Agree to Agree

In the course of trying to slodge through a new Quanta article about the importance of modular forms (what are they? Hey, I said "slodge" for a reason), I ran across a different article by about how mathematical proofs have a social dimension.

Quanta writer Jordana Cepelewicz interviews mathematician Andrew Granville of the University of Montreal about this in an August Q&A. The jumping off point is the claim by reclusive Japanese mathematician Shinichi Mochizuki to have created a proof solving something called the "ABC Conjecture" that has do to with a relationship between addition and multiplication. Mochizuki's 2012 proof was 500 pages long and pretty dense, even for a mathematical proof. After two other mathematics professors visited Mochizuki in 2018 and found out what seemed to be fatally flawed gaps in the proof, Mochizuki dismissed their claims by saying they did not understand his work.

While "I'm right and you're too dumb to know it" might work in conversations with politicians and many celebrity figures, it's not an acceptable way of discussing mathematical proofs. In order to be useful, they must be held to be valid by a large group of mathematically knowledgeable people, so that when those people rely on the proof in their own work it won't fall apart.

According to Granville, Mochizuki's response hit on a key feature of mathematics and proof writing. The only way a math person may prove a proof is by convincing other math people their answer is accurate and not missing anything. Now, the other math people obviously have knowledge everyday folks lack. Ask me to evaluate a complex proof and my answer will be a single word: Hellifino.

But even the math people start with understandings that might be different from the proof writer.  They are the people who can say, "You cannot use that squiggly line in this spot! It must go here instead, and if you don't see that you breathe through your mouth and your knuckles drag the ground when you walk." They may say that because they know better. Granville points out that they may also say that because that's how they learned or because that's how their equations work properly.

The upshot of his understanding is that a social factor among mathematicians plays a much larger role than anyone might have thought it would in this discipline seen as the realm of cold logic. Perhaps it once was, but work in the first part of the 20th century opened the door to letting the community of mathematicians put their thumbs on the scale. It could have been a detriment to their work, but Granville sees it as a way to build closer ties among different mathematical disciplines and ideas. 

Bear Math

This article at Quanta highlights a possible way that math can save a person in danger of being attacked by a bear.

I found it intriguing but I don't ever intend to give it a try.

Surprise!

According to this story in Quanta by Steve Nadis, mathematicians investigating large-scale set classification problems have been able to prove something they have suspected for some time. According to a recent paper by Italian mathematician Gianluca Paolini and Israeli mathematician Saharon Shelah, it seems that torsion-free abelian groups are indeed hard to classify.

Did not see that coming.

Q: Is This an Easy Problem or a Hard Problem? A: Yes

I'm not 100% sure of the solution to a famous mathematical conjecture proposed by Paul Erdős some 50 years ago -- by which I mean I'm not sure I understand it. Even this relatively simple explanation at Quanta magazine quickly gets esoteric for one whose math skills drop off once we travel beyond arithmetic.

The thing I thought most interesting about the story was that Erdős and a couple of friends -- Vance Faber and Lászlo Lovász -- dreamed up this problem as intentionally one of the simplest they could think of during a tea party. They dallied with it a little at the party and set it aside to finish the next day. "The next day" turned out to be January 2021, as five mathematicians from the University of Birmingham -- Abishek Methuku, Dong-yeap Kang, Tom Kelly, Daniela Kühn and Deryk Osthus -- finally figured out a way to prove their answer to the Erdős -Faber-Lovász Conjecture.

Although Erdős died in 1996, both Faber and Lovász are still living and congratulated the Birmingham team, which is technically known as the Combinatorics, Algorithms and Probability Team at the university.

The thing that struck me was how the problem was intentionally created to be simple and initially thought to be so by the conjecturing trio, only to turn into a mathematical hairball that took 50 years to figure out. Math, much like life, often winds up with intended simplicity giving way to unintended complexity.

Now as to whether or not I'll ever be able to figure out what any of the 8 mathematicians listed were talking about? I think that problem has a simple answer: Highly unlikely.

Purpose!

So you've slogged through some political and news posts and made yourself depressed, because it seems like the only thing that would be worse than the one guy winning the election would be the other guy winning it. Then you happen across a post for something called Mathemalchemy, in which a couple dozen artists/mathematicians are going to collaborate on a "large multimedia art installation that celebrates the creativity and beauty of mathematics." It'll be unveiled in a little under 300 days.

