Summary
Chapters 1–2
Hardy begins his apology with a declaration: in the very act of "writing about mathematics," he feels he has descended from the elevated realm of a pure mathematician. He compares his position to that of an art critic—a role he considers suited for "second-rate minds"—as opposed to the artist who creates. Hardy recounts a conversation on this topic with the esteemed British poet A. E. Houseman. Moving into chapter 2, Hardy poses the central questions he aims to explore throughout the book: Why is dedicating one's life to mathematics worthwhile? And what truly justifies a mathematician's existence?
Chapters 3–4
Hardy suggests that individuals often select their careers because "it is the one and only thing that [they] can do at all well." Mathematics stands as a uniquely specialized realm, and those who delve into it are not known for wide-ranging skills. In chapter 4, he enumerates several mathematicians whom he deems immortal geniuses, noting that many either reached their intellectual zeniths or passed away before turning forty. Those who ventured into new careers later in life largely met with failure. Hardy uses these observations to explain his current endeavor: he has grown too old for theoretical work and lacks the aptitude for any other vocation.
Chapters 5–9
Hardy proceeds to address the questions from chapter 3. On the matter of why one might choose the path of a mathematician, he refers to a lecture he delivered at Oxford two decades ago, asserting that mathematics is pursued for three reasons. Firstly, it is an inherently "harmless" occupation; secondly, given the universe's vastness, the pursuit of excellence by a few professors is "no overwhelming catastrophe"; and thirdly, mathematics possesses a "permanence" transcending the capabilities of most people. Here, Hardy introduces what he sees as the three driving forces that propel individuals toward their professions: intellectual curiosity, professional pride, and the quest for reputation and its rewards. To bolster his argument, Hardy cites ancient civilizations now remembered solely for their mathematical contributions. He concludes with a haunting dream shared by mathematician and philosopher Bertrand Russell, revealing Russell's deep-rooted anxiety about being forgotten by future generations.
Chapters 10–11
Hardy argues that mathematics, much like art or poetry, holds an aesthetic allure—a notion for which he and this book are famed. He takes aim at contemporary mathematician Lancelot Hogben, renowned for opposing Hardy’s theories. By using chess as an example, Hardy counters Hogben's stance. Chess, adored by many and an exercise in pure mathematics—albeit of a "lowly kind"—holds a form of mathematical beauty in every cleverly executed move. Nonetheless, as the highest form of mathematics requires "seriousness," or "importance," and no chess player or problem has ever shaped the broader scientific landscape, chess is deemed "trivial" in comparison to pure mathematics.
Chapters 12–14
Hardy employs the timeless proofs of Euclid and Pythagoras to showcase the elegance of mathematics, illustrating their significance despite a lack of practicality. These proofs are presented with clarity, requiring only a basic mathematical understanding to follow. This marks the sole occasion in the memoir where Hardy ventures to elucidate mathematical concepts or logic.
Chapters 15–18
Hardy continues refining his notion of mathematical beauty by further exploring the concept of "seriousness." To elucidate this, he introduces the ideas of "generality" and "depth." Generality can be loosely understood as "abstractness," while depth parallels "difficulty." He discusses the assertions of mathematician and philosopher Alfred North Whitehead, quoting, "The certainty of mathematics rests entirely on abstract generality." Hardy partially concurs with this from a logician's perspective but insists that theoreticians must seek distinctions among generalities. Regarding "depth," Hardy employs a metaphor akin to...
(This entire section contains 1480 words.)
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geologic strata: the more intricate the theorem, the more layers it comprises, each linking ideas with those above and below. This discussion culminates in chapter 18 with Hardy explaining why chess fails to achieve "beauty." In essence, the nature of chess allows any move to be countered with numerous countermoves—what Hardy terms "proof by enumeration of cases," which stands as the antithesis of beauty to a mathematician. Instead of a beautiful array of mathematical theorems, a chess game is fundamentally a psychological duel between two minds.
Chapters 19–21
Hardy revisits his Oxford lecture to tackle the question of mathematics' utility. In essence, he firmly states that while some "elementary" mathematics, like calculus, may serve practical purposes, the "pure" mathematics that captivates him defies justification on utilitarian grounds.
