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
zonalandeducation.com 

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 http://www.nytimes.com/2009/12/13/books/review/Hoffman-t.html
• Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century Reviewed by Donal O’Shea NOTICES OF THE AMS VOLUME 58, NUMBER 1 http://www.ams.org/notices/201101/rtx110100056p.pdf
• MAA REVIEWS http://www.maa.org/press/maa-reviews/perfect-rigor-a-genius-the-mathematical-breakthrough-of-the-century [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

PREVIOUSLY ON NECESSARY FACTS

Happy Pi Day of the Century 

Imaginary Numbers are Real. Pegasus is not. 

Turn Left at Orion  

Epigenetics  

IMAGINARY NUMBERS ARE REAL. Pegasus is Not.

Technical language is derived from common language to allow more accurate and precise statements, and more penetrating and perceptive questions. Commonly, when we say “real” we mean “true”, independent of the observer, valid, or verifiable, actual; not imaginary or mythical. Pegasus was not a real animal.  Polyphemus the Cyclops was not real, though Odysseus may well have been. In mathematics, “real” numbers are the set that includes all integers, rational, and irrational numbers. “Imaginary” numbers are those whose square (or other “even” power) is a negative number; in other words, the square roots of negative numbers are imaginary. But imaginary numbers are real in the common sense.

Euler's Equation. As sin^2(x) + cos^2(x) = 1
this can be applied to the analysis of
alternating current electricity. 

Imaginary numbers are important to the design and analysis of alternating current circuits. (See, for instance, the Wikipedia article on Volt-ampere-reactive.)   Without imaginary numbers we could not have complicated electrical power systems, just as we could not have modern commerce without negative numbers.  Overcoming the confusion and ignorance about the reality of numbers has been a historical process.

The Golden Ratio seen in the Parthenon
has many applications.
Geometrically pleasing,
its algebraic expression
(1 + sqrt(5))/2 is irrational

It is said that the followers of Pythagoras killed one of their disciples for revealing that the square root of two is irrational. Apparently, until about 5^th century BCE, the Greeks accepted that every number must be rational, only that not all reductions were known. For example, the Egyptians chose to represent all fractions as sums of fractions with 1 in the numerator: ¾ = ½ + ¼ or 3/7 = 1/3 + 1/14 + 1/42.  

Diophantus of Alexandria (3^rd century CE) denied the reality of negative numbers.  The Nine Chapters on Mathematical Art (Jiu zhang suan-shu) of about the same time accepted their reality. Although early conceptions of zero as a placeholder are known from Babylonian and Egyptian texts, zero was not accepted as a number in the modern sense until about 500 CE.  What is nothing? And in the technical language of metaphysics, nothing is not a different kind of something. “Nothing exists beyond the universe” does not mean that “beyond the universe” is “something else.” The confusion over zero comes from the difference between “nothing” and “none.”  The number 207 has no tens; it does not have metaphysical “nothing” in the tens place. 

But we are not confused by that in daily life.  When Mom asked “What’s going on?” and you replied “Nothing!” she was not thrown into a metaphysical conundrum.

So, too, with imaginary numbers. They have an unfortunate etymology, but we use them every day. If the operators in control rooms of electrical power plants could not manipulate reactive power – expressed in imaginary numbers—with real controls, we would suffer blackouts. 

My motivation here is a post on the Galt’s Gulch Online discussion board.  On April 5, 2016, about 7:00 AM local time, frequent contributor ewv wrote: “Mathematics by itself doesn't describe reality. It is the means by which you relate in terms of concepts what can be measured. Mathematics is a science of method, not about things like physics does.” (Reply here in "What is Science?" here.)

Her keyboading error aside (“physics does” for “physics is”), she is usually a very adept student of Objectivism. As a quip, I once accused her of being Dr. Leonard Peikoff.  Her comment about mathematics was a direct derivation of statements by Ayn Rand in Introduction to the Objectivist Epistemology, as well as elucidations by David Harriman in The Logical Leap. However, mathematics does describe reality, as does any language. 

We can give expression to falsehoods using common language, as when we attempted to deflect Mom’s inquiry about our noisy play. Pegasus and Nike of Samothrace are other examples. They are mathematically impossible. Whatever the wings represent symbolically, they cannot function from the meager muscles on the back of the horse or the girl. Arguments about politics and religion, and Monday morning quarterbacking supply a surfeit of such falsehoods. That silliness is impossible in mathematics.

Patent for application
of the Moebius strip
to a power conveyor.

Can you have a sheet of paper with only one side? Can you have a container with only an inside? The Möbius Strip and the Klein Bottle were inventions of topology, a study in mathematics that contravenes common sense. But they do exist; and they do have practical applications. As an investigation of relationships, topology is based on qualities, not quantities. Topology is nonetheless a study within mathematics. Topology is rigorous and consistent. It does not allow for internal contradictions, just as integer arithmetic does not.  

Mathematics does have unsolved challenges. Science always has frontiers.  However, anything that is proved to be mathematically true must be realizable, even if we have not found one or built one yet.

ALSO ON NECESSARY FACTS

The Man Who Loved Only Numbers 
U.S. Patent Law Does Not Add Up  
Patent Nonsense  
Visualizing Complex Data  
Knowledge Maps