The World Record for the Shortest Math Article: 2 Words

in Math | May 16th, 2025

In 2004, John Con­way and Alexan­der Soifer, both work­ing on math­e­mat­ics at Prince­ton Uni­ver­si­ty, sub­mit­ted to the Amer­i­can Math­e­mat­i­cal Month­ly what they believed was “a new world record in the num­ber of words in a [math] paper.”

Soifer explains: “On April 28, 2004 … I sub­mit­ted our paper that includ­ed just two words, ‘n2 + 2 can’ and our two draw­ings. [See one of them above.]” The sto­ry then con­tin­ues: “The Amer­i­can Math­e­mat­i­cal Month­ly was sur­prised, and did not know what to do about our new world record of a 2‑word arti­cle. Two days lat­er, on April 30, 2004, the Edi­to­r­i­al Assis­tant Mrs. Mar­garet Combs acknowl­edged the receipt of the paper”:

The Month­ly pub­lish­es expo­si­tion of math­e­mat­ics at many lev­els, and it con­tains arti­cles both long and short. Your arti­cle, how­ev­er, is a bit too short to be a good Month­ly arti­cle… A line or two of expla­na­tion would real­ly help.

Soifer writ­ers: “The same day at the cof­fee hour I asked John [Con­way], ‘What do you think?’ His answer was con­cise, ‘Do not give up too eas­i­ly.’ Accord­ing­ly, I replied [to] The Month­ly the same day”:

I respect­ful­ly dis­agree that a short paper in general—and this paper in particular—merely due to its size must be “a bit too short to be a good Month­ly arti­cle.” Is there a con­nec­tion between quan­ti­ty and qual­i­ty?… We have posed a fine (in our opin­ion) open prob­lem and report­ed two dis­tinct “behold-style” proofs of our advance on this prob­lem. What else is there to explain?

Soifer adds: “The Month­ly, appar­ent­ly felt out­gunned, for on May 4, 2004, the reply came from The Month­ly’s top gun, Edi­tor-in-Chief Bruce Pal­ka”:

The Month­ly pub­lish­es two types of papers: “arti­cles,” which are sub­stan­tive expos­i­to­ry papers rang­ing in length from about six to twen­ty-five pages, and “notes,” which are short­er, fre­quent­ly some­what more tech­ni­cal pieces (typ­i­cal­ly in the one-to-five page range). I can send your paper to the notes edi­tor if you wish, but I expect that he’ll not be inter­est­ed in it either because of its length and lack of any sub­stan­tial accom­pa­ny­ing text. The stan­dard way in which we use such short papers these days is as “boxed filler” on pages that would oth­er­wise con­tain a lot of the blank space that pub­lish­ers abhor. If you’d allow us to use your paper in that way, I’d be hap­py to pub­lish it.

Soifer con­cludes: “John Con­way and I accept­ed the ‘filler’, and in the Jan­u­ary 2005 issue our paper was pub­lished.” Vic­to­ry!

Get more of the back­sto­ry here.

Relat­ed Con­tent:

The Short­est-Known Paper Pub­lished in a Seri­ous Math Jour­nal: Two Suc­cinct Sen­tences

John Nash’s Super Short PhD The­sis: 26 Pages & Two Cita­tions

The Map of Math­e­mat­ics: Ani­ma­tion Shows How All the Dif­fer­ent Fields in Math Fit Togeth­er

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John Nash’s Super Short PhD Thesis: 26 Pages & Two Citations

in Math | May 12th, 2025

When John Nash wrote “Non-Coop­er­a­tive Games,” his Ph.D. dis­ser­ta­tion at Prince­ton in 1950, the text of his the­sis (read it online) was brief. It ran only 26 pages. And more par­tic­u­lar­ly, it was light on cita­tions. Nash’s diss cit­ed two texts: John von Neu­mann and Oskar Mor­gen­stern’s The­o­ry of Games and Eco­nom­ic Behav­ior (1944), which essen­tial­ly cre­at­ed game the­o­ry and rev­o­lu­tion­ized the field of eco­nom­ics; the oth­er cit­ed text, “Equi­lib­ri­um Points in n‑Person Games,” was an arti­cle writ­ten by Nash him­self. And it laid the foun­da­tion for his dis­ser­ta­tion, anoth­er sem­i­nal work in the devel­op­ment of game the­o­ry, for which Nash was award­ed the Nobel Prize in Eco­nom­ic Sci­ences in 1994.

