Pure Mathematics: Unveiling the Beauty and Significance of Pure Mathematics in Academic Discourse” **For the first time in 75 years the Godfrey Harold Hardy’s** *The Mathematicatician’s Apologia***continues to fuel the debate on applied and pure math.**

Godfrey Harold Hardy was one of the most influential theorists in this century. Mathematics was the dominant factor in his life and only the sport of cricket was able to compete for his attention.

As he grew older, he lost creativity and the heart attack that occurred at the age of 62 destroyed the strength of his body, Hardy composed *A Mathematician’s Apologia.* It was an *apology* in the manner that Aristotle or Plato could have understood it, and a self-defense against his life’s work.

Second editions of *A Mathematicatician’s Apology* included the now iconic image of Hardy in the dust jacket. For the first edition Hardy sent out postcards asking presentation copies be sent out to colleagues such as C. D. Broad and J. E. Littlewood as well as the scientist Sir Arthur Eddington, chemist and novelist C. P.

Snow as well as Cricketer John Lomas (to whom he gave the book) as well as Gertie, his younger sister. Gertie. The author also requested that copies be given for colleagues from and around the United States. **Postcards photo by the Cambridge University Library.**

“A mathematician” Hardy contended, “like poets or artists is a creator of patterns. …. The mathematician’s designs, similar to the artist’s or poet’s are expected to be gorgeous and the concepts, just like the colors or the language are to be arranged in a way that is harmonious.”

This was an deeply personal and insightful perspective on mathematics for the layman unlike anything else that had been published prior to. The book that is now celebrating an anniversary that marks the 70th anniversary its initial publication, is an excellent and most easily understood description of the mathematical world as it is.

Since its initial appearance, *A Mathematician’s Apology* has been a lightning rod attracted by angry bolts due to its rejection math applied to the real world as boring and irrelevant.

The shaft that sparked the start of a critique in the magazine *Nature* written by Nobel laureate and chemical scientist Frederick Soddy was particularly piercing: “This is a slight book. With such clowning, the world is sickened.”

Hardy’s views on the value of free thinking were strong and uttered with “careful touch and controlled passion” to borrow the words of the well-known novelist Graham Greene. The author continues to attract people who are sympathetic to their ideas in a variety of artistic areas. They were adroit at the time and are still relevant to the present day.

##### Art of Argument: The Art of Argument

Hardy was born the 7th of February 1877 at Cranleigh, Surrey. His parents were very fond of education however neither of them was in a position to pay for the cost of a university.

Hardy developed into an athlete, scholar as well as an atheist and an pacifist, but most of all the latter, he was an individualist. In an obituary for him mathematics professor E. C. Titchmarsh wrote: “If he dined at high tables in tennis clothing, it was due to his desire to dress in tennis clothes rather than because he’d lost track of what he had worn.”

Hardy’s fascination with maths and his apathy to religious beliefs were evident from an young age. His efforts in church were typically focused on the calculation of numbers on the hymnal boards, instead of worship. But his view of religion was much more than mere disinterest. As per Titchmarsch:

*Hardy always spoke of God to be his own personal foe.** **Of course, this was the case but there was something deeper behind this.** **He believed in the religion’s doctrines more seriously than the majority of people would think.*

Hardy displayed his doubt in bizarre ways. For instance, he was unable to go into any church. Mathematical scientist George Polya remembered that whenever he and Hardy passed the church, Hardy would be sure that Polya was standing between him and the church. Polya didn’t know why.

Hardy ( *far right*) and his protégé Srinivasa RAMANUJAN ( *center*) are shown alongside colleagues from Cambridge University. After submitting Hardy many letters with proofs of theorems, Ramanujan worked closely with Hardy for five years working on different aspects of number theory and mathematics, including highly composite numbers that are positive integers that have greater divisors than the smallest positive integer. Ramanujan earned a doctorate degree from Cambridge for his work at the time of 1916. **Wikimedia Commons**

Hardy started a renowned partnership with an researcher J. E. Littlewood in 1911. Two years later, they were able to study strange handwritten mathematical documents that were uninvitedly sent by an untried Indian civil servant named Srinivasa Ramanujan.

