Month: October 2023

Tens of thousands of handwritten pages by one of the 20th century’s greatest mathematicians, Alexander Grothendieck, many of which the eccentric genius penned while living as a hermit, were unveiled in France on Friday.

The unpublished manuscripts, which veer from math to metaphysics, autobiography and even long musings on Satan, offer a unique insight into the enigmatic mind of the French mathematician, according to experts at the Paris library where they were donated.

Grothendieck, who died aged 86 in 2014, is considered by some to have revolutionized the field of mathematics in the way that Einstein did for physics. His work on algebraic geometry earned him the 1966 Fields Medal, known as the Nobel prize of the math world.

At that time Grothendieck was already a radical environmentalist and pacifist. But he withdrew from the world almost entirely in the early 1990s, in part to focus on what he referred to as his “scribblings”.

While living as a hermit in the southern French village of Lasserre he frantically wrote “Reflections on Life and the Cosmos,” one of the two main works added to the collection of the National Library of France (BnF) on Friday.

The massive tome includes 30,000 pages across 41 different volumes covering science, philosophy and psychology—all densely scribbled with a fountain pen.

The second work, “The Key to Dreams or Dialogue with the Good Lord,” is a typed manuscript in which he explores the interpretation of dreams.

These pages, which have previously circulated online, were written between 1987-1988.

‘Completely cut ties’

At that time, Grothendieck remained a professor at the University of Montpellier but had largely withdrawn from the mathematical community.

He became a recluse when he moved to Lasserre.

“He completely cut ties with his family, we could no longer communicate with him,” his daughter Johanna Grothendieck told AFP.

“When we sent him a letter, it was returned to sender,” said Johanna, a 64-year-old ceramic artist who traveled from southwest France to attend the ceremony at the library.

“Writing was his main activity,” she added.

Towards the end of Grothendieck’s life, a neighbour told his family that his health was deteriorating.

Johanna and one of her brothers were finally able to visit their father. It was than that they discovered “Reflections on Life and the Cosmos,” which was meticulously catalogued in his library.

In his 1997 will, Grothendieck left the early sections of the tome to the BnF. Now his children have donated the rest.

“It was an extremely important work in his eyes. He even wanted to create a foundation to look after it,” Johanna Grothendieck said.

‘Ghosts of his past’

Jocelyn Monchamp, a curator an the BnF, said the manuscripts were unique because they covered so many subjects at the same time yet formed a whole with “undeniable literary qualities”.

This is particularly the case for the autobiographical volume “Harvest and Sowing”, which depicts the author “in a metaphysical retreat,” she said.

Monchamp has spent a month poring over the writing, trying to decipher the dense fountain pen text.

“I became used to it,” she said, adding that at least Grothendieck methodically wrote the numbers and dates on all the pages.

In one of the sections, “Structures of the Psyche,” enigmatic diagrams translate psychology into the language of algebra.

In another, “The Problem of Evil,” Grothendieck muses over 15,000 pages on metaphysics and Satan.

One gets the feeling of a man “overtaken by the ghosts of his past,” Johanna Grothendieck said.

The mathematician’s father fled Germany during World War II, only to be handed by the Vichy France government to the Nazis and die at the Auschwitz concentration camp. Experts expect it will take some time to fully understand Grothendieck’s writing. On Friday, the collection joined the manuscript department of the BnF, where it will only be accessible to researchers.

During the donation ceremony, one of the volumes was placed in a glass case next to a manuscript by ancient Greek mathematician Euclid, considered the father of geometry.

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Credit of the article given to Juliette Collen

Understanding how children learn to count can have profound impacts on the kinds of instructional materials used in the classroom. And the way those materials are designed can shape the strategies children use to learn, according to a new paper led by Concordia researchers.

Writing in the journal School Science and Mathematics, the authors study how young children, mostly in the first grade, used a hundreds table to perform age-appropriate counting tasks. Hundreds tables, as the name suggests, are charts divided into rows and columns of 10, with each square containing a number from one to 100. The researchers discovered that the children who counted left-to-right, top-to-bottom outperformed children who counted left-to-right, bottom-to-top.

