I much enjoyed this book: “Models of the mind” by Grace Lindsay, an account of the uses of theory and mathematics in brain science. The book addresses the general reader, equations are mostly relegated to an appendix, and there are copious citations of technical literature as well. Obviously, conveying mathematics without equations is a challenge, and Lindsay handles it very well, finding many effective metaphors where needed. This all reflects her deep mastery of the material.

Beyond the target audience of curious laypeople, **I also recommend this book to professors and students in mathematical neuroscience**. For one, the ten chapters 2-11 would make a perfectly reasonable topic list for a 10-week course. While the course lectures and problem sets will focus on technical content and practice, Lindsay’s accompanying chapter can give a historical account of how the ideas evolved to our present understanding. When I try to do this in class, I always feel pressed for time, and the results aren’t satisfying. This book would make a great companion to a technical lecture series.

Lindsay’s last chapter is dedicated to “grand unifying theories” of the brain. She covers three attempts at such a theory, and politely but unmistakably dismisses them. One is “demonstrably wrong” and the other two are not even wrong. To find out what theories these are you’ll have to read the book; calling them out by name here would already give them too much credit. Lindsay concludes that brains are so “dense and messy” that they don’t lend themselves to physics-style theories that boil everything down to simple principles. On the other hand, her opening chapter makes the case that mathematics is the only language that can ultimately deal with the complexity of the brain. There’s a tension here regarding the future of mathematics in neuroscience.

This contrast between the rigor of mathematics and the squishiness of brains contributes to what I think is a continuing reluctance of many brain scientists to engage with quantitative methods. Here I’m reminded of one of the few question marks I jotted in the margins of the book. Lindsay writes that early in his career “[David] Hubel was actually quite interested in mathematics and physics”. If so, he certainly changed his mind later on. I have heard Hubel say that he “never had to use an equation after high school”. And he liked to ridicule quantitative measurement in neuroscience as “measuring the thickness of blades of grass”. Torsten Wiesel was similarly dismissive of mathematical approaches. I recall an editorial board meeting of the journal Network, which the chief editor, Joe Atick, had organized in a conference room at the Rockefeller University. In mid meeting, Torsten Wiesel – who was president of Rockefeller at the time – popped in, said hello, asked what the meeting was about, then delivered a monologue on how computational neuroscience will never make a useful contribution, and left abruptly. Whatever one thinks of these opinions, Hubel and Wiesel do have a Nobel Prize, as do many other experimenters, and theoretical neuroscientists don’t.

One can argue that Hubel and Wiesel really did not need mathematics to report the remarkable phenomena they discovered. But connecting those phenomena with their causes and consequences does require math. The little napkin sketch of simple cell receptive fields made from LGN neurons is cute but not convincing; it needs to be translated into a model before one can test the idea. Similarly, one can make a hand-waving argument that line detector neurons are useful for downstream visual processing, but understanding the reason is an entirely different matter. Unfortunately our discipline today still values isolated qualitative reports of phenomena. Most of the celebrated articles in glossy journals remain as singular contributions. No-one builds on them, hardly anyone tries to replicate them (and the rare attempts often don’t go well). Meanwhile the accompanying editorials celebrate the “tour-de-force” achievement that delivers a “quantum leap in our understanding”^{1}.

We should resign ourselves to the recognition that no single research paper will pull the veil of ignorance from our eyes and reveal the workings of the brain. The only hope lies in integrating results that come from many researchers with diverse complementary approaches. And one can piece these results together only if they are reported with some quantitative precision^{2} and fit into an overall mathematical model. It is to enable this basic building of a scientific edifice that neuroscientists need to learn and use mathematics, not only for the pursuit of a grand unified theory.

## Footnotes:

- The next time you write this phrase, remember that a “quantum leap” is the smallest possible increase.
- The next time you write a review article, please include some numbers about the magnitude of reported effects. Thank you!