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Posts Tagged ‘history of mathematics

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

 

2D02EB62-9945-4BB8-A5E9507B396FEF67_source

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.

220px-Georg_Cantor2 source

 

Written by LW

January 6, 2019 at 1:01 am

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

 

hilberts_hotel

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

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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 LW

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

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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)

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Written by LW

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.

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Written by LW

April 15, 2012 at 1:01 am

Teach your children well…

Your correspondent is headed again across the Dateline (where he loses all track of what day it actually is), thus (R)D will be in abeyance until March 9 or 10, when “service” will resume again.

In the meantime, a message from the past to the future…

In 1959, philosopher, logician, mathematician, historian, and social critic Bertrand Russell appeared on the BBC interview program Face-to-Face.  As the program came to its final two minutes, he was asked a valedictory question: “What would you tell a generation living 1,000 years from now about the life you’ve lived and the lessons you’ve learned?”…

[TotH to Open Culture]

As we do our level best to take good advice, we might recall that it was on this date in 1611 (81 years before the beginning of a case Russell might have cited, the Salem Witch Trials, on this date in 1692) that John Pell was born.  An English mathematician of accomplishment, he is perhaps most remembered for having introduced the “division sign”– the obelus: a short line with dots above and below– into use in English.  It was first used in German by Johann Rahn in 1659 in Teutsche Algebra; Pell’s translation brought the symbol to English-speaking mathematicians.  Indeed, Pell was an important influence on Rahn, and edited his book– so may well have been, many scholars believe, the originator of the symbol for this use.  (In any case the symbol wasn’t new to them:  the obelus [derived from the word for “roasting spit” in Greek] had already been used to mark passages in writings that were considered dubious, corrupt or spurious…. a use that surely seems only too familiar to legions of second and third grade math students.)

John Pell (source)

Paradoxically…

Suppose there is a town with just one male barber; and that every man in the town keeps himself clean-shaven: some by shaving themselves, some by attending the barber. It seems reasonable to imagine that the barber obeys the following rule: He shaves all and only those men in town who do not shave themselves. Under this scenario, we can ask the following question: Does the barber shave himself?

From Epimenides’ Paradox to the Omnipotence Paradox, more fun-with-logic at “Brain Twisting Paradoxes.”

As we return to first principles, we might wish a carefully-reasoned Joyeux Anniversaire to Félix-Édouard-Justin-Émile Borel, a mathematician and pioneer of measure theory and its application to probability theory; he was born in Saint-Affrique on this date in 1871.  Borel is perhaps best remembered by (if not for) his thought experiment demonstrating that a monkey hitting keys at random on a typewriter keyboard will– with absolute certainty– eventually type every book in the Bibliothèque Nationale (or, as oft repeated, every play in the works of Shakespeare, or…)– that is, the infinite monkey theorem.

Borel (image source)

 

When the facts are just too confusing…

Fake Science to the rescue, with entries both informative…

and useful…

More, at Fake Science.

As we try to remember how to spell “paramecium,” we might recall that this is the birthday of the first woman in the Western world considered to be a mathematician:  Maria Gaetana Agnesi, born this date in 1718.  While she thought and wrote broadly about natural science and philosophy, she is best remembered for her work in differential calculus– perhaps most particularly for her work on the cubic curve now know as the “witch of Agnesi.”

Maria Agnesi

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