Whittle2002

Introduction

One of the most prominent mathematicians of the twentieth century was G.H. Hardy. Hardy is noted for his contributions in analysis and number theory, and for the advancement of pure mathematics in Britain. But these were not the only contributions Hardy made to mathematics. In contrast to most mathematicians, Hardy was also a gifted writer. In fact, Hardy stated that if he were to choose a second career, it would have been journalism. And, if you read biographical works on G.H. Hardy, you will find that authors not only praise his writing ability, but also his communication skills in general (Kanigel, 1991, pp. 145 - 157; Snow, 1967, pp. 9 - 13).¹

Hardy’s literary ability is showcased in his book, A Mathematician’s Apology, first published in 1940 by the Cambridge University Press. It was subsequently reprinted twenty times, most recently in 1994. This was a short book, divided into 29 essay-type chapters, each 1 - 3 pages in length. Hardy’s book was written for the general reader; it contained very little mathematics, none of which was difficult. The book may be described as an expansion of Hardy’s inaugural address at Oxford University in 1920 (Hardy, 1967).

The objective of this work is to examine the impact of A Mathematician’s Apology on the intellectual community. Towards this objective I will compare and contrast five published reviews of Hardy’s book. Following the introduction, the body of the paper will consist of three parts: (a) first, a brief summary of the main ideas presented in A Mathematician’s Apology; (b) secondly, a summarization of each of the five reviews; and (c) conclusions.

Methodology

One means for assessing the reaction of the academic community to a polemical issue is by a comprehensive, even-handed examination of informed opinion from relevant scholars. In the case of a controversial publication, such as G.H. Hardy’s A Mathematician’s Apology, it is convenient to collect and analyse published reviews in a manner that parallels an empirical study. While it is unfortunate that only five published reviews of Hardy’s book were found, the quality² of the reviews offers much towards the objective of this study. In particular, the reviews were authored by leading scholars from both the humanities and the sciences. This is a particularly fortuitous circumstance because such variety permits a balanced examination of the academic community’s response to A Mathematician’s Apology.

A Mathematician’s Apology (Hardy, 1967, pp. 59 - 153)

The purpose of this book, as stated by the author, is to provide a justification for the study of pure mathematics. Hardy felt that the general public, and even some scholars, perceive mathematics as only a tool for the advancement of other disciplines, with no inherit value of its own. Those who are genuine scholars of mathematics find aesthetic value in the discipline, and this, according to Hardy is the real reason for studying pure mathematics. To justify the study of pure mathematics, Hardy then attempted to justify the claim that mathematics is an art³. In turn, to elucidate this claim, Hardy demonstrated two theorems from Euclid’s Elements, pointing to specific examples of what he perceived to be aesthetic qualities.

In chapters 12 - 18, Hardy stated two theorems from Euclid’s Elements and talked, with some degree of specificity, about aesthetic properties he perceived in these theorems. This was a big challenge for two reasons: (1) Whereas mathematical beauty is more easily found beyond the most elementary level of mathematics, Hardy must limit himself to mathematics accessible to the general reader. Hence, Hardy must carefully navigate between the two extremes of selecting topics beyond the comprehension of the reader, and selecting topics that are too elementary to bear beauty. (2) Beauty (particularly mathematical beauty) does not lend itself to precise definition. Rather, that which is beautiful is a highly individual and personal matter, varying from one person to another.

To undertake this enormous challenge, Hardy looked to a classic resource, Euclid’s Elements, and selected two theorems. In chapter 12, Hardy demonstrated a proof that the set of prime numbers is infinite. Following this, in chapter 13, Hardy advanced to the slightly more detailed proof that the square root of 2 is irrational. For both proofs, Hardy employed fundamental definitions and the reductio ad absurdum technique.

While I agree with Hardy that these proofs represent examples of mathematical beauty, a fundamental question remains unsettled: what exactly is mathematical beauty? Hardy talked to some degree about this in the ensuing chapters, but it was not until chapter 18 that he hit the nail on the head:

In both theorems (and in the theorems, of course, I include the proofs) there is a very high degree of unexpectedness, combined with inevitability and economy. The arguments take so odd and surprising a form; the weapons used seem so childishly simple when compared with the far-reaching results; but there is no escape from the conclusions. There are no complications of detail–one line of attack is enough in each case; and this is true too of the proofs of many much more difficult theorems (p. 113).

