A Mathematician’s Apology

I need to update my reading list. Perhaps later this week I will get around to it. We went to a book sale at the Library early last week, or . . . was it a couple weeks ago? I have no idea but I did purchase several books in the genre of biography and historical fiction that interested me.

I just finished reading the first of the books that I purchased there. (A whole grocery bag full for 1 dollar, not bad!)

I just finished A Mathematician’s Apology by G.H. Hardy with a foreward by C.P. Snow. It was a short book that gives a bit of glimpse into the world of a ‘real’ mathematician, as Hardy would say. The foreward was by a fellow who was a good friend of Hardy, one of his few, all the way up until his death. The foreward presents a brief biography of a man who many would consider one of the top, say, five mathematical minds of his day, in company with Einstein, Littlewood, Ramanujan, . . . I knew of Hardy through a single theorem called the Hardy-Littlewood maximal Theorem, but that was the extent of it. I really have not read much by way of history of mathematics, at least not on a personal, biographical level, so this was a first to see some of the day to day interactions and even hobbies of a mathematician. I have fancied myself a mathematician but I am still so early in my career that I can’t say I really know what I am doing with my life.

Some of the interesting things that stood out to me about Hardy’s life:

  • He was shy or self-conscious in many respects, but very clearly knew what he thought and believed and would clearly state it when he felt like doing so.
  • He was a huge cricket fan. (Cricket, the sport, in case you were curious.) He was a British mathematician holding positions at both Cambridge and Oxford.
  • He was a somewhat outspoken atheist. Outspoken, in the sense that those who knew him at all well, would have known this fact.
  • He attempted suicide near the end of his life and failed. He died very much a sad and lonely man, depressed largely by his inability to continue doing what he felt he had been most suited for in life.
  • Most of his serious contributions to mathematics were made after he was forty, very late in life for mathematicians. Those contributions were almost entirely made through collaboration with other mathematicians, such as Littlewood or Ramanujan, both of which were ‘discovered’ by Hardy.

A Mathematician’s Apology, according to Wikipedia, is often considered the layman’s best insight into the mind of a working mathematician. However, it is a bit of a sad account by a mathematician who has come to realize that due to age and the deterioration of mind, he is unlikely to produce any more significant contributions to his field. In his words,

It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done.

Overall, this short insight into Hardy’s perspective on his chosen field is a well-written account of the aesthetics of mathematics. He breaks down, in terms any non-mathematician can understand why there is beauty in mathematics and from where the beauty is derived, such as the generality, depth, and sometimes, utility of mathematics. In his apology, he defines and illustrates with examples these aesthetics that, again, most non-mathematicians would not find beyond their ability to follow. It was all enlightening and I must say that I have had a fairly similar impression of mathematics as did Hardy.

One topic that I found particular interesting was his description of the contrast between pure and applied mathematics. He makes the case that the difference between the two is largely a misconception. He states,

It is quite natural to suppose that there is a great difference in utility between ‘pure’ and ‘applied’ mathematics. This is a delusion: there is a sharp distinction between the two kinds of mathematics, . . . but it hardly affects their utility.”

I bring up this topic out of the book because it is one of my own hobbies to jokingly deride those in the division that are in the ‘pure’ category as I myself am most definitely an applied mathematician. Hardy would definitely have categorized himself as a pure mathematician and his statements on the difference have had the affect of graying the line between the two categories, at least for me.

First, let me give the “off-the-cuff” definitions I have most often used in my classes. An applied mathematician is solving a real world problem, taking a physical phenomena or practical problem and developing a model through certain assumptions, then solves the model to determine an answer to the real-world problem. A pure mathematician does mathematics for the sake of mathematics, generalizing the relationships of the mathematical objects that we consider and deriving new relationships based on the general patterns in mathematical theory.

Hardy believed that mathematics is some external truth that we are discovering not something dependent on the human mind.

The contrast between pure and applied mathematics stands out most clearly, perhaps, in geometry. There is the science of pure geometry, in which there are many geometries, projective geometry, Euclidean geometry, non-Euclidean geometry, and so forth. . . Let us suppose that I am giving a lecture on some system of geometry, such as ordinary Euclidean geometry, and that I draw figures on the blackboard to stimulate the imagination of my audience, rough drawings of straight lines or circles or ellipses. It is plain . . . that the truth of the theorems which I prove is in no way affected by the quality of my drawings. . .

This is the point of view of the a pure mathematician [that pure geometries are independent of the lecture rooms or of any other detail of the physical world]. Applied mathematicians, mathematical physicists, naturally take a different view, since they are preoccupied with the physical world itself, which also has its structure or pattern.

I find it interesting that I have often lauded the fact of mathematical truth being largely independent of any of the mathematicians who discover them and yet, this claim sounds very much like a pure mathematician talking. In fact, the results of the applied mathematician, in general, can only approximate the physical and yet the pure side of mathematics is independent of the world around us.

The last thing I would say about the book is that it is an interesting read for anyone who has ever wondered what in the world a person must be thinking to become a mathematician (and who also has the desire to read about it).

NEXT BOOK: 1812 by David Nevin.

UPDATE: Based on another portion of the book I am planning on adding one more book to my reading list, Cours d’ Analyse by Jordan.

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One thought on “A Mathematician’s Apology

  1. Finally a book written for non-mathematicians to explain about mathematicians. This is defiantly on my to read list. Thank you!! And for future references please oh please let me know when the library is having a book sale. I would gladly drive the distance to get a bag of books for $1. What an exciting buy.

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