Together, they would conclude that this is the product of a genius. After a lot of work, Hardy succeeded in bringing Ramanujan to the University of Cambridge, where Hardy was an academic. The “one unforgettable moment of my entire life” is the way Hardy described his collaboration and discovery with his young protégée who tragically passed away from illness seven years after.

After submitting to Hardy numerous theorems via writings, Srinivasa Ramanujan worked closely with Hardy for five years working on different aspects of the field, which included highly composite numbers which are positive integers that have greater divisors than the smallest positive integer. Ramanujan was awarded a doctorate by Cambridge for his research in the year 1916.**Wikimedia Commons**

The atmosphere in Cambridge was pretty unimaginable and a nightmare for Hardy in his World War I years of 1914 until 1918. A lot of his friends and colleagues including Littlewood were off to fight. Hardy had been a peace-loving man, but not a conscientious objector.

He was a volunteer for military service but was rejected due to medical reasons. The deep respect he had for German culture and his equally strong disdain for politicians added to his angst.

In 1916, his pacifist friend Bertrand Russell was dismissed from his position as a lecturer in Trinity College of Cambridge for printing “statements that could prejudice the recruitment and discipline of His Majesty’s troops.” Hardy felt quite isolated.

The year 1919 saw Hardy was transferred to Oxford University in the UK, where his eccentricities were thriving, and he became satisfied and productive. In his room, he kept the image of Vladimir Lenin.

He was averse to mechanical devices like telephones and would not even look in the mirror, was not willing to allow his photo to be taken and was extremely shy when having conversations with people.

But He was a great conversate, who could continue to talk about a range of topics (including cricket, of course). cricket). Titchmarsh was quoted as saying: “Conversation was one of the games he adored to play. It was often difficult to determine what his true opinions were.”

Polya shared a similar memory: “Hardy liked to shock people by offering unorthodox views. He loved to defend his opinions just to have a convincing argument, since it was his passion to argue.”

It’s evident that Hardy was a fan of teasing his readers and this is something to keep in mind while reading *The Mathematicatician’s Apologia.*

The prospect of the Sadleirian chair in pure maths at Cambridge was a too strong to resist for Hardy. In 1931, he returned the University by the River Cam, once home to Isaac Newton and James Clerk Maxwell.

As his debilitation and illness increased, academic distinctions were accrued such as The Chauvenet Award of the Mathematical Association of America in 1932, and the Sylvester Medal from the Royal Society in 1940. It was the Copley Medal from the Royal Society was due to be awarded to him on the 1st of December on the day of his death.

##### The Work for Second-Rate Minds

A number of reviews for *The Mathematician’s Apology* were published in the first several years of the book’s existence. The majority of them were positive. The writer of one review, which was published within *The Spectator* in 1940 was Graham Greene.

Hardy would have been elated when he read that: “I know no writing–except Henry James’s introduction essays, which conveys in such a clear and concise manner with without exuberance the enthusiasm of a artistic artist.”

The reviews of other reviewers were less positive. As of today there are a variety of reasons to be upset by *A Mathematician’s Apology* particularly if you’re scientist.

If you’re the author of expository writing (such such as the one below) and you’re a writer, then you won’t need to wait for long for insults to be hurled at you. The Hardy book started:

*It’s a sad and lonely moment for any professional discover himself writing about mathematics.** **The job of a mathematician is perform something, to prove new mathematical concepts, and to contribute to math, not to discuss what mathematicians like him have accomplished.** **…Exposition or criticism, as well as appreciation is a job for people with second-rate minds.*

The majority of reviewers commended Hardy for this claim. Felix de Grand’Combe, professor of French at Bristol University, did not. In The *Journal of Education* in August 1943 the former French army officer declared:

*It really is a touching–albeit ostentatious–confession of a local intellectual debility… It is clear that Prof. Hardy is a great mathematician.** **It’s also clear by his own example that one can be a master mathematician, and still not be able to comprehend things that are easily understood by a normal, well-educated mind.*

Art critics are apprehensive about artists while musicians hate music critics. It’s a classic tale that is reflected in the way Grand’Combe informed readers in his long review. But he also said that observation and reformulation is a creative and insightful actions:

*When Linnaeus came up with his fantastic classification of plants, he didn’t “make” any thing, he just discovered a treasure that was already there that explains and makes visible to all, a variety of coherent connections that exist in Nature However, his work reshaped and clarified our view of the world of plants and gave a rational basis to the apparent chaos, and living to what the ancients considered to be the dark, sombre stench that was “non-being.”*

If Hardy believed that exposition, criticism and appreciation were the work of inferior brains, then he would be coming to this conclusion at a later point in his the course of his.

