But, as we will see, Bach was definitely a mathematician in a more general sense, as a composer whose works are replete with patterns, structures, recursions and other precisely crafted features. There are even hints of Fibonacci numbers and the golden ratio in Bach’s music (see below). Indeed, in this larger

Continue reading Bach as mathematician

]]>But, as we will see, Bach was definitely a mathematician in a more general sense, as a composer whose works are replete with patterns, structures, recursions and other precisely crafted features. There are even hints of Fibonacci numbers and the golden ratio in Bach’s music (see below). Indeed, in this larger sense, Bach arguably reigns supreme over all classical composers as a “mathematician,” although Mozart is a close second.

Certainly in terms of sheer volume of compositions (or even the sheer volume of “mathematical” compositions), Bach reigns supreme. The Bach-Werke-Verzeichnis (BWV) catalogue lists 1128 compositions, from short solo pieces to the magnificent Mass in B Minor (BWV 232) and Christmas Oratorio (BWV 248), far more than any other classical composer. Bach regularly garners the top spot in listings of the greatest composers, typically followed by Mozart and Beethoven.

Further, Bach’s clever, syncopated style led the way to twentieth century musical innovations, notably jazz. Contemporary pianist Glenn Gould, for instance, is equally comfortable and adept playing modern jazz and Bach’s The Well-Tempered Clavier.

Just as some of the best musicians and composers are “mathematical,” so too many of the best mathematicians are quite musical. It is quite common at evening receptions of large mathematical conferences to be serenaded by concert-quality musical performers, who, in their day jobs, are accomplished mathematicians of some renown.

Perhaps the best real-life example of a mathematician-musician was Albert Einstein, who was also an accomplished pianist and violinist. His second wife Elsa recalled how Albert, during deep concentration on a mathematical problem, would often sit down with a piano or violin and play for a while, then return to his work. Once, after a two-week period of intense research interspersed with random music playing, Einstein emerged with the first working draft of his paper on general relativity. He later said, “If … I were not a physicist, I would probably be a musician. I often think in music. I live my daydreams in music. I see my life in terms of music.”

So who were Einstein’s two favorite composers? You guessed it — Bach and Mozart.

There does seem to be a credible connection between the sort of mental gymnastics done by a mathematician and by a musician. To begin with, there are well-known mathematical relationships between the pitch of various notes on the musical keyboard. An octave is separated by a factor of two; a fifth interval (say C to G) by the ratio 3/2. A significant innovation in the 17th and 18th century was the rise of the “tempered” musical scale, where two adjacent notes on the keyboard are separated in ratio by the twelfth root of two = 1.059463… Bach himself was the first major composer to employ this tempered scale, a feature he exploited with gusto in The Well-Tempered Clavier (BWV 846-893).

But beyond mere analysis of pitches, it is clear that the arena of musical syntax and structure has a very deep connection to the sorts of syntax, structure and other regularities that are studied in mathematical research. Bach and Mozart are particularly well-known for music that is both beautiful and structurally impeccable. As Salieri explained in the musical “Amadeus,” referring to Mozart’s music, “Displace one note and there would be diminishment. Displace one phrase and the structure would fall.”

Bach was a master of musical structures. His works typically start with a fairly simple theme. In the case of the Brandenburg Concerto #5 (BWV 1050), it is a simple four-note pattern. He then typically combined the theme with offsets (think of a chorus singing “rounds”), reversals, inversions and other variations, all presented with multiple overlapping “voices,” producing a stunning polyphonic effect.

Andrew Qian illustrates Bach’s methods by analyzing Fugue #16 from Book I of The Well-Tempered Clavier, BWV 861 (note however that Qian erroneously says Book II):

Note here that the third bar of the bass clef is merely an inversion of the main theme in the first two bars. In fact, this inversion is a second theme. These three bars (and two themes) are repeated, with variations, six times in the piece, by four voices, which in analogy to a chorus, may be labeled soprano, alto, tenor and bass. Why six? Because the number of combinations of four items taken two at a time is six! In particular, the fugue features the two themes as alto + soprano, then bass + tenor, then tenor + alto, then bass + soprano, then bass + alto, then soprano + tenor. See Qian’s article for additional details.

Another example, visualized using a very nice online tool, is Bach’s Great Fantasia and Fugue in G minor (BWV 542) for the organ. In the fugue movement, the theme is introduced immediately, and then developed into countless polyphonic variations. A third example is Bach’s Prelude and Fugue in E minor (BWV 548), an organ work known as the “Wedge.” It is so named for a strong theme that develops as an expanding sequence of notes in the shape of a wedge in the printed score (in the fugue movement), and then is repeated in polyphonic variations in a sophisticated high-level structure.

Perhaps the most remarkable “mathematics” in Bach’s music are the instances of the golden ratio, usually denoted with the Greek letter ø = (1 + sqrt(5))/2 = 1.6180339887…, together with the Fibonacci numbers, whose limiting ratio is equal to ø.

Loic Sylvestre and Marco Costa pursue this topic at length in a 2010 paper (also available here). They focus on Bach’s The Art of Fugue (BWV 1080), which was composed in the last decade of Bach’s life and was clearly designed as an ultimate expression of Bach’s “mathematical” style. Its partner, The Musical Offering (BWV 1079), which has a similar objective and style, was named by musicologist Charles Rosen as the most significant piano work of the millennium.

Sylvestre and Costa tabulated the number of bars in each of the 19 movements of The Art of Fugue, then carefully analyzed different groupings of the movements. They found a number of intriguing patterns, including the following:

- The total number of bars for counterpoint movements 1 through 7 is 602. Of these, 372 are in counterpoint movements 1 through 4 and 230 in counterpoint movements 5 through 7. Note that 602/372 = ø very closely, and also 372/230 is very close to ø.
- Counterpoint movements 8 through 14 (988 bars in total) can be divided into double and mirror fugues (377 bars) and triple fugues (611 in total). Note that 611/377 and 988/611 are each very close to ø.

Note that in each case, the ratios mentioned above are as close as possible to ø as an integer ratio, given the respective denominator. Several other examples of this sort are given in the Sylvestre-Costa paper (also available here).

Sylvestre and Costa conclude, “we report a mathematical architecture of The Art of Fugue, based on bar counts, which shows that the whole work was conceived on the basis of the Fibonacci series and the golden ratio.”

However, as a word of caution, it should be kept in mind that the evidence cited by Sylvestre and Costa is a bit on the weak side. For example, their observation of the partial Fibonacci sequence 2,3,5,8,13 in the bar counts could be just a coincidence. Even the instances of the golden ratio could simply be due to Bach’s innate sense of natural balance, instead of deliberate numerical design (which itself is rather remarkable). Hopefully additional research in this arena will clarify this matter. See, for example, this this article on difficulties in observations of Fibonacci and the golden ratio in biology.

In short, while it is problematic to claim any equivalence between mathematics and music, it is clear that the two disciplines have a deep commonality in syntax, structure and recursion. Bach, arguably more than any other composer before or since, clearly championed this “mathematical” style, even though Bach never had any formal mathematical training beyond the basics (except that he may have been aware of the golden ratio and Fibonacci numbers).

Some of the more interesting current research work in this area is to program computers to actually compose music in a given style. David Cope, for instance, has written computer programs that can analyze a corpus of music, say by a particular composer, and then create new works in a similar style. He was most successful in replicating and producing variations of the music of Bach and Mozart, which is perhaps not surprising given the sophisticated structures used by these composers.

With the rapid rise of computer technology in general, and artificial intelligence in particular, we may well see some even more astonishing connections between music and mathematics. We may even be able to “resurrect,” in a virtual sense, composers such as Bach and enjoy new musical works that are truly in their style. May your mathematical future also be a musical one!

For further details, see these articles:

- Qian’s article on Bach’s techniques.
- The Sylvestre-Costa paper (also available here).
- Rosen’s article on The Musical Offering.
- Article on Einstein, emphasizing his love of Bach and Mozart.
- Article on Bach and Glen Gould.

Here are a few notable examples of Bach’s music for those who wish to explore further. Each of these is available on CD or download from various sources, such as Amazon.com or Apple Music. Two highly recommended complete collections of Bach’s works are: Bach Edition: Complete Works (155 CDs) and The Complete Bach Edition (153 CDs).

- Major choral-orchestral works:
- Mass in B Minnor (BWV 232). This is regarded as one of the greatest musical works of all time.
- Christmas Oratorio (BWV 248). Three hours of great choral-orchestral music.
- Easter Oratorio (BWV 249). A shorter but highly listenable piece.