So there's something to look forward to after all!

MatheMADics

Writing at New Discourses, James Lindsay covers a recent low-scale dustup among academics about the idea that 2+2 could equal 5.

As Lindsey points out, a number of different academics in different fields attempted to "prove" not that 2+2 did equal 5, but that it could and the idea that it 2+2 always and must equal 4 is a form of hegemonic thinking. Lindsey's piece is long, but the upshot of it is, of course, that 2+2 equaling four is not hegemonic thinking, it's plain old logical thinking. And the idea that 2+2 could equal 5 is not a form of open-minded thinking, it's plain old illogical not thinking.

There were a couple of hilarious examples of folks trying to come up with situations in which 2+2=5. One involved two factories with two machines apiece and an assortment of spare parts. If the factories were merged and the assorted spare parts were assembled into another machine, then 2+2 would equal 5! Except, as Lindsay points out, the example actually proves that 2.5+2.5=5, which is the exact same kind of statement as 2+2=4. Other finagles may have smelled mathier but none of them provided any real case for saying that 2+2=5.

Sure, Kurt Gödel's "Incompleteness Theorem" makes it impossible to prove that 2+2=4 using just plain old arithmetic. But in the history of humanity's use of simple arithmetic it has always done so and every mathematical operation which has assumed basic addition to be true has shown itself to work. 

Lindsey quotes some of the tweets that sniffed down their collective noses at his posts (Jack Dorsey, I don't know if you are a praying man but if you are, a fit subject for your most fervent, ground groveling petitions is that karma is not real, because if it is your invention has loaded you up with enough of the bad kind to keep you reincarnating as a bug long past the heat death of the universe). It emphasizes something that people in my line of work need to remember.

People who follow Jesus and proclaim him as Lord are not, if they obey the instructions, permitted to hate other people. Especially just because of disagreements over worldviews.

But we are not required to pretend they are not stupid as all get-out.

Pi Poetry

The good folk at The Aperiodical created a contest for poetry similar to the well-known Japanese format of haiku. Only instead of the usual 5-7-5 syllable arrangement, they altered it to 3-1-4 -- the first three digits of the mathematical term π or pi -- and called it "pi-ku."

A couple of them are pretty clever. The winner was not only mathematical in composition, but also in subject matter:

Statistics
Lies?
Or, perchance, truth

Shortest Distance

The old saying goes that the shortest distance between two points is a straight line. That's clear the way we usually picture it, with a line linking two points on a flat surface or map.

And it's true on weird shapes as well, as this post at Curiosa Mathematica indicates.

If you were to find yourself on a shape like the above, you should probably first stop doing drugs. But if it turns out that you really were standing on this strange surface and you started walking straight ahead at 0º, you would wind up following the orange line. After which you might need some drugs, specifically dramamine.

Pieces of the Puzzle

So, were you and I waiting by the screen to see when we would finally know whether or not the number 33 could be expressed as the sum of three cubes?

Of course you weren't -- you, O Tolerant Reader, have a life and I have an inability to math very well. Nor, I imagine, did we know that there was a search on for that number. In any event, the solution to k = x³+ y³+ z³ when k = 33 has been found: (8,866,128,975,287,528)³ + (–8,778,405,442,862,239)³ + (–2,736,111,468,807,040)³ = 33.

For some numbers, the equation solves simply. You can write 29 as 3³ + 1³ + 1³. For others, there is no solution. Any number that has either 4 or 5 as a remainder when you divide it by 9 can't be written as the sum of three cubes. So while 33 has this newly-found solution, 32 will never have one. Divide 32 by 9 and you get 27, with 5 left over.

Andrew Booker of the University of Bristol wrote the algorithm which found the number. He figured the supercomputer running it would take six months to solve the problem, but it actually took only three weeks. There only two numbers between 1 and 100 that had never been solved were 33 and 42, and now only 42 remains. Booker will train his algorithm on that next, although the search will involve even larger numbers than the quadrillions that solved for 33.

One reason to find the answers to these so-called "stubborn numbers" is because mathematicians don't really like having unsolved equations laying around. Another is that finding solutions like this can play a role in some future attempts to find proofs for k = x³+ y³+ z³, or proofs that use it.

