This is so utterly ridiculous it has to be passed along: a clown claiming to have trademarked the mathematical symbol pi… this is part of why many Americans come to loathe, or at least distrust, businessmen (and lawyers)… not that this guy is at all typical, but just that his ilk even exists. Read all about it:
Thus far, the story of Malaysian Airline Flight MH 370 seems like one of the greatest mysteries of my lifetime… will they ever find missing MH 370? This story of Bayesian statistics and Air France Flight 447 from 2009 gives some hope:
The Weierstrass function was a remarkable function when introduced, being continuous EVERYwhere, yet differentiable NOwhere... a precursor to fractals. It was presented by Karl Weierstrass in 1872 to the horror of many mathematicians! Many called it Weierstrass's monster and since then many more "monster" mathematical functions have been introduced. R.J. Lipton has a new interesting post up on the monsters and non-monsters that lurk amongst us:
From the piece: "In analysis there are many strange and wonderful functions, for which it seems surprising that they even exist: space-filling curves, discontinuous additive functions, through to differentiable functions that are nowhere monotone... "What we have learned in analysis—and the same lesson applies to many if not all parts of mathematics—is that monsters are often very common."
First, this from Paul Dirac 1939 (quote taken from an earlier post by Joselle Kehoe):
"There is no logical reason why the method of mathematical reasoning should make progress in the study of natural phenomena but one has found in practice that it does work and meets with reasonable success. This must be ascribed to some mathematical quality in Nature, a quality which the casual observer of Nature would not suspect, but which nevertheless plays an important role in Nature’s scheme. . . What makes the theory of relativity so acceptable to physicists in spite of its going against the principle of simplicity is its great mathematical beauty… The theory of relativity introduced mathematical beauty to an unprecedented extent into the description of Nature. . . We now see that we have to change the principle of simplicity into a principle of mathematical beauty. The research worker, in his efforts to express the fundamental laws of Nature in mathematical form, should strive mainly for mathematical beauty."
"If mathematics is, as I tend to see it, the mind itself building structure with the elements of thought, then mathematics’ development is like the development of another sense. It is, after all, the structure we give to any sensory data that creates a meaningful thing perceived. And so it is with mathematics. Mathematics, like story and name, brings meaning, not mechanics, to our experience. It increasingly links what we see to what we think we know, as we try to reconcile our immediate experience with what’s ‘out there.’ The evolution of mathematical concepts allows us to probe deeper and deeper into the universe, as well as into our own nature."
A fun post from Tanya Khovanova last month... if the Langlands Program is a bit too advanced for you than perhaps you can work on "Tanya's Program," uniting jokes and puzzles!:
"The CCSS were created to ensure that all students who graduate from an
American high school do so with the skills and knowledge necessary to
succeed in college, career, and life in the Twenty-First Century,
regardless of where they live." -- K. Devlin
Dr. Keith Devlin addresses the math Common Core debate in his latest posting for Huffington Post (he supports CC):
...just from the first two comments to his piece, I can imagine this is going to generate some lively, effusive discussion!
In all honesty, it's hard for me to see how this whole debate ever ends well, both sides, pro and con, having quite hardened, opposing views. Even if Common Core works well, it might be years, or even a generation, before we are able to fully recognize that. Meanwhile, in the short-term (which seems to be all people focus on these days) there are bound to be difficulties with the implementation of this education overhaul, and both sides are poised to blame each other for whatever travails result.
[None of that, by the way, is meant as a criticism of Keith's piece, which I think is great, but just an acknowledgment, that I don't believe it will be persuasive to the audience it's aimed at: generations of adult parents who have only ever known one way of learning math; or, as Keith writes, "...they were only ever exposed to the algorithmic-skills math instruction
developed for earlier times -- a form of teaching that is hopelessly
inadequate for life in today's world."] I hope I'm wrong, but, from my standpoint, this whole squabble isn't looking pretty heading into the future... :-(
Philosopher/logician Deborah Mayo wrote a week ago about an upcoming conference on "scientism"… the term often loosely used almost as an epithet for the over-reliance on, or over-confidence in, the scientific method. She sees it as especially being tied into the modern reliance on statistics or what she terms "statisticism" or "statistics as window dressing" (and by extrapolation, one can reference the even more recent reverence for "big data"):
"...'getting philosophical' about uncertain inference is not articulating rarified concepts divorced from statistical practice, but providing tools to avoid obfuscating philosophically tinged notions about evidence, induction, testing, and objectivity/subjectivity, while offering a critical illumination of flaws and foibles surrounding technical statistical concepts. To warrant empirical methods of inquiry–both in day-to-day learning or science –- demands assessing and controlling misleading, biased, and erroneous interpretations of data. But such a meta-level scrutiny is itself theory-laden -– only here the theories are philosophical. Understanding and resolving these issues, I argue, calls for interdisciplinary work linking philosophers of science, statistical practitioners, and science journalists."
