## Sunday, May 1, 2016

### Letter To and From a Math Student

For this morning's Sunday reflection, just a 7-month-old Keith Devlin post that was one of my favorites:

http://devlinsangle.blogspot.com/2015/10/letter-to-calculus-student-sequel.html

## Thursday, April 28, 2016

### The Man Who Almost Defies Comprehension

Ramanujan is much in the news lately with release of the new biopic of his life, "

**":**

*The Man Who Knew Infinity*Colm Mulcahy newly-reviews the film here:

http://blogs.scientificamerican.com/guest-blog/a-math-genius-like-no-other-comes-to-the-big-screen/

...and Stephen Wolfram presents this riveting account of Ramanujan (h/t Steven Strogatz):

http://blog.stephenwolfram.com/2016/04/who-was-ramanujan/

And finally here, Ken Ono speaks briefly about his involvement with the film as a consultant:

If it comes to a theater near you, don't miss it!

## Tuesday, April 26, 2016

### Good Science?

via WikiMediaCommons |

Early in my career I did ~6 years of animal research... but couldn't stick with it as I saw too much poor/sloppy science being done. The variables are so many, so complex, so ill-defined and easily overlooked, as to make it almost surprising how often progress is actually made (or at least perceived).

Here's just one more problem:

**https://www.statnews.com/2016/04/19/lab-mice-temperature/**

*IF*temperature affects the results of experiments with lab mice, what about sounds, sights, lighting, diet, human touch, air circulation, altitude, and on and on and on... of course no one truly knows all the effects on physiology and brain chemistry of sensory inputs (anymore than anyone knows what weather events may be affected halfway around the globe eight months after a butterfly flaps its wings). Cause-and-effect, when it comes to living beings, is nothing if not chaotic.

That's not to be too harsh about such work, but simply to prompt a skeptical stance, especially toward initial, and unverified-or-unreplicated results (let alone the hype of headlines).

I've previously voiced dismay here with those who proclaim themselves "skeptics," yet who largely grant a free pass to weak science and methods published routinely in major journals (luckily, now, decades since I experienced my cynicism, such skepticism is creeping into more mainstream outlets).

There is pseudoscience, speculative science, and real or good science... and the lines blur far more than admitted. Even theoretical physics, revered in my youth, today stands accused from many quarters, of bordering on metaphysics or philosophy, and not true empiricism... I'm not judging it one way or the other, except to say that even such an accusation, from bright people, is telling.

No one said good science should be easy... or common... indeed, it is difficult and rare. Mediocre science is the norm. And wrong-headed science is not uncommon... but is correctable. The (scary) anti-science attitudes/backlash of so many Americans today is a direct result of being sold a naive bill-of-goods and never understanding the true tentative, uncertain nature of science, its strengths and too-often-unacknowledged weaknesses. Still, it remains the best, by far, we've got... and its cornerstone, by the way, is mathematics.

On Twitter I've often used the below graphic (sorry I don't know its origination), but with the suggestion that you can replace the word "success" with the word "science" and it remains true:

Bottom line, good science is incredibly complex at a time when many increasingly gravitate toward simple answers (ala the absurd rise of Donald Trump). Science, misunderstood and misused, can destroy us... yet it is also probably the ONLY thing that can save us... from ourselves.

## Monday, April 25, 2016

### Bongard Problems

Good Alex Bellos column to start the week in

**The Guardian**today on "Bongard problems" (visual comparison puzzles):

**http://tinyurl.com/jeyb7y8**

Further interesting reading on philosophical, cognitive, and AI aspects of Bongard problems:

http://app.ebape.fgv.br/comum/arq/Linhares2.pdf

http://www.foundalis.com/res/diss_research.html

## Sunday, April 24, 2016

### Numbers are Very Real

Sunday reflection:

-- John Conway"It’s quite astonishing and I still don’t understand it, having been a mathematician all my life. How can things be there without actually being there? There’s no doubt that 2 is there or 3 or the square root of omega. They’re very real things. I still don’t know the sense in which mathematical objects exist, but they do. Of course, it’s hard to say in what sense a cat is there, too, but we know it is, very definitely. Cats have a stubborn reality but maybe numbers are stubborner still. You can’t push a cat in a direction it doesn’t want to go. You can’t do it with a number either. I’m only using the word number because you’ll have a vague idea in your head as to what I mean. The objects that a mathematician studies are more abstract than numbers but very real.

