Tag Archives: maths

How simple subtractions can stump even mathematicians — and why that matters for understanding our brains

Math is often regarded as the purest and most elegant form of problem-solving. But a new study claims that our mathematical thinking is often muddled by real-life knowledge. As weird as it sounds, our day-to-day information can get in the way of mathematical calculations — and this can happen to anyone, even experienced mathematicians.

When we learn to solve problems in school, we’re often given real-life scenarios. Jake buys a bunch of melons, then loses some of them, how many melons does he have left? Whether it’s melons, apples, or dividing flowers between vases, we’re taught at an early age to think of math in a practical context. While that approach teaches kids the practical applicability of mathematical calculations, it might also be counterproductive in some situations.

In a new study, researchers report that in some cases, worldly knowledge interferes with mathematical reasoning.

“We argue that such daily-life knowledge interferes with arithmetic word problem solving, to the extent that experts can be led to failure in problems involving trivial mathematical notions,” the  study reads.

They designed twelve problems which they presented to two groups. The main focus was the way in which the problems were presented. They were exactly the same problems, but they were presented in a different way.

“We devised six 5th grade subtraction problems (i.e. for pupils aged 10-11) that could be represented by sets, and six others that could be represented by axes”, begins Emmanuel Sander, an FPSE professor. “But all of them had exactly the same mathematical structure, the same numerical values and the same solution. Only the context was different.”

Half of the problems could be viewed as sets. Whether it’s the number of animals in a pack, the price of a meal in a restaurant or the weight of a stack of books, they all involved elements that can be grouped together in sets. For example:

  • Sarah has 14 animals: cats and dogs. Mehdi has two cats fewer than Sarah, and as many dogs. How many animals does Mehdi have?

The second type of problems (the axes ones) asked participants to calculate things like how long it takes to build a cathedral, to which floor an elevator arrives or how tall a Smurf is. Here’s an example:

  • When Lazy Smurf climbs onto a table, he attains 14 cm. Grumpy Smurf is 2 cm shorter than Lazy Smurf, and he climbs onto the same table. What height does Grumpy Smurf attain?

There’s a mental trick to the way these problems were designed. For instance, you can solve them through a simple subtraction: 14 -2 = 12. But when it comes to sets, the same approach doesn’t work.

Take the animal question with Sarah’s cats and dogs. Instinctively, you’d want to calculate how many cats and dogs Mehdi has — but you can’t. You can solve the problem and calculate how many animals he has, but not how they are divided between cats and dogs. The mathematical structure is identical: it’s the same simple subtraction, 14 – 2 = 12.

Scientists had a hunch that these answers would be a bit more difficult to answer, despite their identical mathematical structure. The context, they argue, makes it somewhat harder to process. Some problems were more difficult than others, but they all followed the same line

But even they weren’t expecting the results to be this striking.

In the non-expert adult group, 82% answered correctly for the axis problems, compared to only 47% for the problems involving sets. Surprisingly, in over more than half of the time (53%), respondents thought that there was no solution to the statement, which the team interprets as reflective of their inability to detach themselves from the elements of the problem.

Even expert mathematicians sometimes struggled with this. A total of 95% answered correctly for the axis problems, but that rate that dropped to only 76% for the sets problems. In other words, 1 out of 4 times, the experts thought there was no solution “even though it was of primary school level,” the study reads.

“We even showed that the participants who found the solution to the set problems were still influenced by their set-based outlook, because they were slower to solve these problems than the axis problems”, continues Hippolyte Gros, a researcher in UNIGE’s Faculty of Psychology and Educational Sciences and one of the study authors.

While the sample size was relatively small and the study design has significant limitations, the results are still intriguing. They seem to suggest that even in mathematical thinking, we are highly dependent on context. Even those who have the capacity to address the problems can suffer from these cognitive biases and be tricked into not finding the answer to a simple problem.

