Algebra II--dead?

"Nicholson Baker Argues that Algebra II Shouldn’t Be a Required Course"

August 19th, 2013

Harper’s Magazine

Young people, rejoice, you have a friend in Nicholson Baker (though we do recommend you wait a couple more years before reading his novels). Baker feels your pain — the pain of Algebra II, which, he argues in the September issue of Harper’s Magazine, should be kept out of the Common Core. Students, he writes, “are forced, repeatedly, to stare at hairy, square-rooted, polynomialed horseradish clumps of mute symbology that irritates them, that stop them in their tracks, that they can’t understand.”

Baker calls the course textbook, Algebra 2 Common Core, “a highly efficient engine for the creation of math rage: a dead scrap heap of repellent terminology, a collection of spiky, decontextualized, multistep mathematical black-box techniques that you must practice over and over and get by heart in order to be ready to do something interesting later on, when the time comes.”

He speaks with many who agree with him, even people who are proficient with algorithms. “I’m a math guy, it’s not like I’m some fuzzy-headed humanist,” education reformer Grant Wiggins tells Baker. “You don’t need algebra for the majority of jobs. You need it for the burgeoning field of high-tech, but that’s not all the jobs. I just don’t get it.”

“Good heavens, no,” number theorist Underwood Dudley replies when asked by Baker if Algebra II should be required of all high schoolers. “Forcing people to take mathematics is just terrible. We shouldn’t do it.” Dudley believes that a silent majority of math teachers share this opinion. Steven Strogatz, a mathematician at Cornell, calls for the amount of math to be diminished, and for what is taught to be made more meaningful for the average child. “As someone who is working on the front lines,” he tells Baker, “it’s alarming to me, and discouraging that year after year I see such a large proportion of people really not learning anything — and just suffering while they’re doing it. We spend a lot of time avalanching students with answers to things that they wouldn’t think of asking.”

Baker agrees that less is more. He proposes “a new, one-year teaser course for ninth graders, which would briefly cover a few techniques of algebraic manipulation, some mind stretching geometric proofs, some nifty things about parabolas and conic sections, and even perhaps a soft-core hint of the infinitesimal, change-explaining powers of calculus. Throw in some scatter plots and data analysis, a touch of mathematical logic, and several representative topics in math history and math appreciation.”

Apparently nixing Algebra II is a touchy subject. Dudley warns Baker that he will get in trouble for writing about it. “The entire math department at the University of Tennessee stopped speaking to me,” said Michael Smith of the response to a book he wrote questioning the practical necessity of higher math in schools. “You’ve got a very tough subject to tackle,” says Michael Wiener, who wrote a book about the national obsession with college-prep math courses. “I feel sorry for you. It’s like quicksand. The more you get into this, the more you’ll sink.”

Maria Gaetana Agnesi...mathematician

The Writer's Almanac...

It is the birthday of one of the first well-known female mathematicians of the Western world. Maria Gaetana Agnesi was born in Milan (1718). Her father, Pietro, was a wealthy businessman and her mother, Anna Fortunata Brivio, was an aristocrat whom her father married to raise his status in Milan society.

Maria was a brilliant child. By age five, she spoke French as well as her native Italian. A few years later, she was fluent in Latin, Greek, and Hebrew, and her family called her the "Walking Polyglot." At age nine, she addressed a group of academics in Latin on the subject of women's rights and access to education, and soon she was leading complex philosophical discussions between her father and his scholarly friends. She also began to pursue mathematics.

Maria was shy and devout, and she longed to give up her public speaking and enter a convent. Her religious aspirations were dashed, however, when her mother died and she was left in charge of the household and the care of her many siblings.

She maintained her interest in math and philosophy. In 1738, she published Propositiones Philosophicae, a collection of essays based on the talks she gave to her father's circle of friends. That same year, she began working on a math textbook that she could use to teach math to her siblings. But the book grew into more than just a teaching tool. In it she wrote an equation for a specific bell-shaped curve that is still used today and is known — because of mistranslation of the Italian by a British mathematician — as the "Witch of Agnesi." Analytical Institutions, which was published in 1748, was highly regarded in academic circles for synthesizing complex mathematical ideas with clarity and precision.

Analytical Institutions and the articulation of the Witch of Agnesi earned her a spot in the Bologna Academy of Sciences. But by that time, she had abandoned mathematics and devoted herself to charity work. When asked a decade later what she thought of recent developments in calculus, she said she was "no longer concerned with such interests." She was eventually appointed director of a home for ill and infirm women, and she spent the rest of her life caring for the dying until her own death in 1799.