"I often think of cats. I think of trees. I think of dogs occasionally but I don’t think of them all that much because dogs are agreeable. They do what you want them to do to some extent. Some people believe that mathematics is what we think it is and it’s created by our thoughts. I don’t. I’m a Platonist at heart, although I know there are very great difficulties in that view."

## Wednesday, April 20, 2016

### Three for Wednesday

1) This morning,

*introduced many of us to "Rayo's Number," "*

**Futility Closet***the largest finite number ever written on an ordinary-sized chalkboard*":

http://www.futilitycloset.com/2016/04/20/the-outer-limits/

2) Ben Orlin delights again with his comically insightful characters trying to define math, because it's "

*crazy hard to define*"':

https://mathwithbaddrawings.com/2016/04/20/math-is-crazy-hard-to-define/

3) and h/t to Julie Rehmeyer for pointing out this more general piece on some of the ills of current science, "

*Big Science Is Broken*." One line toward the end reads as follows: "

*Science, at heart an enterprise for mavericks, has become an enterprise for careerists*":

http://theweek.com/articles/618141/big-science-broken

## Sunday, April 17, 2016

## Friday, April 15, 2016

### Beware of Dating Logicians

YES NO ????

Got a hot date this weekend?

For the men out there (...though I s'pose women could turn the tables and use this as well), two questions to pose to your next date after that romantic dinner:

1) Will you answer THIS question the very SAME way you answer the NEXT question?

and

2) Will you come back to my place and make mad, passionate love to me tonight?

(I adapted this specifically from a fine little volume, "

**Paradoxes: Adventures In the Impossible**" by Gary Hayden and Michael Picard, but have run across versions of it in other places.)

## Wednesday, April 13, 2016

### John Baez Ruminates

John Baez is one of those fellows virtually incapable of writing a dull post!

A couple of recent challenging ones from him; not easy reads, but interesting:

**1)**This one on "

*surprises in logic*" focuses on "

*a complexity barrier built into the very laws of logic*," with some Chaitin, some GÃ¶del, some Kolmogorov, some Kritchman-Raz all rolled in together, before closing out with Joel Hamkins and the "computability of incomputable functions" (also includes lots of good links):

**http://math.ucr.edu/home/baez/surprises.html**

(this is mainly for logicians, but generalists can find bits of interest as well)

**2)**and this one, completely different, from his personal blog ("

**") on "**

*Azimuth**Diamonds and Triamonds*" which leads to mention of one of his "favorite entities," the E

_{8}lattice, and closes out with 5 puzzles:

https://johncarlosbaez.wordpress.com/2016/04/11/diamonds-and-triamonds/

(again a challenging read)

## Monday, April 11, 2016

### Searching For Simplicity

*"Simplify, simplify, simplify."*-- Thoreau ;-)

I'll start with an old puzzle many or most will know:

25 basketball teams compete in a single-elimination tournament -- as soon as you lose one game you're out of the tournament, and with 25 teams one team will get a bye in the first round. How many games total will be played by the end in order to crown the champion?Clearly you can reach the correct answer by patiently working out all possible pairings or games and counting up their number; that will take some time...

*BUT*there's a much easier way that gives the answer literally in seconds. It requires a simple insight that cuts through the process and in 'Aha!' fashion let's you see the answer in a moment. For any who don't know it, I give the answer at the bottom of this post.