Left as yet undiscussed is the possibility that solving this polynomial for 42 might just be the way to find three cubes that add up to everything.

Here's Looking at Euclid

The interesting thing about geometry is that it's real in a couple of ways. There's what we call Euclidean geometry, named after an ancient mathematician/philosopher, that we use in everyday life to measure things. In it, parallel lines never converge and the sum of the angles of a triangle always equals 180 degrees.

But we live on the surface of a sphere, which means that the endless plane we imagine when we construct our Eucledean drawings is really curved. And on a curved surface, parallel lines do intersect and the angles of a triangle add up to more than 180 degrees.

As this quote on Math Blab from English mathematician G. H. Hardy suggests, his brother and sister number wonks have constructed several such non-Euclidean geomtries, each of which is perfectly internally consistent despite their significant differences from one another. The only thing that changes between them are their initial assumptions.

For some reason, contemplation of the different descriptions of the world that can hang together and be internally real proves peaceful this evening. Although I presume that if I were a math student attempting to master that understanding for the purpose of an exam or project my serenity might diminish -- just like the distance between two parallel lines drawn on a sphere.

Numbers

David Berlinksi is among the leaders of writing so-called popular books about different aspects of math -- some that is highly advanced, as in The Advent of the Algorithm, and some that is very very basic, as in 2011's One, Two Three.

Berlinksi assigns the basic mathematical functions the group name "AEM" or Absolutely Elementary Mathematics. The four major functions of addition, subtraction, multiplication and division are outlined as the building blocks of far more complicated functions and equations. Berlinksi also digs even deeper, offering ways to think about even the idea of "number."

Berlinski holds doctorates in philosophy and mathematics, so he is a good choice to explain math concepts in terms that don't lean too heavily on equations. His purpose in One, Two, Three is to suggest answers for these simplest and most basic questions about AEM and to show how such answers can be deduced via logic from some very simple assumptions.

One, Two, Three is both aided by and labors under Berlinksi's habit of breezy and almost flippant writing. On the one hand he largely succeeds in getting complex ideas boiled down to terms that most people can understand, and presents his arguments in ways that can be followed without specialized knowledge. But on the other hand, his tone sometimes crosses over into flippancy in ways that can slow readers down while they finish rolling their eyes.

He too often sacrifices some clarity and direction in order to make a witty observation and in more than one place sticks in some jokes for their own sake rather than explanatory value (yes, O Tolerant Reader, this blog does the same thing quite often. However, it's the product of some moke running his mouth and not someone explaining a potentially difficult subject. Judge for yourself what kind of damage that can do to explaining an idea). Whether or not Berlinksi is actually all that impressed with his own wit, he gives a good enough imitation of being so to make several parts of One, Two, Three way more annoying and way less useful than they could have been.

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Of all the people who probably hide their heads at the goofs they have made, the person who named "imaginary" numbers is probably among them as far as the field of mathematics is concerned.

So-called "imaginary" numbers describe the square roots of negative numbers, which are impossible to calculate using plain integers. The square root of 1, for example, is 1 because 1x1=1. But the square root of -1 seems impossible to figure, because the only way to get to -1 is to multiply two different numbers together. A negative number multiplied by another negative number leaves a positive number, not a negative one. At some point, mathematicians decided that there would be a square root of -1, and it would just be a 1 that was on another "axis" than the regular positive-negative line. But since the number didn't seem to have any real-world analogue like positive and negative numbers did, it somehow got hung with the tag, "imaginary." So today we say that the square root of negative 1 is i. The square root of -4 is 2i, and so on.

Retired electrical engineering professor Paul Nahin outlines some of the development of i through the history of mathematics in An Imaginary Tale. Some early cultures refused to acknowledge the existence of a quantity that could be squared to form a negative number, and even into the Renaissance and enlightenment years the so-called "imaginary" numbers were considered at best unimportant. They were not useful except in specialized cases and it seemed even mathematicians had reservations about dealing with numbers that didn't represent any real quantity.

Today, i and its counterparts find widespread use in many areas of math, and the only reservations that seem to continue deal mostly with the use of the word "imaginary." Nahin explores how important i is in many fields of engineering, especially his own. This part of the book -- about the latter two-thirds -- is heavily laden with equations and formulas and is going to be beyond most non-mathematician or non-engineer readers. He probably would have had to lengthen the book considerably to bring that subject matter within the grasp of the lay reader, but that doesn't make the string of equations and engineering language any easier to navigate.

