Go Gowers!

This is waaaay (as in leaps-and-bounds) beyond my brainpower, but still I find the latest post from the always-worth-reading Tim Gowers, fascinating… he proposes to utilize a polymathematical (collaborative) approach to test an idea that he thinks (but also doubts) may have some merit, to the P vs. NP Millennium problem. Before even getting to the idea he wants to consider, he spends considerable time on the pros and cons of such a collaborative undertaking. It's a verrry long, and before the end, a very deeeep read, but if you're particularly interested in P vs. NP, or just in the polymath approach to problems, highly recommended:

http://gowers.wordpress.com/2013/10/24/what-i-did-in-my-summer-holidays/

I'm guessing that RJ Lipton's blog may have some response to Gower's proposal (or maybe he'll just send along a comment to Tim's blog), so if the subject does interest you, might be worth monitoring that as well:

http://rjlipton.wordpress.com/

Laboring Over Formalism

Happy Labor Day to US readers... and here's a little reading you can labor over:

A piece (not sure how old it is???) from non-Platonist Timothy Gowers on philosophy and mathematics, focusing on Platonism, logicism and formalism (good stuff, BUT ONLY if you're already inclined toward philosophical underpinnings):

https://www.dpmms.cam.ac.uk/~wtg10/philosophy.html

It's brimming with interesting ideas, including (in the "#6 Truth and Provability" section) the notion that somewhere in the decimal expansion of pi there ought surely be a string of a million 7's, on the basis of it being a "normal number."

Here's a little bit of his wrap-up to the longread:

"...the point remains that if A is a mathematician who believes that mathematical objects exist in a Platonic sense, his outward behaviour will be no different from that of his colleague B who believes that they are fictitious entities, and hers in turn will be just like that of C who believes that the very question of whether they exist is meaningless...
"So why should a mathematician bother to think about philosophy? Here I would like to advance a rather cheeky thesis: that modern mathematicians are formalists, even if they profess otherwise, and that it is good that they are...

"When mathematicians discuss unsolved problems, what they are doing is not so much trying to uncover the truth as trying to find proofs….

"I also believe that the formalist way of looking at mathematics has beneficial pedagogical consequences. If you are too much of a Platonist or logicist, you may well be tempted by the idea that an ordered pair is really a funny kind of set -- the idea I criticized earlier. And if you teach that to undergraduates, you will confuse them unnecessarily. The same goes for many artificial definitions."

Monday Math Buffet...

Another potpourri of offerings, if you've missed any of these…:

1) I love this relatively brief recent post from a secondary educator:

http://practicaltheory.org/blog/2013/02/28/we-dont-know-how-to-teach-math/

Also have to smile a bit at this particular line from the piece: "Seymour Papert said that math represents the failure of progressive education because the way we teach math always reintroduces coercion back into education."

But every line is good, and it largely reminds me of a fantastic, (looong) older post by Fields Medalist Timothy Gowers (which drew over 160 comments) that deserves frequent re-visiting:

http://gowers.wordpress.com/2012/06/08/how-should-mathematics-be-taught-to-non-mathematicians/

2) Latest (#96) Carnival of Mathematics is now up at "Math Mama Writes" blog:

http://mathmamawrites.blogspot.com/2013/03/carnival-of-mathematics-96.html

...Plenty of variety!

3) "Futility Closet" recently highlighted the delightful (to me) Yablo's Paradox:

http://www.futilitycloset.com/2013/02/22/yablos-paradox/

4) On the same day that I interviewed Evelyn Lamb over at MathTango, she put up a new post at her blog on the four-color theorem:

http://blogs.scientificamerican.com/roots-of-unity/2013/03/01/4-color-map-theorem/

5) Once again Keith Devlin covers some aspects/difficulties of running a MOOC at his "Devlin's Angle" blog (Keith is running at least 3 separate blogs, plus a Huff. Post column!):

http://devlinsangle.blogspot.com/2013/03/can-we-make-constructive-use-of-machine.html

