(Illustration: Animesh Debnath.)
(Illustration: Animesh Debnath.)

The algebra of Alice: 152nd anniversary of Alice’s Adventures in Wonderland

Alice in Wonderland, the evergreen whimsical fairy tale for children, was also a cleverly crafted way in which Oxford don Lewis Carroll poked fun at the new mathematical developments of his time.
By Debkumar Mitra | Hindustan Times
UPDATED ON DEC 02, 2017 11:06 AM IST

Alice’s Adventures in Wonderland by Lewis Carroll continues to attract new readers ever since it was told to three sisters on a summer afternoon during a boat ride on the Thames. The apparently whimsical fairy tale charmed its listeners on its first telling but the story was expanded by Carroll into the Alice of today. On the 152nd anniversary of the classic’s publication on November 26, 1865, as a Christmas release in England, let’s consider the book as a mathematical puzzle.

Lewis Carroll in the preface to the work ‘All in the Golden Afternoon’, claimed to have invented the story on demand from Alice Liddell, and her two sisters, daughters of an Oxford don – Carroll himself taught mathematics at Oxford – during the boat ride. However, the profusion of mathematical puzzles, logical paradoxes and innuendoes throughout the body of the text tell a different story. While there is no doubt about the fact that it was created for, and to be told to children and young adults, what 21st century readers read today is a cleverly crafted tale to poke fun at the mathematics in Carroll’s time and its practitioners.

Carroll, a nom de plume of Charles Lutwidge Dodgson, a mathematics tutor at the Christ Church College in Oxford, was actually not a front-ranking mathematician. He swore by Elements, the famous geometry text by Euclid. Carroll waged a long battle with his peers who were revolutionising Victorian mathematics. Projective geometry, imaginary numbers, quaternion were turning the old-world of algebra and geometry upside down. Mathematics was no longer tied to the ground insofar as it was becoming more abstract, and logic that appealed to Carroll and his ilk could not be used to demystify the new avatar. Carroll was a Euclidean geometry orthodox who did throw the gauntlet at the new kids on the mathematics block but lost out. These were the times when Alice Liddell asked the young mathematics tutor to tell a story.

THE MISS LIDDELLS
Close to a decade and a half later, in 1879, Carroll, under his real name, published Euclid and his Modern Rivals. Written in the form of a play, it was Carroll’s way of telling the world that Euclid’s Elements is the best textbook for teaching geometry. Carroll’s introduction lays out his purpose and why he went about it the way he did. His words on writing for a non-scientific audience still sound particularly relevant. “It is presented in a dramatic form,” writes Charles Dodgson in the introduction, “partly because it seemed a better way of exhibiting in alteration the arguments on the two sides of the question; partly that I feel myself at liberty to treat it in a rather lighter style than would have suited an essay, and thus to make it a little less tedious and little more acceptable to unscientific readers.” Not many now are even aware of this curious publication but this can be seen as an extension of Carroll’s thought process that started with Alice’s Adventures in Wonderland.

There is, however, no direct evidence that Carroll actually planned such a tale. Martin Gardner notes is his book, The Annotated Alice, the definitive edition, that Reverend Robinson Duckworth, who accompanied Carroll and the Liddell sisters on the boat ride, says in his account of the trip: “…when three Miss Liddells were our passengers, and the story was actually composed and spoken over my shoulder for the benefit of Alice Liddell…I remember turning round and saying, “Dodgson, is this an extempore romance of yours?” And he replied, “Yes, I’m inventing as we go along.”” That story, on the insistence of Alice, was turned into a manuscript and presented to her by the Oxford mathematician.

By now, the content of the story is presented in disguised form with the use of riddles, apparently meaningless poems, puzzles, puns, and a lot more that is ostensibly nonsense. Carroll was surely not the first to use such devices.

Several examples of puns and riddles are found in nursery rhymes, and folk tales for children. The mastery of Carroll over this kind of recreational mathematics and logic takes Alice’s Adventures in Wonderland to a different league – it is not without reason that the story continues to inspire mathematical puzzles and word-game designers even today.

