“People often follow a different way of thinking than the one dictated by classical logic,” says Aerts. “The mathematics of quantum theory turns out to describe this quite well.”

It’s a finding that has kicked off a burgeoning field known as “quantum interaction”, which explores how quantum theory can be useful in areas having nothing to do with physics, ranging from human language and cognition to biology and economics.

{ New Scientist | Continue reading }

image { William Hogarth, Analysis of Beauty, 1753 }

Paraconsistent mathematics is a type of mathematics in which contradictions may be true. In such a system it is perfectly possible for a statement A and its negation not A to both be true.

How can this be, and be coherent? (…)

This statement is false.

To be true, the statement has to be false, and vice versa. Many brilliant minds have been afflicted with many agonising headaches over this problem, and there isn’t a single solution that is accepted by all. But perhaps the best-known solution (at least, among philosophers) is Tarski’s hierarchy, a consequence of Tarski’s undefinability theorem.

In a nutshell, Tarski’s hierarchy assigns semantic concepts (such as truth and falsity) a level. To discuss whether a statement is true, one has to switch into a higher level of language. Instead of merely making a statement, one is making a statement about a statement. A language may only meaningfully talk about semantic concepts from a level lower than it. Thus a sentence such as the liar’s sentence simply isn’t meaningful. By talking about itself, the sentence attempts unsuccessfully to make a claim about the truth of a sentence of its own level. (…)

How does a paraconsistent perspective address these paradoxes? The paraconsistent response to the classical paradoxes and contradictions is to say that these are interesting facts to study, instead of problems to solve.

{ +Plus | Continue reading }

Sorbonne in Paris, France, on 8 August 1900 (…) David Hilbert’s address set the mathematical agenda for the 20th century. It crystallised into a list of 23 crucial unanswered questions, including how to pack spheres to make best use of the available space, and whether the Riemann hypothesis, which concerns how the prime numbers are distributed, is true.

Today many of these problems have been resolved, sphere-packing among them. Others, such as the Riemann hypothesis, have seen little or no progress. But the first item on Hilbert’s list stands out for the sheer oddness of the answer supplied by generations of mathematicians since: that mathematics is simply not equipped to provide an answer.

This curiously intractable riddle is known as the continuum hypothesis, and it concerns that most enigmatic quantity, infinity. Now, 140 years after the problem was formulated, a respected US mathematician believes he has cracked it. What’s more, he claims to have arrived at the solution not by using mathematics as we know it, but by building a new, radically stronger logical structure: a structure he dubs “ultimate L”.

{ NewScientist | Continue reading }

artwork { Jean-Michel Basquiat, Flexible, 1984 }

We don’t quite understand small probabilities.

You often see in the papers things saying events we just saw should happen every ten thousand years, hundred thousand years, ten billion years. Some faculty here in this university had an event and said that a 10-sigma event should happen every, I don’t know how many billion years.

So the fundamental problem of small probabilities is that rare events don’t show in samples, because they are rare. So when someone makes a statement that this in the financial markets should happen every ten thousand years, visibly they are not making a statement based on empirical evidence, or computation of the odds, but based on what? On some model, some theory.



So, the lower the probability, the more theory you need to be able to compute it. Typically it’s something called extrapolation, based on regular events and you extend something to what you call the tails. (…)

The smaller the probability, the less you observe it in a sample, therefore your error rate in computing it is going to be very high. Actually, your relative error rate can be infinite, because you’re computing a very, very small probability, so your error rate is monstrous and probably very, very small. (…)

There are two kinds of decisions you make in life, decisions that depend on small probabilities, and decisions that do not depend on small probabilities. For example, if I’m doing an experiment that is true-false, I don’t really care about that pi-lambda effect, in other words, if it’s very false or false, it doesn’t make a difference. (…) But if I’m studying epidemics, then the random variable how many people are affected becomes open-ended with consequences so therefore it depends on fat tails. So I have two kinds of decisions. One simple, true-false, and one more complicated, like the ones we have in economics, financial decision-making, a lot of things, I call them M1, M1+.