...and over at Huffington Post Keith writes about the dropout rates for MOOCs, and why an 80+% dropout rate isn't necessarily a problem:

http://www.huffingtonpost.com/dr-keith-devlin/moocs-and-the-myths-of-dr_b_2785808.html

6) Math educator Maria Droujkova, with a focus on toddlers and youngsters, is Sol Lederman's 24th podcast interviewee here:

http://wildaboutmath.com/2013/03/01/maria-droujkova-inspired-by-math-24/

7) And finally, if you have any mental energy/time left, a long, thought-provoking read from the Simons Foundation on the future of computers in mathematical proofs (can we trust computers, as we turn to them more and more in the future for proofs of highly-complex theorems?):

https://simonsfoundation.org/features/science-news/in-computers-we-trust/

Of Patterns and Publishing...

Two bits today:

1) At some point recently, the following tweet crossed my screen and I copied it down only to later re-look at it and be intrigued…:

"OK. So mathematics is the science of patterns. I can dig that. But what, exactly, is a pattern?"

As someone interested in semantics, words, meaning, and tautology, this struck me as a fair and deeper question than it appeared at first glance… how does one define pattern without falling into some tautological trap? Anyway, that led me to a few pages worth passing along:

First of course, was the always handy Wikipedia, which essentially defined a pattern as 'elements repeating in a predictable manner'… that's probably about as good as it gets, even if it evokes the further questions of what actually constitutes an "element" and when is something precisely corroborated as "predictable"?

Anyway, it turns out that the Mathematics Assoc. of America has also previously tackled this topic a bit in a multi-part series here:

http://mathdl.maa.org/mathDL/46/?pa=content&sa=viewDocument&nodeId=437&bodyId=465

…the theme of the piece is that the notion of math as the 'science of patterns' is actually a modern approach to mathematics (Keith Devlin has certainly been a popular exponent of it) and that early mathematics was quite a different kind of study.

a quick excerpt:

"It is in view of this, I want to consider the often-heard definition of mathematics as the “science of patterns.”   Specifically, I want to show, by comparing Euclid and Steiner, that while this is presented to students as a timeless—that is, non-historical—definition, in fact, it represents a modern view of mathematics.  I shall show that Greek mathematics, for example, is not a search for patterns but for concrete properties of concrete mathematical objects; and I shall show, conversely, that it is when mathematics becomes symbolic that patterns, as such, are suggested to mathematicians and become objects of their thought."

 2) Almost exactly a year ago, renowned mathematician Tim Gowers launched a broadside at (and boycott of) journal publisher Elsevier, and probably, to his own surprise, struck a nerve with a great many other academics (not just mathematicians) who joined the fray with their own pent-up feelings about glossy publishers. I won't replay all the discussion that took place over the matter, except to say that arguments against the stranglehold of expensive professional journals versus more open-access forms of publishing in the digital age continue, and Gowers (with others) remains at the forefront with his announcement of new open-access journals on the way.
This Aperiodical article summarizes what Gowers is up to (with direct links back to Gowers' latest posts on it):

http://aperiodical.com/2013/01/the-good-the-bad-and-gowers/

...and Nature has good coverage of the effort as well, here:

http://www.nature.com/news/mathematicians-aim-to-take-publishers-out-of-publishing-1.12243

Even for those of us not involved in research and its publish-or-perish world, watching this inevitable evolution play out in the open is fascinating.