- Cantatas (choral-orchestral works typically 30-45 minutes in length):
- Cantata #29 (BWV 29). The first movement was featured in Disney’s Fantasia.
- Cantata #32 (BWV 32). A very listenable cantata, particularly movement 5.
- Cantata #51 (BWV 51). Includes several soprano/alto solos and duets.
- Cantata #102 (BWV 102). Has an eerie quality reminiscent of contemporary movie score composer Danny Elfman.
- Cantata #110 (BWV 110). Arguably Bach’s most thrilling cantata.
- Cantata #146 (BWV 146). The first movement is one of Bach’s greatest organ concertos.
- Cantata #147 (BWV 147). An excellent choral example of Bach’s polyphonic style.

- Keyboard compositions (for harpsichord or clavichord, but often played on piano today):
- The Well-Tempered Clavier (BWV 846-893). This collection of 24 pairs of preludes and fugues is an excellent introduction to Bach’s style.
- The English Suites (BWV 806–811). These are among Bach’s most popular keyboard works.
- The Art of Fugue (BWV 1080). This is a single work of 19 movements that represents the epitome of Bach’s “mathematical” style.
- Two- and Three-Part Inventions (BWV 772-801). A collection of 30 short keyboard compositions.

- Instrumental and orchestral works:
- The Brandenburg Concertos (BWV 1046–1051). These are among Bach’s most popular works today. Concerto #5 (BWV 1050) has a long, sophisticated harpsichord solo.
- The Orchestral Suites (BWV 1066–1069). Four very listenable orchestral works, which were the foundations for later composers’ symphonies.
- Harpsichord Concerto in D minor (BWV 1052). One of Bach’s most thrilling instrumental compositions.
- Concertos for 2, 3 and 4 Harpsichords (BWV 1060-1065). A highly listenable set of instrumental-orchestral works with harpsichords.
- The Musical Offering (BWV 1079). As noted above, this is similar to The Art of Fugue, except for an instrumental ensemble, and is regarded as among Bach’s greatest works.

- Organ works:
- Trio Sonatas (BWV 525-530). Six very cheerful, listenable organ works.
- Fantasia & Fugue in G Minor, “Great G Minor” (BWV 542). See description above.
- Prelude & Fugue in B Minor (BWV 544). One of Bach’s grandest organ works.
- Prelude & Fugue in E Minor, “Wedge” (BWV 548). See description above.
- Toccata & Fugue In D Minor (BWV 565). One of Bach’s most popular organ works.
- Passacaglia & Fugue In C Minor (BWV 582). A popular piece that has been featured in movies.
- Organ Concertos in A Minor and D Minor (BWV 593, 596). Eerie pieces that are among Bach’s most memorable.

After several decades of disappointment, effective artificial intelligence (AI) systems emerged in the late 1990s and early 2000s, with the emergence of Bayes-theorem-based methods, combined with steadily advancing computer technology.

One notable milestone came in March 2016, when a computer program named “AlphaGo,” developed by researchers at DeepMind, a subsidiary of Alphabet (Google’s parent company), defeated champion Go master Lee Se-dol, an achievement that many observers had not expected to occur for decades. Then in October 2017, Deep Mind researchers announced a new program, AlphaGo Zero, which was programmed only with the rules

Continue reading AI system finds counterexamples to graph theory conjectures

]]>After several decades of disappointment, effective artificial intelligence (AI) systems emerged in the late 1990s and early 2000s, with the emergence of Bayes-theorem-based methods, combined with steadily advancing computer technology.

One notable milestone came in March 2016, when a computer program named “AlphaGo,” developed by researchers at DeepMind, a subsidiary of Alphabet (Google’s parent company), defeated champion Go master Lee Se-dol, an achievement that many observers had not expected to occur for decades. Then in October 2017, Deep Mind researchers announced a new program, AlphaGo Zero, which was programmed only with the rules of Go and a simple reward function and then instructed to play games against itself. After just three days of training (4.9 million training games), the AlphaGo Zero program had advanced to the point that it defeated the earlier Alpha Go program 100 games to zero.

AI and machine-learning methods are now being used in Apple’s Siri and Amazon’s Alexa voice recognition systems, in Facebook’s facial recognition API, in Apple’s 3-D facial recognition hardware and software, and in Tesla’s “autopilot” facility. See also these earlier Math Scholar blogs: Blog A and Blog B, from which some of the material below was taken.

Since the earliest days of AI, researchers have wondered whether these programs would ever be able to do serious mathematical research. In fact, computer programs that discover new mathematical identities and theorems are already a staple of the field known as experimental mathematics. A typical scenario is that a computer program, typically employing high precision computation software with an integer relation algorithm such as PSLQ, numerically discovers a previously unknown identity and confirms it to some compelling level of precision, such as 1000 or 10,000 digits. However, such programs typically offer only scant assistance as to why the result holds, so rigorous proofs must be done the old-fashioned human way.

So what about computers actually proving theorems?

Actually, this is also old hat at this point in time. Perhaps the best example is Thomas Hales’ 2003 proof of the Kepler conjecture, namely the assertion that the simple scheme of stacking oranges typically seen in a supermarket has the highest possible average density for any possible arrangement. Hales’ original proof met with some controversy, since it involved a computation documented by 250 pages of notes and three Gbyte of computer code and data. So Hales and his colleagues began entering the entire proof into a computer proof-checking program. In 2014 this process was completed and the proof was certified.

In November 2019, researchers at Google’s research center in Mountain View, California, published results for a new AI theorem-proving program. This program works with the HOL-Light theorem prover, which was used in Hales’ proof of the Kepler conjecture, and can prove, essentially unaided by humans, many basic theorems of mathematics. They have provided their tool in an open-source release, so that other mathematicians and computer scientists can experiment with it.

The Google AI system was trained on a set of 10,200 theorems that the researchers had gleaned from several sources, including many sub-theorems of Hales’ proof of the Kepler conjecture. Most of these theorems were in the area of linear algebra, real analysis and complex analysis, but the Google researchers emphasize that their approach is very broadly applicable. In the initial release, their software was able to prove 5919, or 58% of the training set. When they applied their software to a set of 3217 new theorems that it had not yet seen, it succeeded in proving 1251, or 38.9%.

For additional details see the Google authors’ technical paper, and a New Scientist article by Leah Crane.

In a remarkable new development, Adam Zsolt Wagner of Tel Aviv University in Israel employed an AI approach to search for counterexamples to a set of long-standing conjectures in graph theory. In each case, these conjectures were thought to be true, but no one had been able to produce a proof.

For each problem, Wagner’s approach was to devise a measure of how close a given specific construction is to a disproof. Wagner then coded a neural network program to create random constructions and evaluate them according to the measure.

As is common with other neural network research efforts, Wagner’s program advanced step-by-step, discarding low-scoring constructions and replacing them with additional random examples, proceeding over many steps to steadily improve the top-scoring constructions. In the majority of cases, Wagner’s program was unable to find a counterexample construction (perhaps because the conjecture in question may actually be true?). But in five cases the program was successful in discovering a construction that disproved the conjecture.

These were rather modest computations — Wagner ran his program on his five-year-old laptop, which required between two and 48 hours run time to discover the successful counterexamples. Wagner notes that the discovered disproof examples were mostly rather counterintuitive: “I would never have come up with these constructions by myself even if you gave me hundreds of years.”

For additional details see this New Scientist report.

Present-day computerized theorem provers are typically categorized as automated theorem provers (ATPs), which use computationally intensive methods to search for a proof; and interactive theorem provers, which rely on an interplay with humans — verifying the correctness of an argument and checking proofs for logical errors. Researchers in the field, however, acknowledge that both types of software are still a far cry from a completely independent computer-based mathematical reasoning system.

For one thing, these tools have been met with cold shoulders by many present-day mathematicians, in part because they require considerable study and practice to become proficient in using them, and in part because of a general distaste for the notion of automating mathematical thought.

But some mathematicians are embracing these tools. Kevin Buzzard of Imperial College London, for example, has begun to focus his research on computerized theorem provers. But he acknowledges, “Computers have done amazing calculations for us, but they have never solved a hard problem on their own. … Until they do, mathematicians aren’t going to be buying into this stuff.”

