Saturday, December 28, 2013

190: Loving Only Numbers

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Before we start, I'd like to thank listeners sblack4, WalkerTxClocker, and LaxRef for some more nice reviews on iTunes.  Thanks guys!

Now, on to today's topic.   During this holiday season in the U.S., many of you are spending time with your families, and taking pity on those who don't have many relatives to visit.     (Or  taking pity on those with too many relatives visiting.)    Sometimes this might bring to mind the strange story of one  mathematician who chose never to marry or have children, instead devoting his entire life to his mathematics:   Paul Erdos, as profiled in Paul Hoffman's famous biography "The Man Who Loved Only Numbers".   Erdos was a Hungarian Jew born in Budapest in 1913.  He originally left his country out of fear of  anti-Semitism soon after receiving his doctorate, and then spent the rest of his life, until his death in 1996, traveling from university to university taking various temporary and guest positions.

There are many surprising and contradictory aspects to Erdos's life.   You would think that someone who chose not to start a family or even to settle in one place would be some kind of social recluse, but Erdos was the opposite.   He considered mathematics to be a social activity, not the domain of isolated geniuses behind closed doors.   His travels were constantly motivated by the desire to collaborate with other mathematicians, where he would help them solve particularly tough problems.   Though he wasn't the leader in any single field of mathematics, never winning the Fields Medal for example, he co-authored about 1525 papers in his lifetime, with 511 different co-authors.   Despite being very odd, and sometimes coming across like a homeless drug addict due to his lack of social graces, he was very popular and well-liked in the mathematical community. 

Due to his large number of co-authors, the concept of an 'Erdos Number' became a common in-joke in the math world.   If you wrote a paper with Paul Erdos, your number was 1.  If you wrote a paper with a co-author of his, your number was 2, and so on.  It is said that nearly every practicing mathematician in the world has an Erdos number of 8 or less.  Incidentally, I found a cool Microsoft site online (linked in the show notes)  to search for collaboration distances between two authors, and found that despite being an engineer rather than a mathematician, I have the fairly respectable Erdos number of 4.   Perhaps the most famous person with a low Erdos number is baseball legend Hank Aaron.   Since he and Erdos once signed the same baseball, when they were both granted honorary degrees on the same day and thus were sitting next to each other when someone requested an autograph, Aaron's Erdos number is said to be 1. 

But as you would expect with someone who constantly travelled and never settled down, Erdos had a rather quirky personality that people who worked with him would need to get used to.   He had his own unique vocabulary, for example.   Children would be referred to as "epsilons", referencing the Greek letter typically used to refer to infinitesmally small quantities.   Women and men were "bosses" and "slaves", while people who got married or divorced were "captured" and "liberated".    Perhaps this reflected an internal attitude that negatively impacted his potential for dating, even if he had ever given a thought to such things.    If someone retired or otherwise left the field of mathematics, Erdos would refer to them as having "died".   At one point he was very sad about the "death" of a teenage protegee, having to clarify to symathetic friends that the cause of death was his discovery of girls.    The United States and Soviet Union were "Sam" and "Joe".   He referred to God as the "Supreme Fascist" or "SF" for short, apparently in protest at being expected to obey the will of divine beings. 

Related to proving mathematical results, Erdos had one piece of private vocabulary that was very important to him.   He always imagined that somewhere up in the heavens, God had a Book in which were listed the most elegant and direct proofs for every conceivable mathematical result.    He wasn't sure if he believed in God, but he definitely believed in the Book.    So if he heard a solution to a problem that he liked, he would always say, "That's one from the Book".   And if he heard a proof that was valid but seemed very awkward or roundabout, he would acknowledge its validity, but still want to search for the one in the Book.   A classic example of a non-Book proof might be Andrew Wiles's famous proof of Fermat's Last Theorem:  while it was fully valid and a work of genius, it took hundreds of pages and is understood by very few people in the world.   Many still hope that a more elegant solution is out there somewhere, waiting to be found.