Chapters 22–24
Hardy re-engages with the comparison between "applied" and "theoretical" mathematics, asserting that it would be an oversimplification to claim one is useful while the other is not. He argues that pure mathematics is, in fact, closer to "reality" than applied mathematics. He posits that a "mathematical reality" exists, no different from the "physical reality" we all perceive. This mathematical reality is not a mental construct but an objective presence in the world, waiting to be discovered and described. Mathematicians crafting proofs are, in truth, merely documenting their observations.
Hardy vividly illustrates his point by delving into geology, which aims to sketch a "picture" of a segment of mathematical reality. Yet, because geometry fails to encapsulate the ever-changing nature of spatiotemporal reality—changes brought by eclipses and earthquakes, for instance—these "drawings" in a geometer's theorems might abruptly become irrelevant to the tangible world around them. Nonetheless, the truths within these theorems remain unscathed. To draw a simpler analogy, spilling coffee over the pages of a Shakespearean play might render some pages unreadable, but the essence of the play remains untouched. In this analogy, pure mathematicians are akin to those who treasure the play itself, while applied mathematicians focus on the material pages on which the play is inscribed.
In chapter 24, Hardy boldly posits what might initially appear to be a paradox: pure mathematicians have a closer connection to reality than their applied counterparts. He argues that an applied mathematician must grapple with a complex and often debated physical reality. Consider a chair: it could be seen as a whirlwind of electrons or perhaps as "an idea of God." In contrast, the realm of a pure mathematician is crystal clear, untainted by ambiguity. Concepts like "2" or "317" are universally understood, and the prime nature of "317" exists not because it's perceived to be so, but because it inherently is.
Chapter 25
In his exploration of pure versus applied mathematics, Hardy extols the timeless beauty of pure mathematics, which offers a lasting aesthetic pleasure. Its ageless nature provides a deep emotional fulfillment, unlike the contemporary and transient accomplishments of applied mathematics. Hardy edges toward labeling the theories of applied mathematics as "useless."
Chapter 26
Hardy further explores the notion of utility within mathematics, querying, "What part of mathematics is truly useful?" He categorizes various mathematical branches by their practicality. He concludes that the more beneficial a mathematical discipline is to an engineer or physicist, the less aesthetic value it retains. Hardy favors the fertile realms of imagination and artistry over the "mundane" reality of applied mathematics. He muses,
‘Imaginary’ universes are so much more beautiful than this stupidly constructed ‘real’ one; and most of the finest productions of an applied mathematician’s fancy must be rejected, as soon as they have been created, for the brutal but sufficient reason that they do not fit the facts.
Chapters 27–28
Engaging with his critics, particularly Lancelot Hogben, Hardy adopts a tone of sharp superiority as he reiterates his stance on the distinction between pure and applied mathematics. He often describes applied mathematics as "trivial." To Hardy, true mathematicians are akin to artists. He dismisses any defense of applied mathematics, merely suggesting that it might appeal to Hogben. As Hardy writes amidst the looming threat of global conflict, he finds it imperative to reflect on the intersection between mathematics and warfare. He asserts that "real mathematics" doesn't influence war, noting that no bellicose application has been found for theories such as number theory or relativity, and it seems improbable anyone will uncover such uses in the foreseeable future. Hardy penned these thoughts five years before the theories of relativity played a role in developing the atomic bomb.
Chapter 29
As Hardy approaches the conclusion of his memoir, he shifts to a more personal narrative style. This final chapter bears a more autobiographical tone than the preceding ones. Having laid out his theoretical ideas, Hardy feels justified in reflecting on his life—a summation of a man aware of his waning years. Although the tone is tinged with sadness and reflection, he persuades himself that his life held significance. In lines frequently quoted by critics, Hardy remarks, "Well, I have done one thing you could never have done, and that is to have collaborated with both [mathematicians John Edensor] Littlewood and [Srinivasa] Ramanujan on something like equal terms." These oft-repeated lines reveal not only his pomposity but also a profound humility. He acknowledges genius and admits his own limitations. Amidst the subtle melancholy that pervades the work, these words resonate as a powerful affirmation of a brilliant existence.