Emily Riehl, of Johns Hopkins University, teaches students to write mathematical proofs using a computerized tool. She reports that these tools help students to rigorously formulate their ideas. Even for her own research, she says that “using a proof assistant has changed the way I think about writing proofs.”

Vladimir Voevodsky of Princeton University, after finding an error in one of his own published results, was an ardent advocate of using computers to check proofs, until his death in 2017.

Timothy Gowers of Cambridge, who won the Fields Medal in 1998, goes even further, saying that major mathematical journals should prepare for the day when the authors of all submitted papers must first certify that their results have been verified by a computerized proof checker.

Josef Urban of the Czech Institute of Informatics, Robotics and Cybernetics believes that a combination of computerized theorem provers and machine learning tools is required to produce human-like mathematical research capabilities. In July 2020 his group reported on some new mathematical conjectures generated by a neural network that was trained on millions of theorems and proofs. The network proposed more than 50,000 new formulas, but, as they acknowledged, many of these were duplicates: “It seems that we are not yet capable of proving the more interesting conjectures.”

A research team led by Christian Szegedy of Google Research sees automated theorem provers as a subset of the field of natural language processing, and plans to capitalize on recent advances in the field to demonstrate solid mathematical reasoning. He and other researchers have proposed a “skip-tree” task scheme that exhibits suprisingly strong mathematical reasoning capabilities. Out of thousands of generated conjectures, about 13% were both provable and new, in the sense of not merely being duplicates of other theorems in the database.

For additional examples and discussion of recent research in this area, see this Quanta Magazine article by Stephen Ornes.

So where is all this heading? With regards to computer mathematics, Timothy Gowers predicts that computers will be able to outperform human mathematicians by 2099. He says that this may lead to a brief golden age, when mathematicians still dominate in original thinking and computer programs focus on technical details, “but I think it will last a very short time,” given that there is no reason that computers cannot eventually become proficient at the more creative aspects of mathematical research as well.

Some mathematicians are already envisioning how this software can be used in day-to-day research. Jeremy Avigad of Carnegie Mellon University sees it this way:

You get the maximum of precision and correctness all really spelled out, but you don’t have to do the work of filling in the details. … Maybe offloading some things that we used to do by hand frees us up for looking for new concepts and asking new questions.

One implication of all this is that mathematical training, both for mathematics majors and other students, must aggressively incorporate computer technology and teach computer methods for mathematical analysis and research, at all levels of study. Topics for prospective mathematicians should include a solid background in computer science (programming, data structures and algorithms), together with statistics, numerical analysis, machine learning and training in the usage of at least one symbolic computing computing tool such as *Mathematica*, *Maple* or *Sage*.

Mathematical research is hardly the only field that may be revolutionized by advanced computation. Indeed, intelligent computers are destined to transform virtually every aspect and occupation of modern civilization. Thus our modern society must find a way to accommodate this technology, and to deal respectfully with the many people whose lives will be impacted. This will be no picnic, but neither need it be all gloom and doom. The eventual outcome depends on choices we make now.

]]>The standard model of physics is arguably is the most successful physical theory ever devised, explaining all known fundamental particles and all known forces

Continue reading Muon result may rewrite standard model of physics

]]>The standard model of physics is arguably is the most successful physical theory ever devised, explaining all known fundamental particles and all known forces between these particles, except for gravity. It has reigned supreme since it was first formulated in the 1970s. However, physicists have long recognized that it cannot be the final answer. For one thing, it is incompatible with general relativity. Further, it says nothing about the identity of dark matter, which consists of at least 75% of known matter, nor does it identify dark energy that pervades the cosmos. At several junctures physicists have thought they found experimental evidence for physics beyond the standard model, but in each case these results evaporated in the wake of additional, more carefully analyzed data.

Muons are similar to electrons in several ways (negative electrical charge, and a spin), but their mass is approximately 207 times that of an electron. In part due to their much larger mass, muons are very short-lived, decaying into electrons and neutrinos with a half-life of roughly 2.2 microseconds. The Fermilab experiment measures the magnetic moment of the muon, known as g. In 1928, Paul Dirac, one of the founders of quantum mechanics, showed that g should equal 2. But to this value must be added the contribution of a sea of virtual particles.

A careful tabulation of these particles and their forces, done via massive calculations on supercomputers, yields the current best theoretical result that the magnetic moment of the muon should be 2.00233183620(86), where (86) denotes the uncertainty in the last digits of the calculation.

The discrepancy reported in the paper certainly isn’t great. The latest Fermilab experimental value is 2.00233184122(82), yielding a difference of 5.02 x 10^{-9}, or 4.2 standard deviations from the theoretical value. This does not yet meet the 5.0 standard deviation level that most physics experiments strive for, but it is enough to start raising the possibility of a major issue with the theoretical model. Additional runs are planned on the Muon g-2 experimental facility in the coming months that should further clarify the matter.

Theoreticians are already speculating as to the identity of other particles and forces that may be the source of the discrepancy. One possibility is a lightweight particle known as Z’ (Z prime). If such a particle exists, it could explain another nagging anomaly, namely the fact that the universe appears to be expanding slightly faster than standard cosmological models (based on the standard model) predict.

Gordon Krnjaic of Fermilab says the the g-2 result could set the agenda for particle physics for the next generation. “If the central value of the observed anomaly stays fixed, the new particles can’t hide forever. … We will learn a great deal more about fundamental physics going forward.” Fermilab Deputy Director of Research Joe Lykken adds, “this is an exciting time for particle physics research.”

However, physicist Sabine Hossenfelder recommends holding the champagne. After all, as mentioned above the result still does not meet the 5.0 standard deviation level recommended for most major physics experiments. She notes that the Higgs boson was “discovered” in 1996, in the wake of a 4 sigma result at the Large Electron-Positron (LEP) facility at CERN (near Geneva, Switzerland) but then subsequently disappeared after more data was obtained (the Higgs boson remained elusive until 2012, when experiments at the Large Hadron Collider definitely established its existence). Similarly, supersymmetric particles were “detected” at LEP in 2003, but again the evidence later evaporated. So let’s be careful before we jump to conclusions.

What’s more, another paper, also published today (7 Apr 2021), presents results of a different theoretical calculation (see also this report). Its results are closer to the new experimental value, meaning that the muon g-2 experiments might not represent a departure from the standard model. Additional calculations are planned to refine these values.

For additional details, see the researchers’ research paper, this Fermilab press report, and news reports from BBC, Nature, New York Times, Quanta Magazine and Scientific American.

]]>Continue reading Aho and Ullman receive the ACM Turing Award

]]>The 2020 prize was awarded for the Aho and Ullman’s work in “fundamental algorithms and theory underlying programming language implementation,” and also for “synthesizing these results and those of others in their highly influential books, which educated generations of computer scientists.”

Beginning with their work at Bell Labs in 1967, Aho and Ullman laid the foundations for modern compiler technology. Compilers are the software that translate computer programs written in some programming language into efficient machine code that is executed by the computer. Without modern high-level programming languages and the compilers that translate them to machine code, programming computers would be an incredibly labor- and time-intensive task, and would undoubtedly result in code that is many times more prone to failures and many times more difficult to maintain.

Compilers are used in every aspect of modern computing, from apps that run on smartphones, to database query and update software controlling large-scale corporate databases, to sophisticated climate modeling codes running on the world’s most powerful highly parallel supercomputers. Every time one interacts with a smartphone, orders an item online, pays a bill using an online banking facility, or accesses a local weather forecast, one should thank not only the the engineers who designed the hardware chips of your computer and the remote computer, and the programmers who programmed the software on both ends, but also the computer scientists, including Aho and Ullman, who laid the foundations for compilers that make programming these computers possible.

It is worth noting that the principles set forth in the work of Aho and Ullman are being used to design compilers for future quantum computers that rely on quantum superposition of atomic-scale particles to perform certain special tasks far faster than is possible with conventional computers.

Aho and Ullman are also the co-authors of two very widely used textbooks in the field: The Design and Analysis of Computer Algorithms (co-authored with John Hopcroft) and Principles of Compiler Design. The latest edition of the latter work is Compilers: Principles, Techniques and Tools (co-authored with Ravi Sethi and Monica Lam).

Jeff Dean of Google summed up their contributions in these terms:

Aho and Ullman established bedrock ideas about algorithms, formal languages, compilers and databases, which were instrumental in the development of today’s programming and software landscape. … They have also illustrated how these various disciplines are closely interconnected. Aho and Ullman introduced key technical concepts, including specific algorithms, that have been essential. In terms of computer science education, their textbooks have been the gold standard for training students, researchers, and practitioners.