The reward of invent­ing a new field is hav­ing a slim bib­li­og­ra­phy.

If you would like to sign up for Open Culture’s free email newslet­ter, please find it here. It’s a great way to see our new posts, all bun­dled in one email, each day.

If you would like to sup­port the mis­sion of Open Cul­ture, con­sid­er mak­ing a dona­tion to our site. It’s hard to rely 100% on ads, and your con­tri­bu­tions will help us con­tin­ue pro­vid­ing the best free cul­tur­al and edu­ca­tion­al mate­ri­als to learn­ers every­where. You can con­tribute through Pay­Pal, Patre­on, and Ven­mo (@openculture). Thanks!

Note: An ear­li­er ver­sion of this post appeared on our site in June, 2015.

Relat­ed Con­tent:

The Short­est-Known Paper Pub­lished in a Seri­ous Math Jour­nal: Two Suc­cinct Sen­tences

The World Record for the Short­est Math Arti­cle: 2 Words

Doc­tor­al Dis­ser­ta­tion as a Graph­ic Nov­el: Read a Free Excerpt of Nick Sou­sa­nis’ Unflat­ten­ing

How to Dance Your Dis­ser­ta­tion: See the Win­ning Video in the 2014 “Dance Your PhD” Con­test

Umber­to Eco’s How To Write a The­sis: A Wit­ty, Irrev­er­ent & High­ly Prac­ti­cal Guide Now Out in Eng­lish

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The Map of Mathematics: Animation Shows How All the Different Fields in Math Fit Together

in Math | April 8th, 2025

Back in Decem­ber, you hope­ful­ly thor­ough­ly immersed your­self in The Map of Physics, an ani­mat­ed video–a visu­al aid for the mod­ern age–that mapped out the field of physics, explain­ing all the con­nec­tions between clas­si­cal physics, quan­tum physics, and rel­a­tiv­i­ty.

You can’t do physics with­out math. Hence we now have The Map of Math­e­mat­ics. Cre­at­ed by physi­cist Dominic Wal­li­man, this video explains “how pure math­e­mat­ics and applied math­e­mat­ics relate to each oth­er and all of the sub-top­ics they are made from.” Watch the new video above. You can buy a poster of the map here. And you can down­load a ver­sion for edu­ca­tion­al use here.

If you would like to sign up for Open Culture’s free email newslet­ter, please find it here. It’s a great way to see our new posts, all bun­dled in one email, each day.

If you would like to sup­port the mis­sion of Open Cul­ture, con­sid­er mak­ing a dona­tion to our site. It’s hard to rely 100% on ads, and your con­tri­bu­tions will help us con­tin­ue pro­vid­ing the best free cul­tur­al and edu­ca­tion­al mate­ri­als to learn­ers every­where. You can con­tribute through Pay­Pal, Patre­on, and Ven­mo (@openculture). Thanks!

Relat­ed Con­tent 

Com­plex Math Made Sim­ple With Engag­ing Ani­ma­tions: Fouri­er Trans­form, Cal­cu­lus, Lin­ear Alge­bra, Neur­al Net­works & More

John Coltrane Draws a Pic­ture Illus­trat­ing the Math­e­mat­ics of Music

The Ele­gant Math­e­mat­ics of Vit­ru­vian Man, Leonar­do da Vinci’s Most Famous Draw­ing: An Ani­mat­ed Intro­duc­tion

The Math Behind Beethoven’s Music

Free Online Math Cours­es

 

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The Decimal Point Is 150 Years Older Than We Thought, Emerging in Renaissance Italy

in History, Math | March 5th, 2024

His­to­ri­ans have long thought that the dec­i­mal point first came into use in 1593, when the Ger­man math­e­mati­cian Christo­pher Clav­ius wrote an astron­o­my text called Astro­labi­um. It turns out, how­ev­er, that the his­to­ry of the dec­i­mal point stretch­es back anoth­er 150 years–to the work of the Venet­ian mer­chant Gio­van­ni Bian­chi­ni. In his text Tab­u­lae pri­mi mobilis, writ­ten dur­ing the 1440s, Bian­chi­ni used the dec­i­mal point to cal­cu­late the coor­di­nates of plan­ets. In so doing, he invent­ed a sys­tem of dec­i­mal frac­tions, which, in turn, made the cal­cu­la­tions under­pin­ning mod­ern sci­ence more effi­cient and less com­plex, notes Sci­en­tif­ic Amer­i­can.