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Most of the time we use math we do just that: use math. We rarely think about things like why numbers come together the way they do. Or why certain mathematical functions and relationships seem to matter in the world of actual things as well as within the realm of pure equation and solution. But almost every mathematical advance throughout human history has often stemmed from and also sparked some serious thought.

Luke Heaton's A Brief History of Mathematical Thought skims through history and sketches some of the thinking that accompanied the ciphering. He begins as close to the beginning as possible, offering some ideas on how our stone-age ancestors may have begun to progress beyond the simple counting of objects into understanding the numbers behind the counting had relationships that could be regularized and predicted. At what point, for example, did some forgotten genius figure out that two of anything added to three of that same thing would always make five of that thing? If you had two rocks and were handed three, you did not need to count all five of them over again -- you could add the three to your two and know you had five whether you counted them or not. And once people had developed this understanding, how did it change their civilization and culture?

History is better in the earlier sections, such as the one mentioned above and others that deal with numerical development among the ancient Greeks, ancient Indians and other civilizations. It's also a good overview of how the switch to Arabic numerals and the use of the zero as a place-keeper propelled scientific thought far beyond what had been possible with cumbersome systems like Roman numerals. Later sections, though, deal with more esoteric subjects within math and their impact seems less obvious. Heaton offers reasons to spend some time pondering non-Euclidian geometry, for example, but has fewer explanations about how this particular wrinkle affects the way we live and work. Still, History is a good primer on what kind of thought can come from dwelling on even the most mundane of numerical tasks, as well as how that thought has shaped who we are today.

Time Stampin'

At Plus, Antonella Perucca describes the ways that the mathematically hip could write all of the numbers necessary on a clock face by using only one number.

The trick is to put that number into several different math functions, the answers to which shake out to be the proper chronological digits.

This could be a useful way to help students test their knowledge of different equations or functions. But since some schools are ditching clocks with hands because students don't know how to use them -- and the schools apparently don't know how to teach -- it probably won't be a widespread activity.

Equations That Rhyme?

This space recently reviewed a book by a German physicist who spends a lot of time on issues connected to the concept of mathematical beauty. Most of us didn't encounter that idea when we took math, unless we really got into the subject. A mathematician has a slightly different view of beauty when he or she refers to an equation instead of a painting, natural scenic view or a person.

Turns out math can do a lot of things, like be a limerick. Math blogger "zorn-lemon" created the following equation:

The equation, when read in words, can be made into a limerick poem:

The integral of inverse two z
Taken over a circle, size e
Divided by i
Times square root of pi
Gives you gamma of one-sixth times three

My appreciation for this particular move does not mean I in any way understand the equation in either format.

Baby Numbers

The Teaching Company is the organization that sells video and audio lecture series called "The Great Courses." They're lectures by university professors based on courses that the professors may teach during the year. The lectures are usually altered a little bit to make up for the fact that there's rarely any supplemental reading material and certainly no assignments or exams, but most of the content is the same that they would teach a regular classroom of students.

I'm a customer, finding among the many and varied course offerings something now and again that's worth the cost of the download (The DVD's are often quite pricey). One was Dr. Dennis Kung's How Music and Mathematics Relate, which showed exactly that: The many ways in which things we see in music represent mathematical concepts and relationships. It didn't require expert level knowledge of either field, which is good because I have neither.

But the knowledge that I do have may have come hard-wired into my brain, or at least picked up at a very early age. This excerpt from one of Dr. Kung's lectures in the course describes experiments in which babies were shown to grasp simple arithmetical relationships like addition. Babies as young as five months showed they knew what should happen when they saw a second object added to a first object: they should see two objects. If they didn't, they stared longer trying to figure out what went wrong (I have had the same response when listening to elected officials describe how they will offer free stuff. And sometimes even other responses common to babies, such as shrieking and throwing things).

Dr. Kung also notes that experiments showed babies understood music as more than a string of random noises, well before they had begun to process the idea that certain sounds represent certain things. If they turned their heads one way, they heard one song, but if they turned their heads the other direction they heard a different song, and they were able to exhibit preferences early on. If they spit up, they heard Nickelback, and if they had filled their diapers they heard Kanye West. That last phase of the experiment was canceled when the babies in the tests developed constipation.