For additional information, see the ACM Turing Award website and this New York Times report.

]]>The 2021 Abel Prize, arguably the closest equivalent in mathematics to the Nobel prize, has been awarded jointly to Avi Widgerson of the Institute for Advanced Study in Princeton, and László Lovász of the Eötvös Loránd University in Budapest, for their research linking discrete mathematics and computer science. The recipients will split the award, which is approximately USD$880,000.

According to Hanz Munthe-Kaas of the University of Bergen in Norway, who chaired the Abel Prize committee, Widgerson and Lovász “really opened up the landscape

Continue reading Two researchers share Abel prize for work in discrete mathematics and computer science

]]>The 2021 Abel Prize, arguably the closest equivalent in mathematics to the Nobel prize, has been awarded jointly to Avi Widgerson of the Institute for Advanced Study in Princeton, and László Lovász of the Eötvös Loránd University in Budapest, for their research linking discrete mathematics and computer science. The recipients will split the award, which is approximately USD$880,000.

According to Hanz Munthe-Kaas of the University of Bergen in Norway, who chaired the Abel Prize committee, Widgerson and Lovász “really opened up the landscape and shown the fruitful interactions between computer science and mathematics.” He added, “This prize is on the applied side, toward computer science. But it’s deep mathematics.”

Russell Impagliazzo of the University of California, San Diego, who has collaborated with both of the recipients, observes, “In many ways their work is complementary. Avi is on the computer science side and Lovász is on the mathematics side, but a lot of the issues they work on are related.”

Avi Widgerson, a native of Haifa, Israel, began his career at a time when researchers in computer science began investigating complexity theory, namely fundamental assessments of how difficult certain computational problems are. One prominent problem in the field is the P = NP conjecture, namely the question of whether all problems that can be solved in a number of operations that scales at most as $n^c$, for some constant $c$, are also of the class of problems that can be quickly verified by a classical computer once a solution is given. Almost all computer scientists believe the answer is “no,” although no proof is yet known.

Widgerson and his collaborators have researched the computational complexity of the “perfect matching problem” — if one has a set of $N$ computers, each one of which is capable of performing some but not all of a set of $N$ problems, is it possible to assign tasks to these $N$ computers such that all tasks are covered, and each system performs just one task?

Widgerson and collaborators have also investigated how randomness conveys an advantage in many computational problems. In particular, they proved that under certain conditions it is always possible to convert a fast random algorithm into a fast deterministic algorithm. Their result established that the class of computational problems known as “BPP” is exactly equal to the complexity class P, under the assumption that any problem in a related class “E” has a property known as “subexponential circuitry.”

According to Kevin Hartnett, “It tied decades of work on randomized algorithms neatly into the main body of complexity theory and changed the way computer scientists looked at random algorithms.”

László Lovász, a native of Budapest, Hungary, was a precocious mathematician from an early age, earning not just one but three gold medals at the International Math Olympiad. Early on, Lovász met Paul Erdos, the famous Hungarian mathematician whose publication record of 1500 mathematical papers still remains unsurpassed. Erdos introduced Lovász to graph theory and discrete mathematics. Lovász recalls, that at the time, “graph theory was not mainstream mathematics because many of the problems or results arose from puzzles or sort of recreational mathematics.”

That changed with his co-discovery, with Arjen and Hendrik Lenstra, of what is now known as the Lenstra-Lenstra-Lovasz algorithm, or “LLL algorithm” for short. This algorithm addresses a problem about lattices, namely geometric objects whose sets of spatial points have coordinates with integer values, and which are closed under linear combinations — in other words, if $A$ and $B$ are in the lattice, then $mA + nB$ is also in the lattice, for every integer $m$ and $n$.

The LLL algorithm addresses a basic computational problem: What is the shortest vector in the lattice, or, in other words, which point in the lattice is closest to the origin. The problem is particularly difficult to solve in very high dimensions. The LLL algorithm finds a “good approximation” to this shortest point, together with a guarantee that there is no other point much closer to the origin.

Since its discovery, the LLL has been applied in numerous settings, ranging from integer factorization and analysis of public-key cryptosystems to the disproof of the Mertens conjecture. It is one of the most fundamental and widely applicable algorithms of modern computational mathematics.

As a single example, the LLL algorithm can be applied to the problem of integer relation detection, namely the problem of determining, given an $n$-long vector $X = \{x_1, x_2, \ldots, x_n\}$ of real numbers, typically computed to high numeric precision, whether there exist integers $\{a_i\}$ such that $a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = 0$ (to within a comparable numerical precision). One algorithm for this purpose is the PSLQ algorithm (see also here). But the LLL algorithm can also be applied, often producing a faster solution.

One application is the following: If one suspects that a real number $\alpha$ is an algebraic number, namely the root of a polynomial of some degree $n$ with integer coefficients, what are the integer coefficients of the minimal polynomial? One approach is simply to calculate the vector $X = \{1, \alpha, \alpha^2, \ldots, \alpha^n\}$ to an appropriately high level of numeric precision, and then apply an integer relation algorithm such as PSLQ or LLL to $X$, in order to find the integer coefficients $\{a_i\}$, if any exist, such that $a_0 + a_1 \alpha + a_2 \alpha^2 + \cdots + a_n \alpha^n = 0$ (to a corresponding level of precision).

Lovász has published numerous other important results in graph theory. One of these is a proof of the Kneser conjecture, an assertion about whether a certain type of graph is colorable. He has also proposed some conjectures of his own, including the KLS conjecture, which has generated some significant research results just in the past few weeks.

For additional details and background, see this Quanta Magazine article (from which some of the above text was summarized), this New York Times report, and this New Scientist article.

]]>The Covid-19 pandemic has disrupted human life like no other event of modern history. As of the present date (1 Mar 2021), over 114,000,000 confirmed cases and 2.5 million deaths have been recorded worldwide, according to the Johns Hopkins University database. The U.S. has recorded over 514,000 deaths, and the U.K. has recorded over 123,000. The U.S. death toll, for instance, exceeds the combined combat death toll of all wars fought in its 245-year history (save only the civil war). And for every death there are

Continue reading Pandemics, misinformation and pseudoscience

]]>The Covid-19 pandemic has disrupted human life like no other event of modern history. As of the present date (1 Mar 2021), over 114,000,000 confirmed cases and 2.5 million deaths have been recorded worldwide, according to the Johns Hopkins University database. The U.S. has recorded over 514,000 deaths, and the U.K. has recorded over 123,000. The U.S. death toll, for instance, exceeds the combined combat death toll of all wars fought in its 245-year history (save only the civil war). And for every death there are thousands more, particularly among less fortunate populations, who have lost employment, suffered depression, depleted savings or were deprived educationally.

Sadly, alongside the Covid-19 pandemic, and often inextricably intertwined with it, the world has seen a pandemic of misinformation and pseudoscience. While notions that are at odds with facts and science have always circulated in the public arena, the global rise of smartphones and social media has supercharged the proliferation of such material. What’s worse, numerous key issues have become politicized, further hampering progress.

First and foremost, there has been a flood of misinformation on the Covid-19 pandemic. Here are some of the **hopelessly misinformed** claims that have proliferated just in the past year:

- Covid-19 death counts have been greatly exaggerated by the media.
- Bill Gates has inserted microchips in vaccines to monitor the public or cull undesirables.
- 5G cell phone towers cause or exacerbate coronavirus infections.
- The coronavirus was engineered in a Chinese lab.
- Covid-19 is no more serious than influenza.
- Increased case counts are merely due to increased testing.
- Masks are useless to prevent Covid-19.
- Hydroxychloroquine is an effective treatment for Covid-19.

These claims, all which have been thoroughly debunked (see the links above), are not just child’s play. They are having tragic consequences, almost certainly worsening and extending the pandemic, not just in the U.S. but also worldwide. Some of more worrisome developments include:

- Activists have burned numerous cell phone towers in the U.K., Netherlands and Belgium, thinking they are behind Covid-19.
- Paranoia over 5G and Covid-19 may have led to the Christmas Day bombing in Nashville, Tennessee.
- An anti-vaccination movement is growing rapidly in Germany.
- Roughly one sixth of U.K. residents would not receive vaccination, due to anti-vaccination propaganda.
- Whole towns in Mexico are refusing vaccination.
- More than one third of Americans would not receive vaccination, even if offered free.
- Roughly 40% of U.S. nursing home staff are refusing vaccination.
- Roughly one third of the U.S. military are refusing vaccination.

