**[a]**(have been waiting for opportune time to write this up since may.) proofs like these are a complex or “complicated relationship” for mathematicians (aka facebook-speak), a love-hate affair. (did the extraordinary/ breakthrough/ revolutionary/ paradigm-shifting 4-color computer proof ever win any awards? and how much despair/ handwringing and further effort has there been over it over the decades?)

the breakthrough is celebrated but mathematicians would like to see shorter proofs that are human-comprehensible, so there are mixed/ ambivalent feelings about it within the community. have written on this topic quite at length in this blog even since its beginnings, and this latest breakthrough is delightfully affirmationally crosscutting across many of this blogs categories, and think this is the tip of the iceberg of 21st century mathematics in a way not yet fully recognized. its a dramatic, vivid realization/ materialization of an idea suggested a few years ago here called “SAT induction.” think that these types of proofs will lead to new theory that is indeed human comprehensible but some of the isolated theorems will be claimed first by computer analysis before the more thorough theory catches up to integrate them.

my thinking is that there is some basic theory waiting to be uncovered to explain how infinite search problems, ie equivalent to the halting problem, are sometimes reduced to finite problems. some of this is referred to in hilberts old concept of “finitistic” mathematics. (maybe this was never proposed, but it has long seemed quite natural to me to refer to Hilberts finitism in terms of Turings algorithmic decidability and terminating computations.)

what is this general theory of finding the finitistic “core” of infinite problems? what would/ does a general framework look like? how can it be combined with AI? the glimmers of this are now seen in the pythagorean triple problem in a microcosm, waiting to be tied together & connected by some enterprising/ brilliant geniuses/ visionaries. think some of the answer can be found via “reverse engineering” established/ somewhat-toy-like problems, eg say Bertrands postulate as written about here. in another recent heralded breakthrough there seems somewhat vaguely related work in ramsey theory that is analyzing/ connecting the “finite-infinite divide”.**[b18]**

there is some nice new recent empirical work eg by Aaronson on the busy beaver problem.**[b1][b2]** theres some new attention to the LMFDB L-functions/ modular forms database which epitomizes some of the new empirical style.**[b3][b4]** theres continual dialog about empirical approaches esp in CS.**[b5][b8][b9][b10]** alas a leader in the field Borwein died at 65.**[b11]**

have been waiting to write on Mochizuki for years and have collected various links, there are a few refs to him in the blog.**[c]** it looks like the math community has not dismissed his proof so far and is continuing to engage with it. its a great study in the complementary properties of “old-school” human-focused/ virtuoso/ committee work versus computer or empirical proofs. these two styles are crosspollinating in remarkable ways in our era. math has entered an era of unprecedented complexity.

there are many neat profiles in the media of mathematicians for those who are attentive, have collected many great/ fun links on that on some of my favorites.**[d]** notably Simons agreed to be interviewed.**[d4]** numberphile is turning into a one of the top web resources for “popsci” mathematics curiosity/ inquiry, highly recommend it, really great for inquisitive/ precocious kids. (they just did a new segment on Collatz.) other highlights: NFL player Urschel**[d8][d9][d10]**, Chaitin**[d11]**, Conway**[d7]**, Grothendieck**[d1][d2]**, Gardner**[d15]**, erdos.**[3]**

amid the noise/ lament of many students not engaging with math & achieving basic math literacy, there are scattered encouraging stories about the next generation of students learning math.**[f]**

now seems as good a time as any to include my cool list of favorite essays written by mathematicians, aka “the joy of mathematics,” there is very much great writing in this vein.**[g]**

also (bonus links for free!), reactions/ connections to the ramanujan movie**[h]** and recent misc mathematics advances/ breakthroughs.**[i]**

**a. pythagorean triples****b. empirical****c. mochizuki****d. profiles****e. erdos****f. students/ edu****g. essays by mathematicians**

**a. pythagorean triples**

- 1. Two-hundred-terabyte maths proof is largest ever : Nature News & Comment
- 2. Computer cracks 200-terabyte maths proof | Cosmos
- 3. How Hard, Really, is SAT? | Gödel’s Lost Letter and P=NP
- 4. [1605.00723] Solving and Verifying the boolean Pythagorean Triples problem via Cube-and-Conquer