Following the conference she posted this summary update:
"Truth" arising from an "infinite sequence of lies"….
One of my favorite 2012 math books, "Measurement" by Paul Lockhart, is now out in paperback, so in honor of that, some brief passages to ponder therefrom on this Sunday morning:
"The solution to a math problem is not a number; it's an argument, a proof. We're trying to create these little poems of pure reason. Of course, like any form of poetry, we want our work to be beautiful as well as meaningful. Mathematics is the art of explanation, and consequently, it is difficult, frustrating, and deeply satisfying." "Geometry, then, is not so much about shapes themselves as it is about the verbal patterns that define them. The central problem of geometry is to take these patterns and produce measurements -- numbers which themselves must necessarily be given by verbal patterns."
After explaining calculating the area of a circle from an infinite-sided polygon inscribed within, Lockhart writes (in one of my favorite bits from the entire book):
"Something really serious has just happened here. We have somehow obtained an exact description of the area of a circle using nothing but approximations. The point is that we didn't just make a few good approximations, we made infinitely many. We constructed an infinite sequence of increasingly better approximations, and there was enough of a pattern in those approximations that we could tell where they were heading. In other words, an infinite sequence of lies with a pattern can tell us the truth. It is arguable that this is the single greatest idea the human race has ever had."
And finally, this: "Maybe the bottom line is that I don't have that much to say about the real world. Maybe part of it is that I'm not altogether entirely here a lot of the time. Maybe the point of this book is to give you a glimpse of what it is like to live a mathematical life -- to have the better part of one's mentality off in an imaginary world. At any rate, I know that I am by nature permanently isolated from reality -- my brain is alone, receiving only the (possibly illusory) sensory input that it does -- but mathematical reality is me."
Am currently in process of moving a few miles down the road, and will have limited internet time for another 5-6 days, so not doing much posting/tweeting/whatever -- there will be no linkfest on Friday over at MathTango, though maybe I'll find time to get something up here at Math-Frolic on the weekend?
I'd rather be getting a root canal... it would be quicker and less painful!!
For Sunday morning just a few quotes I've come across recently that I liked:
*****************
"It is magic until you understand it, and it is mathematics thereafter." -- Bharati Krishna Tirthaji
"Good mathematics is not about how many answers you know… it's about how you behave when you don't know." -- Anonymous
"To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature… If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.
-- Richard Feynman
*****************
[Meanwhile, within an hour, MathTango should have posted an interview you will all enjoy... and, also contains a lot of nice bits for Sunday reflection.]
Barry Mazur and William Stein are working on a volume entitled "Primes" to explain the Riemann Hypothesis to a larger audience. A working (very lay-friendly) draft of it is available for free download here!:
Another GREAT followup piece from Robert Talbert about (and in support of) the "flipped classroom." Though I'm not a teacher, I've formed some notions about flipped teaching from reading so much on it along the way, and Robert excellently parallels my own thoughts here:
It's a longish piece with too many good points to try to summarize (but it is largely about patiently working through certain "conflicting" aspects of flipped learning), so try to find time to read it yourself if flipped learning is pertinent to you.
Early on he writes, "my first experience with running a flipped classroom was characterized
by conflict. So was the second experience (same class, subsequent year).
And even today I still get a nontrivial amount of pushback from
students in a flipped classroom setting, usually for the same reasons," and then proceeds to explain how to overcome this.
What I will say is that I relate very closely to the intransigence Robert describes of some students who actually prefer memorization over deeper more abstract learning and discussion (and it's nice to see I wasn't alone in that preference). Young minds are like sponges -- youth is a great time for simply sucking up facts and figures to the mind's delight. I remember well wanting teachers to just give 'me the facts' -- tell me, for example, the Pythagorean theorem and let me apply it, but don't bore me with wearisome details of its proof or how it came about… as a youngster, I viewed proof and explanation as minutiae that just got in the way of learning the real stuff, and I didn't have the time or need for it. Boy, was I wrong!….
h/t to William Wu for pointing out a recent (April) submission to arXiv.org from Mingchun Xu (professor from South-China Normal University), claiming a proof of the Riemann Hypothesis (…in 7 pages no less!):
Such Riemannian proofs show up from time-to-time... my
impression is that they are usually sincere attempts, but dispatched
with in fairly short order. Have seen no commentary or other news about this particular one on the Web, so don't know how seriously it's being taken -- but not really expecting to see $1 million from the Clay Institute change hands anytime soon. Also, don't know anything about the author, nor understand the paper's content, but Gary Davis tweets me that "such weak tools will not prove" the Riemann Hyp.
