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Sunday, February 14, 2016

Perspective....


From whimsical Lewis Carroll, a rather different Sunday reflection:
"I may as well just tell you a few of the things I like, and then, whenever you want to give me a birthday present (my birthday comes once every seven years, on the fifth Tuesday in April) you will know what to give me. Well, I like, very much indeed, a little mustard with a bit of beef spread thinly under it; and I like brown sugar — only it should have some apple pudding mixed with it to keep it from being too sweet; but perhaps what I like best of all is salt, with some soup poured over it. The use of the soup is to hinder the salt from being too dry; and it helps to melt it. Then there are other things I like; for instance, pins — only they should always have a cushion put round them to keep them warm. And I like two or three handfuls of hair; only they should always have a little girl’s head beneath them to grow on, or else whenever you open the door they get blown all over the room, and then they get lost, you know."

Thursday, February 11, 2016

Anyone Who'd Visit a Psychiatrist Ought to Have His Head Examined


Fun with self-reference....

I've long thought that "self-reference" and "recursion" are among the most important topics out there (and I'm in good company, since they've been central to a lot of Douglas Hofstadter's work ;-). They straddle, in crucial ways, the fields of linguistics/cognitive science and math/logic. So I was delighted this week to see Ben Orlin do his humorous take on self-reference (and even Gödel's Incompleteness Theorem) and chop it down to size:

http://mathwithbaddrawings.com/2016/02/10/faqs-about-self-reference/comment-page-1/

So much fun! ...but Ben is not the only funny one... surfing around, I found this page of self-reference jokes/humor:
http://web.maths.unsw.edu.au/~jim/selfref.html

And Alexander Bogomolny also has an entertaining page of examples:
http://www.cut-the-knot.org/selfreference/

...and then there's always this xkcd classic:


...Lastly, always remember that, "All generalizations are false, including this one" (Mark Twain).


Wednesday, February 10, 2016

Not even sure what to title this!!?


Wonderful dot patterns....

Wow!. Yesterday, Patrick Honner tweeted out a link to this fabulous, freaky Numberphile video from December that I missed. Of course, all the Numberphile clips are great, but this is already one of my favorites (Patrick called it "mind-blowing")... seems like it perhaps crosses the boundaries of several mathematical/cognitive fields:




Monday, February 8, 2016

Bang?


OK, we'll kickstart the week with a recent problem from the DataGenetics blog... a fairly simple, straightforward (...and morbid ;-) probability conundrum, I've re-written below:

I say "morbid" because it involves playing "Russian roulette" with the following twist... after duct-taping you to an IKEA chair a villain pulls out an empty 6-chamber gun and loads it (in your full view) with just two bullets in ADJACENT chambers. He closes the gun cylinder, gives it a good spin, points the barrel at your carefully-coiffed head, and pulls the trigger. CLIIIICK... no bullet... maybe you've lived to enjoy another day... or... NOT. 
The villain announces he is going to pull the trigger one more time, and if you're still alive you're then free to go merrily home, or to the nearest brew pub. He'll even give you a choice: he can pull the trigger again right where he left off, or he can re-spin the cylinder again first. Which should you have him do?
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answer:   pull the trigger without doing a re-spin... 75% chance of survival in that case (3 of 4 possible remaining chambers being blank)  vs. 2/3 chance of survival with a re-spin (4 out of 6 chambers empty) .  If that's not clear, follow link back to DataGenetics for fuller explanation; and they also worked out several further variations worth checking out.


Sunday, February 7, 2016

Regression to the Mean


For today's Sunday reflection, this passage from a journal reviewer quoted in Gary Smith's "Standard Deviations":
"There are few statistical facts more interesting than regression to the mean for two reasons. First, people encounter it almost every day of their lives. Second, almost nobody understands it.
"The coupling of these two reasons makes regression to the mean one of the most fundamental sources of error in human judgment, producing fallacious reasoning in medicine, education, government, and, yes, even sports."
And speaking of statistics, this ;-):
"The average human has about one breast and one testicle."
-- Statistics 101

Wednesday, February 3, 2016

Strip Tease...


The Möbius strip of course...

I suspect we all think we've read enough about Möbius strips in the last many years (decades?)... but I'd encourage folks to make time for Evelyn Lamb's current read on the topic (one of her "favorite spaces"):

http://blogs.scientificamerican.com/roots-of-unity/a-few-of-my-favorite-spaces-the-moebius-strip/

She relates it back to the four-color-theorem, which is not often done (or six-color-theorem in this case). And these days, if you're expounding on Möbius strips, it's almost obligatory to include Vi Hart's story based on the same. I'm happy to report that Dr. Lamb does so at the end.


