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Posts Tagged ‘infinity

“To Infinity and Beyond!”*…

The idea of infinity is probably about as old as numbers themselves, going back to whenever people first realized that they could keep counting forever. But even though we have a sign for infinity and can refer to the concept in casual conversation, infinity remains profoundly mysterious, even to mathematicians. Steven Strogatz explores that mystery with Justin Moore

No one really knows where the idea of infinity came from, but it must be very ancient — as old as people’s hopes and fears about things that could conceivably go on forever. Some of them are scary, like bottomless pits, and some of them are uplifting, like endless love. Within mathematics, the idea of infinity is probably about as old as numbers themselves. Once people realized that they could just keep on counting forever — 1, 2, 3 and so on. But even though infinity is a very old idea, it remains profoundly mysterious. People have been scratching their heads about infinity for thousands of years now, at least since Zeno and Aristotle in ancient Greece.

But how do mathematicians make sense of infinity today? Are there different sizes of infinity? Is infinity useful to mathematicians? And if so, how exactly? And what does all this have to do with the foundations of mathematics itself?…

All infinities go on forever, so “How Can Some Infinities Be Bigger Than Others?“, from @stevenstrogatz in @QuantaMagazine.

See also: Alan Lightman‘s “Why the paradoxes of infinity still puzzle us today” (source of the image above).

* Buzz Lightyear

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As we envision endlessness, we might send carefully-calculated birthday greetings to Gaspard Monge; he was born on this date in 1746. A mathematician, he is considered the inventor of descriptive geometry, (the mathematical basis of technical drawing), and the father of differential geometry (the study of smooth shapes and spaces, AKA smooth manifolds).

During the French Revolution he was involved in the reform of the French educational system, most notably as the lead founder of the École Polytechnique.

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“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.

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Written by (Roughly) Daily

January 16, 2023 at 1:00 am

“In order for a book to exist, it is sufficient that it be possible. Only the impossible is excluded.”*…

One of your correspondent’s daily pleasures is Rusty Foster‘s newsletter, Today in Tabs. Here, an especially pleasing excerpt…

In 1941, Jorge Luis Borges wrote a short story called “The Library of Babel.” If you haven’t read it, or if it’s been a while, go read it now. It’s only eight pages. If that’s all this email accomplishes for you today, I’ll consider it a success.

In its finite but innumerable books, Borges’ Library contains every possible arrangement of letters. In 2015 Jonathan Basile made LibraryofBabel.info, a website that not only accomplishes this but is even searchable. Here’s one of the 10²⁹ pages that just say “today in tabs.” Here’s the last line of The Great Gatsby. Can you find it? If not, don’t worry, it shows up embedded in 29³¹⁴¹ more pages of gibberish. How about this page? It implicitly existed before I searched for it, which I find kind of upsetting.

But as interesting/disturbing as the Library’s content is, I’m also fascinated by the physical structure of it. Picture a cross between “The Name of Rose” and “House of Leaves.” A sort of infinite scriptorium designed by bees

Enlightened, solitary, infinite, perfectly unmoving, armed with precious volumes, pointless, incorruptible, and secret: “The Library of Babel,” from @fka_tabs.

See also “Visit The Online Library of Babel: New Web Site Turns Borges’ “Library of Babel” Into a Virtual Reality, source of the image above.

* Jorge Luis Borges, “The Library of Babel”

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As we check it out, we might send thoughtfully and warmly observed birthday greetings to David John Lodge; he was born on this date in 1935. An author, critic, and professor of literature, he has written 18 novels, a baker’s dozen works of nonfiction (plus two memoirs), three plays, and four teleplays. He’s probably best remembered for his wonderful “Campus Trilogy” – Changing Places: A Tale of Two Campuses (1975), Small World: An Academic Romance (1984) and Nice Work (1988), the second two of which were each shortlisted for the Booker Prize.

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“If the doors of perception were cleansed everything would appear to man as it is, infinite”*…

For 50 years, mathematicians have believed that the total number of real numbers is unknowable. A new proof suggests otherwise…

Infinity comes in many sizes. In 1873, the German mathematician Georg Cantor shook math to the core when he discovered that the “real” numbers that fill the number line — most with never-ending digits, like 3.14159… — outnumber “natural” numbers like 1, 2 and 3, even though there are infinitely many of both.

Infinite sets of numbers mess with our intuition about size, so as a warmup, compare the natural numbers {1, 2, 3, …} with the odd numbers {1, 3, 5, …}. You might think the first set is bigger, since only half its elements appear in the second set. Cantor realized, though, that the elements of the two sets can be put in a one-to-one correspondence. You can pair off the first elements of each set (1 and 1), then pair off their second elements (2 and 3), then their third (3 and 5), and so on forever, covering all elements of both sets. In this sense, the two infinite sets have the same size, or what Cantor called “cardinality.” He designated their size with the cardinal number 0 (“aleph-zero”).

But Cantor discovered that natural numbers can’t be put into one-to-one correspondence with the continuum of real numbers. For instance, try to pair 1 with 1.00000… and 2 with 1.00001…, and you’ll have skipped over infinitely many real numbers (like 1.000000001…). You can’t possibly count them all; their cardinality is greater than that of the natural numbers.

Sizes of infinity don’t stop there. Cantor discovered that any infinite set’s power set — the set of all subsets of its elements — has larger cardinality than it does. Every power set itself has a power set, so that cardinal numbers form an infinitely tall tower of infinities.

Standing at the foot of this forbidding edifice, Cantor focused on the first couple of floors. He managed to prove that the set formed from all the different ways of ordering natural numbers (from smallest to largest, for example, or with all odd numbers first) has cardinality 1, one level up from the natural numbers. Moreover, each of these “order types” encodes a real number.

