## Effective games from spatial structure

December 7, 2018 1 Comment

For the last week, I’ve been at the Institute Mittag-Leffler of the Royal Swedish Academy of Sciences for their program on mathematical biology. The institute is a series of apartments and a grand mathematical library located in the suburbs of Stockholm. And the program is a mostly unstructured atmosphere — with only about 4 hours of seminars over the whole week — aimed to bring like-minded researchers together. It has been a great opportunity to reconnect with old colleagues and meet some new ones.

During my time here, I’ve been thinking a lot about effective games and the effects of spatial structure. Discussions with Philip Gerlee were particularly helpful to reinvigorate my interest in this. As part of my reflection, I revisited the Ohtsuki-Nowak (2006) transform and wanted to use this post to share a cute observation about how space can create an effective game where there is no reductive game.

Suppose you were using our recent game assay to measure an effective game, and you got the above left graph for the fitness functions of your two types. On the x-axis, you have seeding proportion of type C and on the y-axis you have fitness. In cyan you have the measured fitness function for type C and in magenta, you have the fitness function for type D. The particular fitnesses scale of the y-axis is not super important, not even the x-intercept — I’ve chosen them purely for convenience. The only important aspect is that the cyan and magenta lines are parallel, with a positive slope, and the magenta above the cyan.

This is not a crazy result to get, compare it to the fitness functions for the Alectinib + CAF condition measured in Kaznatcheev et al. (2018) which is shown at right. There, cyan is parental and magenta is resistant. The two lines of best fit aren’t parallel, but they aren’t that far off.

How would you interpret this sort of graph? Is there a game-like interaction happening there?

Of course, this is a trick question that I give away by the title and set-up. The answer will depend on if you’re asking about effective or reductive games, and what you know about the population structure. And this is the cute observation that I want to highlight.

## Plato and the working mathematician on Truth and discourse

December 1, 2018 Leave a comment

Plato’s writing and philosophy are widely studied in colleges, and often turned to as founding texts of western philosophy. But if we went out looking for people that embraced the philosophy — if we went out looking for actual Platonist — then I think we would come up empty-handed. Or maybe not?

A tempting counter-example is the mathematician.

It certainly seems that to do mathematics, it helps to imagine the objects that you’re studying as inherently real but in a realm that is separate from your desk, chair and laptop. I am certainly susceptible to this thinking. Some mathematicians might even claim that they are mathematical platonists. But there is sometimes reasons to doubt the seriousness of this claim. As Reuben Hersh wrote in

Some Proposals for Reviving the Philosophy of Mathematics:What explains this discrepency? Is mathematical platonism — or a general vague idealism about mathematical objects — compatible with the actual philosophy attributed to Plato? This is the jist of a question that Conifold asked on the Philosophy StackExchange almost 4 years ago.

In this post, I want to revisit and share my answer. This well let us contrast mathematical platonism with a standard reading of Plato’s thought. After, I’ll take some helpful lessons from postmodernism and consider an alternative reading of Plato. Hopefully this PoMo Plato can suggest some fun thoughts on the old debate on discovery vs invention in mathematics, and better flesh out my Kantian position on the Church-Turing thesis.

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Filed under Commentary, Preliminary Tagged with algorithmic philosophy, philosophy of math, philosophy of mind, stackexchange