(The ancient Greeks argued that the best life was filled with beauty, truth, justice, play and love. The mathematician Francis Su knows just where to find them.)

Math conferences don’t usually feature standing ovations, but Francis Su received one last month in Atlanta. Su, a mathematician at Harvey Mudd College in California and the outgoing president of the Mathematical Association of America (MAA), delivered an emotional farewell address at the Joint Mathematics Meetings of the MAA and the American Mathematical Society in which he challenged the mathematical community to be more inclusive.

Su opened his talk with the story of Christopher, an inmate serving a long sentence for armed robbery who had begun to teach himself math from textbooks he had ordered. After seven years in prison, during which he studied algebra, trigonometry, geometry and calculus, he wrote to Su asking for advice on how to continue his work. After Su told this story, he asked the packed ballroom at the Marriott Marquis, his voice breaking: “When you think of who does mathematics, do you think of Christopher?”

Su grew up in Texas, the son of Chinese parents, in a town that was predominantly white and Latino. He spoke of trying hard to “act white” as a kid. He went to college at the University of Texas, Austin, then to graduate school at Harvard University. In 2015 he became the first person of color to lead the MAA. In his talk he framed mathematics as a pursuit uniquely suited to the achievement of human flourishing, a concept the ancient Greeks called eudaimonia, or a life composed of all the highest goods. Su talked of five basic human desires that are met through the pursuit of mathematics: play, beauty, truth, justice and love.

If mathematics is a medium for human flourishing, it stands to reason that everyone should have a chance to participate in it. But in his talk Su identified what he views as structural barriers in the mathematical community that dictate who gets the opportunity to succeed in the field — from the requirements attached to graduate school admissions to implicit assumptions about who looks the part of a budding mathematician.

When Su finished his talk, the audience rose to its feet and applauded, and many of his fellow mathematicians came up to him afterward to say he had made them cry. A few hours later Quanta Magazine sat down with Su in a quiet room on a lower level of the hotel and asked him why he feels so moved by the experiences of people who find themselves pushed away from math. An edited and condensed version of that conversation and a follow-up conversation follows.Mark Skovorodko for Quanta Magazine

QUANTA MAGAZINE: The title of your talk was “Mathematics for Human Flourishing.” Flourishing is a big idea — what do you have in mind by it?

FRANCIS SU: When I think of human flourishing, I’m thinking of something close to Aristotle’s definition, which is activity in accordance with virtue. For instance, each of the basic desires that I mentioned in my talk is a mark of flourishing. If you have a playful mind or a playful spirit, or you’re seeking truth, or pursuing beauty, or fighting for justice, or loving another human being — these are activities that line up with certain virtues. Maybe a more modern way of thinking about it is living up to your potential, in some sense, though I wouldn’t just limit it to that. If I am loving somebody well, that’s living up to a certain potential that I have to be able to love somebody well.

And how does mathematics promote human flourishing?

It builds skills that allow people to do things they might otherwise not have been able to do or experience. If I learn mathematics and I become a better thinker, I develop perseverance, because I know what it’s like to wrestle with a hard problem, and I develop hopefulness that I will actually solve these problems. And some people experience a kind of transcendent wonder that they’re seeing something true about the universe. That’s a source of joy and flourishing.

Math helps us do these things. And when we talk about teaching mathematics, sometimes we forget these larger virtues that we are seeking to cultivate in our students. Teaching mathematics shouldn’t be about sending everybody to a Ph.D. program. That’s a very narrow view of what it means to do mathematics. It shouldn’t mean just teaching people a bunch of facts. That’s also a very narrow view of what mathematics is. What we’re really doing is training habits of mind, and those habits of mind allow people to flourish no matter what profession they go into.

Several times in your talk you quoted Simone Weil, the French philosopher (and sibling of the famed mathematician André Weil), who wrote, “Every being cries out silently to be read differently.” Why did you choose that quote?

I chose it because it says in a very succinct way what the problem is, what causes injustice — we judge, and we don’t judge correctly. So “read” means “judged,” of course. We read people differently than they actually are.