This isn’t the first study to suggest that our mathematical or scientific reasoning can be aided or hindered by semantic context. Given the wide scale at which these findings can make a difference in the education system, it seems there is a need to better understand the full impact of the semantical context.

The study has been published in Psychonomic Bulletin & Review.

Renowned mathematician Michael Atiyah claims to have solved the Riemann Hypothesis

Mathematics doesn’t usually make headlines and yet, to say that the announcement from Sir Michael Atiyah caused a stir would be an understatement. The renowned mathematician claimed to have solved the long-standing Riemann Hypothesis, with potentially massive implications for worldwide digital security. Well-aware of the skepticism that would surround his announcement, Atiyah pushed on, announcing that he would present his “simple” proof at the Heidelberg Laureate Forum.

Michael Atiyah, speaking at the International Congress of Mathematicians in 2018.

This is a continuation of a previously published article, which you can read hereFor a brief overview of the Riemann Hypothesis itself, check the bottom of the article.

There’s something inherently romantic about solving a math problem — a sort of man-against-nature kind of endeavor — and although most people seemed skeptical of Atiyah’s announcement, I’d dare say that deep down, we all hoped he would come out a victor. In Heidelberg, Germany, the stage was set.

It’s not just that the hypothesis is inherently challenging to solve — it stood unsolved for almost 160 years — but the fact that Atiyah himself claimed this solution struck people as unusual. It’s not that anyone doubted his ability: having won the two most prestigious awards in math (the Fields Medal and the Abel Prize), he’s one of the most renowned and respected mathematicians alive today. But most people in the field make their big findings in the earlier stretches of their career, before they’re 40. At a ripe 90, Atiyah stands in stark contrast, and he says recent papers tend to get rejected because people doubt his ability due to his old age.

So few people (if any) knew what to expect coming into the presentation, which made things all the more exciting, of course. The presentation itself was nothing if not entertaining — which is something you wouldn’t really expect, although Atiyah prides himself on his ability to explain everything at a fairly simple level.

“Solve the Riemann and you become famous. If you’re famous already, you become infamous,” he quipped in an almost hasty David Attenborough tone.

[panel style=”panel-info” title=”Hypothesis vs Theory” footer=””]Riemann’s hypothesis has been repeatedly observed to be accurate over a wide range of domains. However, it was never fully demonstrated — which is why it’s a hypothesis, rather than a theory. In science, ‘theory’ has a very different (and strong) meaning, as opposed to how ‘theory’ is used in our day to day lives.[/panel]

But as the minutes went by, it became clear that Atiyah wasn’t squeezing any long, compelling proof into the 45-minute presentation. At points, it felt like an introduction to mathematical history, with numerous sidetracks and backtracks. But there was also a sense of anticipation — a sense that the simple slides he was presenting were more like puzzle pieces, inconspicuously falling into place to reveal a much bigger picture.

Atiyah wasn’t even looking to solve the Riemann hypothesis — he was working in physics, trying to derive something called the fine structure constant. But sometimes, he says, when you solve problem A, you might end up solving problem B and not even know about it — this transposition of ideas makes math so great. The Riemann hypothesis was merely a problem B, something that came along almost accidentally.

Then, there it was — the “punchline”, as Atiyah himself referred to it: a relatively simple slide, with only a few lines. This is where all the “meat” of the demonstration is.

Michael Atiyah’s “punchline” for the Riemann Hypothesis.

If you’re not a mathematician (or if you are, but work in a different field), that probably looks like gibberish — and that’s fine, we won’t go into specifics; in all honesty, we couldn’t, even if we wanted to. But here’s the thing: extraordinary claims require extraordinary proof, and extraordinary proof rarely fits into one slide.

There was a sense of bewilderment after the presentation finished. When it was time for the questions, no one stood up and no one raised their hand, and for the next 30 seconds, you could cut the atmosphere with a knife. Timidly, one young man raised his hand — and he spoke the words that were on everyone’s mind: has the Riemann hypothesis been proven?

