Posts Tagged ‘logic’
“Machines take me by surprise with great frequency”*…
In search of universals in the 17th century, Gottfried Leibniz imagined the calculus ratiocinator, a theoretical logical calculation framework aimed at universal application, that led Norbert Wiener to suggest that Leibniz should be considered the patron saint of cybernetics. In the 19th century, Charles Babbage and Ada Lovelace took a pair of whacks at making it real.
Ironically, it was confronting the impossibility of a universal calculator that led to modern computing. In 1936 (the same year that Charlie Chaplin released Modern Times) Alan Turing (following on Godel’s demonstration that mathematics is incomplete and addressing Hilbert‘s “decision problem,” querying the limits of computation) published the (notional) design of a “machine” that elegantly demonstrated those limits– and, as Sheon Han explains, birthed computing as we know it…
… [Hilbert’s] question would lead to a formal definition of computability, one that allowed mathematicians to answer a host of new problems and laid the foundation for theoretical computer science.
The definition came from a 23-year-old grad student named Alan Turing, who in 1936 wrote a seminal paper that not only formalized the concept of computation, but also proved a fundamental question in mathematics and created the intellectual foundation for the invention of the electronic computer. Turing’s great insight was to provide a concrete answer to the computation question in the form of an abstract machine, later named the Turing machine by his doctoral adviser, Alonzo Church. It’s abstract because it doesn’t (and can’t) physically exist as a tangible device. Instead, it’s a conceptual model of computation: If the machine can calculate a function, then the function is computable.
…
With his abstract machine, Turing established a model of computation to answer the Entscheidungsproblem, which formally asks: Given a set of mathematical axioms, is there a mechanical process — a set of instructions, which today we’d call an algorithm — that can always determine whether a given statement is true?…
… in 1936, Church and Turing — using different methods — independently proved that there is no general way of solving every instance of the Entscheidungsproblem. For example, some games, such as John Conway’s Game of Life, are undecidable: No algorithm can determine whether a certain pattern will appear from an initial pattern.
…
Beyond answering these fundamental questions, Turing’s machine also led directly to the development of modern computers, through a variant known as the universal Turing machine. This is a special kind of Turing machine that can simulate any other Turing machine on any input. It can read a description of other Turing machines (their rules and input tapes) and simulate their behaviors on its own input tape, producing the same output that the simulated machine would produce, just as today’s computers can read any program and execute it. In 1945, John von Neumann proposed a computer architecture — called the von Neumann architecture — that made the universal Turing machine concept possible in a real-life machine…
As Turing said, “if a machine is expected to be infallible, it cannot also be intelligent.” On the importance of thought experiments: “The Most Important Machine That Was Never Built,” from @sheonhan in @QuantaMagazine.
* Alan Turing
###
As we sum it up, we might spare a thought for Martin Gardner; he died on this date in 2010. Though not an academic, nor ever a formal student of math or science, he wrote widely and prolifically on both subjects in such popular books as The Ambidextrous Universe and The Relativity Explosion and as the “Mathematical Games” columnist for Scientific American. Indeed, his elegant– and understandable– puzzles delighted professional and amateur readers alike, and helped inspire a generation of young mathematicians.
Gardner’s interests were wide; in addition to the math and science that were his power alley, he studied and wrote on topics that included magic, philosophy, religion, and literature (c.f., especially his work on Lewis Carroll– including the delightful Annotated Alice— and on G.K. Chesterton). And he was a fierce debunker of pseudoscience: a founding member of CSICOP, and contributor of a monthly column (“Notes of a Fringe Watcher,” from 1983 to 2002) in Skeptical Inquirer, that organization’s monthly magazine.
“I think inequality is fine, as long as it is in the common interest. The problem is when it gets so extreme, when it becomes excessive.”*…
Alvin Chang, with a beautifully-told (and beautifully-illustrated) primer on a startling unpacking of the fundamental logic of our market economy…
Why do super rich people exist in a society?
Many of us assume it’s because some people make better financial decisions. But what if this isn’t true? What if the economy – our economy – is designed to create a few super rich people?
That’s what mathematicians argue in something called the Yard-sale model…
Read it and reap: “Why the super rich are inevitable,” by @alv9n in @puddingviz.
* Thomas Piketty, A Brief History of Equality
###
As we ponder propriety, we might recall that it was on this date that Jane Austen‘s [and here] Pride and Prejudice was published. A novel of manners– much concerned with the dictates of wealth (and the lack thereof), it was credited to an anonymous authors “the author of Sense and Sensibility,” as all of her novels were.
Title page of the first edition (source)
“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
###
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)
###
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
###
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.
You must be logged in to post a comment.