New Books in Mathematics

For many cognitive scientists, psychologists, and philosophers of mind, the best current theory of cognition holds that thinking is in some sense computation “in some sense,” because that core idea can and has been elaborated in a number of different ways that are or at least seem to be incompatible in at least some respects. In Unifying the Mind: Cognitive Representations as Graphical Models (MIT Press, 2014), David Danks proposes a version of this basic theory that links the mind closely with the computational framework used in machine learning: the idea that thinking involves manipulation of symbols encoded as graphical models. Danks, who is Professor of Philosophy and Psychology at Carnegie Mellon University, argues that graphical models provide a unifying explanation of why we are able to move smoothly between different cognitive processes and why we are able to focus on features of situations that are relevant to our goals. While the book includes the mathematics behind graphical models, Danks explains his proposal in accessible yet precise terms for the non-mathematically trained reader. He discusses how graphical models work in causal reasoning, categorization, and other processes, how his view is related to more familiar cognitive frameworks, and some implications of his view for modularity and other traditional debates.

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If ever there were a course that needs a book like Raffi Grinberg’s The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs (Princeton University Press, 20170, analysis is unquestionably it, and I only wish that Raffi had gotten into the wayback machine and delivered me a copy when I was taking this course more than half a century ago. I got a C+, and almost certainly would have done a lot better if I’d had this book, and so will present and future students who struggle with this course.

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Matthew L. Jones’s wonderful new book traces a history of failed efforts to make calculating machines, from Blaise Pascal’s work in the 1640s through the efforts of Charles Babbage in the nineteenth century, incorporating an account of both the work and relationships of scholars and artisans, and their reflections on the nature of invention. Innovative in its approach and its form, Reckoning with Matter: Calculating Machines, Innovation, and Thinking about Thinking from Pascal to Babbage (University of Chicago Press, 2016) offers a thoughtful and beautifully-written history of technology that offers an important perspective on a division between two poles of writing the history of technology: “the collective, deterministic account of inventive activity and the individualistic, heroic, creative account (7).” In Jones’s hands, we are offered a third way of understanding cultural production in early modernity, one that did not bifurcate between imitation and originality, social and individual making, or design and production. Central to the story is the history of efforts to mechanize the process of carrying ones in addition, and this fascinating problem persists as a thread through many of the projects discussed in the book. On the pages of Reckoning with Matter, readers will not only enjoy a compelling account of machine calculation through the nineteenth century, but will also find the story of a frog that tears out the eyes of a fish, a man who designed machines for making breakfast, and discussions of the significance of credit and intellectual property, modern programming, sketching, imitation, and debates over the nature of thinking. Highly recommended!

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Brian Clegg’s Are Numbers Real? The Uncanny Relationship of Mathematics and the Physical World (St. Martin’s Press, 2016) is a compact, very readable, and highly entertaining history of the development and use of mathematics to answer the important practical questions involved in advancing civilization. The question “Are Numbers Real?” is a terrific way to attack the problem so compellingly stated by the physicist Eugene Wigner what accounts for the unreasonable effectiveness of mathematics in describing the natural and the man-made Universe? Some of Clegg’s journey from the basics of counting through the intricate mathematical structures currently being used to explore reality will be familiar to the experienced reader. But even such a reader will find new insights and pleasures in the book and for those just starting out on their intellectual journey, Are Numbers Real? Is a superb introduction to mathematics, science, and that branch of philosophy devoted to exploring the nature of reality.

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The book discussed here is Ian Stewart’s Calculating the Cosmos: How Mathematics Unveils the Universe (Basic Books, 2016). If you would like to read a book that in my opinion represents the nicest job of presenting astronomy and cosmology in one volume since Isaac Asimov wrote The Universe half a century ago, this is absolutely the one to get. In addition, for those of us who are lovers of math, this book does a far better job than Asimov of presenting the close relationship between mathematics, astronomy, and cosmology.

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From the title, you might guess that Alfred Posamentier and Stephen Krulik’s Effective Techniques to Motivate Mathematics Instruction (Routledge, 2016) is aimed at mathematics teachers which it is. However, the techniques and strategies discussed in the book can be effectively employed by a much larger group of people, and one who hasconsiderably more influence with students. Those people are parents, who play as large or larger a role in their children’s education than do teachers.

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Alfred S. Posamentier and Robert Geretschlager, The Circle: A Mathematical Exploration Beyond the Line (Prometheus Books, 2016) goes considerably beyond what its modest title would suggest. The circle has played a pivotal role–that’s “role” with an ‘e,’ but its ability to “roll” with an ‘l’–has helped produce our industrial civilization. Moreover, the circle appears in our art, our literature, and our culture as well. This delightful book will not only reacquaint readers with the pleasures of the geometry they once knew, but will show how the circle continues to enchant mathematicians today, who continue to discover new and surprising properties about this most fundamental of shapes.

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Jennifer Beineke and Jason Rosenhouse‘s new book The Mathematics of Various Entertaining Subjects: Research in Recreational Math (Princeton University Press, 2015) covers a multitude of topics and is in many ways as entertaining as the various subjects it describes. Even though the book can be skimmed simply to expose one to various aspects of recreational mathematics, I think it’s fair to say that some mathematical background is needed to fully appreciate it. But even if you’re only willing to skim the book, you’re going to find sections which will make you want to dive in more deeply.

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Adam Kucharski, who won the 2012 Wellcome Trust Science Writing Prize, has delivered another winner in an area rife with both winners and losers. The Perfect Bet: How Science and Math Are Taking the Luck Out of Gambling (Basic Books, 2016) is a brilliant, fascinating, and sometimes slightly terrifying look at how math and science are not just conquering gambling, the algorithms that math has devised and the computerized means of implementing them are paradoxically simultaneously removing risk and creating a lot more of it.

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Romance. Crime. Mathematics. These things do not go together. Or do they? James D. Stein thinks they do, and he admirably shows us how in his wonderful collection of stories L.A. Math: Romance, Crime, and Mathematics in the City of Angels (Princeton University Press, 2016). Jim’s a mathematician, but don’t let that put you off: he’s the author of several popular books and an excellent writer at that. In this interview we talk about writing “clean”, math-phobia, and what everyone should really know, math-wise. Listen in.

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