## 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.

## 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.

## 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’?

## A year in books: philosophy, psychology, and political economy

If you follow the Julian calendar — which I do when I need a two week extension on overdue work — then today is the first day of 2015.

Happy Old New Year!

This also means that this is my last day to be timely with a yet another year-in-review post; although I guess I could also celebrate the Lunar New Year on February 19th. Last year, I made a resolution to read one not-directly-work-related book a month, and only satisfied it in an amortized analysis; I am repeating the resolution this year. Since I only needed two posts to catalog the practical and philosophical articles on TheEGG, I will try something new with this one: a list and mini-review of the books I read last year to meet my resolution. I hope that based on this, you can suggest some books for me to read in 2015; or maybe my comments will help you choose your next book to read. I know that articles and blogs I’ve stumbled across have helped guide my selection. If you want to support TheEGG directly and help me select the books that I will read this year then consider donating something from TheEGG wishlist.

## Cataloging a year of blogging: the philosophical turn

Passion and motivation are strange and confusing facets of being. Many things about them feel paradoxical. For example, I really enjoy writing, categorizing, and — obviously, if you’ve read many of the introductory paragraphs on TheEGG — blabbing on far too long about myself. So you’d expect that I would have been extremely motivated to write up this index of posts from the last year. Yet I procrastinated — although in a mildly structured way — on it for most of last week, and beat myself up all weekend trying to force words into this textbox. A rather unpleasant experience, although it did let me catch up on some Batman cartoons from my childhood. Since you’re reading this now, I’ve succeeded and received my hit of satisfaction, but the high variance in my motivation to write baffles me.

More fundamentally, there is the paradox of agency. It feels like my motivations and passions are aspects of my character, deeply personal and defining. Yet, it is naive to assume that they are determined by my ego; if I take a step back, I can see how my friends, colleagues, and even complete strangers push and pull the passions and motivations that push and pull me. For example, I feel like TheEGG largely reflects my deep-seated personal interests, but my thoughts do not come from me alone, they are shaped by my social milieu — or more dangerously by Pavlov’s buzzer of my stats page, each view and comment and +1 conditioning my tastes. Is the heavy presence of philosophical content because I am interested in philosophy, or am I interested in philosophy because that is what people want to read? That is the tension that bothers me, but it is clear that my more philosophical posts are much more popular than the practical. If we measure in terms of views then in 2014 new cancer-related posts accounted for only 4.7% of the traffic (with 15 posts), the more abstract cstheory perspective on evolution accounted for 6.6% (with 5 posts), while the posts I discuss below accounted for 57.4% (the missing chunk of unity went to 2014 views of post from 2012 and 2013). Maybe this is part of the reason why there was 24 philosophical posts, compared to the 20 practical posts I highlighted in the first part of this catalog.

Of course, this example is a little artificial, since although readership statistics are fun distraction, they are not particularly relevant just easy to quantify. Seeing the influence of the ideas I read is much more difficult. Although I think these exercises in categorization can help uncover them. In this post, I review the more philosophical posts from last year, breaking them down less autobiographically and more thematically: interfaces and useful delusions; philosophy of the Church-Turing thesis; Limits of science and dangers of mathematics; and personal reflections on philosophy and science. Let me know if you can find some coherent set of influences.

## Models and metaphors we live by

George Lakoff and Mark Johnson’s Metaphors we live by is a classic, that has had a huge influence on parts of linguistics and cognitive science, and some influence — although less so, in my opinion — on philosophy. It is structured around the thought that “[m]etaphor is one of our most important tools for trying to comprehend partially what cannot be comprehended totally”.

The authors spend the first part of the book giving a very convincing argument that “even our deepest and most abiding concepts — time, events, causation, morality, and mind itself — are understood and reasoned about via multiple metaphors.” These conceptual metaphors structure our reality, and are fundamentally grounded in our sensory-motor experience. For them, metaphors are not just aspects of speech but windows into our mind and conceptual system:

Our ordinary conceptual system, in terms of which we both think and act, is fundamentally metaphorical in nature. … Our concepts structure what we perceive, how we get around the world, and how we relate to others. Our conceptual system thus plays a central role in defining our everyday realities. … Since communication is based on the same conceptual system that we use in thinking and actiong, language is an important source of evidence for what that system is like.