Sadly, as much as the present author dislikes venturing into political matters, in this discussion it is impossible to ignore “the elephant in the room,” namely the most unfortunate promotion of false and misleading information by former U.S. President Donald J. Trump. To begin with, Mr. Trump repeatedly disseminated false information on the Covid-19 pandemic, which in many cases government officials have found necessary to rebut. This includes:

- Repeated assertions that Covid-19 was a “Chinese epidemic” (it was first detected in Wuhan, China, but it is misleading to blame China for its worldwide spread).
- Repeated claims that Covid-19 is no more serious than influenza.
- Repeated claims that increased case counts were almost entirely due to increased testing.
- Refusal to wear a mask or insist on masks at his public rallies (Stanford researchers estimate 30,000 cases and 700 deaths as a result).
- Repeated claims that the Covid-19 pandemic would very soon “go away”.
- Repeated claims that hydroxychloroquine is effective in treating Covid-19.
- Refusal to disavow wild Covid-19 conspiracy theories by groups such as QAnon.

Even these actions paled in comparison with Trump’s nonstop claims, starting a few months before the 2020 presidential election, that the election was “rigged,” that large numbers of votes were “fraudulent,” that he actually won the election “by a landslide,” and, in total, that the election was “stolen” from him and his followers.

Needless to say, these claims are wholly, utterly false. Joe Biden won the electoral college vote 306 to 232, and won the popular vote by a margin of over 7,000,000 votes. The U.S. Department of Homeland Security, led by Trump-appointed officials, concluded its investigation by announcing that the November 2020 election was the most secure ever. U.S. Attorney General William Barr, a devoted Trump loyalist, confirmed that the Justice Department had uncovered no voting fraud on a scale that could have effected a different outcome in the election. Several states conducted careful audits of their results, finding no significant differences. A statewide audit in Georgia, for instance, found virtually identical counts.

Trump’s legal team filed over 60 lawsuits, in state courts, in U.S. District Courts, and even in the U.S. Supreme Court. Only one of these legal actions was successful, in a Pennsylvania case that covered only about 200 votes. Many of these lawsuits were dismissed on procedural, timing or standing bases, but in numerous key cases the detailed claims were considered, and invariably found wanting. In a Wisconsin case, for instance, the court ruled:

This is an extraordinary case. A sitting president who did not prevail in his bid for reelection has asked for federal court help in setting aside the popular vote based on disputed issues of election administration, issues he plainly could have raised before the vote occurred. This Court has allowed plaintiff the chance to make his case and he has lost on the merits.

For an exhaustive examination of Trump’s claims of election malfeasance, see this Washington Post report.

For several months leading up to early January 2021, many observers, both in the U.S. and elsewhere, had watched Mr. Trump’s patently false claims and quixotic legal moves with a sort of bemused puzzlement, not taking any of it very seriously. This naive mindset was tragically shattered on January 6, when an armed mob of thousands, riled by months of “election was stolen” rhetoric and fiery speeches by Trump and loyalists on the morning of the event, stormed the U.S. capitol building in a violent insurrection, attempting in vain to thwart the U.S. Congress’ final certification of the election results.

For his part in pressuring state officials to change vote counts, promoting the insurrection and refusing to stop it for several hours, Mr. Trump became the first U.S. president to be impeached by the House of Representatives a second time. He was ultimately acquitted in the Senate (57 of 100 senators voted for conviction, short of the 67 required). But the shocking images of the insurrection, displayed in worldwide newscasts and at the trial (see video transcripts) will not soon be forgotten.

One consequence of the insurrection is that the Republican Party is now deeply divided. Figures such as Mitt Romney (the party’s 2012 candidate) have declared that they can no longer support Trump, yet others (including 59% of the rank-and-file party members) endorse him as the leading Republican contender for the 2024 presidential election. 76% of Republicans remain convinced by Trump’s propaganda that the election was “stolen” and that Biden is an illegitimate president.

At the very least, both the Covid-19 pandemic and the January 6 storming of the U.S. capitol have destroyed the claims of those who dismiss the daily parade of misinformation on various news and social media platforms as mere “talk,” of little real long-term importance. To the contrary, **words and facts matter**, and can have life-and-death consequences. Public officials especially must be held accountable for the language they use. We must never again allow such extreme departures from facts and scientific reality to take hold, either in the U.S. or anywhere else.

The events of the past year have also highlighted the role of large social media platforms in the propagation of misinformation. Facebook has 2.8 billion active users, and Twitter has 330 million users. Both platforms have recently increased their efforts to monitor usage for false and misleading posts. Both, for instance, have now banned Mr. Trump. But many observers remain concerned that misinformation still reigns across the social media landscape.

At the present time, the U.S. Congress is considering additional regulations on large tech firms, but it is not clear how this will turn out, since the issue is highly politicized — for example, some legislators have decried the media firms’ recent anti-misinformation efforts as “censorship.” Similarly, the European Union is considering additional regulation to curb “fake news and disinformation,” but again it is not clear how effective these measures will be.

If social media companies and legislators cannot protect us from misinformation, then what can we do?

The answer may be looking at us in the mirror. Maybe each of us needs to rethink our attachment to social media and the amount of time we spend immersed in social media, and, more generally, to rethink the sources of our information.

Michael Caulfield, a digital media scholar at Washington State University Vancouver, urges the public to “resist the lure of rabbit holes, … by reimagining media literacy for the internet hellscape we occupy.” Caulfield encapsulates his recommendations into four principles (with apt acronym “SIFT”):

- Stop.
- Investigate the source.
- Find better coverage.
- Trace claims, quotes and media to the original context.

Charles Weizel adds:

Our focus isn’t free, and yet we’re giving it away with every glance at a screen. But it doesn’t have to be that way. In fact, the economics are in our favor. Demand for our attention is at an all-time high, and we control supply. It’s time we increased our price.

Perhaps the single most effective tool at our disposal is to shift our focus away from social media and instead to internationally recognized news sources. Some recommended platforms include the BBC, the Economist, the New York Times, the Washington Post, and, for scientific news and information, Nature, Science, New Scientist and Scientific American. It is true that viewing more than a handful of articles per month from any one of these sites (except BBC) requires a paid subscription. But isn’t being a well-informed member of the world society worth the modest cost?

After all, we are what we read.

]]>This puzzle conforms to the New York Times crossword conventions. As far as difficulty level, it would be comparable to the NYT Tuesday or Wednesday puzzles (the NYT puzzles are graded each week from Monday [easiest] to Saturday [most difficult]).

If you would like a full-page version

Continue reading PiDay 2021 crossword puzzle

]]>This puzzle conforms to the New York Times crossword conventions. As far as difficulty level, it would be comparable to the NYT Tuesday or Wednesday puzzles (the NYT puzzles are graded each week from Monday [easiest] to Saturday [most difficult]).

If you would like a full-page version of the puzzle suitable for printout, click here: FULL-PAGE VERSION.

A pi-themed gift will be awarded to the first two persons who send me a correct solution (U.S. only). See HERE for contact information.

[23 Feb 2021:] Congratulations to Neil Calkin (Clemson University)! He sent a correct solution within one hour after the puzzle posted.

[3 Mar 2021:] Congratulations to Ross Blocher (cohost of the Oh No, Ross and Carrie podcast) and Morgan Marshall (Stanford Sierra Camp), both of whom have sent correct solutions.

]]>Proteins are the workhorses of biology. A few examples in human biology include actin and myosin, the proteins that enable muscles to work; keratin, which is the basis of skin and hair; hemoglobin, the basis of red blood that carries oxygen to cells throughout the body; pepsin, an enzyme that breaks down food for digestion; and insulin, which controls metabolism. A protein known as “spike” is the key for the coronavirus to invade healthy cells. And for every protein in human biology, there

Continue reading Machine-learning breakthrough in protein folding

]]>Proteins are the workhorses of biology. A few examples in human biology include actin and myosin, the proteins that enable muscles to work; keratin, which is the basis of skin and hair; hemoglobin, the basis of red blood that carries oxygen to cells throughout the body; pepsin, an enzyme that breaks down food for digestion; and insulin, which controls metabolism. A protein known as “spike” is the key for the coronavirus to invade healthy cells. And for every protein in human biology, there are many thousands of other proteins in the rest of the biological kingdom.