Tuesday, February 2, 2016

A New Find (for me)


I've indicated before my love for flea markets and thrift stores. And it's always a good day when I find some math book gem at such a site for a buck or less.
...Today, was a good day!

At a local thrift I stumbled upon a volume I'd not seen before from 1987. It's from Oxford University Press (no slouch of a publisher ;-) so I suspect many of you will be familiar with it: "Discovering Mathematics: The Art of Investigation" by British writer A. Gardiner.**  Dover continues to publish it, so it is still readily available today: http://amzn.to/1UIbmn1

I'm not all that far in yet, but my first impression is that it appears very interesting, engaging, and a fun treatment even though in a slightly textbook-ish format... and, perhaps in line with what a lot of current math education reform is attempting to accomplish.

The author's focus is on "mathematical discovery," similar to what others would call "mathematical thinking."

From the back cover:
"The word 'mathematics' usually conjures up a world of more-or-less familiar problems to be solved by more-or-less familiar techniques. This book examines a very different aspect of mathematics, namely how one can begin to explore unfamiliar, fresh ideas and chance observations, how one can pursue them through various stages until the light eventually begins to dawn, and how this whole process invariably throws up other interesting questions one would otherwise never have thought of."

Anyway, take this as a recommendation... if I change my mind as I get further into it, I'll update this post.

Here's an earlier review from MAA:
http://www.maa.org/press/maa-reviews/discovering-mathematics-the-art-of-investigation

**  I've seen the author listed variously as "Anthony Gardiner" and "Alan Gardiner;" if someone knows for sure the correct designation I'd appreciate the information.



Sunday, January 31, 2016

The Problem of Induction


For Sunday reflection today, a passage from physicist David Deutsch touching on science and philosophy, from his "The Fabric of Reality":
(I've added emphasis)
"In fundamental areas of science, observations of ever smaller, more subtle effects are driving us to ever more momentous conclusions about the nature of reality. Yet these conclusions cannot be deduced by pure logic from the observations. So what makes them compelling? This is the 'problem of induction'. According to inductivism, scientific theories are discovered by extrapolating the results of observations, and justified when corroborating observations are obtained. In fact, inductive reasoning is invalid, and it is impossible to extrapolate observations unless one already has an explanatory framework for them.  But the refutation of inductivism, and also the real solution of the problem of induction, depends on recognizing that science is a process not of deriving predictions from observations, but for finding explanations. We seek explanations when we encounter a problem with existing ones. We then embark on a problem-solving process. New explanatory theories begin as unjustified conjectures, which are criticized and compared according to the criteria inherent in the problem. Those that fail to survive this criticism are abandoned. The survivors become the new prevailing theories, some of which are themselves problematic and so lead us to seek even better explanations. The whole process resembles biological evolution."

Thursday, January 28, 2016

30 Years Ago Today...


"We will never forget them, nor the last time we saw them, this morning, as they prepared for their journey and waved goodbye and 'slipped the surly bonds of Earth' to 'touch the face of God.'"

-- Pres. Reagan speaking of the Challenger astronauts (Jan. 28, 1986)

Another non-math post today, in tribute to the seven astronauts (including famously, America's first teacher-in-space) who perished when the Challenger Space Shuttle exploded shortly after lift-off 30 years ago today -- a moment burned into the memory of most who experienced it.

Below, John Denver's tribute to them (Denver, a lover of flight, would die in his own plane crash in 1997).
And below that, some doggerel tribute that I wrote back at the time.




10 Miles High (to the Challenger Seven)

Hopes and dreams
Ensheathed in steel
Adventure sought,
Death revealed
The sky aflame
Seven souls afire
A missile becomes
A funeral pyre...
There, 10 miles high.

And tears enough
To flood a dam
Burst across
This bereaved land
As we unite
To share the pain
That has been dealt
Our way again...
From 10 miles high.

With disbelief
We shake our heads
In heartfelt grief
We hug the kids
Absorb their hurt
Qualm their fears
Help them help us
Through the tears...
From 10 miles high.

Time will pass
So too our sorrow
Always there is
A tomorrow
But not for seven
Bright and brave
Unknowing heroes
We lost today
Somewhere... 10 miles high.


Sunday, January 24, 2016

"Interpreting Mathematics..."