I found the book incredibly insightful, and in large agreement with many of my recent thoughts on the philosophies of mind and science. After taking a few flights to finish the book, I wanted to take a moment to provide a mini-review. The hope is to convincing you to make the time for reading this short volume.

## Transcendental idealism and Post’s variant of the Church-Turing thesis

One of the exciting things in reading philosophy, its history in particular, is experiencing the tension between different schools of thought. This excitement turns to beauty if a clear synthesis emerges to reconcile the conflicting ideas. In the middle to late 18th century, as the Age of Enlightenment was giving way to the Romantic era, the tension was between rationalism and empiricism and the synthesis came from Immanuel Kant. His thought went on to influence or directly shape much of modern philosophy, and if you browse the table of contents of philosophical journals today then you will regularly encounter hermeneutic titles like “Kant on <semi-obscure modern topic>”. In this regard, my post is in keeping with modern practice because it could have very well been titled “Kant on computability”.

As stressed before, I think that it is productive to look at important concepts from multiple philosophical perspectives. The exercise can provide us with an increased insight into both the school of thought that is our eyes, and the concept that we behold. In this case, the concept is the Church-Turing thesis that states that anything that is computable is computable by a Turing machine. The perspective will be of (a kind of) cognitivism — thought consists of algorithmic manipulation of mental states. This perspective that can often be read directly into Turing, although Copeland & Shagrir (2013) better described him as a pragmatic noncognitivist. Hence, I prefer to attribute this view to Emil Post. Also, it would be simply too much of a mouthful to call it the Post-Turing variant of the Church-Turing thesis.

## Kleene’s variant of the Church-Turing thesis

In 1936, Alonzo Church, Alan Turing, and Emil Post each published independent papers on the Entscheidungsproblem and introducing the lambda calculus, Turing machines, and Post-Turing machines as mathematical models of computation. A myriad of other models followed, many of them taking seemingly unrelated approaches to the computable: algebraic, combinatorial, linguistic, logical, mechanistic, etc. Of course, all of these models were shown to be equivalent in what they could compute and this great heuristic coherence lead mathematicians to formulate the Church-Turing thesis. As with many important philosophical notions, over the last three-quarters of a century, the thesis has gradually changed. In a semi-historic style, I will identify three progressively more empirical formulations with Kleene, Post, and Gandy. For this article, I will focus on the purely mathematical formulation by Kleene, and reserve the psychological and physical variants for next time.

Mathematicians and logicians begat the Church-Turing thesis, so at its inception it was a hypothesis about the Platonic world of mathematical ideas and not about the natural world. There are those that follow Russell (and to some extent Hilbert) and identify mathematics with tautologies. This view is not typically held among mathematicians, who following in the footsteps of Godel know how important it is to distinguish between the true and the provable. Here I side with Lakatos in viewing logic and formal systems as tools to verify and convince others about our intuitions of the mathematical world. Due to Godel’s incompleteness theorems and decades of subsequent results, we know that no single formal system will be a perfect lens on the world of mathematics, but we do have prefered one like ZFC.
Mathematics has a deep and rich history, extending well beyond the 16th century start of the scientific revolution. Much like literature, mathematics has a timeless quality; although its trends wax and wane, no part of it becomes out-dated or wrong. What Diophantus of Alexandria wrote on solving algebraic equations in the 3rd century was still as true in the 16th, 17th, or today. In fact, it was in 1637 in the margins of Diophantus’ Arithmetica that Pierre de Fermat scribbled the statement of his Last Theorem. that the margin was too narrow to contain[1]. In modern notation it is probably one of the most famous Diophantine equations $a^n + b^n = c^n$ with the assertion that it has no solutions for $n > 2$ and $a,b,c$ as positive integers. A statement that almost anybody can understand, but one that is far from easy to prove or even approach[2].