{ Nassim Nicholas Taleb/Edge | Continue reading }

Sleeping Beauty goes into an isolated room on Sunday and falls asleep. Monday she awakes, and then sleeps again Monday night. A fair coin is tossed, and if it comes up heads then Monday night Beauty is drugged so that she doesn’t wake again until Wednesday. If the coin comes up tails, then Monday night she is drugged so that she forgets everything that happened Monday – she wakes Tuesday and then sleeps again Tuesday night. When Beauty awakes in the room, she only knows it is either heads and Monday, tails and Monday, or tails and Tuesday. Heads and Tuesday is excluded by assumption. The key question: what probability should Beauty assign to heads when she awakes?

{ Overcoming Bias | Continue reading }

photo { Man Ray, Solarization, 1929 }

The Rule of 72 deserves to be better known among technical people. It’s a widely-known financial rule of thumb used for understanding and calculating interest rates. But others, including computer scientist and start-up founders, are often concerned with growth rates. Knowing and applying the rule of 72 can help in developing numerical literacy numeracy around growth.

For example, consider Moore’s Law, which describes how “the number of transistors that can be placed inexpensively on an integrated circuit doubles approximately every two years.” If something doubles every two years, at what rate does it increase per month, on average? If you know the rule of 72, you’ll instantly know that the monthly growth rate is about 3%. You get the answer by dividing 72 by 24 (the number of months).

{ Terry Jones | Continue reading }

Villagers belonging to an Amazonian group called the Mundurucú intuitively grasp abstract geometric principles despite having no formal math education, say psychologist Véronique Izard of Université Paris Descartes and her colleagues.

Mundurucú adults and 7- to 13-year-olds demonstrate as firm an understanding of the properties of points, lines and surfaces as adults and school-age children in the United States and France, Izard’s team reports.

These results suggest two possible routes to geometric knowledge. “Either geometry is innate but doesn’t emerge until around age 7 or geometry is learned but must be acquired on the basis of general experiences with space, such as the ways our bodies move,” Izard says.

Both possibilities present puzzles, she adds. If geometry relies on an innate brain mechanism, it’s unclear how such a neural system generates abstract notions about phenomena such as infinite surfaces and why this system doesn’t fully kick in until age 7. If geometry depends on years of spatial learning, it’s not known how people transform real-world experience into abstract geometric concepts — such as lines that extend forever or perfect right angles — that a forest dweller never encounters in the natural world.

{ Science News | Continue reading }

artwork { Richard Serra }

Quantification — describing reality with numbers — is a trend that seems only to be accelerating. From digital technology to business and financial models, we interact with the world by means of quantification.

While we all interact with the world through more-or-less inflexible models, mathematics contributes to this lack of flexibility because it is seemingly precise and objective. Even though mathematical models can be very complex, you can use them without understanding them very well. A trader need not really understand the financial engineering models that he may use on daily basis. This uncritical acceptance amounts to the assumption that reality is identical to our rational reconstruction of reality — for example, that the economy or the stock market is captured by our latest model. (…)

Statistical models are all based on the notion of randomness, but no one can really understand randomness. Many people use the word random without realizing that random means what it says — randomness cannot be predicted or controlled. A model of randomness is no longer true randomness.

Because they are logically consistent, mathematical models screen out ambiguity. Ambiguity is real, but business and financial models have little to no room for it.


{ Harvard Business Review | Continue reading }

Could Einstein’s Theory of Relativity be a few mathematical equations away from being disproved? 12-year-old Jacob Barnett thinks so. And, he’s got the solutions to prove it.

Barnett, who has an IQ of 170, explained his expanded theory of relativity — in a YouTube video. (…)

While most of his mathematical genius goes over our heads, some professors at the Institute for Advanced Study in Princeton, New Jersey — the U.S. academic homeroom for the likes of Albert Einstein, J. Robert Oppenheimer, and Kurt Gödel — have confirmed he’s on the right track to coming up with something completely new. (…)

“I’m impressed by his interest in physics and the amount that he has learned so far,” Institute for Advanced Study Professor Scott Tremaine wrote in an email to the family. “The theory that he’s working on involves several of the toughest problems in astrophysics and theoretical physics.”

“Anyone who solves these will be in line for a Nobel Prize,” he added.

{ Time | Continue reading + video }

Forget passwords, tricky sums are more secure

Classic user identification requires the remote user sending a username and a password to the system to which they want to be authenticated. The system looks up the username in its locally stored database and if the password submitted matches the stored password, then access is granted. This method for identification works under the assumption there exist no malicious users and that their local terminals cannot be infected by malware. (…)

Nikolaos Bardis of the University of Military Education, in Vari, Greece and colleagues there and at the Polytechnic Institute of Kiev, in Ukraine, have developed an innovative approach to logins, which implements the advanced concept of zero knowledge identification.