Despite Erdos's genius and his sociability, there were many aspects of modern life that either baffled him, or were simply considered beneath his notice, and as a result he constantly depended on his many friends to help him get by.   He didn't learn to butter his bread until the age of 21, for example, and always needed help tying his shoes.     If left alone in a public place, he would panic and have a lot of difficulty finding his way back to his university or hotel room.    If he suddenly thought of a solution to a problem he had been working on, he would call his colleagues at any hour of the day or night, with no consideration for whether it might be a convenient time.    He didn't like owning anything, travelling with a single suitcase and requiring his hosts to wash his clothes several times per week.  (It always had to be his hosts doing the washing, since he never bothered to learn how to use a washing machine.)    The last novel Erdos read was in the 1940s, and he did not watch movies since the 1950s.

On the positive side, his lack of concern for money made him quite generous to fellow mathematicians and others in need.   When he won the $50000 Wolf Prize, he used most of the money to establish a scholarship fund in Israel.   He would sometimes give small loans of $1000 to struggling students with strong potential in math, telling them to pay him back whenever they had the money.   At one point he was seen to take pity on a homeless man just after cashing his paycheck:  he took a few dollars out of the envelope to meet his own needs, then handed the rest of the envelope to the stunned beggar.    In addition, Erdos would put out "contracts" on math problems he wanted help solving, ranging from $10 to $3000 depending on his estmates of the difficulty.   Some of his friends pledged to continue honoring the contracts after his death; at the time, it was estimated that he had about $15000 in contracts still outstanding.

Anyway, this short summary just touches on a few of the bizarre personality quirks in the unusual life of Paul Erdos.    If you are as intrigued as I was, be sure to check out Hoffman's biography, "The Man Who Loved Only Numbers".    And may the Supreme Fascist grant you a happy new year.

And this has been your math mutation for today.


Sunday, December 8, 2013

189: Squaring The Circle

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Recently the phrase "squaring the circle" seems to have been popping up more often than usual, perhaps in response to certain problems currently faced by the U.S. government.    I could easily do a very long podcast on some of those topics, but you're probably here to talk about math, not politics.    So let's get back to that phrase, "squaring the circle", which means to solve an impossible problem.   It descends from a problem originating from ancient times:  with no tools except a compass and a straightedge, can we construct a square with the same area as a given circle?   Mathematicians struggled with this problem for thousands of years, until it was definitively proven impossible in 1882.   

Before we discuss why this is impossible, let's review what the problem is.   We start out with a circle of a known radius, drawn on a piece of paper.   For simplicity in this discussion let's assume that the radius is 1, with the radius itself defining our unit of measurement.   We are allowed to use a compass, which lets us draw a circle around any point with a given radius, and a straightedge, which allows us to draw a line connecting any two points.   By the way, no cheating and using markings on the straightedge for measuring distances.   Using just these tools, and without being allowed to do things like create new lines of precise lengths, we want to draw a square on the paper whose area matches that of the circle.    In effect, this means that starting from a circus of radius 1, and thus an area equal to pi,  we need to construct a square with area pi, and thus with each side equal to the square root of pi.

This is one of the most ancient problems known to mathematics, first appearing in some form in the Rhind Papyrus, found in ancient Egypt and dating back to 1650 B.C., though the Egyptians were happy with an approximate solution, treating pi as 256/81 rather than its real irrational value.     The Greeks who followed them, starting with Anaxagoras in the 5th century B.C., had a more sophisticated understanding of mathematics, and were the first to insist on searching for a fully accurate solution rather than an approximation.    Philosopher Thomas Hobbes inaccurately claimed to have squared the circle in the 1600s, leading to an embarrassing public feud between him and mathematician John Wallis.   There were many other discussions and attempts to solve this problem down through the centuries, most amusingly even including two days of frantic scribbling by Abraham Lincoln at one point in the 1850s.    It may have been for the best that he gave up math and returned to politics.