Glen Van Brum­me­len, a his­to­ri­an of math­e­mat­ics, recent­ly recount­ed to NPR how he dis­cov­ered Bian­chini’s inno­va­tion:

I was work­ing on the man­u­script of this astronomer, Gio­van­ni Bian­chi­ni. I saw the dots inside of a table — in a numer­i­cal table. And when he explained his cal­cu­la­tions, it became clear that what he was doing was exact­ly the same thing as we do with the dec­i­mal point. And I’m afraid I got rather excit­ed at that point. I grabbed my com­put­er, ran up and down the dorm hall­way look­ing for col­leagues who still had­n’t gone to bed, say­ing, this per­son­’s work­ing with the dec­i­mal point in the 1440s. I think they prob­a­bly thought I was crazy.

In a new arti­cle appear­ing in the jour­nal His­to­ria Math­e­mat­i­ca, Van Brum­me­len explains the his­tor­i­cal sig­nif­i­cance of the dec­i­mal point, and what this dis­cov­ery means for the his­tor­i­cal devel­op­ment of math­e­mat­ics. You can read it online.

Relat­ed Con­tent 

Trigonom­e­try Dis­cov­ered on a 3700-Year-Old Ancient Baby­lon­ian Tablet

How the Ancient Greeks Shaped Mod­ern Math­e­mat­ics: A Short, Ani­mat­ed Intro­duc­tion

The Map of Math­e­mat­ics: An Ani­mat­ed Video Shows How All the Dif­fer­ent Fields in Math Fit Togeth­er

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A Retired Math Teacher Helps Students Learn Geometry Through Quilting

in Creativity, Design, Education, K-12, Math | February 27th, 2023

Some real talk from retired geom­e­try teacher Wendy Licht­man, above, the author of sev­er­al math-themed YA nov­els:

Not many 15-year-olds care that two par­al­lel lines are crossed by a trans­ver­sal.

“But right here are two par­al­lel lines,” she con­tin­ues, point­ing to a pink and orange quilt. “and these are trans­ver­sals, and they are at a 90º angle and it feels real. You’ve got­ta get it to look right.”

The teenaged par­tic­i­pants in the Oak­land, Cal­i­for­nia pro­gram she found­ed to demys­ti­fy geom­e­try through hands-on quilt­mak­ing get it to look right by plot­ting their designs on graph paper, care­ful­ly mea­sur­ing and cut­ting shapes from bright cal­i­co of their own choos­ing. (Lic­th­man has com­mit­ted to but­ton­ing her lip if their favored print is not to her taste.)

Licht­man came up with this cre­ative approach to help a bright stu­dent who was in dan­ger of not grad­u­at­ing, hav­ing flunked geom­e­try three times.

She details their jour­ney in How to Make a Geo­met­ric Quilt, an essay for­mat­ted as step-by-step instructions…not for quilt­mak­ing but rather how those in the teach­ing pro­fes­sion can lead with humil­i­ty and deter­mi­na­tion, while main­tain­ing good bound­aries.

Some high­lights:

6. Some­time after the sewing has begun, and the math note­book is ignored for weeks, begin to wor­ry that your stu­dent is not real­ly learn­ing geom­e­try.  She’s learn­ing sewing and she’s learn­ing to fix a bro­ken bob­bin, but real­ly, geom­e­try?

