longest single silence on the blog between April 4th and October 26th. This means that the year saw only 29 new entries, 2 indexes cataloging 2014, a report on the EGT reading group, and an update on readership. This post is meant to organize the last year of activity for future reference, and to try to uncover common themes.

If you like lists and TL;DRs then this is for you.

## Abusing numbers and the importance of type checking

What would you say if I told you that I could count to infinity on my hands? Infinity is large, and I have a typical number of fingers. Surely, I must be joking. Well, let me guide you through my process. Since you can’t see me right now, you will have to imagine my hands. When I hold out the thumb on my left hand, that’s one, and when I hold up the thumb and the index finger, that’s two. Actually, we should be more rigorous, since you are imagining my fingers, it actually isn’t one and two, but i and 2i. This is why they call them imaginary numbers.

Let’s continue the process of extending my (imaginary) fingers from the leftmost digits towards the right. When I hold out my whole left hand and the pinky, ring, and middle fingers on my right hand, I have reached 8i.

But this doesn’t look like what I promised. For the final step, we need to remember the geometric interpretation of complex numbers. Multiplying by i is the same thing as rotating counter-clockwise by 90 degrees in the plane. So, let’s rotate our number by 90 degrees and arrive at $\infty$.

I just counted to infinity on my hands.

Of course, I can’t stop at a joke. I need to overanalyze it. There is something for scientists to learn from the error that makes this joke. The disregard for the type of objects and jumping between two different — and usually incompatible — ways of interpreting the same symbol is something that scientists, both modelers and experimentalists, have to worry about it.

If you want an actually funny joke of this type then I recommend the image of a ‘rigorous proof’ above that was tweeted by Moshe Vardi. My writen version was inspired by a variant on this theme mentioned on Reddit by jagr2808.

I will focus this post on the use of types from my experience with stoichiometry in physics. Units in physics allow us to perform sanity checks after long derivations, imagine idealized experiments, and can even suggest refinements of theory. These are all features that evolutionary game theory, and mathematical biology more broadly, could benefit from. And something to keep in mind as clinicians, biologists, and modelers join forces this week during the 5th annual IMO Workshop at the Moffitt Cancer Center.

## Pairing tools and problems: a lesson from the methods of mathematics and the Entscheidungsproblem

Three weeks ago it was my lot to present at the weekly integrated mathematical oncology department meeting. Given the informal setting, I decided to grab one gimmick and run with it. I titled my talk: ‘2’. It was an overview of two recent projects that I’ve been working on: double public goods for acid mediated tumour invasion, and edge
effects in game theoretic dynamics of solid tumours
. For the former, I considered two approximations: the limit as the number n of interaction partners is large and the limit as n = 1 — so there are two interacting parties. But the numerology didn’t stop there, my real goal was to highlight a duality between tools or techniques and the problems we apply them to or domains we use them in. As is popular at the IMO, the talk was live-tweeted with many unflattering photos and this great paraphrase (or was it a quote?) by David Basanta from my presentation’s opening:

Since I was rather sleep deprived from preparing my slides, I am not sure what I said exactly but I meant to say something like the following:

I don’t subscribe to the perspective that we should pick the best tool for the job. Instead, I try to pick the best tuple of job and tool given my personal tastes, competences, and intuitions. In doing so, I aim to push the tool slightly beyond its prior borders — usually with an incremental technical improvement — while also exploring a variant perspective — but hopefully still grounded in the local language — on some domain of interest. The job and tool march hand in hand.

In this post, I want to unpack this principle and follow it a little deeper into the philosophy of science. In the process, I will touch on the differences between endogenous and exogenous questions. I will draw some examples from my own work, by will rely primarily on methodological inspiration from pure math and the early days of theoretical computer science.

## Five motivations for theoretical computer science

There are some situations, perhaps lucky ones, where it is felt that an activity needs no external motivation or justification.  For the rest, it can be helpful to think of what the task at hand can be useful for. This of course doesn’t answer the larger question of what is worth doing, since it just distributes the burden somewhere else, but establishing these connections seems like a natural part of an answer to the larger question.

Along those lines, the following are five intellectual areas for whose study theoretical computer science concepts and their development can be useful – therefore, a curiosity about these areas can provide some motivation for learning about those cstheory concepts or developing them. They are arranged from the likely more obvious to most people to the less so: technology, mathematics, science, society, and philosophy. This post could also serve as an homage to delayed gratification (perhaps with some procrastination mixed in), having been finally written up more than three years after first discussing it with Artem.

## Operationalizing replicator dynamics and partitioning fitness functions

As you know, dear regular reader, I have a rather uneasy relationship with reductionism, especially when doing mathematical modeling in biology. In mathematical oncology, for example, it seems that there is a hope that through our models we can bring a more rigorous mechanistic understanding of cancer, but at the same time there is the joke that given almost any microscopic mechanism there is an experimental paper in the oncology literature supporting it and another to contradict it. With such a tenuous and shaky web of beliefs justifying (or just hinting towards) our nearly arbitrary microdynamical assumptions, it seems unreasonable to ground our models in reductionist stories. At such a time of ontological crisis, I have an instinct to turn — much like many physicists did during a similar crisis at the start of the 20th century in their discipline — to operationalism. Let us build a convincing mathematical theory of cancer in the petri dish with as few considerations of things we can’t reliably measure and then see where to go from there. To give another analogy to physics in the late 1800s, let us work towards a thermodynamics of cancer and worry about its many possible statistical mechanics later.