In his prime when he was writing book reviews in *The Cambridge Review, The Times Literary Supplement, Nature,* and *The Mathematical Gazette.* He was a skilled and enthusiastic lecturer who was always being sought after. The textbook he wrote, *The Course of Pure Mathematics,* published in 1908 and currently available, is completely explicit.

The first printing of the Hardy’s book contained mathematical calculations written by the handwriting of the author on the cover. Certain reviewers thought the illustrations appealing, while others found it difficult to comprehend for the average person.**Photo by Quagga Rare Books and Art, South Africa.**

Greene and Grand’Combe have distinctly different opinions about the mathematical calligraphy written by Hardy in the jacket dust cover of *The Mathematicalist’s Apology.* While Greene found the mystical appeal of “the writer’s tiny handwriting that was just as beautiful as Greek,” Grand’Combe experienced no other reaction than incomprehension

*In a book, the jacket is illustrated with what, I think, is an example of imaginative mathematical calculation that culminates in a formula that encapsulates an array of over 25 figures, interspersed with a myriad of pluses and minuses is encased within the frigid hug of at least three brackets of different types professor G. H. Hardy of Cambridge claims to speak to the average person.*

If you’re an biochemist looking for an effective cure for a terrible illness, you may be irritated by Hardy’s explanation of your motivations. Three motives are intellectual interest, professional pride (including the desire to feel satisfied with one’s accomplishments) and the desire to succeed. In the words of Hardy:

*It is acceptable to feel that, after you’ve accomplished your goal and added to the joy or eased the suffering of others, however, that’s not the reason you chose to do it.*

In *The News Letter* in 1941 in 1941, the English philosopher and physicist of sciences Norman Campbell took Hardy’s assertions as factual. If the motivation for mathematicians is to make a profit for himself rather than society, he inquired “why do we offer …so many better jobs for mathematicians rather than, for instance poets or stamp collectors?”

If you’re a senior mathematician, you may be annoyed by Hardy’s warning: “No mathematician should ever let himself forget that math as much as any other science or art is a man’s games.” After noting that French mathematical genius Evariste Galois died when he was twenty-one, Ramanujan at 33 as well as Bernhard Riemann at 40, Hardy said: “I do not know any instance of a significant mathematical advancement made by a man who was over fifty.”

**Cambridge University philosopher C. D. Broad replied:**

*In the manner that [Hardyis able to do, a list of those who accomplished the most mathematical work and who then passed away old… it is surely not pertinent.** **I think that the stifled assumption lies in the fact that work they accomplished prior to their early death was so incredible that it’s unbelievable to think that they could have done it had they lived.*

Experiments of mathematicians who made important discoveries at the age of 50 can be easily provided. Littlewood was a mathematician who worked until 90, is an example. But, his belief is still prevalent among mathematicians. It is supported by it being the case that Fields Medal, the highest mathematical prize is only awarded for work performed prior to the age of 40.

If you’re an aspiring scientist whose feathers have not yet in a state of ruffle, Hardy’s main point will alter your feathers. “Real” mathematics is, as he claimed, practically “useless” and that mathematics that is useful is “intolerably boring.”

When he said “real” math, Hardy meant pure mathematics which is usually general and abstract, and according to Hardy it has the highest aesthetic value. Contrary to this is the bulk of math that is taught in schools such as elementary algebra, arithmetic elementary geometry integral and differential calculus, maths specifically designed for computation, and possessing the lowest aesthetic value.

Hardy was both defender and prosecutor during an imagined trial that sought to determine whether his life was worth living:

*I’ve never done something “useful.” The discovery that I have made has made or will likely to make either directly or indirectly, for either good or for any difference at all to the general well-being of the world.** **…I only have one chance to avoid a judgement which is completely insignificant, and that I might be deemed to have produced something that is worth making.*

It’s paradoxical that Hardy who believed of pure “useless” math is most well-known by many for his Hardy-Weinberg Principle that quantifies how the frequency of genetic traits are the same across generations, despite there being no mutations, or any other factors that influence evolution.