Maria Gaetana Agnesi [Wikipedia]

THE WITCH OF AGNESI


Hypatia

Science vs religion...Hypatia of Alexandria [science] lost

Hmmmmmmmmmm...better than sex?

Deceased--William P. Thurston

William P. Thurston
October 30th, 1946 to August 21st, 2012

"William P. Thurston, Theoretical Mathematician, Dies at 65"

by

Leslie Kaufman

August 22nd, 2012

The New York Times

William P. Thurston, a mathematician who revolutionized understanding of the structure of three-dimensional spaces and won the Fields Medal, often described as the equivalent of the Nobel Prize for mathematics, died on Tuesday in Rochester. He was 65.

The cause was cancer, his son Dylan said.

Dr. Thurston’s fields of expertise were geometry and topology, the study of different possible shapes for multidimensional space.

Perhaps his greatest accomplishment in a lifetime of breakthroughs was his Geometrization conjecture, which postulated that all possible three-dimensional spaces are made up of eight types of geometric pieces, a discovery he likened to finding eight outfits that could fit anybody in the world.

For most of his professional life, Dr. Thurston was among a very rarefied group in his field that thinks deep theoretical thoughts with no particular practical application, a luxury he reveled in.

“I don’t do it for the bottom line,” Dr. Thurston told The Wall Street Journal in 1983. “The inner force that drives mathematicians isn’t to look for applications; it is to understand the structure and inner beauty of mathematics.”

John Milnor, co-director of the Institute for Mathematical Sciences at Stony Brook University on Long Island, acknowledged that Dr. Thurston delighted in working in a very esoteric realm. But he added that Dr. Thurston’s work had made “a tremendous difference in the way we look at many problems.”

Without that work, a Russian mathematician, Grisha Perelman, would not have been able in 2003 to solve the Poincaré conjecture, which asserts that the sphere is the only three-dimensional shape in which every loop in its structure can be shrunk to a single point, without ripping or tearing either the loop or the space. The problem had challenged mathematicians for 100 years.

In addition, cosmologists have drawn on Dr. Thurston’s discoveries in their search for the shape of the universe.

On a more unlikely note, his musings about the possible shapes of the universe inspired the designer Issey Miyake’s 2010 ready-to-wear collection, a colorful series of draped and asymmetrical forms. The fashion Web site Style.com reported that after the show, the house’s designer and Dr. Thurston “wrapped themselves for the press in a long stretch of red tubing to make the point that something that looks random is actually (according to Thurston) ‘beautiful geometry.’ ”

To his colleagues, Dr. Thurston’s most unusual gift was his ability to visualize complex shapes and problems. They said he loved nothing more than to sit in a common room and help colleagues or students brainstorm on solutions to vexing issues. “He could look at a problem and see simplicity where nobody else could find it,” said Jeff Weeks, a mathematician who studied with Dr. Thurston at Princeton.

For example, he could think in multiple dimensions. “People don’t understand how I can visualize four or five dimensions,” Dr. Thurston told The Journal. “Five-dimensional shapes are hard to visualize — but it doesn’t mean you can’t think about them. Thinking is really the same as seeing.”

William Paul Thurston was born on Oct. 30, 1946, in Washington, to Paul and Margaret Thurston. His father was a Naval engineer. William showed aptitude in math early on, amazing preschool teachers with his ability to add two- and three-digit numbers in his head, his older brother, Robert, recalled. Asked how he did it, he replied that he “counted on his fingers in his mind,” Robert Thurston said.

Dr. Thurston received an undergraduate degree at New College in Florida and a Ph.D. in mathematics at the University of California, Berkeley, in 1972. He quickly earned a reputation as an original thinker and caught the attention of Princeton’s renowned mathematics department, which recruited him to be a professor at the age of 27.

He was in his mid-30s when he received the Fields Medal in 1982 for his work in deepening the connection between geometry and topology. (The medal is awarded every four years by the International Mathematical Union.) He worked longest at Princeton but also held posts at Berkeley and the University of California, Davis. He was most recently at Cornell.

