Monday, April 10, 2017

Grigory Perelman's "Perfect Rigor" by Masha Gessen

Technical errors in common mathematics and common English from a writer who claims an early love for mathematics, and who is professionally literate in two difficult languages (Russian and English) leave the book suspect. Masha Gressen achieved fame for her success as a journalist. Her specialty is the politics of gender. A strong advocate in Russia for gay rights, she fled (back) to the United States shortly after a personal meeting with Vladimir Putin.  That only makes it harder to understand how she could have put her name on such a weak work as this book. 
Perfect Rigor: A Genius and
the Mathematical Breakthrough
of the Century

by Masha Gessen,
Houghton Mifflin Harcourt, 2009.
Page 135: “Indeed, it is easy to see that on this [spherical] surface, any two straight lines-a straight line being the extension of a segment that connects two points in the shortest possible way-will cross. All straight lines on the apple, or on the Earth, are “great circles” with the centers at the center of the sphere."
Page 135: "Not all of us travel so far all the time, but in the imagination - the very place where mathematics resides - the shortest distance between two points is the trajectory described by an airplane, which generally lies along a geodesic, even if we have never hears the word. These straight lines do not go on forever, but, being circles, inevitably close in on themselves. And, of course, they cross, any two of them."
cal state long beach rodrig geog 140 parallel.jpg
How the parallels of latitude do not contradict that is not clear to me. Indeed, it is possible to draw parallels all over the surface of a sphere. They are concentric circles. They can be any place and of any convenient size. You can start by taking any line on a flat map from Hometown to Smallville and drawing a track parallel to it. Drawing the path on a sphere reveals them to be sections of curves with common origins.  It is easy for me to accept that I am missing some critical piece of common information about spherical geometry. But in that case, this book then has another problem: too little is explained. 
Riemann sphere maps to a plane (Encyclopedia Britannica)
Homeomorphic parallel lines are obvious by inspection.
[13 April 2017 - Still spending time with this ...  After I posted the article, I continued googling and found proofs that the shortest distance between two points on the surface of a sphere is a segment of a great circle.  See Wikipedia here and Wolfram Mathworld here. Driving back and forth to work, I visualized two points on an "arctic circle" and then imagined connecting them with an arc segment of a great circle. Rational proofs are nice, especially in mathematics, but I would like to try it with string and a soccer ball.]

Page 143: "Think about a simple function of the sort you studied in high school. Say, 1/x. A graph of this function would look like a smooth line until it got to the point where x=0. Then things would get crazy because you cannot divide by zero. The line of your graph would suddenly soar toward eternity. This is called a singularity."

First, while colloquial writing is fine for common communication, the expression 1/x is not a function. The proper statement - and it is a statement - is of the form f(x) = 1/x or y = 1/x.
Furthermore, the line would still be "smooth" i.e., continuous all along its path. It would not "suddenly" soar; and you could change the apparent "soar" just by changing the scale of the graph. And, in any case, while half of the lines would rise up or down - and down is diving not soaring - the other halves would creep ever closer to the horizontal positive or negative. Finally, the distinction between eternity and infinity might matter most only to philosophers and theologians, but the difference exists nonetheless.
Graph of f(x) = 1/x 
And no one else seemed to have my difficulties.  I found glowing reviews for this book from the New York Times, the American Mathematical Society, and the Mathematical Association of America. 
  • Grigori Perelman’s Beautiful Mind by Jascha Hoffman: SUNDAY BOOOK REVIEW, Dec. 10, 2009
  • Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century Reviewed by Donal O’Shea NOTICES OF THE AMS VOLUME 58, NUMBER 1
  • MAA REVIEWS [Reviewed by Darren Glass on 01/17/2010]

However, despite all of that, or perhaps because of it, I was motivated to search for “Poincaré’s Conjecture” on YouTube and I found several explanations. The best was by Rendell Heyman but each of them helped in some way.  I first found Heyman in an archived panel discussion of the Poincaré  Conjecture from the World Science Festival. Heyman offers several YouTube channels dedicated to explaining mathematics and some technology and science. His website is here

And the UT Libraries shelve several books on Poincaré s Conjecture.  So, I have some reading to do. 
  • Ricci flow and the Poincaré conjecture  by John W. Morgan
  • The Poincaré conjecture: in search of the shape of the universe by Donal O'Shea.
  • Poincaré's prize : the hundred-year quest to solve one of math's greatest puzzles by George Szpiro

1 comment:

  1. I wonder if she's saying the shortest path lines an aircraft would take cross, not that it's impossible to draw circles that don't cross.