Each protein is specified as a string of amino acids, typically several hundred or several thousand long. There are 20 different amino acids, each specified by a particular three-letter word from the basic DNA letters A, C, T, G. Thanks to the recent dramatic drop in the cost of DNA sequencing technology, sequencing proteins is fairly routine.

The key to biology, however, is the three-dimensional shape of the protein once it is created by cell machinery — how a protein “folds.” Hydrophobic amino acids tend to congregate in the middle of the structure, so as to avoid the aqueous environment. Amino acids with a prevailing negative charge attract amino acids with a prevailing positive charge. Hydrogen bonds in the amino acid chain may lead to the formation of spirals or sheets.

Protein shapes can be investigated experimentally, using x-ray crystallography, but this is an expensive, error-prone and time-consuming laboratory operation, certainly not a task that can be economically performed on each of the millions of proteins whose structures are not yet known.

Because of these difficulties, numerous teams of researchers worldwide have been pursuing computational protein folding. Just a few of the many potential applications of this technology include studying the misshapen proteins thought to be the cause of Alzheimer’s disease, or studying the proteins behind various genetically-linked disorders such as cystic fibrosis and sickle-cell anemia. Only a tiny fraction of known proteins (roughly 170,000 of over 180,000,000) have had their structures determined, so there is much to be done. Obviously, the stakes are very high.

A protein folding calculation is typically constructed as a large global energy minimization problem, which, since a straightforward exhaustion of possibilities is computationally infeasible, is typically decomposed into a set of smaller local minimization problems, with results assembled to construct the larger structure. Some approaches to the problem are completely computational, from first principles; others employ, in part, tables of previously computed basic protein shapes.

The underlying computational task in protein folding calculations is often termed molecular dynamics. One team of many in the molecular dynamics field is led by David E. Shaw, founder of the D.E. Shaw mathematical hedge fund. His research team, which has designed a special-purpose system (“Anton”) for molecular dynamics computation, has twice been awarded (2009 and 2014) the Gordon Bell Prize from the Association for Computing Machinery.

While steady progress on protein folding has been reported by numerous teams worldwide, the consensus in the field is that the best programs and hardware platforms are still not up to the task of reliably solving full-scale, real-world protein folding problems in reasonable run times.

Given the daunting challenge and importance of the protein folding problem, in 1994 a community of researchers in the field organized a biennial competition known as Critical Assessment of Protein Structure Prediction (CASP). At each iteration of the competition, the organizers announce a set of problems, to which worldwide teams of researchers then apply their best current tools to solve. When CASP started in 1994, the average score was only 4%; by 2016 the average score had risen to 36% — promising, but still far from acceptable.

In 2018, the CASP competition had a new entry: AlphaFold, a machine-learning-based program developed by DeepMind, a division of Alphabet (Google’s parent company). AlphaFold achieved a score of roughly 56%. DeepMind’s program was by no means the first to apply machine learning techniques to the protein folding problem, but it was clearly the most effective in the 2018 competition. We summarized this achievement and potential applications in a previous Math Scholar blog.

The DeepMind team certainly has credentials for state-of-the-art machine learning work. In March 2016, a DeepMind computer program named “AlphaGo” defeated Lee Se-dol, a South Korean Go master, 4-1 in a 5-game tournament, an achievement that many observers had not expected to occur for decades, if ever. Then in October 2017, DeepMind researchers developed from scratch a new program, called AlphaGo Zero, which was programmed only with the rules of Go and a simple reward function; then it was instructed to play games against itself. After just three days of training, the AlphaGo Zero program had advanced to the point that it defeated the earlier AlphaGo program 100 games to zero, and after 40 days of training, AlphaGo Zero’s performance was as far ahead of champion human players as champion human players are ahead of amateurs. For additional background, see this Economist article, this Scientific American article and this Nature article.

For the 2020 CASP competition, the DeepMind team decided to scrap its original AlphaFold program, and instead developed a new program, known as AlphaFold 2, from scratch. In the just-completed 2020 CASP competition (conducted virtually due to the Covid-19 pandemic), AlphaFold 2 achieved a 92.4% average score, far above the 62% achieved by the second-best program in the competition.

“It’s a game changer,” exulted German biologist Andrei Lupas, who has served as an organizer and judge for the CASP competition. “This will change medicine. It will change research. It will change bioengineering. It will change everything.”

Lupas mentioned how AlphaFold 2 helped to crack the structure of a bacterial protein that Lupas himself has been studying for many years. “The [AlphaFold 2] model … gave us our structure in half an hour, after we had spent a decade trying everything.”

Nobel laureate Venki Ramakrishnan of Cambridge University added:

This computational work represents a stunning advance on the protein-folding problem, a 50-year-old grand challenge in biology. It has occurred decades before many people in the field would have predicted.

DeepMind approached the protein folding problem as a “spatial graph,” where the nodes and edges connect the residues in close proximity, and then applied a neural network that attempts to interpret the structure of the graph. They then trained the system on a set of 170,000 publicly available protein structures. Some additional details and analyses are given in this DeepMind report. DeepMind promises a more detailed technical report in the coming months.

For additional perspectives, see this Scientific American article, this Economist article and this Ars Technica article.

With AlphaFold 2, the DeepMind team has achieved a significant breakthrough in a computational application that, unlike playing Go or chess, is indisputably of great potential importance to human health and society. We can only assume that the DeepMind team will further improve their software, and that their efforts and similar efforts by other research teams, coupled with increasingly powerful hardware targeted to machine learning applications, will ultimately achieve a capability to solve large-scale protein structures on a routine basis.

Further, it is inevitable that the machine learning technology at the heart of these research projects will be effectively applied in other arenas of modern science and technology. Already, significant advances have been achieved in mathematics, physics and finance, to name but three.

So where is all this heading? A recent Time article featured an interview with futurist Ray Kurzweil, who has predicted an era, roughly in 2045, when machine intelligence will meet, then transcend human intelligence. Such future intelligent systems will then design even more powerful technology, resulting in a dizzying advance that we can only dimly foresee at the present time. Kurzweil outlined this vision in his book The Singularity Is Near.

Futurists such as Kurzweil certainly have their skeptics and detractors. Sun Microsystem founder Bill Joy is concerned that humans could be relegated to minor players in the future, if not extinguished. There is already concern that AI systems, in many cases, make decisions that humans cannot readily understand or gain insight into.

But even setting aside such concerns, there is considerable concern about the potential societal, legal, financial and ethical challenges of machine intelligence, as exhibited by the current backlash against science, technology and “elites.” However these disputes are settled, it is clear that in our headlong rush to explore technologies such as machine learning, artificial intelligence and robotics, we must find a way to humanely deal with those whose lives and livelihoods will be affected by these technologies. The very fabric of society may hang in the balance.

]]>A perfect number is a positive integer whose divisors (not including itself) add up to the integer. The smallest perfect number is $6$, since $6 = 1 + 2 + 3$. The next is $28 = 1 + 2 + 4 + 7 + 14$, followed by $496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248$ and $8128 = 1 + 2 + 4 + 8 + 16 + 32

Continue reading Do odd perfect numbers exist? New results on an old problem

]]>A perfect number is a positive integer whose divisors (not including itself) add up to the integer. The smallest perfect number is $6$, since $6 = 1 + 2 + 3$. The next is $28 = 1 + 2 + 4 + 7 + 14$, followed by $496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248$ and $8128 = 1 + 2 + 4 + 8 + 16 + 32 + 64$ $+ 127 + 254 + 508 + 1016 + 2032 + 4064$.

The notion of a perfect number is at least 2300 years old. In approximately 300 BCE, Euclid showed (using modern notation) that if $2^p – 1$ is a prime number (which implies, by the way, that $p$ itself is prime), then $2^{p-1} (2^p – 1)$ is a perfect number. For example, when $p = 5$, we have $2^5 – 1 = 31$, which is prime, and $2^4 (2^5 – 1) = 16 \cdot 31 = 496$ is perfect. In approximately 100 CE, the Greek mathematician Nicomachus noted that 8128 is a perfect number, and stated, without proof, that every perfect number is of the form $2^{n-1} (2^n – 1)$ (also omitting the clear condition that $n$ must be prime). In the first century CE, the Hebrew theologian Philo of Alexandria asserted that Earth was created in $6$ days and the moon orbits Earth in $28$ days, since $6$ and $28$ are perfect. In the early fifth century CE, the Christian theologian Augustine of Hippo repeated the assertion that God created Earth in $6$ days because $6$ is the smallest perfect number.