Dr. Thurston’s first marriage, to Rachel Findley, ended in divorce. In addition to his son Dylan and his brother Robert, he is survived by his wife, Julian Muriel Thurston; their children Hannah Jade and Liam; two children from his first marriage, Nathaniel and Emily; his mother, Margaret; a sister, Jean Baker; a brother, George; and two grandchildren.

Dylan Thurston, also a mathematician, said that despite working in a realm of rather cold abstractions, his father was personally very warm.

“Growing up there were many beautiful mathematical pictures in the house,” he said. “He was a very visual thinker; he had powers to see spaces that no one before him could, and he was always drawing pictures of what he could see and doodles in notebooks, and we would talk about it.

“Math was always very fun for him.”

School of Mathematics and Statistics University of St Andrews, Scotland...

Bill Thurston studied at New College, Sarasota, Florida. He received his B.S. from there in 1967 and moved to the University of California at Berkeley to undertake research under Morris Hirsch's and Stephen Smale 's supervision. He was awarded his doctorate in 1972 for a thesis entitled Foliations of 3- manifolds which are circle bundles. This work showed the existence of compact leaves in foliations of 3-dimensional manifolds.

After completing his Ph.D., Thurston spent the academic year 1972-73 at the Institute for Advanced Study at Princeton. Then, in 1973, he was appointed an assistant professor of mathematics at Massachusetts Institute of Technology. In 1974 he was appointed professor of mathematics at Princeton University.

Throughout this period Thurston worked on foliations. Lawson ( ) sums up this work:

    It is evident that Thurston's contributions to the field of foliations are of considerable depth. However, what sets them apart is their marvellous originality. This is also true of his subsequent work on Teichmüller space and the theory of 3-manifolds.

In Wall describes Thurston's contributions which led to him being awarded a Fields Medal in 1982. In fact the1982 Fields Medals were announced at a meeting of the General Assembly of the International Mathematical Union in Warsaw in early August 1982. They were not presented until the International Congress in Warsaw which could not be held in 1982 as scheduled and was delayed until the following year. Lectures on the work of Thurston which led to his receiving the Medal were made at the 1983 International Congress. Wall , giving that address, said:

    Thurston has fantastic geometric insight and vision: his ideas have completely revolutionised the study of topology in 2 and 3 dimensions, and brought about a new and fruitful interplay between analysis, topology and geometry.

Wall goes on to describe Thurston's work in more detail:

    The central new idea is that a very large class of closed 3-manifolds should carry a hyperbolic structure - be the quotient of hyperbolic space by a discrete group of isometries, or equivalently, carry a metric of constant negative curvature. Although this is a natural analogue of the situation for 2-manifolds, where such a result is given by Riemann 's uniformisation theorem, it is much less plausible - even counter-intuitive - in the 3-dimensional situation.

Kleinian groups, which are discrete isometry groups of hyperbolic 3-space, were first studied by Poincaré and a fundamental finiteness theorem was proved by Ahlfors . Thurston's work on Kleinian groups yielded many new results and established a well known conjecture. Sullivan describes this geometrical work in , giving the following summary:

    Thurston's results are surprising and beautiful. The method is a new level of geometrical analysis - in the sense of powerful geometrical estimation on the one hand, and spatial visualisation and imagination on the other, which are truly remarkable.

Thurston's work is summarised by Wall :

    Thurston's work has had an enormous influence on 3-dimensional topology. This area has a strong tradition of 'bare hands' techniques and relatively little interaction with other subjects. Direct arguments remain essential, but 3-dimensional topology has now firmly rejoined the main stream of mathematics.

Thurston has received many honours in addition to the Fields Medal. He held a Alfred P Sloan Foundation Fellowship in 1974-75. In 1976 his work on foliations led to his being awarded the Oswald Veblen Geometry Prize of the American Mathematical Society . In 1979 he was awarded the Alan T Waterman Award, being the second mathematician to receive such an award (the first being Fefferman in 1976). 

 William Thurston [Wikipedia]

Clay Mathematics Institute

Clay Mathematics Institute [Wikipedia]

The world...merely a roll of the dice?

"Interpretations of Probability"

by

Jason Rosenhouse

October 11th, 2011

scienceblog

Here's Timothy Gowers, a Fields Medalist, from his book Mathematics: A Very Short Intorduction:

However, there certainly are philosophers who take seriously the question of whether numbers exist, and this distinguishes them from mathematicians, who either find it obvious that numbers exist or do not understand what is being asked.