His continuum hypothesis asserts that this is exactly the size of the continuum — that there are precisely 1 real numbers. In other words, the cardinality of the continuum immediately follow 0, the cardinality of the natural numbers, with no sizes of infinity in between.

But to Cantor’s immense distress, he couldn’t prove it.

In 1900, the mathematician David Hilbert put the continuum hypothesis first on his famous list of 23 math problems to solve in the 20th century. Hilbert was enthralled by the nascent mathematics of infinity — “Cantor’s paradise,” as he called it — and the continuum hypothesis seemed like its lowest-hanging fruit.

To the contrary, shocking revelations last century turned Cantor’s question into a deep epistemological conundrum.

The trouble arose in 1931, when the Austrian-born logician Kurt Gödel discovered that any set of axioms that you might posit as a foundation for mathematics will inevitably be incomplete. There will always be questions that your list of ground rules can’t settle, true mathematical facts that they can’t prove. As Gödel suspected right away, the continuum hypothesis is such a case: a problem that’s independent of the standard axioms of mathematics.

These axioms, 10 in all, are known as ZFC (for “Zermelo-Fraenkel axioms with the axiom of choice”), and they undergird almost all of modern math. The axioms describe basic properties of collections of objects, or sets. Since virtually everything mathematical can be built out of sets (the empty set {} denotes 0, for instance; {{}} denotes 1; {{},{{}}} denotes 2, and so on), the rules of sets suffice for constructing proofs throughout math.

In 1940, Gödel showed that you can’t use the ZFC axioms to disprove the continuum hypothesis. Then in 1963, the American mathematician Paul Cohen showed the opposite —you can’t use them to prove it, either. Cohen’s proof, together with Gödel’s, means the continuum hypothesis is independent of the ZFC axioms; they can have it either way.

In addition to the continuum hypothesis, most other questions about infinite sets turn out to be independent of ZFC as well. This independence is sometimes interpreted to mean that these questions have no answer, but most set theorists see that as a profound misconception.

They believe the continuum has a precise size; we just need new tools of logic to figure out what that is. These tools will come in the form of new axioms. “The axioms do not settle these problems,” said Magidor, so “we must extend them to a richer axiom system.” It’s ZFC as a means to mathematical truth that’s lacking — not truth itself.

Ever since Cohen, set theorists have sought to shore up the foundations of infinite math by adding at least one new axiom to ZFC. This axiom should illuminate the structure of infinite sets, engender natural and beautiful theorems, avoid fatal contradictions, and, of course, settle Cantor’s question…

Two rival axioms emerged that do just that. For decades, they were suspected of being logically incompatible.

In October 2018, David Asperó was on holiday in Italy, gazing out a car window as his girlfriend drove them to their bed-and-breakfast, when it came to him: the missing step of what’s now a landmark new proof about the sizes of infinity. “It was this flash experience,” he said.

Asperó, a mathematician at the University of East Anglia in the United Kingdom, contacted the collaborator with whom he’d long pursued the proof, Ralf Schindler of the University of Münster in Germany, and described his insight. “It was completely incomprehensible to me,” Schindler said. But eventually, the duo turned the phantasm into solid logic.

Their proof, which appeared in May in the Annals of Mathematics, unites two rival axioms that have been posited as competing foundations for infinite mathematics. Asperó and Schindler showed that one of these axioms implies the other, raising the likelihood that both axioms — and all they intimate about infinity — are true…

There are an infinite number of infinities. Which one corresponds to the real numbers? “How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer.”

[TotH to MK]

* William Blake

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As we contemplate counting, we might spare a thought for Georg Friedrich Bernhard Riemann; he died on this date in 1866. A mathematician who made contributions to analysis, number theory, and differential geometry, he is remembered (among other things) for his 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, regarded as one of the most influential papers in analytic number theory.

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“Control of consciousness determines the quality of life”*…

 

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Peter Carruthers, Distinguished University Professor of Philosophy at the University of Maryland, College Park, is an expert on the philosophy of mind who draws heavily on empirical psychology and cognitive neuroscience. He outlined many of his ideas on conscious thinking in his 2015 book The Centered Mind: What the Science of Working Memory Shows Us about the Nature of Human Thought. More recently, in 2017, he published a paper with the astonishing title of “The Illusion of Conscious Thought.”…

Philosopher Peter Carruthers insists that conscious thought, judgment and volition are illusions. They arise from processes of which we are forever unaware.  He explains to Steve Ayan the reasons for his provocative proposal: “There Is No Such Thing as Conscious Thought.”

See also: “An Anthropologist Investigates How We Think About How We Think.”

* Mihaly Csikszentmihalyi, Flow: The Psychology of Optimal Experience

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As we think about thought, we might spare one for Georg Ferdinand Ludwig Philipp Cantor; he died on this date in 1918.  Cantor was the mathematician who created set theory, now fundamental to math,  His proof that the real numbers are more numerous than the natural numbers implies the existence of an “infinity of infinities”… a result that generated a great deal of resistance, both mathematical (from the likes of Henri Poincaré) and philosophical (most notably from Wittgenstein).  Some Christian theologians (particularly neo-Scholastics) saw Cantor’s work as a challenge to the uniqueness of the absolute infinity in the nature of God – on one occasion equating the theory of transfinite numbers with pantheism – a proposition that Cantor, a devout Lutheran, vigorously rejected.

These harsh criticisms fueled Cantor’s bouts of depression (retrospectively judged by some to have been bipolar disorder); he died in a mental institution.

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Written by (Roughly) Daily

January 6, 2019 at 1:01 am

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