7. Remind your­self that this kid needs a quilt as much as she needs geom­e­try.

8. Remem­ber, also, the very, very old woman who taught you hat-mak­ing one night long ago.  She had gone to school only through 5th grade because, she said, she was a Black child in the deep south and that’s how it was back then.  Think about how she explained to the hat-mak­ing class that to fig­ure out the length of the hat’s brim, you need­ed to mea­sure from the cen­ter to the edge with a string and then do “three of those and a lit­tle bit more,” and remem­ber how you sat in awe, because three radii and a lit­tle bit more is the def­i­n­i­tion of pi, and this hat-mak­er had evi­dent­ly dis­cov­ered for her­self the for­mu­la for cir­cum­fer­ence.

As the two become bet­ter acquaint­ed, the stu­dent let her guard down, reveal­ing more about her sit­u­a­tion while they swapped sto­ries of their moth­ers.

But this was no easy A.

In addi­tion to expect­ing reg­u­lar, punc­tu­al atten­dance, Lict­man stip­u­lat­ed that in order to pass, the stu­dent could not give the fruits of her labor away.

(Sol­id advice for cre­ators of any craft project this ambi­tious. As Deb­bie Stoller, author of Stitch ‘n Bitch: The Knit­ter’s Hand­book coun­sels:

…those who have nev­er knit some­thing have no idea how much time it took. If you give some­one a sweater, they may think that you made that in an evening when you were watch­ing a half-hour sit­com. It’s only when peo­ple actu­al­ly attempt to knit that they final­ly get this real­iza­tion, this light bulb goes on over their heads, and they real­ize that, “Wow, this actu­al­ly takes some skill and some time. I’ve got new­found respect for my grand­ma.”)

Ulti­mate­ly, Licht­man con­cludes that the five cred­its she award­ed her stu­dent could not be reduced to some­thing as sim­ple as geom­e­try or quilt-mak­ing;

You are giv­ing her cred­it for some­thing less tan­gi­ble.  Some­thing like pride.  Five cred­it hours for feel­ing she can accom­plish some­thing hard that, okay, is slight­ly relat­ed to geom­e­try.

Exam­ples of the cur­rent cohort’s work can be seen on Rock Paper Scis­sors Col­lec­tive’s Insta­gram.

Once com­plet­ed, these quilts will be donat­ed to Bay Area fos­ter chil­dren and pedi­atric patients at the local Chil­dren’s Hos­pi­tal.

via Boing­Bo­ing

Relat­ed Con­tent 

The Solar Sys­tem Quilt: In 1876, a Teacher Cre­ates a Hand­craft­ed Quilt to Use as a Teach­ing Aid in Her Astron­o­my Class

17-Year-Old Ade­line Har­ris Cre­at­ed a Quilt Col­lect­ing 360 Sig­na­tures of the Most Famous Peo­ple of the 19th Cen­tu­ry: Lin­coln, Dick­ens, Emer­son & More (1863)

Bisa Butler’s Beau­ti­ful Quilt­ed Por­traits of Fred­er­ick Dou­glass, Nina Simone, Jean-Michel Basquiat & More

Via Boing Boing

– Ayun Hal­l­i­day is the Chief Pri­ma­tol­o­gist of the East Vil­lage Inky zine and author, most recent­ly, of Cre­ative, Not Famous: The Small Pota­to Man­i­festo and Cre­ative, Not Famous Activ­i­ty Book. Fol­low her @AyunHalliday.

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Why Algorithms Are Called Algorithms, and How It All Goes Back to the Medieval Persian Mathematician Muhammad al-Khwarizmi

in History, Math, Technology | April 12th, 2022

In recent decades, a medieval Per­sian word has come to promi­nence in Eng­lish and oth­er major world lan­guages. Many of use it on a dai­ly basis, often while regard­ing the con­cept to which it refers as essen­tial­ly mys­te­ri­ous. The word is algo­rithm, whose roots go back to the ninth cen­tu­ry in mod­ern-day Greater Iran. There lived a poly­math by the name of Muham­mad ibn Musa al-Khwariz­mi, whom we now remem­ber for his achieve­ments in geog­ra­phy, astron­o­my, and math­e­mat­ics. In that last field, he was the first to define the prin­ci­ples of “reduc­ing” and “bal­anc­ing” equa­tions, a sub­ject all of us came to know in school as alge­bra (a name itself descend­ed from the Ara­bic al-jabr, or “com­ple­tion”).