This is especially important in applications of evolutionary game theory where assumptions abound. These assumptions aren’t just about modeling details like the treatments of space and stochasticity or approximations to them but about if there is even a game taking place or what would constitute a game-like interaction. However, to work toward an operationalist theory of games, we need experiments that beg for EGT explanations. There is a recent history of these sort of experiments in viruses and microbes (Lenski & Velicer, 2001; Crespi, 2001; Velicer, 2003; West et al., 2007; Ribeck & Lenski, 2014), slime molds (Strassmann & Queller, 2011) and yeast (Gore et al., 2009; Sanchez & Gore, 2013), but the start of these experiments in oncology by Archetti et al. (2015) is current events[1]. In the weeks since that paper, I’ve had a very useful reading group and fruitful discussions with Robert Vander Velde and Julian Xue about the experimental aspects of this work. This Monday, I spent most of the afternoon discussing similar experiments with Robert Noble who is visiting Moffitt from Montpellier this week.

In this post, I want to unlock some of this discussion from the confines of private emails and coffee chats. In particular, I will share my theorist’s cartoon understanding of the experiments in Archetti et al. (2015) and how they can help us build an operationalist approach to EGT but how they are not (yet) sufficient to demonstrate the authors’ central claim that neuroendocrine pancreatic cancer dynamics involve a public good.

## Truthiness of irrelevant detail in explanations from neuroscience to mathematical models

Truthiness is the truth that comes from the gut, not books. Truthiness is preferring propositions that one wishes to be true over those known to be true. Truthiness is a wonderful commentary on the state of politics and media by a fictional character determined to be the best at feeling the news at us. Truthiness is a one word summary of emotivism.

Truthiness is a lot of things, but all of them feel far from the hard objective truths of science.

Right?

Maybe an ideal non-existent non-human Platonic capital-S Science, but at least science as practiced — if not all conceivable versions of it — is very much intertwined with politics and media. Both internal to the scientific community: how will I secure the next grant? who should I cite to please my reviewers? how will I sell this to get others reading? And external: how can we secure more funding for science? how can we better incorporate science into schools? how can we influence policy decisions? I do not want to suggest that this tangle is (all) bad, but just that it exists and is prevalent. Thus, critiques of politics and media are relevant to a scientific symposium in much the same way as they are relevant to a late-night comedy show.

I want to discuss an aspect of truthiness in science: making an explanation feel more scientific or more convincing through irrelevant detail. The two domains I will touch on is neuroscience and mathematical modeling. The first because in neuroscience I’ve been acquainted with the literature on irrelevant detail in explanations and because neuroscientific explanations have a profound effect on how we perceive mental health. The second because it is the sort of misrepresentation I fear of committing the most in my own work. I also think the second domain should matter more to the working scientist; while irrelevant neurological detail is mostly misleading to the neuroscience-naive general public, irrelevant mathematical detail can be misleading, I feel, to the mathematically-naive scientists — a non-negligible demographic.

## What makes a discipline ‘mathematical’?

While walking to work on Friday, I was catching up on one of my favorite podcasts: The History of Philosophy without any Gaps. To celebrate the podcast’s 200th episode, Peter Adamson was interviewing Jill Kraye and John Marenbon on medieval philosophy. The podcasts was largely concerned with where we should define the temporal boundaries of medieval philosophy, especially on the side that bleeds into the Renaissance. A non-trivial, although rather esoteric question — even compared to some of the obscure things I go into on this blog, and almost definitely offtopic for TheEGG — but it is not what motivated me to open today’s post with this anecdote. Instead, I was caught by Jill Kraye’s passing remark:

[T]he Merton school, which was a very technical mathematical school of natural philosophy in 14th century England; they applied mechanical ideas to medicine

I’ve never heard of the Merton school before — which a quick search revealed to be also known as the Oxford Calculators; named after Richard Swinehead‘s Book of Calculations — but it seems that they introduced much more sophisticated mathematical reasoning into the secundum imaginationem — philosophical thought experiments or intuition pumps — that were in vogue among their contemporaries. They even beat Galileo to fundamental insights that we usually attribute to him, like the mean speed theorem. Unfortunately, I wasn’t able to find sources on the connection to medicine, although Peter Adamson and Jill Kraye have pointed me to a couple of books.

Do you have pointers, dear reader?

But this serendipitous encounter, did prompt an interesting lunchtime discussion with Arturo Araujo, Jill Gallaher, and David Basanta. I asked them what they thought the earliest work in mathematical medicine was, but as my interlocutors offered suggestion, I kept moving the goalposts and the conversation quickly metamorphosed from history to philosophy. The question became: What makes a discipline ‘mathematical’?