Everyone knows there is friction between scientists and philosophers of science. Richard Feynman spoke for many scientists when he quipped that, “Philosophy of science is as useful to scientists as ornithology is to birds.” From the other side, it is not uncommon for philosophers to lament the philosophical naivete of scientists (for example, in this recent book review.)

I am not aware of any similar tension between mathematicians and philosophers of mathematics, for the simple reason that I do not know any mathematicians who take any interest at all in the philosophy of their discipline. Perhaps this reflects badly on us as a community, but it is what it is. In my own case, every once in a while I get motivated to dip my toe into the philosophical literature, but it's rare that I find myself enriched by the experience.

There have been exceptions, however. While writing the BMHB (that's The Big Monty Hall Book) I found myself moved to read some of the literature about Interpretations of Probability. The reason was that in writing the book's early chapters I found myself very casually making use of three different approaches to probability. In discussing the most elementary methods for solving the problem I used the classical interpretation, in which probabilities record the ratio of favorable outcomes to possible outcomes, assuming the possibilities are equiprobable. Later I discussed the use of Monte Carlo simulations to determine the correctness of our abstract reasoning, and this suggested a frequentist approach to probability. In this view a probability is something you measure from the data produced by multiple trials of some experiment. Later still I discussed matters from the perspective of a contestant actually playing the game. In this context it was convenient to take a Bayesian view of probability, in which a probability statement just records a person's subjective degree of belief in some proposition.

The literature I found about interpreting probability was fascinating, and I certainly found plenty of food for thought. But for all of that I'm still not really sure what people are doing when they speak of interpreting probability. Probability theory is an abstract construction no different from anything else mathematicians study. No one talks about interpreting a perfect circle; instead we ask whether the idea of a perfect circle is useful in a given context. Frankly, as a pure mathematician I say that if you run into philosophical difficulties when applying the theory to a real-world situation, that just serves you right for trying to apply it to anything.

More seriously, the most important criterion for assessing any particular model of probability must surely be usefulness. That my Monty Hall experience led so naturally to three different interpretations suggests that no one interpretation can capture everything we have in mind when we use probability language. For that reason I tend to favor an ecumenical approach to probability: If your interpretation is helpful and leads to correct conclusions, then you just go right ahead and stick with it. The existence of other situations where your interpretation does not work so well is neither here nor there. Why should we even expect one interpretation to cover every facet of probability?

In perusing some of the literature on interpretations of probability, I noticed a bit of a cultural difference between defenders of rival schools of thought. In particular, Bayesians, to a greater degree than their rivals, really really care about this. They also tend to be a bit contemptuous of other approaches, especially the poor frequentists, who they regard with great pity. A case in point is this post by Ian Pollock, over at Rationally Speaking. He writes:

Stop me if you've heard this before: suppose I flip a coin, right now. I am not giving you any other information. What odds (or probability, if you prefer) do you assign that it will come up heads?

If you would happily say “Even” or “1 to 1” or “Fifty-fifty” or “probability 50%” -- and you're clear on WHY you would say this -- then this post is not aimed at you, although it may pleasantly confirm your preexisting opinions as a Bayesian on probability. Bayesians, broadly, consider probability to be a measure of their state of knowledge about some proposition, so that different people with different knowledge may correctly quote different probabilities for the same proposition.

If you would say something along the lines of “The question is meaningless; probability only has meaning as the many-trials limit of frequency in a random experiment,” or perhaps “50%, but only given that a fair coin and fair flipping procedure is being used,” this post is aimed at you. I intend to try to talk you out of your Frequentist view; the view that probability exists out there and is an objective property of certain physical systems, which we humans, merely fallibly, measure.

My broader aim is therefore to argue that “chance” is always and everywhere subjective -- a result of the limitations of minds -- rather than objective in the sense of actually existing in the outside world.

It's hard to see how this could be true. It is simply a fact that a great many physical systems produce outcomes with broadly predictable relative frequencies. A fair coin flipped in a fair way really does land heads about half the time and tails about half the time. The ball in an honest roulette wheel finds each number roughly one thirty-eighth of the time. Those are objective properties of those systems, and it seems perfectly reasonable to use probability language to discuss those objective properties.