THE COMMUNITY OF ALICE
Raymond S Smullyan wrote a delightful little book titled Alice in Puzzle-Land: a Carrollian Tale for Children Under Eighty in which Alice and her friends return for another trip through Wonderland and the Looking-Glass. The book has 88 engaging puzzles, paradoxes, and logic problems. Smullyan’s characters speak and behave like the Carroll creations, and their puzzles abound in typical Carrollian word-play, logic problems, and dark philosophical paradoxes.

The rich tapestry of puzzles and paradoxes in Alice’s Adventures in Wonderland was a lifelong fascination for Carroll that in some way brought his ‘fairy tales’ closer to Austrian-British philosopher Ludwig Wittgenstein. In his 1965 essay “Wittgenstein, Nonsense, and Lewis Carroll,” philosopher George Pitcher’s talks about striking similarities between the philosophical writings of Wittgenstein and the children’s stories of Carroll. According to Pitcher, both were concerned with nonsense and language puzzles. While Wittgenstein was tortured by these things, Carroll appeared to be delighted by them.

Reverend Dodgson had a playful approach to mathematics that he imported into the Alice stories. He was known to use little puzzles in his lessons to make mathematics class more engaging. For instance, here is one of his classics (many versions of this puzzle now can be found all over the web): A cup contains 50 spoonfuls of brandy, and another contains 50 spoonfuls of water. A spoonful of brandy is taken from the first cup and mixed into the second cup. Then a spoonful of the mixture is taken from the second cup and mixed into the first. Is there more or less brandy in the second cup than there is water in the first cup? (If you are scratching your head for an answer, it is equal.)

FIGURE IT OUT
In that famous conversation with the Cheshire Cat, who wants to convince Alice that they both are mad, the feline tells her that she “…must be, or you wouldn’t have come here”, but Alice refuses to believe him and in turn asks how the cat knows that he is mad. The next set of conversations that appears in Chapter IV of the book shows how deep is the logic play in this work. Here Carroll has employed the so-called modus ponens, or affirming the antecedent logic.

“To begin with,” said the Cat, “a dog’s not mad. You grant that?”

“I suppose so,” said Alice.

“Well, then,” the Cat went on, “you see a dog growls when it’s angry, and wags its tail when it’s pleased. Now I growl when I’m pleased, and wag my tail when I’m angry. Therefore I’m mad.”

One can read the above dialogue without even realising that one is trapped in a logic web spun by Carroll. Here, the Cheshire Cat’s argument may appear sound but it is invalid. Here is how Carroll constructed the trap.

Suppose P and Q are two sentences; here, P is ‘an animal growls when angry and wags its tail when pleased’ and Q is ‘it is not mad’. Let us see what the cat says: ‘If an animal growls when angry and wags its tail when pleased, it is not mad.’ This means, if sentence P is true, then Q is also true.

‘I growl when pleased, and wag my tail when angry.’

Here the cat is not saying what P says.

‘Therefore, I am mad.’

So if the cat’s statement does not agree with P then how can it say Q is true?

One interesting aspect of Carroll’s work is that in the world of literature, especially literary criticism, a lot of emphasis has been on the psychoanalytic aspects of characters. There have been critiques highlighting Carroll’s own personal psychological and sexuality issues but almost nothing on reading the tale as a mathematical text. In 2009, Melanie Bayley, of the University of Oxford, published an article in the popular science magazine New Scientist titled “Alice’s Adventures in Algebra: Wonderland Solved”.

In the article Bayley says that Carroll added a lot of material to the illustrated manuscript he personally made for Alice before it was sent for publication. It is in these parts that Carroll took on the proponents of new mathematics, ridiculing their methods and questioning their rigour. The Cheshire Cat becoming a grin, according to the Oxford researcher, was Carroll’s way of portraying increasing and damaging abstraction in mathematics. In the Mad Hatter’s tea party, Bayley discovered the writer’s satire on Irish mathematician William Rowan Hamilton’s discovery – the quaternion.

There are other similar discoveries made by the Oxford researcher. In the scene where Alice is troubled by growing taller or shorter and meets the hookah-smoking Caterpillar, the creature tells Alice “keep your temper.” This Alice interprets as keeping cool but here Carroll is using an older meaning of the word ‘temper’ which was used for “the proportion in which qualities are mingled.” Bayley interprets this as the Caterpillar telling Alice irrespective of her body size she should maintain her body in proportion. If that is true, this reflects Carroll’s love of Euclidean geometry. In this geometry, absolute magnitude does not matter, it’s important to know the ratio of one length to another.

