The Man Who Loved Only Numbers

Paul Erdős was easily the most influential mathematician of the 20^th century, and arguably so for all time.  He published 1475 papers almost all in collaboration.  Mathematicians have Erdős Numbers.  Your number is 1 if you co-authored with him, 2 if you co-authored with a co-author, and so on.  Movie star Natalie Portman has an Erdős number of 7.  Danika McKellar’s 4 is lower than her Kevin Bacon number. 

Erdős’s work was beyond prolific. He knew how to offer motivating challenges to people working at all levels of mathematics from his academic peers to children.  In that, Paul Erdős was responsible for hundreds of proven insights that extended the frontiers of number theory. 

The fact that Erdős’s life (1913-1996) intersected so many others allowed Paul Hoffman’s biography  to explore the domain and range of the history of mathematics.  The Greeks, Fibonacci, and pi are here along with Hardy, Ramanujan, and  transfinite numbers, as well a bit of graph theory, and “what’s behind door number two?”

The Man Who Loved Only Numbers by Paul Hoffman (Hyperion, 1998) explains most of the mathematics with integers.  After all, God created the integers and we built the rest – or so it was claimed by Leopold Kronecker (1821-1893) and echoed by Stephen Hawking.  As a result, many of these puzzles could be explained to a child in third through ninth grade.  The fact is, though, that few would be.  The stampede for standardized testing in K-12 education forces teachers to focus on the examinations to the detriment of the true understanding that comes from the artful competence of leisure and play.

Consider Ramsey theory.  Among the pursuits of Frank Plumpton Ramsey (1903-1930) was the question of the smallest possible "universe" that contains some element.  How many ordinary people would you have to fetch at random in order to be guaranteed one of each sex (not gender)?  Three, right?  If you want to plan a party, what is the smallest number of guests that guarantees that three of them must know each other?  Six; no proof is offered, so it must be hard.  (The opposite problem that no three of them will know each other is the same problem, again stated without proof.)  If you wrote out the first so-many integers in any order you wanted, how many would you need to guarantee a run of eleven in a row ascending (or descending)? 101, but 100 might work for special cases. To find a string of length n+1, you must have a universe of n^2 + 1.  Anyone in a first-term computer programming class could write a "Ramsey generator." 

Reading the book while commuting to work on the city busses, I misread one of the problems and worked a different one entirely.  It so happens that any odd positive integer raised to any integer power can always be expressed as the sum of two consecutive integers:  9^2 = 81; 81= 40+41.  7^5 = 16807 = 8403+8404. 

Suggestive as isolated cases may be, in mathematics, we need proof, and the more general the proof, better.  Best of all is a simple proof.  And a proof must reveal not merely that something is true, but why it must be true.  As abstract as mathematics is, when you work with integers these necessary truths became necessary factual truths because party guests and anything else we count are sensible evidentiary empirical objects.

So, one morning, I started with 2n+1, the common form of an odd number.  (2n+1)^2 = 4n^2 + 4n + 1.  That can be written as (2n^2 + 2n) + (2n^2 + 2n + 1), clearly some number and the next higher.  The next night, I did the same for cubes.  The following day, I had to open up a math book to see how to write out the expansion for any power n, an algebraic statement for Pascal’s Triangle.  I was pretty sure that I could complete the proof.  Then I realized that if an odd number can be expressed as 2n+1, that 2n is always some integer that admits to the existence of n and the next number would be one more than that.  More to the point, no matter what power (2n+1) is raised to, the last term of the polynomial will be 1.  You always will be able to find half the number and the integer next to it.  (I called it "Proving Gershon's Theorem" after the Sidney Harris cartoon: "You can't call it Gershon's Equation if everyone has known it for centuries.")

Then I tried it with negative integers.  They only work with odd powers.  And I can prove why.  It is child’s play, really.  But few people ever approach mathematics that way. Paul Erdős did.  In fact, he exhibited neoteny, never having any intimate relationships, being cared for by his mother into his sixties, being unwilling to cook for himself or otherwise look after the simplest daily tasks.  Instead, he was in constant motion, traveling to visit colleagues, imposing on their hospitality, in return for which, he gave them the impetus to publish over 1400 significant new ideas in mathematics.

