Review of Mathematics without Apologies


Michael Harris’ book is definitely one that mathematicians shouldn’t miss. It’s also a must-read for people who aren’t professional mathematicians but do have a deep appreciation for the subject and have more than a passing interest in understanding what mathematics is “about”. It will not, however, win any converts among people who think they “don’t like math” and would require serious persuasion to change their minds.

As the title should make clear, the book is intended as a counterpoint to G. H. Hardy’s A Mathematician’s Apology. Hardy poses the question “[W]hy is it really worthwhile to make a serious study of mathematics? What is the proper justification of a mathematician’s life?” Hardy’s answer is somewhat defensive and hedged. He rejects a justification based on practical usefulness of mathematics, which he doesn’t defend robustly. Instead, he concludes that the justification for the life of any “real” mathematician is to “have added something to knowledge, and have helped others to add more; and that these somethings have a value which differs in degree only, and not in kind, from that of the creations of the great mathematicians, or of any other artists, great or small, who have left some kind of memorial behind them.”

Harris, on the other hand, rejects a defensive stance and aims to justify mathematics and the work of mathematicians “without apology”. He stands with Hardy in likening mathematics to other fine arts. But he goes further to justify both mathematics and other art forms not in utilitarian or moralistic terms, but in aesthetic terms of the pleasure from creating and appreciating art of high quality. His argument is first set out in Chapter 3 (“Not Merely Good, True, and Beautiful”). And after excursions through diverse mathematical, philosophical, biographical, artistic, and literary topics, the argument concludes in Chapter 10 (“No apology”). Along the way, it’s noted that other top tier scientists besides mathematicians have also justified their work in terms of pleasure. There is, for instance, Richard Feynman, one of whose books is The Pleasure Of Finding Things Out. Although the things studied in physics and mathematics are rather different (though some like Max Tegmark (Our Mathematical Universe) disagree), there is a distinct pleasure associated with important discoveries in any scientific field. Creative art also involves discovery and “finding things out”. Both the creative artist/scientist and others who simply learn to appreciate their work experience the associated pleasure.

Michael Harris’ most noteworthy mathematical accomplishment is his work within a vast generalization of number theory (in more classical forms of which Hardy also made his mark many decades ago). This work is part of an extensive series of related conjectures called “The Langlands Program”, after Robert Langlands who conceived it and has contributed to its continual expansion. The LP is highly abstruse, technical mathematics, and Harris makes no serious attempt in this book to sketch its outlines. He touches only briefly on some of the more concrete manifestations of the program in connection with what are known as “elliptic curves”. A very concise way to describe the LP is in terms of elaborate “correspondences” between seemingly unrelated mathematical objects such as “Galois representations” and “automorphic forms”. There’s a more complete semi-technical discussion of these correspondences in Edward Frenkel’s Love and Math, and a technical introduction (accessible only to math graduate students and professionals) is An Introduction to the Langlands Program.

Some people have found Harris’ book frustrating because it seems to lack “structure”. In fact, however, it does have structure, but as is often true in advanced mathematics, the structure isn’t always readily apparent. As Harris would point out, that is precisely what appeals to mathematicians about their vocation and avocation: its pleasure lies in discovering hidden structures that aren’t obvious on the surface. The Langlands Program is an apt metaphor for mathematics itself. Earlier examples of the discovery of rich hidden structures in mathematics abound – from Galois theory to Hilbert spaces.

Many topics are discussed in the book, some only cursorily, some in more depth. Here are just a few of them; mathematical Platonism (Harris, like most mathematicians, is an adherent), category theory, Hindu and Buddhist philosophy, literature (such as Thomas Pynchon’s), mathematical tricks, the mind-body problem, economics and finance, mathematical “charisma”, life in Paris, mathematics and sex (in reference to Frenkel’s Love and Math – and much more. Some of this is autobiographical, but there’s almost always some connection, anecdotal or otherwise, with the book’s main themes. It’s a smörgåsbord.

The book definitely isn’t light reading. It’s also heavily footnoted – an average of about 60 per chapter. (Most are worth reading.) There’s also a running series of imagined dialogues – at a very elementary level – on “How to Explain Number Theory at at Dinner Party”. So don’t expect to finish it in a day or two. A better approach is to take it a chapter at a time, and allow that to sink in before proceeding.

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4 Responses to Review of Mathematics without Apologies

  1. Thank you for your generous comments about my book. I hope you don’t mind if I correct a mistaken impression: I certainly don’t consider myself an adherent of mathematical Platonism. Insofar as I have a position on the matter, it is summarized by the sentence on the top of p. 199: “Virtual reality offers an appealing metaphorical alternative to the futile opposition between platonism and nominalism, but it will not soon make the word ‘exist’ obsolete in mathematics.”

    • cgd02 says:

      Thanks for taking the time to respond.

      I may not have used the term “Platonism” correctly. I’m not a great admirer of abstract philosophy, including as it applies to mathematics. My simplistic view is that if one believes mathematical truths are “discovered”, that’s Platonism, while if one believes mathematical truths are “created”, that’s the main alternative… and I’m not even clear what it should be called. I hope that’s not hopelessly naive, but I’m not all that interested in the argument to begin with. As you point out, mathematics is a pleasurable pursuit, and that is justification enough.

      It does seem to me that the whole Langlands program, which is looking for very general correspondences between diverse mathematical structures, is amimed at “discovering” something, because the whole problem is trying to pin down rigorously exactly what it “is” that’s being searched for. Mathematicians keep trying to posit different structures in the hope that everything will then fall into place. That’s a “creative” phase, but in mathematics and other endeavors, most creative ideas don’t actually work, or don’t work well enough. Nothing much is accomplished unless the idea makes real progress on a problem, sort of how Edison kept trying things until he had an incandescent lamp that worked well enough. And that was the actual “discovery”. Given the technology of his time, Edison discovered a good solution to the problem; he didn’t create it. The solution was there before he thought of it and tried it. Better solutions have since been discovered, by a long process of trial and error and the development of quite different approaches.

      To paraphrase Bill Clinton, it all depends on what one means by “is” (or “exists”). Debating that issue has kept philosophers in business for thousands of years. Mathematics has a way of settling its important questions sooner or later. Philosophy doesn’t seem to. Physics has a similar attitude towards philosophy. Do quarks exist? Who cares? The QFT works, very well, up to a point anyhow (and despite lack of a rigorous mathematical foundation). Someday a theoretical physicist will try something new that will actually be an improvement.

  2. David says:

    Hello, thanks for the review. Do you recommend this book before or after “Love and Math” by Frenkel. (I’m asking regarding the concepts explained, mostly about the Langlands program).

    • cgd02 says:

      They are very different books. Frenkel’s book has much more about Langlands. The Harris book is very good, but has little about Langlands. I strongly recommend both.

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