The comments which could be heard in some sections of the room were unflattering at best, and don’t belong in any scientific environment. But the concerns raised are, in a general sense, valid. Atiyah made several assumptions which permitted his leap. For instance, he worked with something called infinite iterations — something which is extremely dangerous; it’s the mathematical equivalent of treading on thin ice. Atiyah says he wouldn’t trust himself enough to take this step, but he builds on the work of John von Neumann, widely regarded as the foremost mathematician of his time (first part of the 20th century). Empowered by von Neumann’s work, Atiyah felt confident enough to claim the solution to this problem.

There was also an issue regarding the scope of his demonstration — had he really “solved” the problem, or has he merely tackled a particular aspect about its application? Atiyah’s answer to the question from the audience was, like the entire speech, charming, but left a bit more to be desired:

“This is just the first step on a long road. But yes, the first step, the solution to the problem, I proved that,” he stated, right after he said that he thinks he deserves the Millennium Prize for solving the problem.

Naturally, the problem of verification popped up afterward. Mathematicians worldwide would like to poke and prod around this proof, and they will have the chance to do so. This is where things took an unpleasant turn.

Atiyah wants to be believed — of course he does. A mathematician of his caliber should not settle for anything else and communication is, after all, the end game of any study. But it can be quite hard to get your message across when you’re 90 years old.

“I do care who believes it. Mathematics involves two steps: creation and dissemination. If you don’t disseminate your ideas, you don’t get anywhere,” he said. But when you’re 90, it’s extremely difficult to publish, he adds.

“When you’re my age, people don’t really want to publish my papers. You’re too old, they say.”

As a knee-jerk reaction, you’d want to say this can’t possibly be true. Surely, his work is subjected to the same scrutiny as all others, regardless of age. But the inherent bias is hard to deny, even in a room full of Atiyah’s peers. In the aftermath of his presentation, age seemed to be the most common topic of discussion — perhaps even more so than the math.

Sexism is a big problem in science, Atiyah rightly points out, but so is ageism, he adds. At the end of the day, one can only feel that his paper should be judged by its own value — regardless of whether it comes from an Abel Prize winner or a 90-year-old man — as a paper coming from a mathematician, simple as that.

At this point, it’s not clear whether his papers were reviewed “double-blind” or if the reviewers were aware of who submitted the paper. We don’t know if age really was the decisive factor in the initial rejection of the paper, or if it’s simply invalid — but age shouldn’t be a decisive criterion.

Now, at the very least, his paper (see here) will receive attention and scrutiny. The few people who can truly attest to its worth will presumably review it thoroughly, and while it may take a while (math proofs are often difficult to confirm or invalidate), when the dust will settle, we’ll find out whether we finally have a solution for the Riemann Hypothesis or not. Regardless of that outcome, Atiyah made us all think about the largely overlooked problem of ageism in science.

His presentation was charming and entertaining — which in this context, almost feels like a sin. The stage was brilliantly set, but the final outcome is, as of yet, undecided, and the lead actor’s performance not entirely convincing. Has he really identified a solution? We’ll likely know soon enough. But he’s certainly highlighted a new problem:

[panel style=”panel-default” title=”The Riemann hypothesis” footer=””]The Riemann hypothesis starts with prime numbers — rather strange numbers which can’t be divided by other numbers; one of the oddities of prime numbers is that their distribution is irregular — there’s no precise method to predict where the next prime number will occur. This unpredictability has been widely used in developing digital security systems.

While looking at prime numbers, Berhard Reimann, one of the most prolific German mathematicians, realized something interesting: the distribution of these prime numbers isn’t random at all, it’s very similar to a function, called the Riemann Zeta Function, described below.

ζ(s) = 1/1s + 1/2s + 1/3s + 1/4s + …. up to infinity

This is where it starts to get tricky. The variable can take any value, and Riemann’s work gives us an explicit formula for the number of primes in a given interval. This is done in terms of the so-called ‘zeros’ of the function — the values of under which the function becomes 0. You can think of this function as a way to predict the distribution of prime numbers.