For a little more than 155 years after the story was first told to Alice, Lewis Carroll’s bestseller continues to throw new conundrums. No one can be absolutely sure whether Carroll actually plays those devious games with his readers. The reverend who stammered a lot and enjoyed the company of young girls did love his logic and Euclid like a fanatic. He is not remembered for his mathematics but for puzzles, logic games and biting satire. It is therefore not surprising that some of it made its way into his boat-ride story.

Debkumar Mitra is a Kolkata-based science writer and the author of Mindstretch, a book on mathematics, puzzles and stories.

SHARE THIS ARTICLE ON
Close
This image provided by Megan Telfer shows the wide selection of antique Pyrex dishes she displays at her Texas home. Telfer has more than 300 pieces of vintage Pyrex, displayed on three large bookcases. Telfer's 5-year-old daughter has some vintage Pyrex, too. "We don't use 90 percent of it," Telfer said. "I display it." (AP)
This image provided by Megan Telfer shows the wide selection of antique Pyrex dishes she displays at her Texas home. Telfer has more than 300 pieces of vintage Pyrex, displayed on three large bookcases. Telfer's 5-year-old daughter has some vintage Pyrex, too. "We don't use 90 percent of it," Telfer said. "I display it." (AP)

Pyrex and Pink Daisies: Midcentury vintage cookware is back in style

AP, New York
UPDATED ON MAR 03, 2021 06:15 PM IST
Vintage kitchenware is back in style -– pieces from the mid-20th century painted with flowers, bright colors, and specific functions, such as bracketed chip and dip bowls or four-piece refrigerator storage sets.
Close
Annie McNamara, left, and Sullivan Jones wear costumes designed by Dede Ayite during a performance of "Slave Play" in New York. Ayite has earned two 2021 Tony Award nominations for costume design, one for "Slave Play" and another for “A Soldier's Story.” (Matthew Murphy/DKC O&M Co. via AP)(AP)
Annie McNamara, left, and Sullivan Jones wear costumes designed by Dede Ayite during a performance of "Slave Play" in New York. Ayite has earned two 2021 Tony Award nominations for costume design, one for "Slave Play" and another for “A Soldier's Story.” (Matthew Murphy/DKC O&M Co. via AP)(AP)

Tonywatch: Dede Ayite's costumes always 'build up layers'

AP, New York
PUBLISHED ON MAR 03, 2021 05:56 PM IST
For “A Soldier's Play,” which explores racism within a Black U.S. Army unit, Ayite created special padding in the elbows and knees for actor David Alan Grier, who was frequently pummeled onstage. The soldiers' boots had to look broken in so she handed them out at the beginning of rehearsals.
Close
Sunil Dutt and his handwritten letter in Urdu to Mehrunissa Najma(Twitter)
Sunil Dutt and his handwritten letter in Urdu to Mehrunissa Najma(Twitter)

Personal letters, autographed photos: How Golden Era superstars connected

By Alfea Jamal, Hindustan Times, Delhi
UPDATED ON MAR 03, 2021 03:32 PM IST
Back in the 50s and 60s a young starstruck Bollywood fan, Mehrunissa Najma wrote to her favourite celebrities including legends Sunil Dutt, Saira Bano, Sadhana among others in hopes of autographs. The late cinema lover's hobby has now turned into an invaluable collection of India's cinematic history.
Close
The play will be the first to be staged at SRC since the coronavirus-induced lockdown on March 25.(Wikimedia Commons)
The play will be the first to be staged at SRC since the coronavirus-induced lockdown on March 25.(Wikimedia Commons)