Today, a good few of us have come to resent algo­rithms even more than alge­bra. This is per­haps because algo­rithms are most pop­u­lar­ly asso­ci­at­ed with the deep, unseen work­ings of the inter­net, a sys­tem with ever increas­ing influ­ence over the things we do, the infor­ma­tion we receive, and even the peo­ple with whom we asso­ciate.

Pro­vid­ed suf­fi­cient data about us and the lives we lead, so we’re giv­en to under­stand, these algo­rithms can make bet­ter deci­sions for us than we can make for our­selves. But what exact­ly are they? You can get one answer from “Why Algo­rithms Are Called Algo­rithms,” the BBC Ideas video at the top of the post.

For West­ern civ­i­liza­tion, al-Khwarizmi’s most impor­tant book was Con­cern­ing the Hin­du Art of Reck­on­ing, which was trans­lat­ed into Latin three cen­turies after its com­po­si­tion. Al-Khwarizmi’s Latinized name “Algo­rit­mi” gave rise to the word algo­ris­mus, which at first referred to the dec­i­mal num­ber sys­tem and much lat­er came to mean “a set of step-by-step rules for solv­ing a prob­lem.” It was Enig­ma code­break­er Alan Tur­ing who “worked out how, in the­o­ry, a machine could fol­low algo­rith­mic instruc­tions and solve com­plex math­e­mat­ics. This was the birth of the com­put­er age.” Now, much fur­ther into the com­put­er age, algo­rithms “are help­ing us to get from A to B, dri­ving inter­net search­es, mak­ing rec­om­men­da­tions of things for us to buy, watch, or share.”

The algo­rithm giveth, but the algo­rithm also taketh away — or so it some­times feels as we make our way deep­er into the twen­ty-first cen­tu­ry. In the oth­er BBC Ideas video just above, Jon Stroud makes an inves­ti­ga­tion into both the nature and the cur­rent uses of this math­e­mat­i­cal con­cept. The essen­tial job of an algo­rithm, as the experts explain to him, is that of pro­cess­ing data, these days often in large quan­ti­ties and of var­i­ous kinds, and increas­ing­ly with the aid of sophis­ti­cat­ed machine-learn­ing process­es. In mak­ing or influ­enc­ing choic­es humans would once have han­dled them­selves, algo­rithms do present a risk of “de-skilling” as we come to rely on their ser­vices. We all occa­sion­al­ly feel grat­i­tude for the bless­ings those ser­vices send our way, just as we all occa­sion­al­ly blame them for our dis­sat­is­fac­tions — mak­ing the algo­rithm, in oth­er words, into a thor­ough­ly mod­ern deity.

Relat­ed con­tent:

Algo­rithms for Big Data: A Free Course from Har­vard

Advanced Algo­rithms: A Free Course from Har­vard Uni­ver­si­ty

This Is Your Kids’ Brains on Inter­net Algo­rithms: A Chill­ing Case Study Shows What’s Wrong with the Inter­net Today

The Prob­lem with Face­book: “It’s Keep­ing Things From You”

The Com­plex Geom­e­try of Islam­ic Art & Design: A Short Intro­duc­tion

How Youtube’s Algo­rithm Turned an Obscure 1980s Japan­ese Song Into an Enor­mous­ly Pop­u­lar Hit: Dis­cov­er Mariya Takeuchi’s “Plas­tic Love”

Based in Seoul, Col­in Mar­shall writes and broad­casts on cities, lan­guage, and cul­ture. His projects include the Sub­stack newslet­ter Books on Cities, the book The State­less City: a Walk through 21st-Cen­tu­ry Los Ange­les and the video series The City in Cin­e­ma. Fol­low him on Twit­ter at @colinmarshall or on Face­book.

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Bach on a Möbius Strip: Marcus du Sautoy Visualizes How Bach Used Math to Compose His Music

in Math, Music | November 18th, 2021

“A math­e­mati­cian’s favorite com­pos­er? Top of the list prob­a­bly comes Bach.” Thus speaks a reli­able source on the mat­ter: Oxford math­e­mati­cian Mar­cus du Sautoy in the Num­ber­phile video above. “Bach uses a lot of math­e­mat­i­cal tricks as a way of gen­er­at­ing music, so his music is high­ly com­plex,” but at its heart is “the use of math­e­mat­ics as a kind of short­cut to gen­er­ate extra­or­di­nar­i­ly com­plex music.” As a first exam­ple du Sautoy takes up the “Musi­cal Offer­ing,” and in par­tic­u­lar its “crab canon,” the genius of which has pre­vi­ous­ly been fea­tured here on Open Cul­ture.