So let's see what Pollock has in mind:

The canonical example from every textbook is a coin flip that uses a fair coin and has a fair flipping procedure. “Fair coin” means, in effect, that the coin is not weighted or tampered with in such a way as to make it tend to land, say, tails. In this particular case, we can say a coin is fair if it is approximately cylindrical and has approximately uniform density. ??How about a fair flipping procedure? Well, suppose that I were to flip a coin such that it made only one rotation, then landed in my hand again. That would be an unfair flipping procedure. A fair flipping procedure is not like that, in the sense that it's ... unpredictable? Sure, let's go with that. (Feel free to try to formalize that idea in a non question-begging way, if you wish.)

I don't know what level of description Pollock wants here. If he would care to come to my office, I will simply show him what I mean by a fair flipping procedure. But he knows what I would show him, since it's the same procedure everyone uses when they are not deliberately trying to cheat someone. The case of a roulette wheel is perhaps even clearer. By a fair procedure I mean, “The way it's done in your classier casinos, you know, with the ball going in one direction and the wheel going in the other.”

Let's move on:

Given these conditions, frequentists are usually comfortable talking about the probability of heads as being synonymous with the long-run frequency of heads, or sometimes the limit, as the number of trials approaches infinity, of the ratio of trials that come up heads to all trials. They are definitely not comfortable with talking about the probability of a single event -- for example, the probability that Eugene will be late for work today. Will Feller said: “There is no place in our system for speculations concerning the probability that the sun will rise tomorrow. Before speaking of it we should have to agree on an (idealized) model which would presumably run along the lines 'out of infinitely many worlds one is selected at random...' Little imagination is required to construct such a model, but it appears both uninteresting and meaningless.”

The first, rather practical problem with this is that it excludes altogether many interesting questions to which the word “probability” would seem prima facie to apply. For example, I might wish to know the likelihood of a certain accident's occurrance in an industrial process -- an accident that has not occurred before. It seems that we are asking a real question when we ask how likely this is, and it seems we can reason about this likelihood mathematically. Why refuse to countenance that as a question of probability?

As it happens, I am among those who are uncomfortable with applying probability language to one-off situations. It's fine to speak informally about the likelihood (or odds, or probability) of a one-off event, but if the idea is to assign actual numbers to events and then apply the formal theory of probability to them, then I no longer understand what you are doing. It's unclear to me what it means to say, “Given the information I have I believe the probability of this one-off event is one-third,” unless we can view the event as one among a long sequence of trials.

Let's consider Pollack's examples. Informally I might say that, given what I know about Eugene, it's highly likely that he will be late to work today. But it's hard to imagine what it would mean to assign an actual number to the probability that Eugene will be late, unless we have long experience with Eugene's habits on days that are comparable to this one. Likewise, I could make an informal assessment of how likely it is that an industrial accident will occur, but I don't know how to assign an actual number to the probability of it occurring. Of course, we might look at a specific mechanical part used in the industrial process and say something like, “This part has been used in tens of thousands of industrial processes and empirically it fails roughly one time in five thousand...” Now I know what we're talking about! But if we're truly talking about a one-off event that is completely divorced from any possible long sequence of trials, then I just don't know what it means to assign a probability to its occurrence.

Moving on:

The second, much deeper problem is as follows (going back to coin flipping as an example): the fairness (i.e., unpredictability) of the flipping procedure is subjective -- it depends on the state of knowledge of the person assigning probabilities. Some magicians, for example, are able to exert pretty good control over the outcome of a coin toss with a fairly large number of rotations, if they so choose. Let us suppose, for the sake of argument, that the substance of their trick has something to do with whether the coin starts out heads or tails before the flip. If so, then somebody who knows the magicians' trick may be able to predict the outcome of a coin flip I am performing with decent accuracy -- perhaps not 100%, but maybe 55 or 60%. Suppose that a person versed in such tricks is watching me perform what I think is a fair flipping procedure. That person actually knows, with better than chance accuracy, the outcome of each flip. Is it still a “fair flipping procedure?”

I'm afraid I don't see the problem. I certainly agree that a skillful magician can fool me into thinking he is using a fair procedure when he really isn't. The fact remains that there are flipping procedures that produce stable relative frequencies of heads and tails. If I know you are using one of those, then I can make an objective statement about what will happen in a long-run of trials.

You might retort that I can never really know what procedure you're using, and that is where the subjectivity comes in. But that same argument could be used against any claim to objective knowledge. It's hardly a weakness unique to probability. Any fact you assert is inevitably based on a pile of assumptions about how the world is, and a determined skeptic could challenge you on any of those assumptions. But if we're ever comfortable talking about objective knowledge, then I don't see why, “A fair coin flipped in a fair way will land heads roughly half the time in a long sequence of trials,” should not be considered objective.