Riemann observed this in action, but he was never really able to prove it. This is why this is still a hypothesis, and not a theory (which, in science and math, has a much stricter meaning than in regular talk). It still remains to be seen whether this will still be the case.

[/panel]

Mathematicians solve old mystery about spaghetti breaking

Here’s an experiment: grab a dry spaghetti noodle on both ends. Bend it more and more, until it breaks. Intuitively, you’d think it breaks into two pieces, but that’s almost never the case — it typically breaks into 3 or 4 different pieces. Try again and again, as many times as you want, you’ll likely never end up with two pieces.

Experiments (above) and simulations (below) show how dry spaghetti can be broken into two or more fragments, by twisting and bending.

The Spaghetti Conundrum

If that confuses you, don’t worry — you’re in good company. This spaghetti conundrum has flummoxed scientists for decades. Even the renowned and ever-curious physicist Richard Feynman was fascinated by this. By his own account, he spent the better part of an afternoon breaking spaghetti in halves and wondering why they don’t snap in two. He couldn’t come up with a satisfying explanation, and the mystery remained unsolved until 2005 when physicists from France came up with a working theory.

They found that when a spaghetti — and for that matter, any long rod — is bent at the ends, it will break near the center, where it is most curved. But as it breaks, triggers a “snap-back” effect, producing a bending wave, or vibration, which further breaks the rod. The theory was demonstrated, and as a reward for their trouble, the French physicists received an Ig Nobel Prize, a parody of the Nobel Prize, which celebrates unusual or trivial findings.

But even after this, a question remained: is it never possible to break a spaghetti in two? The answer is ‘yes’, with a twist — as in if you twist them, you can break them in only two. In a paper published this week in the Proceedings of the National Academy of Sciences, researchers report that if you also twist the spaghetti, this dampens the shock wave and reduces the chance of breaking into several pieces. Essentially, if a stick is twisted past a critical degree, then slowly bent in half, it will break in two.

However, researchers say, this could have far-reaching implications, going way beyond culinary curiosities. The findings could be used to control fractures and increase toughness in rod-like materials such as multifiber structures, engineered nanotubes, or even microtubules in cells.

“It will be interesting to see whether and how twist could similarly be used to control the fracture dynamics of two-dimensional and three-dimensional materials,” says co-author Jörn Dunkel, associate professor of physical applied mathematics at MIT. “In any case, this has been a fun interdisciplinary project started and carried out by two brilliant and persistent students — who probably don’t want to see, break, or eat spaghetti for a while.”

The spaghetti can be broken in two by adding a 270 degree twist, due to the snap-back and twist-back effects working together.

Pasta maths

The two students Dunkel is referring to are Ronald Heisser, now a graduate student at Cornell University, and Vishal Patil, a mathematics graduate student in Dunkel’s group at MIT. Their co-authors are Norbert Stoop, instructor of mathematics at MIT, and Emmanuel Villermaux of Université Aix Marseille. They designed a device that can controllably bend and twist spaghetti ends, focusing on two types of spaghetti: Barilla No. 5 and Barilla No. 7, which have slightly different diameters.

“They did some manual tests, tried various things, and came up with an idea that when he twisted the spaghetti really hard and brought the ends together, it seemed to work and it broke into two pieces,” Dunkel says. “But you have to twist really strongly. And Ronald wanted to investigate more deeply.”

Meanwhile, Patil developed a mathematical model to explain this behavior, building on the previous work done by French scientists Basile Audoly and Sebastien Neukirch, who first studied this behavior. Putting all the hard work together, they finally solved this unusual puzzle — but there is one caveat.

Their study works on the assumption of cylindrical shapes — in other words, it only works for “classic” pasta. Other types of pasta, like fussili or linguini will have a different behavior because they also have a different geometry.

The study has been published in PNAS.