This all makes me a bit curious to know if there is any consensus as to which RH "proof" from the past might be considered to have been the most promising, or alternatively, which proof withstood the longest scrutiny before succumbing to a discovered flaw? Any ideas?
In other matters, h/t to Colin Beveridge for passing along this 3 7-month-old Jeremy Kun mini-rant on high school math teaching (it again fleshes out the same idea Keith Devlin and others have pushed):
A conjecture both deep and profound Is whether the circle is round. In a paper of Erdös Written in Kurdish A counterexample is found.
-- by a colleague of Paul Erdös
This Sunday I've been thinking about that worshiper of the Supreme Fascist, Paul Erdös…
I often recall Doug Hofstadter's words in tribute to Martin Gardner, noting how peculiar it was that someone who wrote such a large, excellent body of work as Martin, and had such a devoted following, was nonetheless unknown to the vast majority of Americans. Although they were widely different individuals, the same could be said of Paul Erdös, who I suspect is unknown to 99% of Americans. Even if his contributions to mathematics were more average (instead of being the most prolific math paper author in history), he ought be better known for the sheer entertaining eccentricity of his life. A novelist couldn't invent Paul Erdös and have him be a believable figure, so odd/unique a character was he.
Anyway, just a few reflective quotes from Paul for this Sunday morn:
"God may not play dice with the universe, but something strange is going on with the prime numbers."
"Why are numbers beautiful? It's like asking why is Beethoven's Ninth Symphony beautiful. If you don't see why, someone can't tell you. I know numbers are beautiful. If they aren't beautiful, nothing is."
"The purpose of life is to prove and to conjecture."
By the way, the epitaph Erdös purportedly suggested for his grave was, "I've finally stopped getting dumber" -- now, ya gotta love that! [...can't you just imagine that as the new American euphemism for dying... instead of the NY Times reporting that "So-and-So passed away last week at the age of...." it would be "So-and-So stopped getting dumber last week at the age of...."]
For all his brilliance, Erdös was also one of those famous mathematicians who had difficulty with the "Monty Hall puzzle" when it first hit the scene, believing Marilyn vos Savant had blown the answer. It took computer simulations to finally convince him of her answer's correctness.
Erdős was also known for his own idiosyncratic terminology which included the following:
Children were "epsilons"
Women were "bosses"
Men were "slaves"
(...ahhh, so bosses and slaves sometimes had epsilons together)
People who stopped doing mathematics had "died"
Music (other than classical) was "noise"
People who married were "captured"
People who divorced were "liberated"
Giving a mathematics lecture was "preaching"
And, as indicated above, God was "the Supreme Fascist" who retained "the BOOK" of great math proofs (Erdös was actually an agnostic-atheist)
As we celebrate the Centennial of Martin Gardner's birth this year, worth noting that a little over a year ago was the centennial of Erdös' birth (he died in 1996... and no doubt by now has peeked at, and committed to memory, all chapters of the BOOK ;-).
Lastly, here's a 10-min. clip from an Erdös documentary:
[p.s… I tend to just stumble across topics I use here for Sunday "meditations" or "reflections"… But if YOU have a favorite passage, quotation, or subject to share that you think mathematics lovers might enjoy contemplating on a Sunday morning please send it along to me (sheckyr@gmail.com) for consideration, and if used, I'll note the contributor.]
quick ADDENDUM: Fawn Nguyen just passed along to me mention of a second Erdös biography that I was unaware of, but looks great: "My Brain Is Open" by Bruce Schecter (title coming from a famous phrase of Erdös).
There once was a blogger named Shecky A math buff and amateur techie He blogged just for fun 'til his fingers went numb From trying to rhyme Fibonacci. :-(
….okay, sorry, that's pretty lame!
Anyway, feeling a tad silly today, so just some limericks for your delectation:
And if you can't find at least a few amongst all these that appeal to you, visit an orthopedist to have your funny-bone evaluated. (or if you've got a life-long favorite, not included here, offer it up in the comments).
[...finally, if you do insist on math content this morning, then check out the freshly-posted Friday potpourri list over at MathTango.]