And how does that apply to the math community?

We do this in lots of different ways. I think part of it is that we have a picture of who actually can succeed in math. Some of that picture has been developed because the only examples we’ve seen so far are people who come from particular backgrounds. We’re not used to, for instance, seeing African-Americans at a math conference, although it’s become more and more common.

We’re not used to seeing kids from lower socioeconomic backgrounds in college or grad school. So what I was trying to say is: If we’re looking for talent, why are we choosing for background? If we really want to have a more diverse set of people in mathematical sciences, we have to take into account the structural barriers that make it hard for people from disadvantaged backgrounds to succeed in math.

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Under ancient Jewish law, if a suspect on trial was unanimously found guilty by all judges, then the suspect was acquitted. This reasoning sounds counterintuitive, but the legislators of the time had noticed that unanimous agreement often indicates the presence of systemic error in the judicial process, even if the exact nature of the error is yet to be discovered. They intuitively reasoned that when something seems too good to be true, most likely a mistake was made.

n a new paper to be published in The Proceedings of The Royal Society A, a team of researchers, Lachlan J. Gunn, et al., from Australia and France has further investigated this idea, which they call the “paradox of unanimity.”

“If many independent witnesses unanimously testify to the identity of a suspect of a crime, we assume they cannot all be wrong,” coauthor Derek Abbott, a physicist and electronic engineer at The University of Adelaide, Australia, told Phys.org. “Unanimity is often assumed to be reliable. However, it turns out that the probability of a large number of people all agreeing is small, so our confidence in unanimity is ill-founded. This ‘paradox of unanimity’ shows that often we are far less certain than we think.”

Unlikely agreement

The researchers demonstrated the paradox in the case of a modern-day police line-up, in which witnesses try to identify the suspect out of a line-up of several people. The researchers showed that, as the group of unanimously agreeing witnesses increases, the chance of them being correct decreases until it is no better than a random guess.

In police line-ups, the systemic error may be any kind of bias, such as how the line-up is presented to the witnesses or a personal bias held by the witnesses themselves. Importantly, the researchers showed that even a tiny bit of bias can have a very large impact on the results overall. Specifically, they show that when only 1% of the line-ups exhibit a bias toward a particular suspect, the probability that the witnesses are correct begins to decrease after only three unanimous identifications. Counterintuitively, if one of the many witnesses were to identify a different suspect, then the probability that the other witnesses were correct would substantially increase.

The mathematical reason for why this happens is found using Bayesian analysis, which can be understood in a simplistic way by looking at a biased coin. If a biased coin is designed to land on heads 55% of the time, then you would be able to tell after recording enough coin tosses that heads comes up more often than tails. The results would not indicate that the laws of probability for a binary system have changed, but that this particular system has failed. In a similar way, getting a large group of unanimous witnesses is so unlikely, according to the laws of probability, that it’s more likely that the system is unreliable.

The researchers say that this paradox crops up more often than we might think. Large, unanimous agreement does remain a good thing in certain cases, but only when there is zero or near-zero bias. Abbott gives an example in which witnesses must identify an apple in a line-up of bananas—a task that is so easy, it is nearly impossible to get wrong, and therefore large, unanimous agreement becomes much more likely.

On the other hand, a criminal line-up is much more complicated than one with an apple among bananas. Experiments with simulated crimes have shown misidentification rates as high as 48% in cases where the witnesses see the perpetrator only briefly as he runs away from a crime scene. In these situations, it would be highly unlikely to find large, unanimous agreement. But in a situation where the witnesses had each been independently held hostage by the perpetrator at gunpoint for a month, the misidentification rate would be much lower than 48%, and so the magnitude of the effect would likely be closer to that of the banana line-up than the one with briefly seen criminals.

Read more at: http://phys.org/news/2016-01-evidence-bad.html#jCp

posted by f.sheikh

A Japanese mathematician claims to have solved one of the most important problems in his field. The trouble is, hardly anyone can work out whether he’s right.