A Bit of Serendipity

Since the prior post was heavy with some Tim Gowers material I can't help but point out another cosmic synchrony ;-) that materialized on the Web… A short while back Patrick Honner (who I just interviewed recently) posted this note on his blog:

http://mrhonner.com/2012/10/24/this-is-not-a-trig-function/

...basically, he notes that a purported sine wave curve on a NY State math exam simply didn't look right to him at a glance ("too rounded"), and then goes on to show that his visual hunch was indeed correct upon closer inspection. The ensuing comments to his brief post are fascinating as well, but the capper is that about 10 days later, in a bit of serendipity, Tim Gowers writing on Google+ noted his immediate sense that a British air traffic control tower he'd passed by recently did not conform to a "mathematically natural shape," and wondered aloud "how acute this sense that mathematicians have is," at which point he stumbled upon Honner's post:

https://plus.google.com/u/0/103703080789076472131/posts/VhHE8TuJhcm

…it reminds me a wee bit of the old retort (applied when you succeed at something, and someone else says it was 'just luck'), that it seems 'the more I practice the luckier I get!' ...or, in-other-words, it's a skill developed from experience.
Having said that though I s'pose it's a bit difficult to know which is the cart and which is the horse here: i.e., do mathematicians have a special knack for recognizing certain patterns and forms because it's something their minds have experienced repeatedly (practiced)… or, is it that people who have a special or inborn aptitude for form and pattern recognition are more likely to end up in mathematics???


[p.s... if anyone hears the result of Tim Gowers' recent heart surgery and how he is doing, please report to us in the comments; I'm sure many would like to hear that all went well.]

ADDENDUM 11/10/12: Tim now has a detailed post up that he is back home recovering (with some discomfort) from the hopefully successful surgery:
http://gowers.wordpress.com/2012/11/09/what-actually-happened/

Mochizuki, Gowers, Alan November

First, a wonderful article from The Boston Globe emphasizing how difficult it will be to check Shinichi Mochizuki's lengthy claimed proof of the ABC conjecture, because of the complexity and newness of the math involved:

http://tinyurl.com/a8srze4

The piece ends thusly:

"And this, [Minhyong] Kim thinks, might pose the greatest challenge of all. 'When you’ve been wrapped up in your own research program for a long time sometimes you lose a sense of what it is that other people don’t understand,' he says. 'Other people feel quite mystified as to what he’s [Mochizuki] doing and part of him, I suspect, doesn’t quite understand why.'"

On a simpler note, do you wish to engage your own math students… well, so does Tim Gowers...
British Field Medalist Gowers (who deserves to be read/discussed whenever possible) has a piece in The Spectator regarding math education:

http://www.spectator.co.uk/features/8744071/should-alice-marry-bob/

from it: "...rather than explaining mathematical ideas (about statistics, say) and then discussing how they can be applied to the real world, a teacher should instead start with a question that is interesting for non-mathematical reasons and keep a completely open mind about what mathematics has to contribute to the discussion."

Gowers' take is that teachers need to be utilizing real 'real-life' examples in the classroom, not the 'if 2 painters can paint 2 houses in 5 days how many houses can 4 painters paint in 15 days' sort of story problem that often gets passed off as an application.

This is more-or-less a followup to a fantastic earlier piece he did on same subject:

http://gowers.wordpress.com/2012/06/08/how-should-mathematics-be-taught-to-non-mathematicians/

[On a side note, just yesterday, Gowers put up another interesting post about probabilities and surgery he is undertaking for an atrial fibrillation condition… hopefully, we'll soon hear that all went well!]:
http://gowers.wordpress.com/2012/11/05/mathematics-meets-real-life/

Finally, hat tip to John Golden for leading me to this TED video (~16 mins.) by Alan November that piggy-backs nicely onto Tim Gowers' views, in describing student engagement in learning:

…and Now the Joy of Gowers

If/when you have the time, this older post from "Math blog" includes a wonderful hour-long talk (broken into 8-min. segments), entitled "The Importance of Mathematics" from Fields Medalist Tim Gowers... worth a watch:

http://math-blog.com/2008/03/31/on-the-importance-of-mathematics/

Gowers On Math

A recent LOOOONG, rich post from the ever-thoughtful/insightful Tim Gowers on math education here:

http://tinyurl.com/6q6dfln

It's chockfull of LOTS of examples and ideas to chew on and think about (...and debate over). Also, LOTS of comments... don't expect to read it all in one sitting!