Posts Tagged ‘logic’
“The opposite of a correct statement is a false statement. But the opposite of a profound truth may well be another profound truth.”*…
Zeno shows the Doors to Truth and Falsity (Veritas et Falsitas). Fresco in the Library of El Escorial, Madrid (source)
As Joel David Hamkins explains, an ancient puzzle leads ultimately to a remarkable observation on the malleable nature of infinite sums…
The Greek philosopher Zeno of Elea (c. 490–430 BC) argued in antiquity that all motion is impossible. It is simply impossible to walk through town or even across the room, to go from here to there. What? We know, of course, that this is possible—we walk from here to there every day. And yet, Zeno offers us his proof that this is an illusion—we simply cannot do it.
Zeno argued like this. Suppose it were possible for you to move from some point A to another distinct point B.
Before you complete the move from A to B , however, you must of course have gotten half way there.
But before you get to this half-way point, of course, you must get half way to the half-way point! And before you get to that place, you must get half way there.
And so on, ad infinitum.
Thus, to move from A to B , or indeed anywhere at all, one must have completed an infinite number of tasks—a supertask. It follows, according to Zeno, that you can never start moving—you cannot move any amount at all, since before doing that you must already have moved half as much. And so, contrary to appearances, you are frozen motionless, unable to begin. All motion is impossible.
Is the argument convincing? On what grounds would you object to it? Do you think, contrary to Zeno, that we can actually complete infinitely many tasks? How would that be possible?
It will be no good, of course, to criticize Zeno’s argument on the grounds that we know that motion is possible, for we move from one point to another every day. That is, to argue merely that the conclusion is false does not actually tell you what is wrong with the argument—it does not identify any particular flaw in Zeno’s reasoning. After all, if it were in fact an illusion that we experience motion, then your objection would be groundless…
Learning from an enigma– plus “the most contested equation in middle school” and more: “Zeno’s paradox,” from @JDHamkins.
* Niels Bohr
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As we interrogate infinity, we might send-well-groomed birthday greetings to Frank Joseph Zamboni, Jr.; he was born on this date in 1901. An engineer and inventor, he is best known for the modern ice resurfacer, seen at work at hockey games and figure skating competitions (completing its rounds, Zeno notwithstanding); indeed, his surname is the registered trademark for these devices.
“Oops, I did it again”*…
(Roughly) Daily has contemplated game theory a number of times (e.g., here). The Generalist Academy offers a particularly poignant example…
Football [or, as it’s called in the U.S., soccer] has a lot of strange rules – like Ted Lasso, I still don’t understand exactly how the offside rule works. But the basic game is pretty simple: get the ball into your opponent’s goal, and prevent them from getting the ball into your goal. Scoring a goal against your own side is a rare and accidental embarrassment. Usually.
The qualification round for the 1994 Caribbean Cup had some unusual rules. No match could end in a draw; if the teams were tied at the end of regular time, they would go into sudden death extra time. But! Any goal scored in extra time would count as two goals. This was presumably done because this tournament, like many, used goal difference to break ties in the qualifying groups. (Goal difference = total number of goals they’ve scored minus the number of goals they’ve conceded.) So that extra time “golden goal” would give a team an edge in the overall competition. Little did the organisers know that it would also lead to one of the strangest football games ever seen…
A truly remarkable match: “Own-Goal Football,” from @GeneralistAcad.
* Britney Spears (Songwriters: Martin Max / Rami Yacoub)
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As we work backwards, we might send carefully-calculated birthday greetings to Félix Édouard Justin Émile Borel; he was born on this date in 1871. A mathematician (and politician, who served as French Minister of the Navy), he is remembered for his foundational work in measure theory and probability. He published a number of research papers on game theory and was the first to define games of strategy.
But Borel may be best remembered for a thought experiment he introduced in one of his books, proposing that a monkey hitting keys at random on a typewriter keyboard will – with absolute certainty – eventually type every book in France’s Bibliothèque Nationale de France. This is now popularly known as the infinite monkey theorem.
“All the world is made of faith, and trust, and pixie dust”*…
Beyond the Prisoner’s Dilemma— an interactive guide to game theory and why we trust each other: The Evolution of Trust, from Nicky Case (@ncasenmare), via @frauenfelder@mastodon.cloud in @Recomendo6.