But, if one is willing to accept Hardy’s claim that mathematics is an art, why did Hardy feel the art-like qualities of mathematics merit study? In response to this, Hardy rambled off on a philosophical bend about the duty of man. Man’s first duty, according to Hardy is to be ambitious. Hardy felt that ambition is the driving force behind the best work of the world, and the noblest ambition is to leave behind something permanent. Hardy claimed that an individual does not select a vocation because of its value to society, but rather because of intellectual curiosity. And, as you would expect, Hardy claimed that mathematics offers the greatest opportunity to satisfy this drive: In the words of Hardy:

If intellectual curiosity, professional pride, and ambition are the dominant incentives to research, then assuredly no one has a fairer chance of gratifying them than a mathematician. His subject is the most curious of all–there is none in which truth plays such odd pranks. It has the most elaborate and the most fascinating technique, and gives un-rivalled openings for the display of sheer professional skill. Finally, as history proves abundantly, mathematical achievement, whatever its intrinsic worth, is the most enduring of all (p. 80).

Hence, Hardy sees mathematics as offering the opportunity to not only display intellectual skill, but to display skill in a manner that will not be forgotten. Hardy acknowledged that some mathematics is of value in a utilitarian sense, such as in engineering. However, he referred to this as the “crude” utility of mathematics. And, Hardy made the bold and astonishing claim that the defining characteristic of real mathematics is uselessness.⁴

En route to presenting a case for the study of pure mathematics based on aesthetic criteria, Hardy criticized the “educational” value of applied disciplines. In chapter 20, Hardy made what sounded like a self-contradictory statement: “. . . the most ‘useful’ subjects are quite commonly just those which it is most useless for most of us to learn” (p. 117). In other words, Hardy recognized the practical value of such disciplines as physiology, economics and chemistry to the good of society. However, a knowledge of such disciplines is of value only to individuals who are professionals within the respective trades.

Another major point in A Mathematician’s Apology is Hardy’s belief that mathematics is harmless. Hardy claimed that the elements of pure mathematics exist in a mathematical reality, separate from the physical world, and even separate from the human mind. This separation, according to Hardy, afforded pure mathematics two advantages: First, pure mathematics could not be held accountable for war; and secondly, this separation allows mathematics to serve as a sanctuary from the world. In the words of Hardy:

When the world is mad, a mathematician may find in mathematics an incomparable anodyne. For mathematics is, of all the arts and sciences, the most austere and the most remote, and a mathematician should be of all men the one who can most easily take refuge where, as Bertrand Russell says, ‘one of the least of our nobler impulses can best escape from the dreary exile of the actual world’ (p. 143).

In the process of making this claim, Hardy again lashed out at applied mathematics and empirical science. Whereas pure mathematics is innocent of war, Hardy claimed that applied disciplines have contributed to war by way of their roles in the production of munitions.

Finally, I’d like to make a few comments about the use of the word “apology” in the title. In terms of the main content, the use of “apology” obviously means “justification.” However, the very beginning of chapter one suggests the other, more common meaning of apology. Hardy (1967, p. 61) began his book in a depressed tone:

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 already done. . . Exposition, criticism, appreciation, is work for second-rate minds (p. 61).

A Mathematician’s Apology is a product of Hardy’s latter years. Hardy recognized that, in advanced age and failing health, his mathematical productivity had dropped off. Although still competent enough as a researcher and lecturer, his creative genius of earlier years was gone (Snow, 1967, p. 50).

Having summarized the main points of “A Mathematician’s Apology,” I would now like to examine the question: “What impact did this book have on intellectual society?” To organize the reviews, I would like to divide them into two groups; those that were positive, and those that were negative. The longest and most informative review was in the negative group, so I will present that group first.

Frederick Soddy, 1941

The first source is a book review by Frederick Soddy, appearing in the January, 1941 issue of Nature. Frederick Soddy, who worked with Ernest Rutherford in the field of radio chemistry, won a Nobel prize in 1921 for the discovery of isotopes. Soddy’s review is by far the longest of the five reviews, consisting of 2700 words, including an illustration and postscript. Frederick Soddy and G.H. Hardy shared some things in common. Both were the same age, having been born in 1877. Both took positions at Oxford in 1919, and were colleagues there until Hardy’s return to Cambridge in 1931. But, as will become evident from Soddy’s review, they had some sharp contrasts⁵ (Kanigel, 1991, p. 145 - 157; Merricks, 1996, p. 84).