Hardy was first introduced to the issue by the Geneticist Reginald Punnett, with whom he played cricket. In an article in 1908 addressed to *Science,* Hardy declared his mathematical explanation of this issue “very easy.”**Barbara Aulicino**

In the past, as more than one person has pointed out, it is absurd to think that Hardy is perhaps best recognized for his discovery of genetics. The theorem that he along with Wilhelm Weinberg independently proved is widely known as the Hardy-Weinberg concept.

However, for Hardy who fought through two world wars at the time, number theory offered a way to escape that was thankfully not useful to military planners. Hardy was against the entry of England into the First World War, a deadly conflict that was made even more dangerous due to technological advancements and science.

*The Mathematicalist’s Apology* was first published in 1940, while England was once more in war. The article was based on his own, “Mathematics in Wartime,” published in the journal *Eureka* in 1940, Hardy wrote in *Apology* in the same year “When the world goes mad mathematicians may discover mathematical an extraordinary anodyne. Because mathematics is, out of all sciences and arts one of the most primitive and distant.”

According to Hardy the mathematical world is directly connected to the real world. Theorems cannot be negotiated. However, he claims that science’s reality is just a sketch. “A chair could be the result of spinning electrons or an idea that is in the brain of God,” he declared.

“Each one of these explanations might be valid but they do not conform at any level to the notions from common sense.” The mathematician who is pure is not bound by physical truths. In the words of Hardy: “‘Imaginary’ universes are far more beautiful than the’real one.”

Frederick Soddy, who had helped to make radioactivity understood by the world The scientist was disgusted by these opinions. In a review published of *Nature,* he stated that If Hardy was taken seriously, it would be”the “real mathematical genius” would be the “religious madman.”

Soddy posted a rebuke on his Twitter:

*It is surely in this age an appreciation of benefits of the military, and not the usual slandering of the arms trade is a must for people of all religions and especially because their form of “education” is at the heart of the entire tragedy and the chemist’s explosives and poison-gas are an appropriate whipping boy of their own.*

Hardy knew about the review by Soddy. He could have been amused by the review. A letter from R. J. L. Kingsford at Cambridge University Press, dated January 1941, said: “I quite agree that Soddy’s remarkable article of the book in *Nature* can be described as an extremely effective advertisement. I’ve enclosed a copy of the review herewith.”

The idea that mathematicians are an autonomous priesthood protected by their own religious beliefs was very well-known. It is a theme throughout Lancelot Hogben’s *Mathematics for the Million,* a highly successful book on math that has been a huge seller since 1937. After demonizing Pythagoras as well as Plato for their awe-inspiring love of abstraction Hogben wrote:

*Mathematicians tend to be like this could be the reason they are likely to keep the esoteric secrets of their Pythagorean family to themselves.*

In their respective reviews, Soddy and Broad suggested that the book may be an ode to Hogben. Hogben critiques Hardy within *Mathematics for the Million,* which adds credence to the theory.

Another rebuke of *A Mathematicatician’s Apology* was issued by E. T. Bell who was a mathematician and science fiction writer most famous for his 1937 work *men of Mathematics.* In his review published within *The Scientific Monthly* in 1942,

Bell recommended Hardy’s book to “solemn young men who feel they are on the responsibility of preaching higher mathematics to mathematicians.” Bell concluded that “Congenital followers will adore the book with awe perhaps as a way of compensating in the wake of losing their faith from their youth.”

Hardy is the leader of a group of mathematicians during an cricket match during an British Association meeting in Oxford in the month of August 1926. The team comprised E. C. Titchmarsh, Bernard Bosanquet, Edward Linfoot and William Ferrar.**Photo by the University of Oxford Mathematical Institute.**

##### It’s Not Going to Make a Dime for anyone

Hardy was planning to publish *The Mathematicalist’s Apology* together with Cambridge University Press at his own cost. The reality was that Press Secretary S. C. Roberts acknowledged the merits of the 90-page paper and was adamant to endorse it to Syndics, which is the governing body for the Press.