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India Two Point Oh: Amita and Raj

Both Aristotle and William Graham Sumner pointed out that laws are only expressions of tradition.  To cite the terminology of Thomas Kuhn, only a paradigm shift can change a folkway.  A hundred years ago, Bhagat Singh Thind (a veteran of the U.S. Army in World War I) was denied citizenship on racial grounds: though admittedly a speaker of an “Aryan” language, the U.S. Supreme Court found that he was “Asian” and therefore not qualified to become a citizen, despite his honorable service in the armed forces.  Almost at that very moment, the self-taught Srinivasa Ramanujan was astounding the mathematics faculty at Cambridge.

Therefore, it is fitting beyond coincidence that Navi Rawat portrayed Dr. Amita Ramunjan in the six seasons of Numb3rs (CBS 2005-2010).  Rawat herself is quite interesting, but actresses often are.  More to the point is the character created for her by the writers.

Though nominally an “Indian” it comes out in the show that she was born and raised in California.  She recognizes one word of Tamil but speaks even fewer.  Off-screen Ramanujan goes to her grandmother for insight into their “native” culture.  (Her own native culture is Californian.)  In addition to her helping Charley Epps in his consultations with the FBI, she announced her intention to earn a second doctorate, this one in astrophysics.  (All that and more on Wikipedia about “Amita Ramanujan” here. )

Kumal Nayyar plays  Raj Koothrappali in Big Bang Theory.  Although he is a comic character, they all are.  His quirks and foibles are not qualitatively different or quantitatively less than those of Sheldon Cooper, Leonard Hofstadter, or Howard Wolowitz – or Dr. Amy Farah Fowler, Dr. Bernadette Rostankowski,. Dr. Leslie Winkle, or Penny, the normal foil for all these nerds. Raj earns his achievements. 

Dr. Rajesh Koothrappali is an astrophysicist and faculty member at Caltech's Physics Department. For his discovery of a planetary object beyond the Kuiper belt, 2008 NQ17 (which he calls "Planet Bollywood"), he is included in People magazine's "30 Under 30 to Watch", granting him some form of celebrity status and a larger office for his work. He is also known for a publication on Kuiper belt object size distribution, having run a simulation to correct for the observational efficiency, and was awarded the Newcomb Medal. His expertise even extends to searching for Trojan asteroids at the Earth-Sun L5 Lagrange point. (See the BBT fan wiki here about Raj here.)

 With the possible exception of Ronald Reagan, actors are not their roles.  Nonetheless, the biographies of Navi Rawat and Kumal Nayyar do echo their popular characterizations if only because life and art are two sides of a relationship. What constant or factor, if any, makes the relationship an equation may remain debatable.  The cast of NUMB3RS (CBS 2005-2010) may have been the exception, as Dylan Bruno holds a BS from MIT, and Judd Hirsch’s degree is in physics. Hirsch later appeared in Big Bang Theory as Leonard's father. Also among the Big Bang cast, Mayim Bialik (“Amy Farah Fowler”) really has a doctorate in neuroscience. The exception tests the rule.

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Where All the Children are Above Average: Advocating for the Gifted

If the Flynn Effect is valid, then public education as we have known it must change – and probably changed already. Identifying the problem (if it exists) and then marketing solutions requires the selfsame creative thinking for which the gifted have been identified.  The first challenge is that the definition of “gifted” is highly nuanced and might be intractable.  Creating a collective concept for a wide assortment of very many unusual individual cases might be counterproductive.  Two recent books demonstrate that experts who study gifted children find the results they seek. In other words, when it comes to social prejudices, scientific research is just confirmation bias supporting the attribution fallacy.

Growing Up Gifted (Seventh Edition) by Barbara Clark (Pearson 2008) is basically a textbook for teachers.  It also can serve parents, and perhaps gifteds, as an educated, intelligent mainstream guide.  The book opens with a section of five chapters on understanding gifted education and talent development.  The last two sections present existing educational processes in school and suggest models for creating and maintaining school programs.  Prof. Clark is widely recognized as an advocate for gifted education.  However, she also accepts the basic premises of public education.  Prof. Clark never questions the pseudo-science of Asperger’s Syndrome. Also, if a gifted child is an “underachiever” then the solution is to find ways to motivate them to do more and better of what others have decided is best. 

And that might be true in many cases.  But those are cases, not populations. 