Sometime on the morning of 30 August 2012, Shinichi Mochizuki quietly posted four papers on his website.

The papers were huge — more than 500 pages in all — packed densely with symbols, and the culmination of more than a decade of solitary work. They also had the potential to be an academic bombshell. In them, Mochizuki claimed to have solved the abc conjecture, a 27-year-old problem in number theory that no other mathematician had even come close to solving. If his proof was correct, it would be one of the most astounding achievements of mathematics this century and would completely revolutionize the study of equations with whole numbers.

Mochizuki, however, did not make a fuss about his proof. The respected mathematician, who works at Kyoto University’s Research Institute for Mathematical Sciences (RIMS) in Japan, did not even announce his work to peers around the world. He simply posted the papers, and waited for the world to find out.

Probably the first person to notice the papers was Akio Tamagawa, a colleague of Mochizuki’s at RIMS. He, like other researchers, knew that Mochizuki had been working on the conjecture for years and had been finalizing his work. That same day, Tamagawa e-mailed the news to one of his collaborators, number theorist Ivan Fesenko of the University of Nottingham, UK. Fesenko immediately downloaded the papers and started to read. But he soon became “bewildered”, he says. “It was impossible to understand them.”

Fesenko e-mailed some top experts in Mochizuki’s field of arithmetic geometry, and word of the proof quickly spread. Within days, intense chatter began on mathematical blogs and online forums (see Nature http://doi.org/725; 2012). But for many researchers, early elation about the proof quickly turned to scepticism. Everyone — even those whose area of expertise was closest to Mochizuki’s — was just as flummoxed by the papers as Fesenko had been. To complete the proof, Mochizuki had invented a new branch of his discipline, one that is astonishingly abstract even by the standards of pure maths. “Looking at it, you feel a bit like you might be reading a paper from the future, or from outer space,” number theorist Jordan Ellenberg, of the University of Wisconsin–Madison, wrote on his blog a few days after the paper appeared.

Three years on, Mochizuki’s proof remains in mathematical limbo — neither debunked nor accepted by the wider community. Mochizuki has estimated that it would take a maths graduate student about 10 years to be able to understand his work, and Fesenko believes that it would take even an expert in arithmetic geometry some 500 hours. So far, only four mathematicians say that they have been able to read the entire proof.

Adding to the enigma is Mochizuki himself. He has so far lectured about his work only in Japan, in Japanese, and despite being fluent in English, he has declined invitations to talk about it elsewhere. He does not speak to journalists; several requests for an interview for this story went unanswered. Mochizuki has replied to e-mails from other mathematicians and been forthcoming to colleagues who have visited him, but his only public input has been sporadic posts on his website. In December 2014, he wrote that to understand his work, there was a “need for researchers to deactivate the thought patterns that they have installed in their brains and taken for granted for so many years”. To mathematician Lieven Le Bruyn of the University of Antwerp in Belgium, Mochizuki’s attitude sounds defiant. “Is it just me,” he wrote on his blog earlier this year, “or is Mochizuki really sticking up his middle finger to the mathematical community”.

Now, that community is attempting to sort the situation out. In December, the first workshop on the proof outside of Asia will take place in Oxford, UK. Mochizuki will not be there in person, but he is said to be willing to answer questions from the workshop through Skype. The organizers hope that the discussion will motivate more mathematicians to invest the time to familiarize themselves with his ideas — and potentially move the needle in Mochizuki’s favour.

In his latest verification report, Mochizuki wrote that the status of his theory with respect to arithmetic geometry “constitutes a sort of faithful miniature model of the status of pure mathematics in human society”. The trouble that he faces in communicating his abstract work to his own discipline mirrors the challenge that mathematicians as a whole often face in communicating their craft to the wider world.

Primal importance

The abc conjecture refers to numerical expressions of the type a + b = c. The statement, which comes in several slightly different versions, concerns the prime numbers that divide each of the quantities a, b and c. Every whole number, or integer, can be expressed in an essentially unique way as a product of prime numbers — those that cannot be further factored out into smaller whole numbers: for example, 15 = 3 × 5 or 84 = 2 × 2 × 3 × 7. In principle, the prime factors of a and bhave no connection to those of their sum, c. But the abc conjecture links them together. It presumes, roughly, that if a lot of small primes divide a and b then only a few, large ones divide c.