In the 12th century, the Egyptian mathematician Ismail ibn Fallūs calculated the 5th, 6th and 7th perfect numbers $(33550336, 8589869056$ and $137438691328$), plus some additional ones that are incorrect. The first known mention of the 5th perfect number in European history is in a manuscript written by an unknown writer between 1456 and 1461. The 6th and 7th were identified by the Italian mathematician Pietro Cataldi in 1588; he also proved that every perfect number obtained using Euclid’s formula ends in $6$ or $8$.

In the 18th century, Euler proved that Euclid’s formula $2^{p-1} (2^p – 1)$, for prime $2^p – 1$, yields all even perfect numbers. He also introduced the $\sigma (N)$ notation (also known as the divisor function) for the sum of the divisors of $N$, including $N$ itself. Thus we may write his result as follows: If $N$ is a positive even integer, then $\sigma(N) = 2N$ (i.e., $N$ is perfect) if and only if $N = 2^{p-1} (2^p – 1)$ for some prime of the form $2^p – 1$. This result is now known as the Euclid-Euler theorem. For additional history and background on perfect numbers, see this Wikipedia article.

The Euclid-Euler theorem pretty well wrapped up the case for even perfect numbers. But what about odd perfect numbers (OPNs)? Do any such integers exist? This question has remained stubbornly unanswered for centuries. Numerous restrictions are known on the properties of any possible OPN, with more restrictions being proved with every passing year. Present-day mathematicians echo the sentiment of James Sylvester, who wrote

a prolonged meditation on the subject has satisfied me that the existence of any one such [odd perfect number] — its escape, so to say, from the complex web of conditions which hem it in on all sides — would be little short of a miracle.

Here are some of the known restrictions on any possible OPN $N$, listed in roughly chronological order:

- $N$ must not be divisible by 105 (Sylvester, 1888).
- If $N$ is not divisible by $3, 5$ or $7$, it must have at least $27$ prime factors (Norton, 1960).
- $N$ must have at least seven distinct prime factors (Pomerance, 1974).
- $N$ must have at least $75$ prime factors and at least $9$ distinct prime factors (Hare, 2005 and Nielsen, 2006). This was later extended to $101$ prime factors (Ochem and Rao, 2012).
- $N$ must be congruent to either $1 (\bmod 12)$ or $117 (\bmod 468)$ or $81 (\bmod 324)$ (Roberts, 2008).
- The largest prime factor $p$ of $N$ must satisfy $p > 10^8$ (Goto and Ohno, 2008).
- $N \gt 10^{1500}$ (Ochem and Rao, 2012).

(See this MathWorld article for references to these and other items.)

Recently mathematicians have taken a slightly different tack on the problem — searching for spoof odd perfect numbers, namely integers that resemble odd perfect numbers but don’t quite meet all the requirements. There is historical precedent here: in 1638 Rene Descartes observed that $198585576189 = 3^2 \cdot 7^2 \cdot 11^2 \cdot 13^2 \cdot 22021$ almost satisfies Euclid’s formula.

To see this, first note two properties of the sigma function: (a) $\sigma(pq) = \sigma(p)\sigma(q)$, if $p$ and $q$ are relatively prime; and (b) $\sigma(p^m) = 1 + p + p^2 + \cdots p^m$ if $p$ is prime and $m \gt 0$. With regards to Descartes’ number, one might be tempted to write $$\hat{\sigma}(198585576189) = \sigma(3^2) \sigma(7^2) \sigma(11^2) \sigma(13^2) \sigma(22021)$$ $$= (1 + 3 + 3^2)(1 + 7 + 7^2) (1 + 11 + 11^2) (1 + 13 + 13^2) (1 + 22021) = 397171152378 = 2 \cdot 198585576189.$$ However, this calculation is incorrect, since $22021$ is not prime; in fact $\sigma(22021) = 23622$, not $22022$ as in the above. Thus the above calculation, which appears at first glance to certify that $198585576189$ is an OPN, is invalid. The correct sigma value for Descartes’ number is $\sigma(198585576189) = 426027470778$.

In 1999, John Voight of Dartmouth University discovered a different variety of spoof odd perfect number: $−22017975903 = 3^4 \cdot 7^2 \cdot 11^2 \cdot 19^2 \cdot (−127)$. Here again one might be tempted to write $$\hat{\sigma}(−22017975903) = \sigma(3^4) \sigma(7^2) \sigma(11^2) \sigma(19^2) \sigma(-127)$$ $$= (1 + 3 + 3^2 + 3^3 + 3^4) (1 + 7 + 7^2) (1 + 11 + 11^2) (1 + 19 + 19^2) (1 + (-127)) = -44035915806 = 2 \cdot (−22017975903).$$ But again, this calculation is invalid, in this case because the integer in question is negative, and $\sigma(-127) = -128$ not $-126$ as used above. But in both cases, these “spoofs” are definitely interesting, and possibly might inspire a line of attack to certify that OPNs simply cannot exist.

The most recent development here is due to a team led by Pace Nielsen and Paul Jenkins of Brigham Young University, subsequently joined by Michael Griffin and Nick Andersen, who embarked on a computer search for additional spoof odd perfect numbers. After roughly 60 CPU-years of computing, the team found 21 spoofs with six or fewer prime bases. Here the team relaxed the criteria in several ways, allowing non-prime bases (as with Descartes), negative bases (as with Voight), and also spoofs whose bases share the same prime factors (by the rules, a given prime may appear only once in the factorization list).

Why the interest in spoofs? According to Nielsen,

Any behavior of the larger set has to hold for the smaller subset. … So if we find any behaviors of spoofs that do not apply to the more restricted class, we can automatically rule out the possibility of an OPN.

So far, they have discovered some interesting facts about the spoofs, but none of these properties would preclude the existence of real odd perfect numbers. For example, the team has found that every one of the 21 spoofs that they have uncovered, with the exception of Descartes’ example, has at least one negative base. If this numerical observation could be proven, then this would prove that no odd perfect numbers exist, since by definition the bases of odd perfect numbers must be both positive and prime numbers.

So the search continues. Will this approach work? Voight, for instance, is not sure that even with the BYU team’s result that we are close to a final attack on the problem. Paul Pollack of the University of Georgia adds,

It would be great if we could stare at the list of spoofs and see some property and somehow prove there are no OPNs with that property. That would be a beautiful dream if it works, but it seems too good to be true.

For additional details and background, see this Quanta article by Steve Nadis, from which some of the above information was taken.

]]>The modern field of artificial intelligence (AI) began in 1950 with Alan Turing’s landmark paper Computing machinery and intelligence, which outlined the principles of AI and proposed the Turing test. Although early researchers were confident that AI systems would soon be a reality, inflated promises and expectations led to disappointment in the 1970s and again in the 1980s.

A breakthrough of sorts came in the late 1990s and early 2000s with the emergence of Bayes-theorem-based methods, which quickly displaced the older methods based mostly on formal reasoning. When combined with steadily advancing

Continue reading Can computers do mathematical research?

]]>The modern field of artificial intelligence (AI) began in 1950 with Alan Turing’s landmark paper Computing machinery and intelligence, which outlined the principles of AI and proposed the Turing test. Although early researchers were confident that AI systems would soon be a reality, inflated promises and expectations led to disappointment in the 1970s and again in the 1980s.

A breakthrough of sorts came in the late 1990s and early 2000s with the emergence of Bayes-theorem-based methods, which quickly displaced the older methods based mostly on formal reasoning. When combined with steadily advancing computer technology, a gift of Moore’s Law, practical and effective AI systems finally began to appear.

One notable milestone in modern AI technology came in March 2016, when a computer program named “AlphaGo,” developed by researchers at DeepMind, a subsidiary of Alphabet (Google’s parent company), defeated Lee Se-dol, a South Korean Go master, 4-1 in a 5-game tournament, an achievement that many observers had not expected to occur for decades, if ever. Then in October 2017, Deep Mind researchers developed from scratch a new program, called AlphaGo Zero, which was programmed only with the rules of Go and a simple reward function and then instructed to play games against itself. After just three days of training (4.9 million training games), the AlphaGo Zero program had advanced to the point that it defeated the earlier Alpha Go program 100 games to zero, and after 40 days of training, AlphaGo Zero’s performance was as far ahead of champion human players as champion human players are ahead of amateurs. See this Economist article, this Scientific American article and this Nature article.

