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Posts Tagged ‘set theory

“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

From the Not-Sure-I-Really-Want-To-Know Department…

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As readers know, some physicists believe that the universe as we know it is actually a giant hologram, giving us the illusion of three-dimensions, while in fact all the action is occurring on a two-dimensional boundary region (see here, here, and here)… shadows on the walls of a cave, indeed.

But lest one mistake that for the frontier of freakiness, others (c.f., e.g., here and here) believe that the existence we experience is nothing more (or less) than a Matrix-like simulation…

A common theme of science fiction movies and books is the idea that we’re all living in a simulated universe—that nothing is actually real. This is no trivial pursuit: some of the greatest minds in history, from Plato, to Descartes, have pondered the possibility. Though, none were able to offer proof that such an idea is even possible. Now, a team of physicists working at the University of Bonn have come up with a possible means for providing us with the evidence we are looking for; namely, a measurable way to show that our universe is indeed simulated. They have written a paper describing their idea and have uploaded it to the preprint server arXiv…

Phys.Org has the whole story at “Is it real? Physicists propose method to determine if the universe is a simulation“; the paper mentioned above can be downloaded here.

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As we reach for the “reset” button, we might send carefully-calculated birthday greetings to Paul Isaac Bernays; he was born on this date in 1888.  A close associate of David Hilbert (of “Hilbert’s Hotel” fame), Bernays was one the foremost philosophers of mathematics of the Twentieth Century, who made important contributions to mathematical logic and axiomatic set theory.  Bernays is perhaps best remembered for his revision and improvement of the (early, incomplete) set theory advanced by John von Neumann in the 1920s; Bernays’s work, with some subsequent modifications by Kurt Gödel, is now known as the Von Neumann–Bernays–Gödel set theory.

Lest, per the simulation speculation above suggest that cosmology has a hammerlock on weirdness:  Set theory is used, among other purposes, to describe the symmetries inherent in families of elementary particles and in crystals. Materials such as a liquid or a gas in equilibrium, made of uniformly distributed particles, exhibit perfect spatial symmetry—they look the same everywhere and in every direction… a condition that “breaks” at very low temperature, when the particles form crystals (which have some symmetry, but less)…  Now Nobel Laureate Frank Wilczek has suggested that there may exist “Time Crystals“– whose structure would repeat periodically, as with an ordinary crystal, but in time rather than in space… a kind of “perpetual motion ‘machine'” (weirder yet, one that doesn’t violate the laws of thermodynamics).

Paul Bernays

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Written by (Roughly) Daily

October 17, 2012 at 1:01 am

It’s kind of like a Sofia Coppola film…

Reader’s may remember Bacon Ipsum, the down-home alternative to Lorem Ipsum, used by designers for “greeking” in placeholder text.

Well, now there’s a cooler, more moderne choice: Hipster Ipsum: “Artisanal filler text for your site or project.”  A sample…

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Head on over to Hispster Ipsum for the coolest of layouts…

As we affect a posture of amused indifference, we might wish a symbolically-logical Happy Birthday to mathematician Guiseppe Peano; he was born on this date in 1858.  The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much of the symbolic notation still in use.

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Written by (Roughly) Daily

August 27, 2011 at 1:01 am

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