Zero knowledge user identification solves these issues by using passwords that change for every session and are not known to the system beforehand. The system can only check their validity.

{ ScienceText | Continue reading }

Many people like mathematics because it gives definite answers. Things are either true or false, and true things seem true in a very fundamental way.

The fact is, however, that mathematics has been moving on somewhat shaky philosophical ground for some time now. With the work of the logician Kurt Gödel and others in the 1930s it became clear that there are limits to the power of mathematics to pin down truth. In fact, it’s possible to build different versions of mathematics in which certain statements are true or false depending on your preference. This raises the possibility that mathematics is little more than a game in which we choose the rules to suit our purpose.

{ Plus mag | Continue reading }

related { Researchers have found a fractal pattern underlying everyday math. In the process, they’ve discovered a way to calculate partition numbers, a challenge that’s stymied mathematicians for centuries. | Wired | full story }

(…)

{ A Paradoxical Property of the Monkey Book | Continue reading }

There is a curious text, of an author who, I don’t know why, isn’t read anymore. A psychiatrist, son of an abominable historian of philosophy of the 19th century. He was called Pierre Janet. He used to be very well-known. He was more or less contemporary to Freud, his career is quite parallel to Freud’s. And neither of them understood the other. It’s very curious, there were endeavors to get them in touch but they didn’t get along. Their starting points were the same, it was hysteria; Janet initiated a very important conception of hysteria and he did a quite curious psychology which he proposed to name “Psychology of the Conduct,” even before Americans propounded the “Behavior Psychology.”

Roughly the method was: a psychological determination given, look for the type of conduct it represents. It was very interesting; he said: memory. The memory. Well it bears no interest, it doesn’t mean anything to me. I ask myself: what is the type of conduct one can hold when one remembers? And his answer was: the narration.

Hence, the famous definition of Janet: the memory is a conduct of narration. The emotion, he said, the emotion, one can’t feel if one can’t set down. You see, he used the conduct as a system of coordinates for all things. Everything was conduct.

I have a childhood memory which has impressed me forever. We all have childhood memories like this. It was during the holidays, my father used to give me Mathematics lessons. I was panic-stricken and it was all settled. That is to say, up to a point, I suspect we both did it already resigned, since we knew what was going to happen. In any case, I knew, I knew what was going to happen beforehand, because it was all settled, regular as clockwork. My father for that matter knew not much of Mathematics but he thought he had, above all, a natural gift for enunciating clearly. So he started, he held the pedagogical conduct, the pedagogical conduct. I was doing it willingly because it was no kidding subject at all; and I held the taught conduct. I showed every signs of interest, of maximal understanding, but all very soberly, and very fast there came a derailment. This derailment consisted in this: five minutes later, my father was yelling, set to beat me and I found myself in tears, I have to say, I was really small, and weeping. What was it? It is clear, there were two emotions. My deep grief, his deep anger. What did they respond to? Two failures. He has failed in his pedagogical conduct, he didn’t manage to explain at all. Of course he didn’t, he wanted to explain it to me with algebra, as he always said, because it was simpler and clearer this way. Then if I protested… and there it derailed. I protested arguing the teacher would never let me do algebra because when a six-year-old is given a problem, he hasn’t got the right, he is not supposed to do algebra. So the other was maintaining that it was the only clear way. Well, therefore, we both got into a tizzy. Misfire in the pedagogical conduct: anger; misfire in the taught conduct: tears.

All right. It was a failure. Janet said: emotion, it’s very simple, it’s a failure of conduct. You are upset when there is, when you hold a conduct and this conduct fails; then there is emotion.

{ Gilles Deleuze, Courses at Vincennes, 1980 | Continue reading }

A Gömböc (pronounced [ˈɡømbøts], simplified to Gomboc) is a convex three-dimensional homogeneous body which, when resting on a flat surface, has just one stable and one unstable point of equilibrium. Its existence was conjectured by Russian mathematician Vladimir Arnold in 1995 and proven in 2006 by Hungarian scientists Gábor Domokos and Péter Várkonyi. (…)

The balancing properties of the Gömböc are associated with the “righting response”, their ability to turn back when placed upside down, of shelled animals such as turtles and beetles.

