Sunday, November 23, 2014

Mathematics: "A Growing Organism"... "A Connected Web"

This Sunday's Reflection:

"Mathematics is a living and growing organism; within it are intricate and delicate structures of strong aesthetic appeal.  It offers opportunities for surprise as unexpected vistas open the mind to new lines of thought...
"Mathematics was created by all manner of people. There were religious bigots and atheists, political reactionaries and wild revolutionaries, snobs and egalitarians; some were people of great charm, some odious. If there is any common denominator, it is a driving curiosity, a desire to understand, a need to build, even if the structures be abstract. Admirably suited though mathematics is to modelling the real world, it can be developed totally without dependence on anything outside itself. Parts of it are simply mind creations, owing nothing to the physical world. It is a playground for the mind...

"Regrettably, many of us have never been allowed to see what mathematics is. It has been obscured by pointless emphasis on routines rather than ideas. This failure to distinguish what is important has led many people to see mathematics as a collection of totally arbitrary rules which have to be learnt by rote, and performed with the exactness and precision of a religious rite. Ask a person if there is much to be remembered in mathematics; if they speak of an overwhelming mass of material, their education in this area has been counter-productive; not merely neutral. Mathematics, properly seen, is a connected web; grasp at one piece and all the surrounding region comes to mind.

-- Laurie Buxton from the Introduction to  "Mathematics For Everyone" (1984)

Thursday, November 20, 2014

Euler's Alchemy

                              e + 1 = 0 

Nice Lee Simmons piece today for Wired, rhapsodizing on "Euler's identity":

Love this wormhole analogy toward the end:
"But the weirdest thing about Euler’s formula—given that it relies on imaginary numbers—is that it’s so immensely useful in the real world. By translating one type of motion into another, it lets engineers convert messy trig problems (you know, sines, secants, and so on) into more tractable algebra—like a wormhole between separate branches of math. It’s the secret sauce in Fourier transforms used to digitize music, and it tames all manner of wavy things in quantum mechanics, electronics, and signal processing; without it, computers might not exist."
Give it a read....

Wednesday, November 19, 2014

A Geometry Wednesday

Today, a simple, practical guide for estimating the height of a tree from Ian Stewart's recent "Professor Stewart's Casebook of Mathematical Mysteries."

Stewart describes this as "an old forester's trick (the trick is old, not the forester) for estimating the height of a tree without climbing it or using surveying equipment."

and continues:

"Stand at a reasonable distance from the tree, with your back towards it. Bend over and look back at it through your legs. If you can't see the top, move away until you can. If you can see it easily, move closer until it's just visible. At that point, your distance from the base of the tree will be roughly equal to its height."

The angle your line-of-sight is forming with the ground is now roughly 45 degrees, and thus the line-of-sight itself is the hypotenuse of an isosceles right triangle with the base-distance and tree height equal side values (thus, measure or walk off the base, and you have the height).

...As for those of us of an age where bending over and looking through our legs isn't such a practical affair, I guess we're out-of-luck :-(

Anyway, Stewart's volume is his usual mix of older and fresher, easier and more technical, math entertainment. Give it a gander.

One of my old favorites that shows up in Stewart's book is what he calls the "Square Peg Problem" (it goes by some different names) -- another one of those seemingly easy, yet exquisitely difficult-to-prove (100+ year-old) conjectures. It asks whether one or more squares can always be fitted upon every closed planar curve by selecting four points on that curve, or stated more succinctly as "Does every simple closed curve have an inscribed square?" For a fuller treatment of it see here:

and here's a 2014 article on it (pdf) from AMS:

via Wikipedia

Monday, November 17, 2014

"Women In Maths"

The University of Nottingham has been doing a delightful series of short videos of "Women in Maths" (apparently inspired, in part, by Maryam Mirzakhani as the first woman to receive a Fields Medal last August). The introductory video is here:

And the other videos (focusing on individual mathematicians in ~2 minutes or less) can be viewed here:

The on-site video description reads in part:

"Although around 40% of the UK undergraduates in mathematics are women, there is a well-documented leaking pipeline when it comes to women choosing to do a PhD and then choosing an academic career path. The proportion of women mathematicians declines rapidly the higher one looks on the academic ladder. Unfortunately, this often makes women who do choose this career path invisible, to students who are about to choose their A-levels, even to students who are already pursuing a maths degree. Therefore the women at the School of Mathematical Sciences at University of Nottingham have made some videos to: become more visible and thus hopefully inspire others; fight stereotypes; talk about what it is like to be a mathematician in academia today and why they chose academia; communicate the passion they feel for what they do and what they love about it; describe the creativity needed for research. And yes -- it can be combined with having a family!"

Sunday, November 16, 2014

Think About It...

short and sweet...

"If my mental processes are determined wholly by the motion of atoms in my brain, I have no reason to believe that my beliefs are true... and hence I have no reason for supposing my brain to be composed of atoms."

--- J.B.S. Haldane, "Possible Worlds" (1927)

Friday, November 14, 2014

'To Thine Own Self Be True' ...A Legend Passes

"This above all -- to thine own self be true,
And it must follow, as the night the day,
Thou canst not then be false to any man.
Farewell. My blessing season this in thee!

 -- Polonius (in Hamlet)

As most know by now Alexander Grothendieck passed away yesterday at age 86. 
This news aggregator has a lot of good material:

And here are some of the older pieces various Twitterers linked to overnight:

from Grothendieck himself, in Récoltes et Semailles

"...I’ve had the chance in the world of mathematics that bid me welcome, to meet quite a number of people, both among my 'elders' and among young people in my general age group who were more brilliant, much more ‘gifted’ than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle -- while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things I had to learn (so I was assured) things I felt incapable of understanding the essentials or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates almost by sleight of hand, the most forbidding subjects.

 "In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still from the perspective of thirty or thirty five years, I can state that their imprint upon the mathematics of our time has not been very profound. They’ve done all things, often beautiful things in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they’ve remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have to rediscover in themselves that capability which was their birthright, as it was mine: The capacity to be alone.

R.I.P.  and as is often said (though not that often in mathematics), "Farewell to one of the great ones"....

Thursday, November 13, 2014

Charts, Graphs, Facepalms

(from: )

Nautilus posted a simple, worthwhile piece recently that was a lesson on how easily the public is fooled by manipulated charts:

It reminds one a bit of another site, "Spurious Correlations," that focuses on examples (like the one above) of graphs that show... guess what...: spurious correlations (well, likely spurious, anyway):

It's all a valuable reminder that people aren't just fooled by empirical-sounding numbers, but by easily-misinterpreted visual presentations as well.

Finally, if you prefer your charts with a dose of humor, check these out: 

...and plenty more here:

Wednesday, November 12, 2014

Either Have to Laugh or Cry...

After last week's elections I either need something to laugh at, or, a triple-dose of Prozac....

Soooo... from the LMAO Dept.:  yesterday Steven Strogatz passed along (on Twitter) a site and old post I'd never seen before... if bad words put you off, don't even bother looking... but if you read it (after sending any younguns out of the room) and guffaw you can thank me for passing it along (newly-minted $20 bills would be much-appreciated)... or, if you read it and hate it, you can blame that low-down, good-for-nuthin, wacko Cornell math-obsessed professor:

...and I've already done the heavy-lifting for you to find the few-other math-related posts on the site:

Monday, November 10, 2014

Bevis's Birthday

To kickstart the week, a little algebraic puzzle I adapted from this week's "Ask Marilyn" column in the Sunday Parade Magazine:

Suppose Bevis's birthday is this month (he's an adult), and on his birthday he will be the same age as the 2-digit year in which he was born (i.e. 1940, or whatever). How old will Bevis be on his upcoming birthday?
.Answer below
answer:  57 years old

Sunday, November 9, 2014

Mathematicians As Mavericks (Sunday Reflection)