* J. M. Barrie, Peter Pan
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As we rethink reciprocal reliance, we might send far-sighted birthday greetings to Michel de Nostredame; he was born on this date in 1503. Better known as Nostradamus, he was an astrologer, apothecary, physician, and reputed seer, who is best known for his book Les Prophéties (published in 1555), a collection of 942 poetic quatrains allegedly predicting future events.
In the years since the publication of his Les Prophéties, Nostradamus has attracted many supporters, who, along with some of the popular press, credit him with having accurately predicted many major world events. Other, more critical, observers note that many of his supposed correct calls were the result of “generous” (or plainly incorrect) translations/interpretations; and more generally, that Nostradamus’ genius for vagueness allows– indeed encourages– enthusiasts to “find” connections where they may or may not exist.
“Whoever wishes to keep a secret must hide the fact that he possesses one”*…
… or, as Sheon Han explains, maybe not…
Imagine you had some useful knowledge — maybe a secret recipe, or the key to a cipher. Could you prove to a friend that you had that knowledge, without revealing anything about it? Computer scientists proved over 30 years ago that you could, if you used what’s called a zero-knowledge proof.
For a simple way to understand this idea, let’s suppose you want to show your friend that you know how to get through a maze, without divulging any details about the path. You could simply traverse the maze within a time limit, while your friend was forbidden from watching. (The time limit is necessary because given enough time, anyone can eventually find their way out through trial and error.) Your friend would know you could do it, but they wouldn’t know how.
Zero-knowledge proofs are helpful to cryptographers, who work with secret information, but also to researchers of computational complexity, which deals with classifying the difficulty of different problems. “A lot of modern cryptography relies on complexity assumptions — on the assumption that certain problems are hard to solve, so there has always been some connections between the two worlds,” said Claude Crépeau, a computer scientist at McGill University. “But [these] proofs have created a whole world of connection.”…
More about how zero-knowledge proofs allow researchers conclusively to demonstrate their knowledge without divulging the knowledge itself: “How Do You Prove a Secret?,” from @sheonhan in @QuantaMagazine.
* Johann Wolfgang von Goethe
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As we stay sub rosa, we might recall that today (All Saints Day) is the (fictional) birthday of Hello Kitty (full name: Kitty White); she was born in a suburb of London. A cartoon character designed by Yuko Shimizu (currently designed by Yuko Yamaguchi), she is the property of the Japanese company Sanrio. An avatar of kawaii (cute) culture, Hello Kitty is one of the highest-grossing media franchises of all time; Hello Kitty product sales and media licensing fees have run as high as $8 billion a year.
“Advantage! What is advantage?”*…
Pradeep Mutalik unpacks the magic and math of how to win games when your opponent goes first…
Most games that pit two players or teams against each other require one of them to make the first play. This results in a built-in asymmetry, and the question arises: Should you go first or second?
Most people instinctively want to go first, and this intuition is usually borne out. In common two-player games, such as chess or tennis, it is a real, if modest, advantage to “win the toss” and go first. But sometimes it’s to your advantage to let your opponent make the first play.
In our February Insights puzzle, we presented four disparate situations in which, counterintuitively, the obligation to move is a serious and often decisive disadvantage. In chess, this is known as zugzwang — a German word meaning “move compulsion.”…
Four fascinating examples: “The Secrets of Zugzwang in Chess, Math and Pizzas,” from @PradeepMutalik.
* Fyodor Dostoyevsky, Notes from Underground
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As we play to win, we might recall that it was on this date in 2011 that scientists involved in the OPERA experiment (a collaboration between CERN and the Laboratori Nazionali del Gran Sasso) mistakenly observed neutrinos appearing to travel faster than light. OPERA scientists announced the results with the stated intent of promoting further inquiry and debate. Later the team reported two flaws in their equipment set-up that had caused errors far outside their original confidence interval: a fiber optic cable attached improperly, which caused the apparently faster-than-light measurements, and a clock oscillator ticking too fast; accounting for these two sources of error eliminated the faster-than-light results. But even before the sources of the error were discovered, the result was considered anomalous because speeds higher than that of light in a vacuum are generally thought to violate special relativity, a cornerstone of the modern understanding of physics for over a century.
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