{ Wikipedia | Continue reading | Thanks to James T. }

For someone who died at the age of 32 the largely self-taught Indian mathematician Srinivasa Ramanujan left behind an impressive legacy of insights into the theory of numbers—including many claims that he did not support with proof.

One of his more enigmatic statements, made nearly a century ago, about counting the number of ways in which a number can be expressed as a sum, has now helped researchers find unexpected fractal structures in the landscape of counting.

Ramanujan’s statement concerned the deceptively simple concept of partitions—the different ways in which a whole number can be subdivided into smaller numbers. Ken Ono of Emory University and his collaborators have now figured out new ways of counting all possible partitions, and found that the results form fractals—namely, structures in which patterns or shapes repeat identically at multiple different scales.

“The fractal theory we’ve discovered completely answers Ramanujan’s enigmatic statement,” Ono says. The problems his team cracked were seen as holy grails of number theory, and its solutions may have repercussions throughout mathematics.


{ Scientific American | Continue reading }

Michael Hartl says it’s time to lose pi in favor of a new symbol: tau. We asked him to explain why pi has to go.

Hartl is the author of The Tau Manifesto, which argues that, quite simply, pi is wrong. He’s also a physicist who has previously both studied and taught at Harvard and Caltech.

So what is tau? Well, simply enough, it’s 2*pi, or 2π.

{ io9 | Continue reading }

As a trained statistician with degrees from MIT and Stanford University, Srivastava was intrigued by the technical problem posed by the lottery ticket. In fact, it reminded him a lot of his day job, which involves consulting for mining and oil companies. A typical assignment for Srivastava goes like this: A mining company has multiple samples from a potential gold mine. Each sample gives a different estimate of the amount of mineral underground. “My job is to make sense of those results,” he says. “The numbers might seem random, as if the gold has just been scattered, but they’re actually not random at all. There are fundamental geologic forces that created those numbers. If I know the forces, I can decipher the samples. I can figure out how much gold is underground.”

Srivastava realized that the same logic could be applied to the lottery. The apparent randomness of the scratch ticket was just a facade, a mathematical lie. And this meant that the lottery system might actually be solvable, just like those mining samples. (…)

The North American lottery system is a $70 billion-a-year business, an industry bigger than movie tickets, music, and porn combined. (…)

It took a few hours of studying his tickets and some statistical sleuthing, but he discovered a defect in the game: The visible numbers turned out to reveal essential information about the digits hidden under the latex coating. Nothing needed to be scratched off—the ticket could be cracked if you knew the secret code.

{ Wired | Continue reading }

artwork { Erik Bulatov }

Statistics is what people think math is. Statistics is about patterns and that’s what people think math is about. The difference is that in math, you have to get very complicated before you get to interesting patterns. The math that we can all easily do – things like circles and triangles and squares – doesn’t really describe reality that much. Mandelbrot, when he wrote about fractals and talked about the general idea of self-similar processes, made it clear that if you want to describe nature, or social reality, you need very complicated mathematical constructions. The math that we can all understand from high school is just not going to be enough to capture the interesting features of real world patterns. Statistics, however, can capture a lot more patterns at a less technical level, because statistics, unlike mathematics, is all about uncertainty and variation. (…)

Bill James once said that you can lie in statistics just like you can lie in English or French or any other language. Sure, the more powerful a language is the more ways you can lie using it. There are a bunch of great quotes about statistics. There’s another one, sometimes attributed to Mark Twain: ‘It ain’t what you don’t know that hurts you, it’s what you don’t know you don’t know.’

{ The Browser | Continue reading }

photo { Flemming Ove Bech }

{ Few recent thinkers have woven such a beautiful braid of art and science as Benoît B Mandelbrot, who has died aged 85 in Cambridge, Massachusetts. (The B apparently doesn’t stand for anything. He just felt like adding it.) Mandelbrot was a provocative mathematician, a subversive geometer. He left a beautiful legacy in visual art, for Mandelbrot was the man who named and explained fractals – those complex, apparently chaotic yet geometrically ordered shapes that delight the eye and fascinate the mind. They are icons of modern understanding of the universe’s complexity. The Mandelbrot set, one of the most famous fractal designs, is named after him. With its fizzing fringe of crystal-like microforms blossoming out of a conjunction of black circles, this fractal pattern looks crazy but is the outcome of geometrical calculations. | The Guardian | full story }