In a meeting in July 1940 the Syndics determined that 4,000 copies *The Mathematician’s Apology* were printed. The price for sale would be 3 shillings or 6 pence each, or roughly 8 pounds in the current British currency.

Postcards from Hardy asked that copies of the presentation be distributed to colleagues which included Broad and Littlewood and the physical physicist Sir Arthur Eddington, chemist and novelist C. P. Snow and Cricketer John Lomas (to whom he committed the novel) and his sister Gertrude who were very close. He also requested that copies of the book be delivered to his colleagues from America. United States.

A letter to him that was sent on 29 May 1941 is a sign of the destruction that was caused by conflict: “…copies which were being bound for a subsequent shipment were destroyed by the enemy’s action on one of our binding machines.”

*The Mathematicatician’s Apology* proved to be a hit. Another printing of 2500 copies was printed in 1941. Another 2000 copies were printed by 1948, the year after the death of Hardy.

Then, in June of 1952 the sister of Hardy addressed the Press:

As a Mathematician’s Apology is no longer available either first or second-hand, I anticipate that you’ll eventually be printing it again; I believe it’s an excellent idea to include the image of my brother within it, provided that it does not cause it to be too costly. It is the original negative which the attached photo is an enlargement. It is a photo taken by an amateur and very distinctive.

The image she shared was later featured in the book’s dust-jacket and it has since become the most well-known photo of her brother, dressed in an all-white suit and sitting in a wicker seat. Hardy looking through his glasses, appears to be gazing at the photographer. Secretary R. W. David who was handling the project, responded to Gertrude with enthusiasm. Gertrude wrote back the next day: “I think that it is by far the most beautiful picture ever shot by the brother of mine.”

Reissuing *The Mathematicians Apology* could be a difficult task. The high rate of inflation in Britain has made it difficult to reprint such a small book at a fair cost. The book would require some kind of material. be required to expand it.

It was the Rouse Ball Lecture “Mathematical proof,” that Hardy had given in 1928 was thought to be a good idea. The same was “What is Geometry What is Geometry?” Hardy’s presidential address for the Mathematical Association of America in 1925. There was nothing that seemed to be appropriate.

In 1959, just 11 years later, chemist journalist C. P. Snow suggested that he create an introduction. It seemed like a brilliant idea. Snow is most famous for his essay in 1959 *the Two Cultures,* describing the supposed struggle between humanists and scientists.

He was an advisor to Cambridge University Press during the period of war. He was also an intimate friend of Hardy and had given his advice on the novel. But, Snow would not commit to a timeframe. A memo to internal staffers written by David in September of 1966 remarked about the fact that “we have been seeking Snow for copies at about annually.”

In 1967, word spread the news that Snow (now Lord Snow) had completed his work. The problem was that he had created it for a publication, *Variety of Men,* which Macmillan in London and Scribner in New York would soon publish. In addition to his biography it would also include biographical profiles of H. G. Wells, Einstein, Churchill, Stalin and many more notables.

Can the Syndics obtain permission to reproduce the Snow article? *The Variety of Men* would also be published in magazines, adding another layer of legal complications. The idea of finding a different creator to launch Hardy was considered, but the idea was the idea was rejected. A pamphlet that was privately printed by Hardy on the dismissal of Bertrand Russell as a member of Trinity was also thought of. David stated that “it was not appropriate for marriage to the *Apologizing Church.”*

David’s determination to find Snow was never diminished. On the 17th of October the 17th of October, 1966, he wrote:

*We’re all in agreement that the best option is to stick with the original idea.** **Snow’s introduction is fluid and anecdotal. It’s a “lively sketch” like one that could be used during a radio program in honor of a notable man.** **Some may find it a bit uninspiring and feel the fact that the man of great stature has being introduced to a lesser.** **However… people who want Hardy to be well-known must not shun the genuine populariser.*

After a few days, all issues appeared to be resolved. David noted: “You will see from the minutes of the Syndics’ meeting on 21 October that it appears the way is now clear for the next step.”