Nowadays, the basic concept of why circle-squaring can't be done is not that hard to grasp, with an elementary knowledge of high-school level math.   Ultimately it stems from the equivalence between geometry and algebra shown by the field of analytic geometry.   View the paper on which you drew the starting circle as a coordinate plane, with every point described by an x and y coordinate.   Recall that both lines and circles are described by simple types of equations in such a plane.   Lines are described by linear equations of the form y = mx + b, and circles are essentially polynomials in the form x^2 + y^2 = r^2.     As a consequence, compass and straightedge constructions are essentially computing a combination of linear and square root functions of your starting lengths.    This means that any multi-step construction is building up a set of these linear and square root operations.

But remember the target we were shooting for:  we want a square of the same area as our starting circle, so each side of our square measures the square root of pi.   However, pi is a transcendental number:   this means you cannot get to this value by solving any set of  polynomial equations with integer coefficients.   Note that this is a stronger condition than being irrational:  many irrationals are non-transcendental.   The square root of 2, for example, is irrational, but can be derived from the simple equation "x^2 = 2."   If a number is transcendental, this implies that no combination of linear and square-root operations, starting with an integer length, could ever reach a value of pi or its square root, and thus our construction of a length pi starting from a circle off radius 1 is impossible.    This last part of the argument, the fact that pi is trancendental, was the hardest part of the proof, and the most recent to be put in place, proven in 1882 by German mathematician Ferdinand von Lindemann.       While we can create constructions for arbitrarily precise approximations, we can never correctly derive a square with the exact same area as a given circle.

We should also point out that this impossibility is implicitly assuming we are talking about ordinary, Euclidean geometry.   If our discussion is including curved spaces, as we discussed in episode 35, all bets are off.    These curved spaces have many properties that upend our usual notions of geometry, such as triangles whose angles sum to more or less than 180 degrees.   If instead of a flat plane we are sitting on the surface of a curved saddle or a sphere, then we can essentially get a 'free' pi by drawing a line that becomes a curve due to the space's curvature, eliminating a key component of the impossibility proof.     I think we can all agree, though, that the ancient Greeks who posed the problem would consider this type of solution to be cheating.

What is most amusing about squaring the circles is that even after its impossibility was very solidly proven in 1882, enthusiastic amateurs kept on trying to "solve" the impossible problem for many years after.   This has some analogues in government as well...  but I'll try again to keep off that tangent.     You may recall that back in episode 20, I talked about an 1897 attempt by a confused Indiana resident to legislate that pi was really equal to 3.2-- this was partially based on his supposedly successful attempts to square a circle using this value.    Before you laugh too hard at him or the legislators who actually listened to him, we should note that somehow he got his bogus circle-squaring method published in the American Mathematical Monthly, which will forever be an embarrassment to that publication.    In 1911, British professor E.W. Hobson wrote, "Every Scientific Society still receives from time to time communications from the circle squarer and the trisector of angles, who often make amusing attempts to disguise the real character of their essays...  The statement is not infrequently accompanied with directions as to the forwarding of any prize of which the writer may be found worthy by the University or Scientific Society addressed, and usually indicates no lack of confidence that the bestowal of such a prize has been amply deserved as the fit reward for the final solution of a problem which has baffled the efforts of a great multitude of predecessors in all ages. "       The circle squaring craze seems to have mostly died off in the latter part of the 20th century, but a 2003 article by NPR's "Math Guy" Keith Devlin mentioned that he still regularly received crackpot letters from circle-squarers.   And even today, if you look up squaring the circle on Yahoo Answers, you'll find a post by some misguided soul who refuses to believe that the problem is unsolvable, stating that " Its possible we lack the mathematical and conceptual understanding to construct it."   I guess anything is possible, but if you trust modern mathematics at all, it's pretty clear that the circle will never be squared. 

And this has been your math mutation for today.