I once had an undergraduate professor, charmingly bespectacled and pleasantly ill-suited for his chosen career of mathematics, who said that no matter how high one goes in math, he or she would still need from time to time to take in some broad mathematical reading. His view was that this is necessary to renew our sense of perspective, and to provide a reminder of the inherent worthiness of the discipline. We were talking, I think, about Simon Singh’s not especially good book Fermat’s Last Theorem, which had just come out. Having spent the last decade and more around mathematicians, I doubt that what he said is exactly true.
But his apothegm applied accurately to himself, and it applies to me as well, and possibly others, and so I thought I would share a couple of books that I’ve found particularly motivational, and which could also have pedagogical uses.
The first is the amazing Princeton Companion to Mathematics, edited by Timothy Gowers. This thousand page tome is a desert island book (for me at least), and should be buried on the moon in case of a terrestrial calamity. It summarizes, in terms that everyone with a BA in mathematics can understand, approximately the whole mathematical firmament as it currently exists. The quality of the articles, each written by preeminent scholar in the relevant field, makes Wikipedia look like dross by comparison. It includes a summary (but highly insightful) treatment of every major branch of mathematics. The authors generally have a gift for exposition, and provide a visceral sense of the real significance and excitement of their areas – a feeling that would otherwise be impossible to experience without serious commitment. There is a listing of individual theorems and conjectures of importance, and several hundred brief biographies. There are philosophical discussions about the nature of mathematics, and a fair amount of general mathematical history. To read the whole volume would be an incredible education. At about $35 for a used copy, the value is impressive as well. It’s true that the corresponding information on the internet is free, but the quality of that material is many times lower than that of the essays in this printed book (also available in ebook.)
A work which is similarly broad but challenging by non-mathematical standards is a 2 volume series called Musimathics by Gareth Loy. Like the above book, this is something I have only nosed through – reading it will take a casual reader approximately a year, I would think. It begins with what you might expect – a mathematical treatment of tones and scales, which, though well done, is the least interesting part of the book. What really makes this book special is the combination of historicity, mathematical exactness, and a complete lack of prerequisites. All of the mathematics needed to understand the concepts is cogently and briefly introduced and then immediately applied to a concrete problem in music. I have never seen anything more digestible and equivalently sophisticated except maybe Frank Shu’s The Physical Universe, to which much of my praise for this book also applies.
It is daunting to try to characterize what a devoted reader of Loy’s book would learn, because the answer seems to be everything. It puts me in mind of a maxim I absorbed from another undergraduate professor: If you really want to understand philosophy, try to deeply understand just one thinker. The implication is that you would then understand everything else through some kind of reflection of the universal in the particular (not coincidentally this was in a class on Hegel.) To understand one thing is to understand its connection to everything else, which is in fact to have total understanding.
Let me give a few samples from Loy’s work:
“Algorithm is the most highly qualified methodology. The word comes from algorism, which means to calculate with Arabic numerals (footnote leading to a brief biography of al-Khorezmi). According to Donald Knuth (1973) algorithm is a broader concept, covering any set of rules or sequence of operations for accomplishing a task or solving a problem so long as it demonstrates each of the following five characteristics: [discussion of the nature of an algorithm]”
“The crucial characteristic of useful random processes is that chance events must be independent of each other. By independent I mean that even knowing a very large set of outcomes does not help us guess any other outcomes. If the outcomes of a random process are absolutely independent, then…[discussion of elementary probability]”
And later still:
“How many seven-note scales are there in the 12 pitches of the dodecaphonic system? This is like taking N unordered objects R at a time. It seems reasonable to expect that there will be fewer scales of seven pitches than melodies of seven pitches because melodies can repeat a note, whereas scales cannot. We must divide the pitches into two groups…”
What? This is a book about music (of which there is plenty), but suddenly we are learning concepts from computer science, number theory, wave mechanics, and seemingly everything else under the sun. Here is a (non-comprehensive) summary of the mathematical concepts a student would learn by reading the first volume: Cardinality, basic set theory, combinatorics, wave mechanics, trigonometry, probability distributions, fractal geometry, Brownian motion, filtering a signal, Euclid’s algorithm, Markov chains, Graph theory, higher order Markov processes, formal grammars, artificial neural networks (including back propagation), genetic programming,…
And this barely scratches the surface in the first volume. Note well that while these topics are presented in an accessible way, all of them are used for something practical, and none of them are treated trivially.
This is not mentioning the various subjects in the humanities covered, such as the nature of learning, time, aesthetics, atonal music, elements of musical and mathematical history, connectionism, calculating beauty, and many more.
I think this quote nicely summarizes the philosophy the book presents:
“Comparing Euclid’s method with Guido’s method, we saw that they are distinguished only by the role of subjective choice—of nondeterminism—in art. Art is not science, but their methods are more alike than different.”
Loy’s book would make an excellent reference in a liberal arts themed mathematics course.