Then there was more trouble. The sister of Hardy died she left all rights to the work of Hardy in the hands of the London Mathematical Society. The Society opposed “filling the pages of” *A Mathematician’s Apology* with any writings by any other person than Hardy.

In a memo dated the 3rd of January, 1967 and signed “Urgent,” David noted that Cambridge University Press owned the copyright to the book along with the control over the book. “Yet I am confident,” he added, “that the Syndics do not intend to violate the desires of [London Mathematical society[London Mathematical Society]. … This plan should be placed on hold until I receive a response to the London Mathematical Society.”

It appears that there was a deal reached which is why the second volume of *A Mathematical Apology* was available in stores by the end of 1967. Snow’s contributions added literary appeal. It started:

*It was a normal evening at Christ’s table, with the exception that Hardy was seated as a guest. ….** **The year was 1931, and the term was not yet being used in English use, however in the future, people would have claimed that in some way Hardy was a star.*

Like Cambridge University Press anticipated, the new version of *The Mathematicatician’s Apology* was well-received across The United States. Byron Dobell, an author and editor based in New York who helped many young writers, such as Tom Wolfe and Mario Puzo and seasoned his admiration with a dash of caution:

*It’s the type of book that you’d like to be being read by all of your colleagues right now that you’re reading it.** **It’s one of those rare perfect books that make many writings seem like the mixture of mush and lead.** **This is the book we’re promoting this month.** **It’s not concerned with anything else but the enjoyment of life and the mind.** **It costs $2.95 and it’s a good thing because with a title such as that, it’s unlikely to earn any money for anyone.*

##### Physical Connections

Second editions of *A Mathematicatician’s Apology* came out as maths became more abstract. Many mathematicians were thrilled by the shift in direction of their area. Some lamented. Should the current trend continue, as some believe that mathematics will become obsolete.

One mathematician who was celebrated is University of Chicago professor Marshall Stone. His essay “The The Revolution of Mathematics” that first appeared within the magazine *Liberal Education* in 1961 and later reprinted that following year in the *American Mathematical Monthly,* observed abstraction as bridging the mathematical disciplines which were previously separated from thought. The recognition of logic and mathematics the author argued, was largely the reason for this:

*Mathematical sciences are now believed as not having any necessary connection to the physical world, other than the mysterious and ambiguous one that is implied by the claim that thought takes place within the brain.** **The fact that this is the case can be considered to be one of the biggest advancements in the field of science and technology throughout the history of humanity.*

Stone observed a paradox: increasing abstraction was creating new applications. He cited his mathematical theories of genetics as well as game theory along with mathematical theories of communication and linguistics.

An entirely different view was voiced the next year the following year in “Applied mathematics: What’s required in education and research,” published in *SIAM Review.* It was the report of a symposium that was chaired by the mathematician H. J. Greenberg.

The panel was comprised of mathematics experts George Carrier, Richard Courant, Paul Rosenbloom, and the physicist C. N. Yang. Stone’s essay with its emphasis on abstraction was discussed with a sense of alarm. Panel members advocated for an older-fashioned view of mathematics which takes its inspiration from the sciences. Courant’s warning was reminiscent of an overview of *A Mathematician’s Apology.*

*We cannot accept the blasphemous old saying that the sole reason for mathematics is “the beauty of the human brain.” Mathematics should not be allowed to divide and diverge toward an “pure” as well as an “applied” range.*

Despite the warning of Courant, a line between applied and pure mathematics is evident in a lot of universities in the present. In many cases, it’s an uneasy line that is witness to battles involving scarce resources and damaged self-esteems.

This line may have been blurred a little by mathematicians’ extensive use of computers, and the immediate need for sophisticated algorithms. Mathematicians who agree with Hardy’s ideas might be reluctant to share these views due to the rising costs of higher education.

Students who have debts that are mounting are becoming increasingly frustrated with teachers who diverge from subjects that are directly relevant to their examinations. Administrators are enthralled by research grants in cyber-security and medicine as they struggle to fill the tiny grants given for pure mathematics.

The distinction between applied and pure mathematicians may blur, but it won’t disappear. As long as it exists, G. H. Hardy’s *A Mathematician’s Apology* will be read and–usually–enjoyed. A more enlightening summary could be given than the one provided by J. F. Randolph in his review of 1942:

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