The temptation, of course, is to ask why elected officials seem unable to grasp certain things about math that are evident to us at the earliest stages of our development, such as what happens when numbers are added together and what happens when they are subtracted. But they understand it quite well: They take your money and my money, and then add it together and call it their money. It's not math they don't get; it's pronouns.

Figuring Out the World

You don't have to study too much modern science before you figure out the world is weird. Realizing that most of what we see in the objects around us is actually empty space, learning that at its most basic level matter has an uncertainty about it that can't be overcome, and so on and so on.

Scientists have operated under the idea that this weird world is understandable and that even if language can't explain it, math can. As our knowledge of the universe expands past the limits of what scientific instruments can detect, that math becomes more and more important. A theory about what matter is like at its most fundamental level may not be experimentally provable because it deals with forces or particles beyond our ability to detect. But it can make some predictions about things which are observable that, if true, would point in that theory's direction, putting a more solid foundation under the esoteric math and conjecture presented. If the experiments don't pan out, then that math may need to be discarded in favor of other possible explanations.

Sabine Hossenfelder, a theoretical physicist and research fellow at the Frankfurt Institute for Advance Studies, wonders if physicists and researchers have gotten a little too dependent on certain kinds of math or math with certain features. She wonders if that dependence has made it difficult for them to continue to explore the universe around us because they are looking for answers only in places that will confirm what they have already suggested is true. The math they follow seems to have congealed around the ideas of "naturalness" and "beauty," leaving whole areas of inquiry unexplored if they lack those two qualities. In Lost in Math: How Beauty Leads Physics Astray, Hossenfelder explores how those ideas came to hold the power they do and why, exactly, that may be a problem.

"Naturalness" roughly means that scientists prefer certain kinds of answers to questions and certain measurements. For example, if two different experiments on related matters produce very long answers that are only different in their smallest digits, scientists are uneasy. Pi, for example, is the ratio of the circumference of a circle to its diameter. It's a never-repeating, never ending number. If some other geometric ratio were found that matched pi almost exactly, not deviating until the 20th decimal place then scientists would be suspicious of the similarity, which they call "fine-tuning." Either there is a connection that they missed or there is another principle more basic than the two being considered. In the interests of full-disclosure: A person such as myself, mired in my traditional Christian theism, has much less of a problem with a fine-tuned universe than do most scientists.

Scientists are also suspicious of numerical solutions to natural equations and ratios that are very large or very small. If an equation describes a natural process, like the force of gravity, then the solution when some of the variables are replaced with real values needs to be an ordinary kind of number rather than 47 quintillion or so.

The preference for naturalness combines with another scientific preference when it comes to equations, which scientists often call "beauty." That word is often a short-hand for equations that are simply written, symmetrical in appearance, and significant in describing the world. Equations that have values which can't themselves be reduced into other equations are simple. Ones which involve basic rather than complex math on either side of an equals sign are more symmetrical. And ones that describe the actual world around us are significant.

Hossenfelder has no real problem with either of these concepts, noting that they have sometimes functioned as good criteria for evaluating scientific hypotheses. Her first few chapters outline how the concepts developed and how they have been used to advance knowledge. The problem comes when naturalness and beauty become the gatekeepers that decide which theories will be tested by experiment and which ones won't.

She says this has been especially true as scientists explore what is called the Standard Model of Physics. It's successfully described much of the real world on the very smallest scales and offered predictions which later tested out to be true. But it has some gaps and as scientists try to rope the force of gravity in with the other three fundamental forces of the universe, those gaps loom large.

One of the theories that would help bridge the gap is super-symmetry. It has, Hossenfelder says, elegant equations and avoids the perception of fine-tuning. Without going into detail I certainly don't understand, one thing super-symmetry has predicted are certain subatomic particles which have never yet been observed. Even more and more powerful experiments at the Large Hadron Collider have failed to show any confirmable evidence the particles exist. Hossenfelder wonders why that fact has spawned more doubling down on confirming super-symmetry through more expensive and elaborate experiments instead of a flurry of, "Well, what else might be true?"

On the one hand Lost in Math could be seen as a book-length gripe about confirmation bias. Scientists have become so certain that the best and truest descriptions of the universe have more naturalness and beauty that they now assume that situation to be true instead of question whether or not it is.

