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Posts Tagged ‘history of mathematics

“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


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.


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



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


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.

220px-Georg_Cantor2 source


Written by (Roughly) Daily

January 6, 2019 at 1:01 am

“I am incapable of conceiving infinity, and yet I do not accept finity”*…



Suppose you’re working at a hotel with infinitely many rooms in it, numbered 1, 2, 3, 4, 5, … all the way up forever and ever. (This is known as a Hilbert Hotel.) One evening when every single room is occupied, a traveler arrives and requests to be accommodated too. You’re the manager. What do you do to help the traveler?

Simple. You just ask each occupant to one room forward. 1 goes to 2, and 2 goes to 3, and so on. Every previous occupant gets a new room. And the first room is now open for the traveler.

The procedure above is characterized by an infinite number of actions or tasks to be carried out in a finite amount of time. Procedures with this character are known as supertasks…

More on the ins and outs of infinities at “Introducing Supertasks.” (More fun musings on infinity here and here; and more on Hilbert’s Hotel here.)

* Simone de Beauvoir, La Vieillesse


As we muse on many, we might spare a thought for Hermann Hankel; he died on this date in 1873.  A mathematician who worked with Möbius, Riemann, Weierstrass,  and Kronecker (among others), he made important contributions to the understanding of complex numbers and quaternions… and to work begun by Bernard Bolzano on infinite series.

220px-Hankel source


Written by (Roughly) Daily

August 29, 2018 at 1:01 am

“Common sense is calculation applied to life…”*


Josh Orter writes…

I updated my iPhone to the latest iOS version last week. Doing so required that “I agree” to Terms and Conditions amounting to 6,114 words: more than 37% longer than The Constitution of the United States (4,447). Although hardcore legalese enthusiasts may curl up with this masterpiece before the fire, mug of hot cocoa in hand, it will be read by virtually no one else. I clicked immediately, blissfully ignorant of my acquiescence…

Or maybe not so blissful. I was, in fact, vaguely annoyed at having just blindly accepted what amounts to a 24-page term paper (12-pt Times Roman, double-spaced). What, then, would fully informed consent cost?

$482,894,368, as it turns out.

With various reputable sources suggesting an average reading speed of 300 words per minute, 6,114 words would suck up 20.38 minutes of life for each user. Multiply by the average American’s hourly earnings of $24.09/hour (.4015/minute) and it’s $8.18257 in labor per updater.

A recently released study estimated Apple’s share of the 145 million-unit American  smartphone market at 40.7%. 59 million iPhoners.

59,015,00 x $8.18257 = $482,894,368

Perhaps more unsettling than money is the cumulative man-hours this particular endeavor would consume: 20-million.

59,015,000 x users x 20.38 minutes per= 1,202,725,700 minutes= 20 million hours…

Find other arithmetic adventures at Josh’s wonderful Stupid Calculations (“Where practical facts get rendered into utterly meaningless ones”).

[TotH to the ever-illuminating Flowing Data]

* Henri Frédéric Amiel


As we carry the 1, we might recall that it was on this date in 1807 that Joseph Fourier’s monograph On the Propagation of Heat in Solid Bodies, was read to the Paris Institute.  An important mathematical work containing what we now call Fourier series, it was reviewed by renowned respondents including Lagrange, Laplace, Monge, and Lacroix– who objected to his innovation (the expansion of functions as trigonometrical series: the Fourier series).  Indeed, his work wasn’t published (in a revised and somewhat expanded form, as The Analytic Theory of Heat) until 1822.  Fourier’s contributions were ultimately judged so important that, in addition to the series, Fourier’s Law and Fourier’s transform were named in his honor.  And indeed, Fourier’s work, having first helped scientists and engineers understand heat transfer and vibrations, went on to help Watson and Crick discover the structure of DNA and to provide the underpinnings for quantum physics, radio astronomy, MP3 and JPEG compression, X-ray crystallography, voice recognition, and PET or MRI scans.

(Fourier was something of a polymath:  just before developing his mathematical advances, he went with Napoleon Bonaparte on his Egyptian expedition in 1798, and was made governor of Lower Egypt and secretary of the Institut d’Égypte, where he was involved in the discovery and decoding of the Rosetta Stone. And after the publication of The Analytic Theory of Heat, he discovered the Greenhouse Effect.)

True greatness is when your name is like ampere, watt, and fourier—when it’s spelled with a lower case letter.

– Richard Hamming (in a 1986 Bell Labs Colloquium)



Written by (Roughly) Daily

December 21, 2013 at 1:01 am

Spiraling into control…

 The Fibonacci spiral (source)

As we remark that math really is beautiful, we might send elegantly parsimonious birthday greetings to one of Fibonacci’s spiritual descendants, a father of Pure Mathematics, Leonhard Euler; he was born on this date in 1707.  While crafting “the most remarkable formula in mathematics,” Euler made foundational contributions to number theory, graph theory, mathematical logic, and applied math; he originated many commonly-used figures of mathematical notation, and invented the concept of the “mathematical function.”  And he was no slouch in physics either, making renowned contributions in work in mechanics, fluid dynamics, optics, and astronomy.


Written by (Roughly) Daily

April 15, 2012 at 1:01 am

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