The title of Soddy’s review immediately sets the tone for his position on Hardy’s book. Qui S’accuse A’acquitte, translated from French, means “who accuses oneself acquits oneself.” I believe, by way of this title, Soddy is insinuating that Hardy is being deceptive; his objective is not to offer an apology in the conventional meaning of the word, but rather to offer a defence of mathematics.

Soddy opened his review with these two sentences: “This is a slight book. From such cloistral clowning the world sickens. (p. 3)” Throughout the review, Soddy’s mode of operation was characterized by mockery. He would present a section of text from A Mathematician’s Apology and follow it with criticism. Soddy occasionally used specialized language and ideas, referring to aspects from the history of mathematics, or to technical aspects of modern math and physics, not accessible to the general reader.

Early in the review, it becomes obvious that Soddy was outraged with Hardy’s comments regarding the dull and trivial nature of applied math and empirical science. In defence of applied mathematics, Soddy said that counting and calculation, like art and music, are capable of giving pleasure to those who have the right kind of mind. And, in the words of Soddy such skills are the “bloody masters (p. 4)” in a world that is ruled by counting tables.

Soddy attacked Hardy on several fronts. His single most aggressive attack was on Hardy’s notion that pure mathematics is innocent of war, and that applied math and the sciences are contributors to war. In particular, Soddy said that Hardy wrote of “evil” chemists, without specifying by evil whether he meant making good ammunition for the bad guys, or bad ammunition for the good guys.

At three points in the review, Soddy claimed that Hardy depicted pure mathematics as a religion. For example, in one instance Soddy said: “. . . the ‘real’ mathematician is a religious maniac, and who can be expected to subscribe to the view that a religion, whatever it is, is ‘harmless?’ (p. 4)”

I believe this charge stems from Hardy’s claim that real mathematics is separate from the physical world, and can therefore serve as an escape “when all the world is mad.” In reaction to this, Soddy pointed to the anecdotal story about the death of Archimedes. While deep in concentration on a circle he had constructed on the ground, Archimedes was speared to death by a Roman soldier. While this story may or may not be true, Soddy’s objective was to question Hardy’s claim that pure mathematics can serve as a sanctuary from the real world. According to this vignette, mathematics surely did not save Archimedes from death, and I think that was Soddy’s point.

In closing his review, Soddy depicted mathematics as anything but art. Specifically, he characterized pure mathematics as unstimulating and boring, claiming that the pure mathematician’s emphasis on rigour is “paralysing to the inventive capacity. (p. 5)”

Arthur Waley, 1941

Arthur Waley, a British poet renowned for translating Chinese works, presented a short, negative review in the February 15, 1941 issue of New Statesman and Nation. It appears that Waley perceived Hardy’s work as a failed attempt at an autobiography. This book was written, Waley felt, because Hardy was past his prime and had nothing to add to mathematics.

Hardy’s comments about poetry are “disquieting” says Waley, and suggest that Hardy has not had much contact with poetry. Waley indicated a failure to see beauty in the mathematical theorems Hardy presented as prime examples. In response to this, he offered two explanations: (1) It may be that there is no parallel between math and poetry. Hardy only thinks there is because he does not understand poetry. (2) As an alternative explanation, it may be that there is a parallel between mathematics and poetry, as Hardy claims. Waley was opened-minded enough to acknowledge that perhaps it is he (Waley) who does not understand mathematics well enough to see the parallel that Hardy speaks of.

Waley closed by saying that Hardy’s book “. . . reflects on every page the dignity and integrity of traditional Cambridge academic life.” This closing statement struck me as abrupt, because of it’s sharp contrast to the preceding negative content of Waley’s review. In fact, I wondered if Waley made this closing statement in a sarcastic tone. But, it may be that, while Waley disagreed with Hardy’s main ideas, he respected the work for its clarity and style of expression.

Graham Greene, 1940

On the positive side, the British novelist Graham Greene published a short review of Hardy’s work in the December 20, 1940 issue of Spectator. Specifically, Greene noted that Hardy cited combinations of various aesthetic qualities, such as generalization and depth, as examples of mathematical beauty. Greene felt this was sufficient evidence to justly call mathematics an art. And, in contrast to Arthur Waley, Greene felt that Hardy had successfully demonstrated a close parallel between mathematics and poetry. Perhaps Greene’s greatest compliment was his comparison of Hardy’s work to that of Henry James. According to Greene, no writing, except possibly that of Henry James, conveys so clearly, and with such an absence of fuss, the excitement of the creative artist. And, Greene also drew an analogy between Henry James’ contempt for journalism, and Hardy’s distaste for applied mathematics.