**b. empirical**

- 1. Shtetl-Optimized » Blog Archive » The 8000th Busy Beaver number eludes ZF set theory: new paper by Adam Yedidia and me
- 2. A Relatively Small Turing Machine Whose Behavior Is Independent of Set Theory/ Yedidia, Aaronson
- 3. Exploring the mathematical universe: Uncovering new worlds along the way — ScienceDaily
- 4. LMFDB – The L-functions and modular forms database
- 5. Process Algebra Diary: Yuri Gurevich’s advice to the young theoretical computer scientist
- 6. Prime After Prime | bit-player
- 7. Post-Doc Ergo Propter Hoc: The Fall of the Twin Prime Gap
- 8. Computational Complexity: Karp v Wigderson 20 Years Later
- 9. Computational Complexity: The Relevance of TCS
- 10. 15-853: Algorithms in the Real World
- 11. Jonathan Borwein dies at 65 « Math Drudge
- 12. 19th century math tactic gets a makeover—and yields answers up to 200 times faster
- 13. Computer Scientist Tells Mathematicians How To Write Proofs
- 14. HoTT awarded a MURI
- 15. Post-human mathematics / Ruelle
- 16. Toward an exploratory medium for mathematics/ Nielsen
- 17. Behind Wolfram|Alpha’s Mathematical Induction-Based Proof Generator
- 18. Mathematicians Bridge Finite-Infinite Divide
- 19. QED manifesto

**c. mochizuki**

- 1. Mathematician’s anger over his unread 500-page proof – physics-math – 07 January 2015 – New Scientist
- 2. An ABC proof too tough even for mathematicians – Ideas – The Boston Globe
- 3. The ABC Conjecture has not been proved | mathbabe
- 4. The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof : Nature News & Comment
- 5. Biggest mystery in mathematics in limbo after cryptic meeting : Nature News & Comment
- 6. Hope Rekindled for ABC Proof | Quanta Magazine
- 7. A New Hope For A Perplexing Mathematical Proof / Quanta
- 8. Mathematician Solved a Nearly Impossible Maths Problem | The Science Explorer
- 9. Mathematicians finally starting to understand epic ABC proof | New Scientist
- 10. Monumental proof to torment mathematicians for years to come : Nature News & Comment

**d. profiles**

- 1. Grothendieck biography
- 2. Grothendieck, eccentric maths genius, dies in France (Update)
- 3. The Man Who Invented Modern Probability – Issue 4: The Unlikely / Kolmogorov – Nautilus
- 4. Billionaire Mathematician – Numberphile – YouTube
- 5. Beale Conjecture 1 Million Dollar Prize – Business Insider
- 6. Meet the Guy Who Sorts All the World’s Numbers in His Attic/ Sloane, OEIS | WIRED
- 7. John Conway | Genius Behind The Numbers – 52 Insights
- 8. One of the Baltimore Ravens Just Published an Insanely Complex Study in a Math Journal – Bloomberg Business
- 9. NFL Player John Urschel Single-Handedly Kills The ‘Dumb Jock’ Stereotype With Complex Math Paper
- 10. Baltimore Ravens Offensive Lineman John Urschel Publishes Paper In Math Journal | ThePostGame
- 11. Chaitin, Conversations with a Mathematician
- 12. Edward Frenkel – Edward Frenkel – The Colbert Report – Video Clip | Comedy Central
- 13. Arthur Benjamin – The Colbert Report – Video Clip | Comedy Central
- 14. Are Mathematicians Past Their Prime at 35?
- 15. The Top 10 Martin Gardner Scientific American Articles | Guest Blog, Scientific American Blog Network
- 16. Math doesn’t get the media attention it deserves – Columbia Journalism Review
- 17. How Mathematicians Make Breakthroughs — Medium
- 18. Last Bastion of Purity in a Corrupt World?
- 19. Maths whizz solves a master’s riddle (Tao/ Erdos Discrepancy Problem)
- 20. You’ve heard of string theory. What about knot theory? / Bill Menasco
- 21. Feynman on Fermat’s Last Theorem

**e. erdos**

- 1. Reflections on Paul Erd˝os on His Birth Centenary, Part II
- 2. Reflections on Paul Erd˝os on His Birth Centenary
- 3. MuckRock • “Nothing to indicate the subject had any interest in any matter other than Mathematics” Paul Erdős FBI file

**f. students/ edu**

- 1. The maths problem set for Singapore teenagers that has left people across the world stumped | Daily Mail Online
- 2. Stepping up your Math Game – 35 Summer Projects for Post-Secondary Math Students
- 3. A Group of American Teens Are Excelling at Advanced Math – The Atlantic
- 4. The U.S.’s Math Olympiad Win Breaks China’s Dominance | FiveThirtyEight
- 5. 2015 Wolfram Summer Camps Exceed Expectations—Wolfram Blog
- 6. How our 1,000-year-old math curriculum cheats America’s kids – LA Times
- 7. Teaching Math to People Who Think They Hate It – The Atlantic

