## Posts Tagged ‘**history of mathematics**’

## “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.”

*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.

## “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*

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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.

## “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)

## 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.