In this study, children used the tables on a screen to solve addition problems. One group of children used a top-down table, where the top left corner was marked 1 and bottom left corner was marked 100. Another group of children used a bottom-up, where one occupied the bottom left and 100 the top right. A third group of children used a bottom-up table with a visual cue of a cylinder next to it. The cylinder was designed to show the “up-is-more” relation as it filled with water when the numbers increased when moving up in the table.

“We found that children using the top-down chart used a more sophisticated strategy of counting by 10 and moving vertically, rather than using the more simplistic strategy of counting by one and moving horizontally,” says Vera Wagner. She co-authored the paper with Helena Osana, a professor in the Department of Education in the Faculty of Arts and Science and Jairo Navarrete-Ulloa of O’Higgins University in Chile.

The authors believe the benefits of the top-down table could be related to the way children learn to read and that they are applying the same approach to base-ten concepts.

“We were working with young children, so reading instruction is likely at the forefront of their attention,” says Wagner, who now teaches elementary students at a Montreal-area school. “The structure of moving in that particular way might be more ingrained.”

The power of spatial configuration

Osana notes that the practice of counting by 10s rather than by ones—which is a more efficient method of arriving at the same answer—is an example of unitizing, in which multiples of one unit form a new unit representing a larger number.

“From a theoretical perspective, the study shows that the spatial configuration of instructional materials can actually support this more sophisticated understanding of numbers and the unitizing aspect that goes along with it,” she says.

While the researchers are not suggesting children will automatically gravitate toward the top-down chart under every circumstance, they do think the study’s results provide educators with a sense of the ways their students process numbers and addition.

“It is important for teachers to be aware of how children are thinking about the tools we are giving them,” says Osana, principal investigator of the Mathematics Teaching and Learning Lab. “We are not saying that teachers have to use the top-down hundreds chart every time, but they should think about the strategies their students are using and why they use them with one particular instructional tool and not another.”

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Credit of the article given to Patrick Lejtenyi, Concordia University

It seems there’s a five-letter word describing what many players of the wildly popular Wordle puzzle do daily as they struggle to find a target word within six tries.

According to one mathematics expert, that word is “cheat.”

James P. Dilger, who by day is professor emeritus at Stony Brook University in New York specializing in the mechanisms of anesthetic action, and by night is a Wordle junkie, says the numbers behind published Wordle success rates don’t quite add up.

Wordle was developed by a software engineerto pass the time during the early days of COVID restrictions. Players must determine a target five-letter word in six or fewer attempts. With each guess, the player is provided with three bits of information: correct letters in the correct position are displayed in green, correct letters placed in incorrect spots are displayed in yellow, and incorrect letters are displayed in black.

In the beginning, Wordle was played mainly among family and friends of the developer, Josh Wardle. Wordle’s popularity soared, reaching 3 million users after The New York Times purchased the game in January 2022. Today, some 2 million play Wordle daily. It is recreated in 50 languages globally.

Dilger’s suspicions arose while studying the game’s statistics published daily by The Times.

“I noticed one day an awful lot of people answered with one guess and thought, ‘that’s strange,'” Dilger said. “And then I paid attention to it and it was happening day after day. Well, I’m a science nerd and wanted to know what’s going on.”

Dilger imported statistics covering four months of user guesses into an Excel spreadsheet. His report, “Wordle: A Microcosm of Life. Luck, Skill, Cheating, Loyalty, and Influence!” appeared in the preprint server arXiv Sept. 6.

The game has a data bank containing 2,315 words, good for five years of play. (There actually are more than 12,000 five-letter words in the English language, but The Times weeded out the most obscure ones.)

Dilger calculated that the odds of randomly guessing the day’s word at 0.043%, totaling 860 players. Yet, Times statistics show that the number of players making correct first guesses in each game never dipped below 4,000.