**g. essays by mathematicians**

- 1. The Two Cultures of Mathematics. W. T. Gowers
- 2. A Mathematician’s Lament by Paul Lockhart
- 3. WHAT IS GOOD MATHEMATICS? TERENCE TAO
- 4. IS OUR MATHEMATICS NATURAL? THE CASE OF EQUILIBRIUM STATISTICAL MECHANICS DAVID RUELLE
- 5. ON PROOF AND PROGRESS IN MATHEMATICS WILLIAM P. THURSTON
- 6. Edward Frenkel on Love and Math: What is it like to be a mathematician?
- 7. MATHEMATICS: ART AND SCIENCE MICHAEL ATIYAH
- 8. The Power of Mathematics JOHN CONWAY
- 9. A Mathematician’s Apology G. H. Hardy
- 10. The Ideal Mathematician/ David, Hersh
- 11. The Unreasonable Effectiveness of Mathematics / Hamming
- 12. Ten Lessons I Wish I Had Been Taught/ Rota
- 13. The Perfect Language / Chaitin | Articles | Inference: International Review of Science
- 14. How mathematicians are storytellers and numbers are the characters / Du Sautoy | Books | The Guardian
- 15. The great mystery of mathematics is its lack of mystery/ Aaronson | Aeon Ideas
- 16. Formal Proof—The FourColor Theorem, Notices AMS, Georges Gonthier
- 17. Large cardinals: maths shaken by the ‘unprovable’ / Richard Elwes – Telegraph
- 18. career – What’s a mathematician to do? – MathOverflow
- 19. Why the history of maths is also the history of art | Science | The Guardian
- 20. Wandering in the heavens: how mathematics explains Saturn’s rings / Ian Stewart
- 21. The Princeton Companion to Mathematics: Timothy Gowers, June Barrow-Green, Imre Leader: 9780691118802: Amazon.com: Books
- 22. Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

**h. ramanujan**

- 1. The Ramanujan Movie | Blog on math blogs
- 2. The Man Who Knew Infinity (Mathematical Movie!)
- 3. A Math Biopic, With Dev Patel, Applies a Different Calculus – The New York Times
- 4. The Man Who Knew The Man Who Knew Infinity | PhD + epsilon
- 5. eScienceCommons: Math shines with the stars in ‘The Man Who Knew Infinity’
- 6. Math shines with the stars in ‘The Man Who Knew Infinity’ | Emory University | Atlanta, GA
- 7. An overview of ramanujans notebooks/ Berndt
- 8. eScienceCommons: Doing math with movie stars
- 9. eScienceCommons: New theories reveal the nature of numbers

**i. advances**

- 1. Prize awarded for largest mathematical proof | New Scientist
- 2. New derivation of pi links quantum physics and pure math: Researchers stumbled upon a famous pre-Newtonian formula for pi while computing the energy levels of a hydrogen atom — ScienceDaily
- 3. New derivation of pi links quantum physics and pure math
- 4. eScienceCommons: A suprise dimension to adding and counting
- 5. eScienceCommons: How a hike in the woods led to a math ‘Eureka!’
- 6. 40-year math mystery and four generations of figuring — ScienceDaily
- 7. Sphere Packing Solved in Higher Dimensions | Quanta Magazine
- 8. Math Quartet Joins Forces on Unified Theory
- 9. Mathematicians Chase Moonshine, String Theory Connections | Quanta Magazine
- 10. Mathematicians Solve 45-Year-Old Kervaire Invariant Puzzle | Simons Foundation
- 11. Using Chaos Theory to Predict and Prevent Catastrophic ‘Dragon King’ Events – Wired Science
- 12. 12-Page Math Paper Supposedly Explains Insanely Elaborate Penis Joke

gentzenWhy do you want to blog on Mochizuki? Is his proof really the only proof of a famous problem that other mathematicians have issues verifying? Or is it just the only proof where mathematicians point to the impenetrable nature of the writting, without also hinting that the proof itself is probably flawed? If a mathematician of lesser fame would present an impenetrable proof of a famous problem, his proof would just get rejected based on the fact that missing clarity is a sure sign of obvious or hidden flaws in the proof itself.

Maybe one should separate the validation of his inter-universal Teichmüller theory from his proof of the ABC conjecture. Then it makes sense again that other mathematicians should try to validate his work. However, this sort of validation should check whether his theory is useful and helps to generate new mathematical answers, insights, and questions. It should not try to check whether the proof of the ABC conjecture itself is correct or not. And it should not mix up the usefulness of the theory with the correctness of the given proof. A one hit wonder still doesn’t make a great band. And some good bands never scored a number one hit, and still delivered great concerts.

vznvznPost authorthink you have all valid points. the Mochizuki proof attempt is notable for its vast span/ complexity, the reputation of its author, and it highlights the sociological aspects of mathematics. there are very few other efforts in math history that have triggered as much collaborative work. it shows that even great mathematicians can get caught up in “crank-like efforts”. another case study on this blog, grothendieck. its a recurring theme on the blog that sometimes the line between genius and cranks can be razor )( thin etc… also, it surely ties in with new efforts to make human proofs more amenable to computer verification also referenced a lot on this blog. etc