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Calculus for cognitive scientists : derivatives, integrals and models
Peterson J., Springer International Publishing, New York, NY, 2016. 507 pp. Type: Book
Date Reviewed: Aug 8 2016

We hear sometimes that most people don’t need to learn calculus, that other branches of mathematics are better choices. Statistics is often suggested, along with discrete mathematics for computer science students. We even hear that no one really needs math at all. Calculus, though, is a good way to develop mathematical maturity and can be very interesting and useful. It is also foundational for much of mathematics, including statistics, and some of the methods are used in very similar ways in computer science. The book’s title implies that it will teach the bits of calculus useful for cognitive science.

The text has five parts (22 chapters):

(1) “Introductory Remarks” has a single chapter that includes several reviews of texts from biology and other fields. It also has a section on “How Should You Study” that includes verbatim text from another text on mathematical models.

(2) “Using One Variable” covers the basic calculus in one variable including limits (though without the usual epsilon/delta formulation), differentiation, integration, logarithms and exponentials, and the basic trigonometric functions. There are also several chapters devoted primarily to models, including two chapters on protein models, a heat transfer model, and a logistic model.

(3) “Using Multiple Variables” covers matrices and vectors briefly, as well as a cancer model and one on evolutionary biology.

(4) “Summing It All Up”

(5) “Advise [sic] to the Reader”

This is clearly a set of class notes that have been assembled into a text, and with little editing. A number of typographical errors are evident, many in mathematical derivations. It is easy enough to skim over “advise” rather than “advice.” Errors in derivations, however, can be confusing and distracting, especially for readers lacking confidence in their mathematical abilities.

There are a number of plots, clearly produced by a number of plotting programs and varying greatly in both expressiveness and quality. In one section, which covers the tangent line, there are plots of derivatives (good), but no plots of tangent lines. In the section on exponentials, there are hand-drawn plots of various exponential and logarithmic curves. While most students should know how to draw a plot, the process of doing that is not discussed, nor do the hand-drawn plots add noticeably to the content.

In the section on the exponential, I don’t see any mention of the fact that ab = eb ln a. This is odd as that is the reason we call it the exponential function. To be fair, this is hinted at, but only just. The exponential function is also discussed in the section on Taylor approximations, but then only a few terms of the Taylor series are given.

Second derivatives are not mentioned until very briefly in the chapter on extremal problems, which has a quick definition of maximum and minimum values and then proceeds to examine a cooling model in which there are no maxima or minima.

In the section on matrices, the three components of a vector are labeled a, c, and e instead of x, y, and z. Why? Indeed the author finds it necessary to say that e is not the e in the exponential function.

The writing often feels a bit patronizing, with lots of exclamation points: “We will do this by example!”

This text is titled Calculus for cognitive scientists. I suspect from what is said in the book (for instance, that it is one in a series of “Calculus for Biologists”) and from the models discussed that it is actually aimed at biologists. Perhaps the “cognitive science” label was tacked on (and added to the text in a few places) to make it appeal to library acquisitions people in universities with cognitive science programs. It is also part of a series with two more books in what would seem to be the same style.

The book is not completely without merit and probably has worked well as class notes for the author because it is very much a product of the author’s personality and class experiences. It is unlikely to be as successful for other instructors. Equally, it is neither a good supplementary text for students nor a good text for self-learning.

Reviewer:  Jeffrey Putnam Review #: CR144673 (1610-0741)
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