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.

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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:

MertonCollege
[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’?

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Change, progress, and philosophy in science

Bertrand_Russell“Philosophy of science is about as useful to scientists as ornithology is to birds” is a quote usually attributed to Feynman that embodies a sentiment that seems all too common among scientists. If I wish to be as free as a bird to go about my daily rituals of crafting science in the cage that I build for myself during my scientific apprenticeship then I agee that philosophy is of little use to me. Much like a politician can hold office without a knowledge of history, a scientist can practice his craft without philosophy. However, like an ignorance of history, an ignorance of philosophy tends to make one myopic. For theorists, especially, such a restricted view of intellectual tradition can be very stifling and make scientific work seem like a trade instead of an art. So, to keep my work a joy instead of chore, I tend to structure myself by reading philosophy and trying to understand where my scientific work fits in the history of thought. For this, Bertrand Russell is my author of choice.

I don’t read Russell because I agree with his philosophy, although much of what he says is agreeable. In fact, it is difficult to say what agreement with his philosophy would mean, since his thoughts on many topics changed through his long 98 year life. I read his work because it has a spirit of honest inquiry and not a search for proof of some preconceived conclusion (although, like all humans, he is not always exempt from the dogmatism flaw). I read his work because it is written with a beautiful and precise wit. Most importantly, I read his work because — unlike many philosophers — he wrote clearly enough that it is meaningful to disagree with him.
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Infographic history of evolutionary thought

Most of you are probably familiar with some variant of George Santayana’s aphorism: “Those who cannot remember the past are condemned to repeat it”. The quote is common to the point of cliche for a reason, in fact if we look at cliodynamics then we can even mathematically demonstrate the cyclic nature of history. This is especially true with the history of thought, and an even easier mistake to make when I am working in an interdisciplinary setting. To avoid interdisciplinitis as I delve deeper into models of evolution, I am always eager to learn more about the progress of evolutionary thought. As such, I was happy to see this new infographic from Tania Jenkins, Miriam Quick and Stefanie Posavec for the European Society for Evolutionary Biology:

EvolutionPoster
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Cliodynamics: A Future for History?

HistoriaWhat is history? And what, if any, are its practical uses? These are the questions I’ve been pondering since being introduced to Cliodynamics – which claims to make history into  “an analytical, predictive science.” To that end, I wish to address two questions: is it possible to make history into “an analytical, predictive science?” And is it desirable, for the purposes of attaining greater knowledge or understanding, to do this?
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