"Mathematicians are mavericks -- inventors and explorers of sorts; they create new things and discover novel ways of looking at old things; they believe things hard to believe, and question what seems to be obvious. Mathematicians also disrupt patterns of entrenched thinking; their work concerns vast streams of physical and mental phenomena from which they pick the proportions that make up a customized blend of abstractions, glued by tight reasoning and augmented with clues glanced from the natural universe. This amalgam differs from one mathematician to another; it is 'purer' or 'less pure,' depending on how little or how much 'application' it contains; it is also changeable, flexible, and adaptable, reflecting (or reacting to) the social intercourse of ideas that influences each of us…

"And here comes a peculiar aspect that distinguishes mathematics among other intellectual domains: Mathematicians seek validation inside their discipline and community but feel little need (if any) for validation coming from outside. This professional chasm surrounding much of the mathematics profession is inevitable up to a point because of the nature of the discipline. It is a Janus-faced curse of the ivory tower, and it is unfortunate if we ignore it."

-- Mircea Pitici from the Introduction to "The Best Writing On Mathematics 2013"

(p.s. -- I believe the 2014 edition of "The Best Writing On Mathematics" will be appearing in stores by the end of this month.)

[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know ( If I use one submitted by a reader, I'll cite the contributor.]


Thursday, November 6, 2014

It's a Wonderful Subject

Another short clip today of Steven Strogatz (from about a year ago at the World Science Festival), explaining simply why he's motivated to know and teach math:

I recently finished reading Dr. Strogatz's 2009 "The Calculus of Friendship" for the second time (having loved it the first time around), and oddly, perhaps because of the holidays approaching, it caused me to think of the old Jimmy Stewart movie, "It's a Wonderful Life," a film often viewed as a tradition for this festive (and nostalgic) time of year. I've never made a point of re-watching particular movies at particular seasons, but have to admit I'm now thinking of making Dr. Strogatz's book my own personal tradition to re-read each year as the holidays approach. It's a wonderful, if subdued, tale, and one wishes that everyone, whatever field you're in, could have a "Joff" in their lives... or... be a "Joff" to others (..."Joff" being Steven's inspiring high school math teacher).

ADDENDUM Now, coincidentally, The Aperiodical blog has just put up a (15-min.) student-made documentary about doing/teaching mathematics:

Monday, November 3, 2014

The Monster That Lurks

Not sure which I love more about this Numberphile offering... Tim Burness's step-by-step explanation of the "Monster Group" (with it's almost 200,000 dimensions) or John Conway's sheer astonishment that it exists at all:

Sunday, November 2, 2014

Contemplating Riemann

"In [his 1859 paper], Riemann made an incidental remark -- a guess, a hypothesis. What he tossed out to the assembled mathematicians that day has proven to be almost cruelly compelling to countless scholars in the ensuing years...

" is that incidental remark -- the Riemann Hypothesis -- that is the truly astonishing legacy of his 1859 paper. Because Riemann was able to see beyond the pattern of the primes to discern traces of something mysterious and mathematically elegant at work -- subtle variations in the distribution of those prime numbers. Brilliant for its clarity, astounding for its potential consequences, the Hypothesis took on enormous importance in mathematics. Indeed, the successful solution to this puzzle would herald a revolution in prime number theory. Proving or disproving it became the greatest challenge of the age...

"It has become clear that the Riemann Hypothesis, whose resolution seems to hang tantalizingly just beyond our grasp holds the key to a variety of scientific and mathematical investigations. The making and breaking of modern codes, which depend on the properties of the prime numbers, have roots in the Hypothesis. In a series of extraordinary developments during the 1970s, it emerged that even the physics of the atomic nucleus is connected in ways not yet fully understood to this strange conundrum. ...Hunting down the solution to the Riemann Hypothesis has become an obsession for many -- the veritable 'great white whale' of mathematical research. Yet despite determined efforts by generations of mathematicians, the Riemann Hypothesis defies resolution.

 -- John Derbyshire, from "Prime Obsession"

[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know ( If I use one submitted by a reader, I'll cite the contributor.]