Writ­ten out, Bach’s crab canon “looks like just one line of music.” But “what’s curi­ous is that when you get to the end of the music, there’s the lit­tle sym­bol you usu­al­ly begin a piece of music with.” This means that Bach wants the play­er of the piece to “play this for­wards and back­wards; he’s ask­ing you to start at the end and play it back­wards at the same time.” His com­po­si­tion thus becomes a two-voice piece made out of just one line of music going in both direc­tions. It’s the under­ly­ing math­e­mat­ics that make this, when played, more than just a trick but “some­thing beau­ti­ful­ly har­mon­ic and com­plex.”

To under­stand the crab canon or Bach’s oth­er math­e­mat­i­cal­ly shaped pieces, it helps to visu­al­ize them in uncon­ven­tion­al ways such as on a twist­ing Möbius strip, whose ends con­nect direct­ly to one anoth­er. “You can make a Möbius strip out of any piece of music,” says du Sautoy as he does so in the video. “The stun­ning thing is that when you then look at this piece of music” — that is the fifth canon from Bach’s Gold­berg Vari­a­tions — “the notes that are on one side are exact­ly the same notes as if this thing were see-through.” (Nat­u­ral­ly, he’s also pre­pared a see-through Bach Möbius strip for his view­ing audi­ence.)

In 2017 du Sautoy gave an Oxford Math­e­mat­ics Pub­lic Lec­ture on “the Sound of Sym­me­try and the Sym­me­try of Sound.” In it he dis­cuss­es sym­me­try as present in not just the Gold­berg Vari­a­tions but the twelve-tone rows com­posed in the 20th cen­tu­ry by Arnold Schoen­berg and even the very sound waves made by musi­cal instru­ments them­selves. Just this year, he col­lab­o­rat­ed with the Oxford Phil­har­mon­ic Orches­tra to deliv­er “Music & Maths: Baroque & Beyond,” a pre­sen­ta­tion that draws math­e­mat­i­cal con­nec­tions between the music, art, archi­tec­ture, and sci­ence going on in the 17th and 18th cen­turies. Bach has been dead for more than a quar­ter of a mil­len­ni­um, but the con­nec­tions embod­ied in his music still hold rev­e­la­tions for lis­ten­ers will­ing to hear them — or see them.

Relat­ed Con­tent:

Take an Intel­lec­tu­al Odyssey with a Free MIT Course on Dou­glas Hofstadter’s Pulitzer Prize-Win­ning Book Gödel, Esch­er, Bach: An Eter­nal Gold­en Braid

The Genius of J.S. Bach’s “Crab Canon” Visu­al­ized on a Möbius Strip

Visu­al­iz­ing Bach: Alexan­der Chen’s Impos­si­ble Harp

How a Bach Canon Works. Bril­liant

The Math Behind Beethoven’s Music

Based in Seoul, Col­in Mar­shall writes and broad­casts on cities, lan­guage, and cul­ture. His projects include the Sub­stack newslet­ter Books on Cities, the book The State­less City: a Walk through 21st-Cen­tu­ry Los Ange­les and the video series The City in Cin­e­ma. Fol­low him on Twit­ter at @colinmarshall or on Face­book.

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Take an Intellectual Odyssey with a Free MIT Course on Douglas Hofstadter’s Pulitzer Prize-Winning Book Gödel, Escher, Bach: An Eternal Golden Braid

in Computer Science, Creativity, Math, MIT, Music, Neuroscience, Philosophy, Psychology | August 2nd, 2021

In 1979, math­e­mati­cian Kurt Gödel, artist M.C. Esch­er, and com­pos­er J.S. Bach walked into a book title, and you may well know the rest. Dou­glas R. Hof­s­tadter won a Pulitzer Prize for Gödel, Esch­er, Bach: an Eter­nal Gold­en Braid, his first book, thence­forth (and hence­forth) known as GEB. The extra­or­di­nary work is not a trea­tise on math­e­mat­ics, art, or music, but an essay on cog­ni­tion through an explo­ration of all three — and of for­mal sys­tems, recur­sion, self-ref­er­ence, arti­fi­cial intel­li­gence, etc. Its pub­lish­er set­tled on the pithy descrip­tion, “a metaphor­i­cal fugue on minds and machines in the spir­it of Lewis Car­roll.”