******

I mention this puzzle as an intro to the post from "

*" that I'm linking to, to launch the week. It starts with chess, then moves to the Reimann Hypothesis and then P vs. NP, while generally wondering out loud about the "easiest possible kinds" of proof or solutions:*

**GÃ¶del's Last Letter...****https://rjlipton.wordpress.com/2016/04/09/missing-mate-in-ten/**

An interesting read. The gist of it is pondering whether there could be simpler, overlooked solutions to some of the exceedingly difficult problems that appear in mathematics... it's like wondering, as folks did for many years, if Fermat could have actually had a simple proof for his "Last Theorem." In most cases, the answer is likely 'no,' but somewhere out there, who knows? What might our human logic be overlooking, or our limited, biased minds be completely missing? The post includes links to a couple of recent "proofs" offered for the Riemann Hypothesis, one of which is relatively simple (or short) -- though a further commenter, "Gentzen," IDs it is a "fake proof," for reasons that exceed my pay grade :-(

And as long as we're talking about the difficulty of proofs, I'll also point you to this nice, succinct exposition on the Continuum Hypothesis, re-published in

**Nautilus**this weekend (a case where, to this point at least, the only "proof" has been to show that there are

*no*proofs, under current set theory):

http://nautil.us/blog/how-a-hypothesis-can-be-neither-true-nor-false

------------------------------------

******

*the answer is 24, because in a single elimination tournament of 25 teams there must, by the end, be 24 LOSERS (and one champion) -- and if there are 24 losers, there must be 24 games played.*

## Sunday, April 10, 2016

### Of Mathematics and Diamonds

*"No diamond can compete with the clarity of a mathematical sentence. When you successfully prove a new sentence, you hold in your hands a clear fragment of truth. It is an exceedingly rare privilege to devote your adult life to the search for truth. I reflect on this day and night."*

-- mathematician Dr. Nathan Linial (from his 2016 Rothschild Prize acceptance speech)

## Saturday, April 9, 2016

### Talking Politics...

Off-topic today, for another detour into political La-La Land... because, why not!

I don't usually have much luck with political prognostication, but fairly satisfied with my efforts this year thus far....

I wrote here

**that only 3 people had a real chance at the Republican presidential nomination: Cruz, Rand Paul, and Paul Ryan. Then, after 3 debates it was clear that R. Paul didn't have the charisma I'd imagined, and I substituted Kasich, as the main alternative to Cruz, since Ryan hadn't announced... the large contingent I'd expected Rand Paul to draw from was instead going to Donald Trump, who I never took seriously, AND STILL DON'T -- between convention rules, the RNC & establishment Republicans, the IRS, CIA, operatives from Russia, China, Europe (seriously), and a 65+% disapproval rating, I've never granted Trump any real chance at the nomination (more likely he'd be indicted for something than nominated)... nor of completing a campaign if he did somehow get nominated (in fact, he can't hardly afford the pay cut of being Presdent!) -- if he did end up running, he'd lose, and finally if somehow he won the popular vote, the Electoral College simply won't select him... that final arbiter of our democracy the Founding Fathers instituted, now suddenly seeming rather useful and insightful! ...And God help us if I'm wrong!!**

*last June*So, still betting on Cruz, Kasich, or Ryan (who I suspect has been running all along and laughing diabolically at how well his strategy has played out); probably on a 3rd or 4th ballot (though even five wouldn't be a surprise, followed by gunfights in the breezeways). In any event, this has been the most interesting, bizarre primary season of my lifetime... and, I hope I never EVER see another like it.

## Friday, April 8, 2016

### "The Heart of Fermat's Last Theorem"

Another video today... this time a wonderful

**Numberphile**effort to pass along a sense of the proof at "

*The Heart of Fermat's Last Theorem*":

## Wednesday, April 6, 2016

### The Math Behind Beethoven

But first, let me re-try a question for

__Mac__users I posted on Twitter and only got a couple of responses to:

Does anyone have a favorite, freely-downloadable anti-virus/malware program (there are several) for Macs (actually for an older MacBook Pro still on Lion OS X)? How about Mac "tune-up"/cleanup software... favorites?.