The basic flaw in her thinking is confusing a taxonomy with a remedy.  Ultimately, she has many good ideas to explain her topic, for instance these presented to the Texas Association for the Gifted and Talented.  But who will implement them? Even this association is for the gifted not of the gifted.  I believe that gifted and talented teachers are best able to provide for the exceptional or advanced learner. 

Nerds: Who They Are and Why We Need More of Them by David Anderegg (Jeremy P. Tarcher/Penguin 2007) slays an array of little dragons that plague nerds.  Dr. Anderegg obviously is an advocate. At the same time, though, he also seeks to deconstruct the concept of “nerd.”  Make no mistake: it is a schoolyard insult.  Whether it (and “geek”, etc.) can subsume other meanings is not clear.  So, the book is ambivalent, at once saluting nerds yet also denying that they exist.  Anderegg covers a lot of ground.  The book reads like transcribed lecture notes.  Unlike Clark, he does not buy into the theory of “Asperger’s Syndrome”. 

Citing Richard Hofstadter, Anderegg ties the prejudice against nerds to the broad anti-intellectual streak in American culture perhaps born (if not conceived) in the Jackson-Adams presidential campaign of 1824, Emerson's "American Scholar" address and Irving’s “Legend of Sleepy Hollow.”  Coming into the present, the boy mechanic who works on cars is not a nerd; the one who builds computers is.  The American male is man of action, not a man of thought. 

David Andergg, Ph.D

That reveals the autobiographical theme of this book, if the congruence between the cover and the picture of the author do not suggest this.  Anderegg’s work is about boys.  The Boy Scouts have four citations in the Index; the Cubs Scouts even get one. The Girls Scouts are mentioned once in passing (along with Boy Scouts); the Brownies do not appear. 

Anderegg does offer hope for change in the broader culture.  He cites the television shows NUMB3RS and The Beauty and the Geek. (Launched in 2007, The Big Bang Theory could not appear here.)  He also holds Harry Potter up as a paradigm.  Potter looks the part, of course; but Potter is an athlete.  His prowess at Quidditch is commanding.  (Anderegg ignores Potter’s contests against Voldemort and the forces of darkness.)

On CNN’s “Schools of Thought” blogs for November 14, 2012, Carolyn Coli presented “Ten Myths About Gifted Students and Programs for the Gifted.”  Her Myth #9: It never hurts gifted students to teach others what they already know. She pointed out correctly: “Most gifted students do not know how to tutor others. They often are frustrated that struggling students don’t understand what they perceive as easy.”  Among the replies was this two days later from Andrea Saylor:

“They wanted to test my daughter for "gifted" and I opted out. I didn't want her to be "different" from the other students. She was shy by nature. She tutored otther students and went on to be valedictorian of her graduating class. She went on to get her Masters in Psychology. She now tests students for "gifted" "special needs" "autism" etc. Funny how things turn out.”

That exchange is an example of the complexity of the issue.  Both writers were correct.  The overarching explanation is that we are all individuals and generalizations about us necessarily fail at some level.  With children who are especially talented, this is starkly apparent.  “One size fits all” is not the problem: one size may not even fit two.

As Carolyn Coli pointed out, contrary to myth not all children are gifted.  However, children do learn; and many will rise to an intellectual challenge.

What happens if you give a thousand Motorola Zoom tablet PCs to Ethiopian kids who have never even seen a printed word? Within five months, they'll start teaching themselves English while circumventing the security on your OS to customize settings and activate disabled hardware. Whoa. "Ethiopian kids hack OLPCs in 5 months with zero instruction" on Dv!ce here. 

 That story raises basic questions about the nature of public schooling in America, for gifteds or normals alike. 

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The Simple Origins of Complexities

A New Kind of Science by Stephen Wolfram (Wolfram Media 2002)..

“Three centuries ago science was transformed by the dramatic new idea that rules based on mathematical equations could be used to describe the natural world.  My purpose in this book is to initiate another such transformation, and to introduce a new kind of science that is based on the much more general types of rules that can be embodied in simple computer programs.” 

For example:

1. All squares are white.
2. Start with a black square.
3. A cell is made black when it or either of its neighbors were black on the step before.

Or, for example:

1. All squares are white.
2. Start with a black square.
3. A cell is black if either of its neighbors was black on the step before.
4. A cell is white if both of its neighbors were white on the step before.

 Beginning with rules like these (and hundreds of others approximately no more complex), fantastic images, patterns, random and pseudo-random runs evolve. 