GEB attempt­ed to reveal the mind at work; the minds of extra­or­di­nary indi­vid­u­als, for sure, but also all human minds, which behave in sim­i­lar­ly unfath­omable ways. One might also describe the book as oper­at­ing in the spir­it — and the prac­tice — of Her­man Hesse’s Glass Bead Game, a nov­el Hesse wrote in response to the data-dri­ven machi­na­tions of fas­cism and their threat to an intel­lec­tu­al tra­di­tion he held par­tic­u­lar­ly dear. An alter­nate title (and key phrase in the book) Mag­is­ter Ludi, puns on both “game” and “school,” and alludes to the impor­tance of play and free asso­ci­a­tion in the life of the mind.

Hesse’s eso­teric game, writes his biog­ra­ph­er Ralph Freed­man, con­sists of “con­tem­pla­tion, the secrets of the Chi­nese I Ching and West­ern math­e­mat­ics and music” and seems sim­i­lar enough to Hof­s­tadter’s approach and that of the instruc­tors of MIT’s open course, Gödel, Esch­er, Bach: A Men­tal Space Odyssey. Offered through the High School Stud­ies Pro­gram as a non-cred­it enrich­ment course, it promis­es “an intel­lec­tu­al vaca­tion” through “Zen Bud­dhism, Log­ic, Meta­math­e­mat­ics, Com­put­er Sci­ence, Arti­fi­cial Intel­li­gence, Recur­sion, Com­plex Sys­tems, Con­scious­ness, Music and Art.”

Stu­dents will not study direct­ly the work of Gödel, Esch­er, and Bach but rather “find their spir­its aboard our men­tal ship,” the course descrip­tion notes, through con­tem­pla­tions of canons, fugues, strange loops, and tan­gled hier­ar­chies. How do mean­ing and form arise in sys­tems like math and music? What is the rela­tion­ship of fig­ure to ground in art? “Can recur­sion explain cre­ativ­i­ty,” as one of the course notes asks. Hof­s­tadter him­self has pur­sued the ques­tion beyond the entrench­ment of AI research in big data and brute force machine learn­ing. For all his daunt­ing eru­di­tion and chal­leng­ing syn­the­ses, we must remem­ber that he is play­ing a high­ly intel­lec­tu­al game, one that repli­cates his own expe­ri­ence of think­ing.

Hof­s­tadter sug­gests that before we can under­stand intel­li­gence, we must first under­stand cre­ativ­i­ty. It may reveal its secrets in com­par­a­tive analy­ses of the high­est forms of intel­lec­tu­al play, where we see the clever for­mal rules that gov­ern the mind’s oper­a­tions; the blind alleys that explain its fail­ures and lim­i­ta­tions; and the pos­si­bil­i­ty of ever actu­al­ly repro­duc­ing work­ings in a machine. Watch the lec­tures above, grab a copy of Hofstadter’s book, and find course notes, read­ings, and oth­er resources for the fas­ci­nat­ing course Gödel, Esch­er, Bach: A Men­tal Space Odyssey archived here. The course will be added to our list, 1,700 Free Online Cours­es from Top Uni­ver­si­ties.

Relat­ed Con­tent: 

How a Bach Canon Works. Bril­liant.

Math­e­mat­ics Made Vis­i­ble: The Extra­or­di­nary Math­e­mat­i­cal Art of M.C. Esch­er

The Mir­ror­ing Mind: An Espres­so-Fueled Inter­pre­ta­tion of Dou­glas Hofstadter’s Ground­break­ing Ideas

Josh Jones is a writer and musi­cian based in Durham, NC. Fol­low him at @jdmagness

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