AI and machine-learning methods are being used for more than playing Go. They are used in Apple’s Siri and Amazon’s Alexa voice recognition systems, in Facebook’s facial recognition API, in Apple’s 3-D facial recognition hardware and software, and in Tesla’s “autopilot” facility. See this earlier Math Scholar blog for additional discussion.

The present author recalls discussing the future of mathematics with Paul Cohen, who in 1963 proved that the continuum hypothesis is independent from the axioms of Zermelo-Fraenkel set theory. Cohen was convinced that the future of mathematics, and much more, lies in artificial intelligence. Reuben Hersch recalls Cohen saying specifically that at some point in the future mathematicians would be replaced by computers. So how close are we to Cohen’s vision?

In fact, computer programs that discover new mathematical identities and theorems are already a staple of the field known as experimental mathematics. Here is just a handful of the many computer-based discoveries that could be mentioned:

- A new formula for pi with the property that it permits one to rapidly calculate binary or hexadecimal digits of pi at an arbitrary starting position, without needing to calculate digits that came before.
- The surprising fact that if the Gregory series for pi is truncated to 10
^{n}/2 terms for some n, the resulting decimal value is remarkably close to the true value of pi, except for periodic errors that are related to the “tangent numbers.” - Evaluations of Euler sums (compound infinite series) in terms of simple mathematical expressions.
- Evaluations of lattice sums from the Poisson equation of mathematical physics in terms of roots of high-degree integer polynomials.
- A new result for Mordell’s cube sum problem.

In most of the above examples, the new mathematical facts in question were found by numerical exploration on a computer, and then later proven rigorously, the old-fashioned way, by mostly human efforts. So what about computers actually proving theorems?

Actually, this is also old hat at this point in time. Perhaps the best example is Thomas Hales’ 2003 proof of the Kepler conjecture, namely the assertion that the simple scheme of stacking oranges typically seen in a supermarket has the highest possible average density for any possible arrangement, regular or irregular. Hales’ original proof met with some controversy, since it involved a computation documented by 250 pages of notes and 3 Gbyte of computer code and data. So Hales and his colleagues began entering the entire proof into a computer proof-checking program. In 2014 this process was completed and the proof was certified.

In November 2019, researchers at Google’s research center in Mountain View, California, published results for a new AI theorem-proving program. This program works with the HOL-Light theorem prover, which was used in Hales’ proof of the Kepler conjecture, and can prove, essentially unaided by humans, many basic theorems of mathematics. They have provided their tool in an open-source release, so that other mathematicians and computer scientists can experiment with it.

The Google AI system was trained on a set of 10,200 theorems that the researchers had gleaned from several sources, including many sub-theorems of Hales’ proof of the Kepler conjecture. Most of these theorems were in the area of linear algebra, real analysis and complex analysis, but the Google researchers emphasize that their approach is very broadly applicable. In the initial release, their software was able to prove 5919, or 58% of the training set. When they applied their software to a set of 3217 new theorems that it had not yet seen, it succeeded in proving 1251, or 38.9%.

Mathematicians are already envisioning how this software can be used in day-to-day research. Jeremy Avigad of Carnegie Mellon University sees it this way:

You get the maximum of precision and correctness all really spelled out, but you don’t have to do the work of filling in the details. … Maybe offloading some things that we used to do by hand frees us up for looking for new concepts and asking new questions.

For additional details see the Google authors’ technical paper, and a New Scientist article by Leah Crane.

Present-day computerized theorem provers are typically categorized as automated theorem provers (ATPs), which use computationally intensive methods to search for a proof; and interactive theorem provers, which rely on an interplay with humans — verifying the correctness of an argument and checking proofs for logical errors. Researchers in the field, however, acknowledge that both types of software are still a far cry from a completely independent computer-based mathematical reasoning system.

For one thing, these tools have been met with cold shoulders by many present-day mathematicians, in part because they require considerable study and practice to become proficient in using them, and in part because of a general distaste for the notion of automating mathematical thought.

But some mathematicians are embracing these tools. Kevin Buzzard of Imperial College London, for example, has begun to focus his research on computerized theorem provers. But he acknowledges, “Computers have done amazing calculations for us, but they have never solved a hard problem on their own. … Until they do, mathematicians aren’t going to be buying into this stuff.”

Emily Riehl, of Johns Hopkins University, teaches students to write mathematical proofs using a computerized tool. She reports that these tools help students to rigorously formulate their ideas. Even for her own research, she says that “using a proof assistant has changed the way I think about writing proofs.”

Vladimir Voevodsky of Princeton University, after finding an error in one of his own published results, was an ardent advocate of using computers to check proofs, until his death in 2017. Timothy Gowers of Cambridge, who won the Fields Medal in 1998, goes even further, saying that major mathematical journals should prepare for the day when the authors of all submitted papers must first certify that their results have been verified by a computerized proof checker.

Josef Urban of the Czech Institute of Informatics, Robotics and Cybernetics believes that a combination of computerized theorem provers and machine learning tools is required to produce human-like mathematical research capabilities. In July 2020 his group reported on some new mathematical conjectures generated by a neural network that was trained on millions of theorems and proofs. The network proposed more than 50,000 new formulas, but, as they acknowledged, many of these were duplicates: “It seems that we are not yet capable of proving the more interesting conjectures.”

A research team led by Christian Szegedy of Google Research sees automated theorem provers as a subset of the field of natural language processing, and plans to capitalize on recent advances in the field to demonstrate solid mathematical reasoning. He and other researchers have proposed a “skip-tree” task scheme that exhibits suprisingly strong mathematical reasoning capabilities. Out of thousands of generated conjectures, about 13% were both provable and new, in the sense of not merely being duplicates of other theorems in the database.

For additional examples and discussion of recent research in this area, see this Quanta Magazine article by Stephen Ornes.

So where is all this heading? With regards to computer mathematics, Timothy Gowers predicts that computers will be able to outperform human mathematicians by 2099. He says that this may lead to a brief golden age, when mathematicians still dominate in original thinking and computer programs focus on technical details, “but I think it will last a very short time,” given that there is no reason that computers cannot eventually become proficient at the more creative aspects of mathematical research as well.

Futurist Ray Kurzweil predicts that at an even earlier era (roughly 2045), machine intelligence will first meet, then transcend human intelligence, leading to even more powerful technology, in a dizzying cycle that we can only dimly imagine (a singularity). Like Gowers, Kurzweil does not see any reason that creative aspects of human thinking, such as mathematical reasoning, will be immune from these developments.

Not everyone is overjoyed with these prospects. Bill Joy, for one, is concerned that in Kurzweil’s singularity, humans could be relegated to minor players, if not ultimately extinguished. However, it must be acknowledged even today, AI-like systems already handle many important decision-making processes, ranging from finance and investment to weather prediction, using decision-making processes that humans only dimly understand.

One implication of all this is that mathematical training, both for mathematics majors and other students, must aggressively incorporate computer technology and teach computer methods for mathematical analysis and research, at all levels of study. Topics for prospective mathematicians should include a solid background in computer science (programming, data structures and algorithms), together with statistics, numerical analysis, machine learning and symbolic computing (or at least the usage of a symbolic computing tool such as *Mathematica*, *Maple* or *Sage*).

In a similar way, university departments of engineering, physics, chemistry, finance, medicine, law and social sciences need to significantly upgrade their training in computer skills — computer programming, machine learning, statistics and graphics. Large technology firms such as Amazon, Apple, Facebook, Google and Microsoft are already aggressively luring top mathematical, computer science and machine learning talent. Other employers, in other fields, will soon be seeking the same pool of candidates.

In short, one way or the other intelligent computers are coming, and are destined to transform fields ranging from mathematical research to law and medicine. Society in general must find a way to accommodate this technology, and to deal respectfully with the many people whose lives will be affected. But not all is gloom and doom. Mathematician Steven Strogatz envisions a mixed future:

]]>Maybe eventually our lack of insight would no longer bother us. After all, AlphaInfinity could cure all our diseases, solve all our scientific problems and make all our other intellectual trains run on time. We did pretty well without much insight for the first 300,000 years or so of our existence as Homo sapiens. And we’ll have no shortage of memory: we will recall with pride the golden era of human insight, this glorious interlude, a few thousand years long, between our uncomprehending past and our incomprehensible future.