'Giraftari': Shri Ram Centre to welcome back theatre lovers

PTI, New Delhi
PUBLISHED ON MAR 03, 2021 12:57 PM IST
In good news for theatre enthusiasts, the Shri Ram Centre (SRC) will finally reopen its doors to the public on Friday with 'Giraftari', a play based on Franz Kafka's popular novel "The Trial", said the organisers in a statement.
Close
About 100 vehicles gathered in a Khartoum parking lot across from giant screens showing Sudanese and European films on Friday, the start of a week-long festival organized by the British Council. (Representational Image) (Unsplash)
About 100 vehicles gathered in a Khartoum parking lot across from giant screens showing Sudanese and European films on Friday, the start of a week-long festival organized by the British Council. (Representational Image) (Unsplash)

Festival gives Sudanese film lovers drive-in cinema

Reuters, Khartoum
PUBLISHED ON MAR 03, 2021 12:02 PM IST
Sudanese moviegoers are enjoying what organisers are saying is their first drive-in cinema after a festival showcasing the country's resurgent, post-uprising film scene moved outdoors this year due to the Covid-19 pandemic.
Close
This photo, provided by Sotheby's, in New York, on Tuesday, March 2, 2021, shows a small porcelain bowl bought for $35 at a Connecticut yard sale that turned out to be a rare, 15th century Chinese artifact worth between $300,000 and $500,000. The bowl will be offered in Sotheby's Auction of Important Chinese Art, in New York, on March 17. (AP)
This photo, provided by Sotheby's, in New York, on Tuesday, March 2, 2021, shows a small porcelain bowl bought for $35 at a Connecticut yard sale that turned out to be a rare, 15th century Chinese artifact worth between $300,000 and $500,000. The bowl will be offered in Sotheby's Auction of Important Chinese Art, in New York, on March 17. (AP)

Yard sale find turns out to be artifact worth up to $500,000

AP, Hartford, Connecticut
PUBLISHED ON MAR 03, 2021 10:13 AM IST
A small porcelain bowl bought for $35 at a Connecticut yard sale turned out to be a rare, 15th century Chinese artifact worth between $300,000 and $500,000 that is about to go up for auction at Sotheby's.
Close
FILE - In this Sept. 24, 2013, file photo, Courtney Keating, education coordinator of The Literacy Center in Evansville, Ind., reads "If I Ran the Zoo," By Dr. Seuss, to passersby during an event to promote literacy along the Evansville Riverfront. Dr. Seuss Enterprises, the business that preserves and protects the author and illustrator's legacy, announced on his birthday, Tuesday, March 2, 2021, that it would cease publication of several children's titles including "And to Think That I Saw It on Mulberry Street" and "If I Ran the Zoo," because of insensitive and racist imagery.(AP)
FILE - In this Sept. 24, 2013, file photo, Courtney Keating, education coordinator of The Literacy Center in Evansville, Ind., reads "If I Ran the Zoo," By Dr. Seuss, to passersby during an event to promote literacy along the Evansville Riverfront. Dr. Seuss Enterprises, the business that preserves and protects the author and illustrator's legacy, announced on his birthday, Tuesday, March 2, 2021, that it would cease publication of several children's titles including "And to Think That I Saw It on Mulberry Street" and "If I Ran the Zoo," because of insensitive and racist imagery.(AP)

Hurtful and wrong: Dr Seuss books pulled from publication due to racist imagery

Reuters, New York
UPDATED ON MAR 03, 2021 08:42 AM IST
The six books - "And to Think That I Saw It on Mulberry Street," "If I Ran the Zoo," "McElligot's Pool," "On Beyond Zebra!" "Scrambled Eggs Super!" and "The Cat's Quizzer" - are among more than 60 classics written by Dr. Seuss, the pen name of the American writer and illustrator Theodor Geisel, who died in 1991.
Close
Freshers’s party in Delhi University colleges are being hosted offline as well as an online, but the later is only happening unofficially.
Freshers’s party in Delhi University colleges are being hosted offline as well as an online, but the later is only happening unofficially.