But Hossenfelder's explorations of exactly how these two concepts came to carry such weight are great studies in the history of science, and show just why many scientists hold to them. She also has clear explanations of a lot of the ideas about the universe that get batted around in the media, mostly it seems by people who could stand to read her explanations. Her writing is clear and a lot of fun, and all the more impressive when you realize she's doing it in another language than her everyday one. Each chapter ends with a list of summary bullet-points that help a reader keep the big ideas in mind before forging ahead.

Hossenfelder includes her interviews with a number of physicists, and a couple of them hint that some of those physicists probably consider her something of a grind when it comes to these ideas. But that's probably a good description of how scientists do their best work: They keep asking questions about stuff, especially the stuff everyone thinks is most certainly true.

Lost in Math has the rare and enjoyable quality of explaining a current scientific situation and what leads to it, as well as the underlying concepts, in layperson's terms while still communicating some of the complicated concepts underneath. Despite the title, it's a book that does not automatically leave a non-scientist reader lost, either in math or theoretical physics.

Cipherin' Champs!

This entry at Curiosa Mathematica offers a list of the winners of some of mathematics top awards for 2018, including four people who won the Fields Medal, sometimes labeled the Nobel Prize of math. Fields Medal recipients probably like to phrase that the other way around, but the Nobel came first so it gets the notoriety.

University of Cambridge professor Caucher Birkar had to be awarded a replacement medal; the briefcase containing his original one was stolen just minutes after it had been given to him, while pictures were being taken. Given that Birkar is an Iranian Kurd who applied for and received asylum in England while studying at Tehran, he will probably take the theft in stride.

The likelihood that the thief or thieves will recognize much profit from their action is small -- although there's about $4,000 worth of gold in the medal it will probably have to be melted down in order to be sold and one art theft expert estimates it will wind up bringing in only about a quarter of that value.

The Numbers Game

People who think that ridding humanity of religion and such will remove the weirdness and episodes of random oddball wackiness from the world will probably continue to be disappointed as long as there's math around. Case in point is an irrational number that goes by the name phi or the Greek letter Φ. Like its more famous cousin π, Φ comes from a geometric relationship. It's a way to divide a line in such a way that the ratio of the two unequal pieces added together to the longer piece is the same as the larger piece to the smaller one. And like the other one, it never repeats and never ends, starting out as 1.618033 and going on from there.

The ancient Greeks calculated the number, called the Golden Ratio because of its aesthetically pleasing quality. But things started to get weird when Φ started showing up in nature. Such as the spiral shell of  a nautilus, which spiraled inward along the same ratio as the line. Botanists found it showing up in the distribution of leaves on a tree branch -- if not exactly, close enough often enough to be significant. Modern researchers have found it in different qualities on the molecular level.

Astrophysicist Mario Livio, in 2003's The Golden Ratio, reviews some of the places Φ is supposed to have shown up across history. He finds that in a lot of those cases, Φ's either not really there or the similarity to it is something of a coincidence. It probably didn't influence the construction of the pyramids or the way Da Vinci painted Mona Lisa, for example. But it does show up in enough different places to be weird enough.

The chapters get a little repetitive and it's possible that Livio could have dropped one or two suspected appearances of the Ratio that turned out to be incorrect. But he's a gifted science writer with a real knack for moving complicated concepts into the realm of lay understanding, and he leaves plenty of room for a readers to figure out for themselves what they think about the prevalence of Φ in the universe and why this particular mathematical expression shows up as often as it does in the real world.

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Irrationality in math is not the same as in other arenas. In fact, if the old definition of insanity is doing the same thing over and over again while expecting different results, mathematical irrationality is its opposite. Irrational numbers are ratios that can't be expressed as a fraction of integers -- they never repeat, no matter how many times the division in the ratio is carried out. Which is one wrinkle with them already -- how do we know a decimal never ends and never repeats? Simple math alone can't get us there; irrationality requires a mathematical proof of its existence.

Some irrationals are common and famous -- π as the ratio of a circle's circumference to its diameter is one. Some showed up when mathematicians got curious about what might happen when they played this or that game with numbers or equations, like the mathematical constant called e. In his 2017 book The Irrationals, mathematician Julian Havil offers some of the history of irrationals, first discovered by ancient Greek and some Hindu mathematicians. He explains how some of the better-known were first discovered and how new ones appear even in math today. The ability of computers and their ability to calculate immense strings of digits mean mathematicians are less sure than they used to be about the non-repeating aspect of irrationals -- they probably don't repeat, but there may be some wiggle room.