...and now to the main feature:

H/T to "

*" blog which passed along this 4-min. YouTube video on music and math (watch it for the math, or for the Beethoven!):*

**3 Quarks Daily**## Monday, April 4, 2016

### 3 To Launch the Week With

Might as well kickstart the week with a couple of pieces from folks I recognized yesterday at

*(NOT that they need any recognition from ME):*

**MathTango****1)**Scott Aaronson has a nice essay (and less technical than much of his stuff) over at

**Aeon**on "

*the great mystery of mathematics*":

https://aeon.co/opinions/the-great-mystery-of-mathematics-is-its-lack-of-mystery

(this was posted a few days ago, though I believe it's a re-post of an earlier essay)

**...And from Keith Devlin this new piece on the history of algebra and the nature of mathematics (in part a response to a certain book and author I shall not mention):**

2)

2)

**http://devlinsangle.blogspot.com/2016/04/algebraic-roots-part-1.html**

**Finally, not math, but a quite fun 13-min. talk from physicist Sean Carroll on his career and how Stephen Hawking figures into it:**

3)

3)

**https://soundcloud.com/the-story-collider/sean-carroll-what-would-stephen-hawking-do**

(again, I think this is much older, but I only came across it today)

Enjoy....

## Sunday, April 3, 2016

### Interconnected Phenomena

The Sunday reflection:

"Science loves to know the direct links between causes and effects, but it does not require us to know that there are such links. Scientists may suspect a correlation between two complex phenomena. The real problem is that humans naturally tend to make connections where there are none, and also tend to ignore connections that are too complex to predict. We see coincidences as events that are mysteriously fated by some deeply significant design. That might be true, and it might not be. In a highly complex world of interconnected phenomena, some connections are so subtly coupled through long chains of indirect links that we can never envision the effect of one on another."

-- Joseph Mazur in "Fluke"

[...and by a stroke of coincidence, according to Google blogspot stats, 2 milestones today: the above marks the

**posting in the history of**

__1500th__*, while over at*

**Math-Frolic***the*

**MathTango****entry has just been posted this morning!]**

__200th__## Thursday, March 31, 2016

### Public Service Announcement...

Just a public service warning today:

*TOMORROW*IS APRIL

__1st__... otherwise known as APRIL FOOL'S DAY...

With that said, my recommendation is to NOT take too seriously

*any*blog entries published tomorrow with the following phrases in the subject line:

1) "the Riemann Hypothesis"

2) "P vs. NP"

3) "squaring the circle"

4) "Goldbach Conjecture"

5) "finally solved"

6) "Elvis Presley"

7) "jet contrails"

8) "A Nigerian Prince"

9) "Your lottery winnings"

10) "Donald Trump"

Otherwise, have a good weekend.

...now I have to get back to writing out an exception I've discovered to the Collatz Conjecture; perhaps I can get it done and posted by tomorrow morning....

## Wednesday, March 30, 2016

### Of Marrieds and Singles...

In case you missed it, there was an Alex Bellos puzzle in

**The Guardian**recently that almost 70% of responding readers missed (and supposedly 80+% of the general public usually miss) -- I was astounded by those numbers, since it doesn't appear terribly difficult, given just a modicum of thought. It ran as follows:

*************************

Jack is looking at Anne, but Anne is looking at George.

Jack is married, but George is not.

Is a married person looking at an unmarried person?

**A)**Yes

**B)**No

**C)**Cannot be determined

*************************

James Grime covers it well here (and he also throws in a second, completely unrelated, problem worth seeing):

After this puzzle began making

*much*easier than the Bellos problem -- a strong indication of how words/language can get in the way of clearly understanding relationships or structure, or simply what is being asked.

.

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*"A coin is flipped 3 times. 1st flip comes up heads and 3rd flip is tails. Is heads ever followed directly by tails?"*

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