We know that truth is self-supporting.  No truth in biology can contradict a truth in chemistry, physics, psychology, or economics.  If they are, indeed, truths, statements support each other.  Here, too, the simplicity suggested by Stephen Wolfram has been explored by Benoit Mandelbrot.  Fractals of the form

z_n+1 = z_n² + c

produce complex geometries based on reiteration of a basic pattern.  These fractals are related the Julia sets, which are also complicated yet generated from simple initial conditions expressed by terse functions.

Julia set (in white)
for the rational function associated to
Newton's method for ƒ:z→z3−1.
Coloring of Fatou set
according to attractor (the roots of ƒ)
(Wikpedia)

Wolfram is associated with two major websites, Wolfram Science and Wolfram Mathematica.  Both offer you the opportunities to download samples, purchase home editions or professional tools.  Wolfram Mathematics was one of the supporters of the television series NUMB3RS.  Having written a couple of fractal programs for the IBM-PC and DEC VAX many years ago, I had some identification for Wolfram when I saw them go by in the credits for NUMB3RS.  My interest this time was sparked by an off-hand comment from Fred Bartlett, a consulting engineer whom I know from Objectivist discussion boards.

Wolfram Science -

Wolfram Mathematica: Professional, Home, and Student editions.

Stephen Wolfram's personal website.His Wikipedia biography.
Sorry now that I was so close and never knew...

Nerd Nation

The jocks have been shoved aside.  While the Super Bowl has not withered from disinterest,  the fact remains that a significant (if unmeasured) population of  America know themselves to be geeks or nerds --  and are proud of it. 

Big Bang Theory:
Wolowicz, Leonard, Penny, Sheldon and Raj

 I  point to mainstream television programs such as CSI, NCIS, Big Bang Theory, NUMB3RS, and Psych, as well as to the foundation media such as NPR's "Science Friday" (1.3 million listeners), and the long established commercial success of the Discovery Channel and the recently rebranded Syfy ("SciFi") Channel.   It is not just that Big Bang Theory is a prime time television comedy about physicists.  To that must be added Big Blog Theory, the website that offers real physics in support of the comedy.  This is in the instant tradition of the Wolfram Mathematica pages to supplement NUMB3RS when it was on the air, 2005-2010.  On an NCIS episode, laboratory technician Abby Sciuto is waxing on about Bill Nye, the Science Guy.  Her boss, former Marine sniper Jethro Gibbs, does not get the reference.  She goes on.  "Oh," he says, "Just like Mr. Wizard."  She replies with a question mark. 


When Hugo Gernsback launched his science fiction magazines, he intended that they be for everyone.  Still, he insisted that his writers pad their stories with expositions from textbooks, to ensure that the science (however fanciful) was credible.  When John W. Campbell took over, he identified his audience specifically as educated in science, and he courted writers such as Robert Heinlein and Isaac Asimov because of their education. 

At least one good reason to take any science class.

 Fully 28% of Americans claim to have four-year degrees.  In a funny way, the show NUMB3RS reflects that.  While David Krumholtz learned his lines well, his supporting actress, Navi Rawat (Dr. Amita Ramanujan) says that she surfed the web to read popular math to help her delivery.  Beyond that, co-star Judd Hirsch actually majored in physics.  Even closer to the limit, supporting actor Dylan Bruno (Agent Colby Granger) has a bachelor's from MIT in environmental engineering.  Pauley Perrette (Abbey Sciuto of NCIS)'s degree in sociology exposed her to the related study of criminology.

Cellphones and computers have morphed into smartphones and pads.  Clearly, comedy and drama consonant with our culture must reflect that.


Also reflected in that broader culture, is the "CSI Effect" in real courtrooms.  At first, it was a complaint from prosecutors.  Lacking physical evidence, some found it hard to win a conviction.  Or so they claimed.  The problem interested criminologist Gregg Barak of Eastern Michigan University who specializes in the mass media affects on our perceptions of crime.  He enlisted his colleagues Young Kim and Donald Shelton.  They found, broadly, that in fact, the "CSI Effect" is really just a "tech effect."  We expect scientifically valid physical evidence because we live in a technological culture.  Beyond that, however, their numbers showed that jurors with less education demand more evidence.  Jurors with more education tend to take the prosecution's word for it.  Whether this correlates to time spent watching television, or solidarity within social classes is a different question. 

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