DU diaries: Offline versus online freshers ka mahayudh

By Aprajita Sharad, New Delhi
PUBLISHED ON MAR 02, 2021 08:31 PM IST
Freshers’ parties have mostly shifted online for DU students, but some seniors keep hosting unofficial offline freshers party for college students who are in Delhi-NCR.
Close
And artwork from Jatin Das's Exodus 2020.(Instagram)
And artwork from Jatin Das's Exodus 2020.(Instagram)

Famed artist Jatin Das captures migrants' lockdown ordeal in dozens of paintings

Reuters, New Delhi
UPDATED ON MAR 02, 2021 07:01 PM IST
Jatin Das, one of India's best known contemporary artists, was so moved by the plight of migrant workers trekking out of the cities during lockdown, he felt compelled to depict their ordeal.
Close
Mick Rock collaborates with urban artist Fin DAC, fuses photography and painting(Instagram/therealmickrock)
Mick Rock collaborates with urban artist Fin DAC, fuses photography and painting(Instagram/therealmickrock)

Mick Rock collaborates with urban artist Fin DAC, fuses photography and painting

Reuters
UPDATED ON MAR 02, 2021 06:24 PM IST
Legendary photographer Mick Rock to mark 51 years of working in the music industry with a new project, in collaboration with urban artist Fin DAC, to create a series of limited edition prints and canvas artworks.
Close
The house at 30 Fifeshire Rd. is built like a 17th century French palace. (Property Vision)
The house at 30 Fifeshire Rd. is built like a 17th century French palace. (Property Vision)

Schitt’s Creek mansion returns to market with price reduction

Bloomberg
PUBLISHED ON MAR 02, 2021 03:07 PM IST
And with its frescoed ceilings, crystal chandeliers, marble staircase, and grand domed cupola, that’s exactly the look Van Lapoyan was going for when he built his Toronto mansion.
Close
Churchill painting owned by Angelina Jolie sells for USD 11.5M(Reuters)
Churchill painting owned by Angelina Jolie sells for USD 11.5M(Reuters)

Landscape painted by Churchill and owned by Angelina Jolie sells for USD 11.5M

AP, London
PUBLISHED ON MAR 02, 2021 09:15 AM IST
A painting of a Moroccan landscape which was made by Britain's World War II leader Winston Churchill and gifted to US President Franklin D Roosevelt was bought by Angelina Jolie in 2011. She recently sold the painting for a whooping amount of USD 11.5 million.
Close
"EVERYDAYS: THE FIRST 5000 DAYS" is a collage, by a digital artist BEEPLE, that is on auction at Christie's. (REUTERS)
"EVERYDAYS: THE FIRST 5000 DAYS" is a collage, by a digital artist BEEPLE, that is on auction at Christie's. (REUTERS)

Photos: Digital artist Beeple's work auctioned at Christie's for $6.6 million

Reuters
PUBLISHED ON MAR 01, 2021 07:42 PM IST
A recent Christie's auction offered the first-ever purely digital work of art: Everydays - The First 5000 Days by Mike Winkelmann, also known as, Beeple. Within the space of an hour the bidding amount of $100 has jumped to $1 million. Beeple's digital art is known to carry an NFT (non-fungible token), which is a unique digital token that carries the artist's signature. Beeple's work focuses on 'society's alternating obsession with and fear of technology.
Close
Ranjan Kamath recording the life story of Kannada filmmaker Girish Kasaravalli. Kamath has published 20 such histories on YouTube. They include celebrities and ordinary folks.
Ranjan Kamath recording the life story of Kannada filmmaker Girish Kasaravalli. Kamath has published 20 such histories on YouTube. They include celebrities and ordinary folks.

Putting life on the record with the Mitra Tantra Archive

By Natasha Rego
PUBLISHED ON MAR 01, 2021 06:03 PM IST
Filmmaker Ranjan Kamath is recording oral history of luminaries and lay people. Anyone with a story to tell, he says, is worth hearing out
Close
This photo provided by the New Orleans Museum of Art shows "Peking University, Beijing, China," 1991, by Lois Conner. The print is a gift from Cherye R. and James F. Pierce to the museum.(AP)
This photo provided by the New Orleans Museum of Art shows "Peking University, Beijing, China," 1991, by Lois Conner. The print is a gift from Cherye R. and James F. Pierce to the museum.(AP)

New Orleans Museum of Art announces 3 big gifts in 1 month

AP, New Orleans
PUBLISHED ON MAR 01, 2021 03:39 PM IST
The New Orleans Museum of Art has announced a major gift to its photography department - the third such announcement this month.
Close
SHARE
Story Saved
OPEN APP