As in some of his other books Havil is not shy about using mathematical formulas and equations, many of which are blank space to people who didn't progress much beyond pre-calculus and have forgotten large swaths of that. It may be unavoidable but it's an unfortunate feature of what is a really interesting set of ideas about our weird ol' universe.

Doing Your Sums

Ordinarily math shows things fitting together in a predictable and orderly fashion. One object added to another object makes a pair of objects. Our intuition and experience of the world suggest the outcome of this operation, and when we apply math to it we find out that our intuition was right.

Except when it isn't. Mathematician Julian Havil offers several scenarios in which the "usual thing" produces the exact opposite of what intuition and experience suggest in his 2007 book Nonplussed! He also outlines the mathematics behind events that should be impossible -- like, for example, why a cone can roll uphill, unaided by outside forces.

Oddball things like these generally involve much more complicated math than simple arithmetic. Havil, who taught math at Winchester College for more than 30 years, is quite well equipped to lay out the various formulas that show why, for example, the 13th of a month is more likely to be a Friday than any other day of the week. Or why a bad sports team can improve its performance by adding worse players than its opponents do.

Nonplussed! is, in fact, pretty formula heavy. Havil has written on this topic in some other books and also on other numerical and mathematical topics, but this particular volume is aimed at readers with a basic working knowledge of calculus. He sets up the problems verbally, but in explaining how the counterintuitive results come about he leans much more heavily on equations that will make little sense to those without such knowledge. It doesn't really harm the book but it does significantly limit the potential audience.

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Very broadly speaking, there are two kinds of math problems in the world: Ones which can easily be solved by a computer and one whose solutions can be easily checked by a computer. "Solved" in this sense means "proven to be true for any value of the variable." Checked means that if a number is plugged into the equation, it can be seen if the equation still makes sense or if it produces an impossible answer.

The set of solvable equations is called P. The set of checkable equations is called NP. And one of the biggest questions in mathematics, one whose solution could make the world a completely different place, is does P = NP? Georgia Tech computer science professor Lance Fortnow has spent much of his career considering this problem and writes about some of the history of its investigation in 2013's The Golden Ticket: P, NP and the Search for the Impossible.

Current mathematical thought suggests that P ≠ NP, although it can't yet prove that to be true. Fortnow outlines what kind of solutions to world problems might happen if P = NP, such as a wide range of cures for cancer not now possible. It would also make some things significantly more difficult, such as online security. Current online security relies on equations that can't be easily solved because of the number of digits they use and immense number of possible number combinations. But in a P = NP world, those equations could be solved and so computer security would become much more complicated.

Fortnow also describes some of the history of different attempts to determine whether we live in a world where P = NP or where P ≠ NP. He offers some thoughts on what the development of quantum computing, which is projected to be immensely faster than current computing, might mean about getting a definitive answer either way. He keeps the formula and equation use to a minimum, saving much of it for appendices for the readers interested in that part of the story.

The Golden Ticket is a great introduction to a math problem that few people know about and even fewer understand, and a good way to try to start thinking about it and its implications. Although Fortnow doesn't delve much into the philosophical implications of the P, NP problem, he provides enough of the basic tools for the curious to begin that part of the journey themselves.

How I Spent My Christmas Vacation

Electrical engineer Jonathan Pace of Tennessee lent his computer to a project called GIMPS -- the Great Internet Mersenne Prime Search. It's a kind of crowd-sourced project that links multiple computers to increase their processing power. And on Dec, 26, Pace's computer was the one that found the largest known prime number and 50th Mersenne Prime, after six solid days of computing.

Volunteers download a software package to run that searches for the prime numbers, which are numbers that aren't divisible by any number other than themselves and one. They have no pattern and so the only way to check if a number is prime is to start dividing it by all of the numbers smaller than it is. This can work early on, but by the time we get into large numbers it takes longer and longer, requiring calculations done at computer-only speed. Even then, Pace's six-day run shows that the job is not easy.

The number was given the name M77232917 because it is 2 raised to the 77,232,917th power, minus one. It has more than 23 million digits. The computer found it by multiplying 77,232,917 2s and then subtracting 1.