C.P. Snow, 1967

In 1967, A Mathematician’s Apology was reprinted for the third time, with a lengthy forward by C.P. Snow. While the forward may be described as a biographical tribute to Hardy, Snow included some passing comments about the book that are relevant to the present study. While Snow’s comments are few, they are important because of Snow’s status in the academic community.⁶

Snow pressed two points about Hardy’s book: First, Snow felt that A Mathematician’s Apology was well written, especially for the intended audience. This is a point shared by all the other reviews, with the exception of Soddy. Secondly, Snow described Hardy’s work as an expression of sadness, written by an author who longs for better days:

. . . A Mathematician’s Apology is, if read with the textual attention it deserves, a book of haunting sadness. Yes, it is witty and sharp with intellectual high spirits: yes, the crystalline clarity and candour are still there: yes, it is the testament of a creative artist. But it is also, in an understated stoical fashion, a passionate lament for creative powers that used to be and that will never come again. I know nothing like it in the language: partly because most people with the literary gift to express such a lament don’t come to feel it: it is very rare for a writer to realize, with the finality of truth, that he is absolutely finished (pp. 50-51).

Snow’s positive comments about Hardy’s writing style are somewhat attenuated by Hardy’s endnote following the last chapter in A Mathematician’s Apology. At this point, Hardy acknowledged that Snow had reviewed the manuscript and suggested some revisions. In particular, Snow suggested that Hardy not be too obsessed with his notions about the role of mathematics and warfare. Soddy had been highly critical of Hardy’s ideas about mathematics and warfare; This suggestion is concurrent with Soddy’s attack on Hardy, though not nearly as vitriolic towards Hardy’s general objective.

Times Literary Supplement, 1967

Following the 1967 reprint of A Mathematician’s Apology, a positive review appeared in the December 28, 1967 issue of the Times Literary Supplement. The review, which was entitled “People who Count”, did not list an author.

Conclusions

Having summarized the five book reviews, I have drawn three conclusions regarding the impact of A Mathematician’s Apology:

1. First, based on the comments from Frederick Soddy and C.P. Snow, I feel Hardy’s book did much to inflame a pair of long-standing feuds in the academic community.

The first of these controversies was that between the pacifists and the war hawks.

At points in A Mathematician’s Apology, Hardy talked of his opposition to war, and, in a more subtle tone, of his distaste for politicians, particularly British. When I take into account the political climate of Hardy’s environment during his professional career, I feel he was using his book as a forum to vent anti-war sentiments. As I mentioned earlier, this drew heavy criticism from Frederick Soddy.

A second long-standing feud that Hardy’s book inflamed was between the pure scientists and the applied scientists. In fact, Soddy labelled Hardy’s book as a “repartee” to Lancelot Hogben’s book, Mathematics for the Million. Hogben’s book, which had been originally published in 1937, emphasized the practical side of mathematics.

2. On the positive side, I believe the book had a desirable impact on the literary world. Collectively, the judgements passed by Graham Greene, C.P. Snow, and the Times Literary Supplement serve as evidence that the author succeeded in justifying the study of pure mathematics as a creative endeavour.

3. My last conclusion is that A Mathematician’s Apology has played a role in the reform of mathematics teaching. An examination of the recent literature reveals significant attention to a link between Hardy’s ideas and mathematics education. Although a complete analysis of this thesis is beyond the scope of the present work, I nevertheless believe that Jessica Sekhon’s review is a sample of what we find on this topic in the literature. The presence of such ideas in the recent literature is also evidence of the permanence of Hardy’s ideas about beauty in mathematics.

Endnotes

1. Excellent resources for general information about G.H. Hardy include C.P. Snow’s lengthy preface in the 1967 edition of A Mathematician’s Apology, and chapter 4 of Kanigel (1991). These sources provide a highly readable account of Hardy’s personality, the social environment in which he lived, and a non-technical description of his contributions to mathematics. Additional insight about G.H. Hardy can be gained from obituaries authored by Titchmarch (1950) and Burkill (1996). In particular, Titchmarch (1950) provided a complete listing of Hardy’s publications; Burkill (1996) provided much the same information as Titchmarch, but also briefly described some of Hardy’s contributions to analysis.