This possibility was first mentioned in 1985, in a rather off-hand remark about a particular class of equations by French mathematician Joseph Oesterlé during a talk in Germany. Sitting in the audience was David Masser, a fellow number theorist now at the University of Basel in Switzerland, who recognized the potential importance of the conjecture, and later publicized it in a more general form. It is now credited to both, and is often known as the Oesterlé–Masser conjecture.

A few years later, Noam Elkies, a mathematician at Harvard University in Cambridge, Massachusetts, realized that the abcconjecture, if true, would have profound implications for the study of equations concerning whole numbers — also known as Diophantine equations after Diophantus, the ancient-Greek mathematician who first studied them.

Elkies found that a proof of the abc conjecture would solve a huge collection of famous and unsolved Diophantine equations in one stroke. That is because it would put explicit bounds on the size of the solutions. For example, abc might show that all the solutions to an equation must be smaller than 100. To find those solutions, all one would have to do would be to plug in every number from 0 to 99 and calculate which ones work. Without abc, by contrast, there would be infinitely many numbers to plug in.

http://www.nature.com/news/the-biggest-mystery-in-mathematics-shinichi-mochizuki-and-the-impenetrable-proof-1.18509

Posted by f.sheikh

( Pigeonhole Principle )

This meme is taken from a scene in the Cohen brother’s 1998 comedy “The Big Lebowski”. During a game of bowling, Walter, in the picture, gets annoyed at the other characters constantly overstepping the line. Drawing a gun, he asks: “Am I the only around here who gives a shit about rules?”[ii]

Considering that there are roughly 7 billion people on earth, a positive answer seems highly unlikely. But it is possible to do better. We can know with certainty, i.e. prove, that the creator of the meme is not the only one. This is a simple and straightforward application of a fascinating, intuitive and yet powerful mathematical principle. It is usually called “pigeonhole principle” (for reasons to be explained below) or “Dirichlet’s principle”.

There exist many formulations of the Dirichlet’s principle, but a very simple one is the following: Suppose you have n holes (where n is a positive integer) and n+1 pigeons. Now, no matter how hard you try, it is impossible to fit all the pigeons into individual holes. There is at least one hole that contains two (or more). Similarly for hairs. The numbers vary of course, but an average blonde person is thought to have about 150’000 hairs on their head[v]. To be on the safe side, let us assume that the hairiest person on earth has 300’000 hairs. For ease of calculation, let us further assume that there are 7 billion people on earth. Then, at least two people will have the same number of hairs. Indeed, at least 23’333 people will do.

To demonstrate the truth of this rather obscure claim, suppose it is false; this means it is not the case that at least two people will have the same number of hairs. Say, we put the 7 billion people into a row, starting with the person of 0 hairs to our left and running up to the Guinness World Record Hairiest person of 300’000 hairs. So, person 1 has 0 hairs, person 2 has exactly one hair, etc., up to person 300’001, who holds the Guinness World Record. Now what about person 300’002? Remember she has to have 0,1,…,300’000 hairs (otherwise the World Record would be broken yet again!). But all those numbers of hairs are already taken by person 1 up to 300’001, so she necessarily has the same number of hairs as someone of them.

Of course, this is a rather silly application, but the principle can be generalised (in semi-mathematical terms): If you have two sets S and T, where S contains more elements than T, there is no way of assigning a single element in T to each element in S.

So far, we have only considered finite sets, but what happens if either S or T (or both) are infinite? The pigeonhole principle still applies, and has implicitly been put to use in some of G. Cantor’s (1845-1918) most beautiful proofs, notably in his diagonal argument.

– See more at:

http://www.3quarksdaily.com/3quarksdaily/2014/10/page/2/#sthash.AAvvsvSc.dpuf

 Posted By F. Sheikh