Back when I was a rube just starting to learn algebraic topology, I started thinking about a unifying platform for mathematics research, a sort of dynamical Wikipedia for math that would converge, given good data, to some global “truth”. (My model: unification programs in theoretical and experimental physics.) The reason was simple—what I really wanted was a unifying platform for AI research, but AI was way, way too hard. I didn’t have a formal language, I didn’t have a type system or a consistent ontology between experiments, I didn’t have good correspondences or representation theorems between different branches of AI, and I certainly didn’t have category theory. Instinctively, I felt it would be easier to start with math. In my gut, I felt that any kind of “unifying” platform had to start with math.

Recently I met some people who have also been thinking about different variants of a “Wikipedia for math” and, more generally, about tools for mathematicians like visualizations, databases, and proof assistants. People are coming together; a context is emerging; it feels like the time is ripe for something good! So I thought I’d dust off my old notes and see if I can build some momentum around these ideas.

• In this post, examples, examples, examples. I will discuss type systems for wikis, Florian Rabe’s “module system for mathematical theories“, Carette and O’Connor’s work on theory presentation combinators, and the pro/con of a scalable “library” of mathematics. I’ll briefly introduce the idea of mathematical experiments.
• In part 2, I will talk about experiments in physics (especially in quantum mechanics), and consider a higher-order model of ensembles of experiments in this setting.
• In part 3, I will talk about mathematical experiments (everything we love about examples, done fancier!), their relationship with “data”, and what they can do for the working mathematician.
• In part 4, I’d like to understand what kind of theoretical foundation would be needed for an attack on mathematical pragmatics (a.k.a. “real mathematics” in the sense of Corfield) and check whether homotopy type theory could be a good candidate.

Continue reading →

Recently, I’ve been working on a project to apply persistent homology to neural spike train data, with the particular goal of seeing whether this technique can reveal “low frequency” relationships and co-firing patterns that standard dimensionality reduction methods like PCA have been throwing away. For example, some neurons fire at a very Hz, around ~75-100 Hz, while in fact most neurons fire at ~10-30 Hz in response to some stimulus. The loud, shouty neurons are drowning out the quiet, contemplative ones! What do the quiet neurons know? More to the point, how do I get it out of them? Continue reading →

Hello! I’m a mathematician / computer scientist doing research on the intersections between geometry, artificial intelligence, and governance. I am currently doing a PhD at Oxford. I also co-founded and lead research at Metagov. For more details, jump to my research page.

To contact me, send me an email at joshua dot z dot tan at gmail dot com (remember the “z” in the middle, otherwise you’ll get someone different!).

Say that we are given a single room in Borges’ library, and that we would like to say something about what those books are about—perhaps we can cluster them by subject, determine a common theme, or state that the selection is rigorously random. One way to start would be to scan every book in the room, represent each one as a long string (an average book has about 500,000 characters, though each book in Borges’ library has 410 pages x 40 lines x 80 characters per line = 1,312,000 characters), then perform some sort of data analysis.

In this case the number of books in each room (32 per bookshelf x 5 shelves per wall x 4 walls = 640 books) is far less than the length in characters of each book, i.e. the dimension of the book if we represent it in vector form. We would like to pare down this representation so that we can analyze and compare just the most relevant features of these books, i.e. those that suggest their subjects, their settings, the language they are written in, and so on.

Typically, we simplify our representations by “throwing out the unimportant dimensions”, and the most typical way to do this to keep just the two to three dimensions that have the largest singular value. On a well-structured data set, singular value decomposition, aka PCA, might leave only 2-3 dimensions which together account for over 90% of the variance… but clearly this approach will never work on books so long as we represent them as huge strings of characters. No single character can say anything about the subject or classification of an entire book.

Another way of simplifying data is to look at the data in a qualitative way: by studying its shape or, more precisely, its connectedness. This is the idea behind persistent homology.

To be continued…