Frederick Soddy was a Nobel laureate in science, C.P. Snow was both a scientist and novelist, while Arthur Waley and Graham Greene were renowned poets.

3. A number of resources are available to the reader who wishes to review the literature on the notion that mathematics is an art: Poincaré (1956) offers a classic account, written for the general reader; Borel (1983) , Reiner (1994), and Tymoczko (1993) offer a more formal, philosophical treatment of the issue; Papert (1988), King (1992), and Flannery (1994) pursue the implications for mathematics education. Finally, reviews of King’s book by Tymoczko (1995) and Whittle (1995) may also be of interest to the reader.

4. This is an astonishing claim, and I am compelled to make two comments: (1) It is easy to find examples of mathematical ideas which initially had no practical application, but eventually appeared in some applied format. King (1992, pp. 116 - 121) points this out, giving a number of specific examples. In particular, King (1992) credited Apollonius of Perga for the discovery of conics around 200 B.C. At this time the theory of conics had no practical application, but was of interest to Apollonius only as a pure discipline. It was not until around 1600 that Kepler utilized the pure mathematics of Apollonius to develop applied ideas in astronomy, which subsequently served as a springboard for the monumental work of Isaac Newton. Because the work of Newton is pervasive in modern science and engineering, we see an example of a link between pure and applied mathematics. If Hardy’s astonishing claim is true, does that mean that this mathematical idea (conic sections) had aesthetic value when originally discovered, but then lost aesthetic value when it found application? (2) Secondly, while Hardy expressed his distaste for applied math in a free and uninhibited manner, it is interesting to note that some of Hardy’s work has found important application. For example, a fundamental tenet of population genetics, the Hardy-Weinberg law, was co-discovered by Hardy and the German physician Weinberg. A historical account of the Hardy-Weinberg Law is provided by Fletcher (1980). Others have noted important contributions by Hardy outside of pure mathematics, such as his role as editor of the Cambridge Tracts in Mathematics and Mathematical Physics from 1914 to 1946 (Todd, 1994); Hardy’s role in international diplomacy following World War I (Dauben, 1980); and finally, Hardy’s role in the reform of the mathematical tripos exam is noted by Mordell (1970). Whereas these latter contributions can not be classified as “applied math” in the sense that the Hardy-Weinberg Law is applied math, they are nevertheless practical, and serve to refute Hardy’s claim that he never produced anything of practical value: “I have never done anything ‘useful’....Judged by all practical standards, the value of my mathematical life is nil; and outside mathematics it is trivial anyhow.” (Hardy, 1967, pp. 150 - 151):

5. It is interesting to compare and contrast the personalities of Hardy and Soddy. While the resources under note 1 give a profile of Hardy’s character, an account of Soddy’s personality can be found in Merricks (1996). From these sources, there is evidence that both were strong-willed men, and both suffered professionally for taking stands on unpopular ideas. This, coupled with the fact that Soddy was outspoken on applications while Hardy advocated pure studies, makes one wonder how well they got along as colleagues at Oxford from 1919 - 1931. Could Soddy’s negative review of A Mathematician’s Apology have been motivated by something akin to sibling-rivalry?

6. C.P. Snow’s commentary on A Mathematician’s Apology is particularly meaningful for two reasons: First, as a renowned scholar in both science and the humanities, Snow possessed an eclectic perspective on the issue of aesthetics in mathematics. For both intellectual and political reasons, Snow would be the ideal translator on Hardy’s book, conveying the mathematical aspects to the humanists, and conveying the artistic aspects to the mathematicians. Snow’s versatility is described by de le Mothe (1992, p. 49) in relation to the “communicative dichotomy” that exists between science and the humanities. Secondly, Snow was a close acquaintance of Hardy, as revealed in the forward of A Mathematician’s Apology, and again in the epilogue of Kanigel (1991). Such a relationship very likely gave Snow a unique insight into Hardy’s true message in A Mathematician’s Apology.

References

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Dauben, J.W. (1980). “Mathematicians and World War I: The International Diplomacy of G.H. Hardy and Gösta Mittag-Leffler as Reflected in their Personal Correspondence”. Historia Mathematica, 7, pp. 261 - 288.

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unknown author (1967). “People Who Count”. [review of the book A Mathematician’s Apology]. Times Literary Supplement.

Whittle, S. (1995). “A Critique of Jerry P. King’s book, The Art of Mathematics”. Georgia Journal of Science, 53 (1) [paper presentation abstract], p. 58.