“Do I mean to tell you that never, not once, was the share percent of the first guess less than 0.2%? Yup!” Dilger asserted.

He went further. His numbers are based on the 2,315-word master list compiled by The Times, but 800 of those words have already been used. Most players are not likely to know that detail, but if they did, and they excluded words already played, their odds of guessing the correct word would rise slightly. Yet, according to Dilger, their odds would still be a low 0.066%.

“Yet, it happens consistently every day,” Dilger said. “Some days it’s as high as 0.5%,” which would be 10,000 players.

He also noted how unlikely it would be that a user would correctly guess such poor first-choice candidates as “nanny” and “igloo.” Players gain maximum advantage when they surmise words with non-repeating characters and as many vowels as possible. “Nanny” repeats one letter three times and uses only two vowels. “Igloo” not only is a relatively rare word, but contains only two vowels, repeating one of them.

“What shall we call these people?” He asked. “‘Cheaters’ comes to mind, so that’s what I call ’em.”

Dilger did not offer any explanation for such nefarious behaviour, other than to say that many players “became frustrated at some point in the game and then felt joy or relief after having surpassed the hurdle with a cheat.”

“We are baffled as to how first-word cheaters actually have fun playing,” Dinger said, “but that does not diminish our enjoyment of the game.”

He might have quoted former wrestler, actor, philosopher and governor of Minnesota Jesse Ventura, who once suggested, “Winners never cheat, and cheaters never win.” Except maybe in Wordle.

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Credit of the article given to Peter Grad, Phys.org

You ordered a pizza for your party, but the restaurant forgot to slice it – these mathematical tricks can help you cut it evenly, says Katie Steckles.

Fairness is important – in life, and in pizza. If you want to cut a pizza into equal-sized pieces, the difficulty will depend on how many people you need to share it between. Luckily, mathematics has some tricks to keep things equal.

For example, if the number of people you are sharing a pizza between is a power of two – one, two, four, eight, 16 – cutting the pizza into as many slices is easy. For one piece, obviously no cuts are needed. For each larger power of two, a cut across through the centre of the pizza – cutting all of the existing pieces exactly in half – will result in pieces of equal size.

Some numbers will be much harder: prime numbers, by definition, can’t be divided easily. Luckily, geometry can help.

If you need to cut a pizza into five equal pieces, first grab a long, thin, rectangular strip of paper. Tie the paper in an ordinary overhand knot, like you would tie in a piece of string. Then, keeping the ends flat, pull gently to tighten the knot. The whole thing will flatten and come together – stop pulling when you can’t go any further without it wrinkling.

The flat shape you are looking at should now be vaguely familiar, if you ignore the two ends of paper sticking out. Fold these ends into the middle, or cut them off, and you will have a shape with five straight edges, created purely by the shape of the knot. Yes, that is right – you have made a perfect regular pentagon, with five equal-length sides and five equal angles at the corners.

It is possible to prove this mathematically by showing that all the folds you make in the paper strip are at 72 degrees to the parallel edges of the strip. But for simplicity, because the paper is the same width everywhere, and weaves in and out five times in the right way, these will be five equal edges. And more importantly, the pentagon’s corners are equally spread around a circle – making it the perfect guide for pizza slicing.

Place your pentagon in the centre of the pizza, then cut along lines radiating out from the centre of the pentagon and through each corner. And presto: you have a pentagonal pizza party for five. This paper-strip method can be used whenever you are in a pentagon-based emergency.

You can use the same technique to produce a shape with any odd number of sides by creating a more complex knot with the strip passing through the middle more times, although the strip of paper needs to be increasingly thin and it takes a lot more patience to pull the ends through and carefully flatten out the shape.

Combined with our existing halving methods, you can now produce any number of slices you like. The same results can be extended to any other round food – thanks to maths, the world is your cheesecake.

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*Credit for article given to Katie Steckles *