A Sunday reflection from Mircea Pitici today -- which I've also included in my review of his latest book, new today at MathTango:
"Interpreting mathematics is a further stage of thinking mathematically -- not a "higher" stage or a "lower" stage, just an essentially different one. Interpreting mathematics is not about mathematical truth (or any other truth); it is a personal take on mathematical facts, and in that it can be true or untrue, or it can even be fiction; it is vision, or it is rigorous reasoning, or it is pure speculation, all occasioned by mathematics; it is imagination on a mathematical theme; it goes back several millennia and it is flourishing today, as I hope this series of books lays clear... To speak about interpreting mathematics sounds odd, but it seems so only because the customary indoctrination served by our school system pervades the common views of mathematics, both among mathematicians and the lay public."

Friday, January 22, 2016

On the Verge of Primaries




No math, just political-ramble today... what-the-hey... realized it was over 7 months ago (time flies when you're having fun) that I posted a political entry here predicting that the only Republicans running (I think ~17 at the time!) with a shot at the nomination, were Ted Cruz and Rand Paul (and Paul Ryan if he entered the race).
Within a couple weeks, Rand Paul's performance in two debates persuaded me that his chances were much lower, and that the main other possibility to Cruz would be Kasich, squeaking through if the right-wingers all ate each other up during the primaries (and they are doing a lot of chomping).

Now, over 7 months later and on the verge of the first tests, not all that much has changed, EXCEPT that I now doubt any Republican can go to the convention with enough delegates... and after multiple ballots, suspect Party powers-that-be may turn to someone not currently even in-the-running as the standard-bearer (who I won't predict).
It's all a car-wreck waiting to happen. POW!, BAAAM!, WHAM!, ZAP!, CRASH!!! And the Republican National Committee, which can (and if necessary, will) de-rail Trump whenever they wish, look powerless to stop it. Or, so it seems to me. Let the games commence.
(Who that Republican standard-bearer will face in the general, I don't know, though I suspect Hillary.)




Thursday, January 21, 2016

Two for Thursday, via Ben Orlin, Keith Devlin


1)  All aboard the number line... I shouldn't even bother linking to this posting of Ben Orlin's from yesterday... because ALL OF YOU SHOULD BE FOLLOWING HIM REGULARLY ON YOUR OWN by now! (if you're not, get with the program... so I can quit shouting).
Great stuff from Ben, taking us on a journey to-and-fro the number line:

http://mathwithbaddrawings.com/2016/01/20/the-number-line-a-journey/

2)  When someone on Twitter asked Keith Devlin about "KIC 5520878 and the golden ratio," he responded, "That one is intriguing. Most GR claims are nonsense. But I'm keeping an open mind on this one, b/c of cont fraction rep of GR."

So I had to look that one up to get the backstory. And here it is from Natalie Wolchover:

https://www.quantamagazine.org/20150310-strange-stars-pulse-to-the-golden-mean/


Tuesday, January 19, 2016

Props to Propp...


Everyone probably knows Jordan Ellenberg's fabulous bestseller from a few years back, "How Not To Be Wrong." A few days ago at "Mathematical Enchantments" Jim Propp put up one of the odder math posts I've seen in quite awhile (and a bit longish, but nonetheless entertaining) entitled (taking off from Jordan), "How To Be Wrong." It has to do with the benefit of teachers, for the sake of teaching, making mistakes and facing them square-on... and, not being too dogmatic or pompous along the way; what he calls "the art of being wrong":
 
https://mathenchant.wordpress.com/2016/01/16/how-to-be-wrong/

At one cogent point Propp writes:
"I stress to my students that the earlier you make your mistakes, the better.  Every mistake you make in the classroom, or on your homework, is a mistake you probably won’t make on the exam, where mistakes can really hurt you.  And, carrying this idea further: a mistake you make in college or graduate school is a mistake you’re less likely to make after you graduate, when you’re building bridges or designing cancer treatment protocols. So my answer to the question 'How to be wrong?' is: 'Early and often!'”
By name or initials he brings a number of interesting folks into the post, as well as many interesting ideas/examples. I'm actually surprised there haven't been more responses in the comments section to the post, though maybe it is so odd that folks aren't sure just how to respond to it.
Anyway, check it out, especially if you're a math teacher.

Also, all of this emphasis on "mistakes" reminds me of an older Keith Devlin post on the importance of "failure":
http://mooctalk.org/2013/12/15/maththink-mooc-v4-part-2/

It too, should NOT be missed, nor should the two commencement addresses Dr. Devlin links to at the end (Steve Jobs and J.K. Rowling). Great stuff!