Wednesday, October 29, 2014

Moravec's Paradox

This isn't exactly math, but it's artificial intelligence (AI), and that's close enough... especially since a few posts back I wrote about IBM's "Deep Blue" and its 1997 defeat of chess grandmaster Gary Kasparov (at the time, a long-held goal of AI). Well, Moravec's paradox is the interesting idea that advanced or high-level reasoning and logic is much more easily mimicked by a computer system than are low-level sensori-motor skills that have evolved over millions of years... it's easier for a computer to learn to play chess, than to recognize human faces. This is one of those things that is fairly obvious when you stop to think about it... but, we often don't stop to think about it!
Here's what Steven Pinker wrote in "The Language Instinct":
“The main lesson of thirty-five years of AI research is that the hard problems are easy and the easy problems are hard. The mental abilities of a four-year-old that we take for granted – recognizing a face, lifting a pencil, walking across a room, answering a question – in fact solve some of the hardest engineering problems ever conceived…. As the new generation of intelligent devices appears, it will be the stock analysts and petrochemical engineers and parole board members who are in danger of being replaced by machines. The gardeners, receptionists, and cooks are secure in their jobs for decades to come.”   
A more recent blog piece applies the paradox to Google's self-driving cars, a creation I've certainly had trouble comprehending, given the countless issues/variables involved:

[p.s. -- actually, where are the dang flying jetpacks I grew up believing we would all have by now... forget the cars Google, I want my personal commuting jetpack!]

anyway, below, another somewhat provocative post applying Moravec's paradox to brain processing:

Tuesday, October 28, 2014

Just Passing This Along

Colin Hegarty, who runs Hegartymaths has been bestowed a "Gold" tech-teaching award for his free math-tutorial site in Britain:

His videos are here:

I've not actually experienced the site or videos, so not directly endorsing it, but just recognizing that others attest to its value. It sounds a lot like (and was indeed inspired by) Khan Academy, which remains controversial in various quarters.

Anyway, check it out if you're looking for adjunct math tools.  Also, Colin tweets here:  @hegartymaths

Sunday, October 26, 2014

Taleb on Randomness

Today, a number of bits from an older Nassim Taleb volume, "Fooled By Randomness":

"Probability is not a mere computation of odds on the dice or more complicated variants; it is the acceptance of the lack of certainty in our knowledge and the development of methods for dealing with our ignorance. Outside of textbooks and casinos, probability almost never presents itself as a mathematical problem or a brain teaser. Mother Nature does not tell you how many holes there are on the roulette table, nor does she deliver problems in a textbook way (in the real world one has to guess the problem more than the solution)."

"This book is about luck disguised and perceived as nonluck (that is skills) and, more generally, randomness disguised and perceived as non-randomness (that is, determinism). It manifests itself in the shape of the lucky fool, defined as a person who benefited from a disproportionate share of luck but attributes his success to some other, generally very precise, reason."

"We are still very close to our ancestors who roamed the savannah. The formation of our beliefs is fraught with superstitions -- even today (I might say especially today). Just as one day some primitive tribesman scratched his nose, saw rain falling, and developed an elaborate method of scratching his nose to bring on the much-needed rain, we link economic prosperity to some rate cut by the Federal Reserve Board, or the success of a company with the appointment of a new president 'at the helm.'"

"Disturbingly, science has only recently been able to handle randomness (the growth in available information has been exceeded only by the expansion of noise). Probability theory is a young arrival in mathematics; probability applied to practice is almost nonexistent as a discipline"

"Indeed, probability is an introspective field of inquiry, as it affects more than one science, particularly the mother of all sciences: that of knowledge. It is impossible to assess the quality of the knowledge we are gathering without allowing a share of randomness in the manner it is obtained and cleaning the argument from the chance coincidence that could have seeped into its construction. In science, probability and information are treated in exactly the same manner. Literally every great thinker has dabbled with it, most of them obsessively.

[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know ( If I use one submitted by a reader, I'll cite the contributor.]