Mersenne Primes take their name from the French monk Marin Mersenne, who in the 17th century offered a theory about certain kinds of prime numbers that today bear his name. M77232917 is just the 50th Mersenne Prime. They're found by multiplying two together x times, where x represents another prime number, and then subtracting 1. So 3 is a Mersenne Prime, because 2 multiplied by itself is 4, minus one is 3.

They also generate what mathematicians call "perfect numbers," which are numbers whose proper divisors add up to the number. The smallest perfect number is 6, because 6 is divisible by 1, 2 and 3, and 1+2+3 = 6. The perfect number from M77232917 has more than 46 million digits. Perfect numbers so far are all even, which is interesting because other than 2 itself, all prime numbers are odd.

These numbers are so huge they exist in complete abstraction -- there is not enough of anything in the universe to require them to actually count it. But immense primes have proven useful in cryptography and internet security, so the search goes on. And for Pace, M77232917 = 3,000, because that's the cash prize he's eligible to share in following the discovery.

Fluids' Dynamic

If you're ever poured a colored liquid into clear water you've seen now it first billows outward before diffusing throughout the container. And you've probably noticed how the amount of liquid poured and how fast it's poured affects the shape of the billowing. Although the action seems to produce similar results from similar amounts and speeds, it would seem impossible to predict with any great accuracy how the two currents would interact.

But believe it or not, there are mathematical equations that describe those changes to a degree that scientists can often predict not just something as simple as two liquids in one container but the interactions of ocean currents and airflows in the atmosphere. They're called the Navier-Stokes equations and they've been around for almost two hundred years. Claude-Louis Navier and George Gabriel Stokes didn't work as a team to develop them, but their development of how to apply Newton's laws of motion to elastic materials linked up and were collected under their names. Navier is one of the 72 names inscribed on the Eiffel Tower and Stokes held the Lucasian Chair in Mathematics at Cambridge -- a job also held by Isaac Newton, Paul Dirac and Stephen Hawking, among others.

Navier-Stokes equations help meteorologists forecast weather changes. Air behaves like a very, very thin fluid so the equations can predict some of its motions. Oceanographers predict changes in sea currents depending on the temperature or relative strength of some motion in the water. Both groups will use computers to build models of likely air or water behavior given starting conditions. Because new factors can change conditions in an instant, those predictions are not necessarily as precise or accurate as they would be in computer simulations.

As their name indicates, the Navier-Stokes equations are mathematical operations. They have proven more than adequate to describing the physical world in which we live. This means that physicists, as well as oceanographers, meteorologists and other scientists who work with fluids are quite satisfied with them. Mathematicians, on the other hand, aren't. Mathematicians deal with equations that may or may not apply to "real world" situations; either way they focus on the numbers and such involved as abstract concepts instead of physical things.

And the mathematicians think that the Naver-Stokes equations may have a problem or two when they are handled outside of their real-world contexts. Under certain conditions, the equations describe two possible states for a fluid at the same time, which is a no-no (unless you're doing quantum mechanics, but that's another beastie). The example in the story at Quanta magazine is of a perfectly still glass of water. When the Navier-Stokes equations are turned loose on it under certain parameters, then you have a glass of water that either stayed still all night or at some time spontaneously erupted in the glass and then returned to its still state. Ghost Hunters and similar shows notwithstanding, that sort of thing doesn't happen. But even if it did, the Navier-Stokes equations should tell an observer which one it was rather than coming up with both answers at the same time.

If the math crowd does figure out that the Navier-Stokes equations are flawed, they probably won't get abandoned. After all, Albert Einstein showed that Newton's own Laws of Motion got a little wrinkly when things were either very fast or very small, but we still use Newton's understanding most of the time. Things rarely move that fast and even though we know the very very small is real, its fuzziness doesn't translate to everyday-sized objects. So the physicists, meteorologists, oceanographers and others will probably keep using them (although the meteorologists on TV will usually choose whichever model allows them to monger the most fear).

The possible dichotomy does provoke interesting possibilities. One of the things that Einstein did with his theories of relativity was explain a kink in Mercury's orbit that plain ol' Newtonian physics couldn't. Could the mathematical inadequacy of the Navier-Stokes equations prompt some new world-flipping paradigm shift? Who knows? But it will be fun to watch.