So it breaks down like this: It is an objective fact that certain physical systems produce outcomes with broadly stable relative frequencies. Probability theory is very useful for understanding such situations. Plainly, then, there is an objective aspect to probability. In practice I can be mistaken about certain facts that are relevant to making correct probability assignments. Thus, there is also a subjective aspect to probability. That is why, depending on the situation, it might be useful to think of probability in terms of the objective properties of physical systems, or in terms of our subjective knowledge of what is taking place.

This problem is made even clearer by indulging in a little bit of thought experimentation. In principle, no matter how complicated I make the flipping procedure, a godlike Laplacian Calculator who sees every particle in the universe and can compute their past, present and future trajectories will always be able to predict the outcome of every coin flip with probability ~1. To such an entity, a “fair flipping procedure” is ridiculous -- just compute the trajectories and you know the outcome!

Generalizing away from the coin flipping example, we can see that so-called “random experiments” are always less random for some agents than for others (and at a bare minimum, they are not random at all for the Laplacian Calculator), which undermines the supposedly objective basis of frequentism.

I disagree. That a godlike Laplacian Calculator can perfectly predict the outcome of any coin toss has no relevance at all to the objective basis of frequentism. The thing that's objective is the stable long-run frequency, not the outcome of any one toss. Our godlike Calculator will presumably predict that heads will occur half the time in a long sequence of trials.

Pollack goes on to discuss quantum mechanics and chaos theory, but I won't discuss that part of his post.

The three interpretations of probability I have mentioned are clearly related to one another. The classical interpretation defines probability without any reference to long runs of trials, but the ratio you compute is understood to represent a prediction about what will happen in the long run. And Bayesians don't think that long run data is irrelevant to probabilistic reasoning. They just treat that data as new information they use to update a prior probability distribution. And no one would deny that our judgements about how likely things are to happen in the future depends on the information we have in the present.

Given that different interpretations are plainly useful in different contexts, I don't understand the mania for trying to squeeze everything about probability into just one interpretation. You have lost something important by declaring that any probability assignment is purely subjective. Let's not forget that probability was invented in the context of games of chance, and in that context it developed models that permit fairly detailed predictions about long-run frequencies. That I can never be absolutely certain, in a given situation, that my model applies does not imply that all probability statements are purely subjective.

[Jason Rosenhouse received his PhD in mathematics from Dartmouth College in 2000. He subsequently spent three years as a post-doc at Kansas State University. Observing the machinations of the Kansas Board of Education led to his unhealthy obsession with issues related to evolution and creationism. Currently he is an Associate Professor of Mathematics at James Madison University, in Harrisonburg, VA.]

Comic book and logical certainty

Logicomix: An Epic Search for Truth

by

Apostolos Doxiadis and Christos H. Papadimitriou

ISBN-10: 0747597200
ISBN-13: 978-0747597209

"Algorithm and Blues"

by

Jim Holt

September 27th, 2009

New York Times

Well, this is unexpected — a comic book about the quest for logical certainty in mathematics. The story spans the decades from the late 19th century to World War II, a period when the nature of mathematical truth was being furiously debated. The stellar cast, headed up by Bertrand Russell, includes the greatest philosophers, logicians and mathematicians of the era, along with sundry wives and mistresses, plus a couple of homicidal maniacs, an apocryphal barber and Adolf Hitler.

Improbable material for comic-book treatment? Not really. The principals in this intellectual drama are superheroes of a sort. They go up against a powerful nemesis, who might be called Dark Antinomy. Each is haunted by an inner demon, the Specter of Madness. Their quest has a tragic arc, not unlike that of Superman or Donald Duck.

So, at least, the creators of “Logicomix” would have us believe. First published last year in Greece (where it became a surprise best seller), the comic book — er, graphic novel? — is the brainchild of Apostolos Doxiadis, previously the author of a not-bad mathematical fiction called “Uncle Petros and Goldbach’s Conjecture.” For expert assistance on logic, Doxiadis called on his friend Christos Papadimitriou, a professor of computer science at Berkeley and the author of a novel about Alan Turing. The art was done by Alecos Papadatos (drawings) and Annie Di Donna (color).

All four collaborators pop up in interludes throughout the book. (Doxiadis, evidently a handsome fellow, is drawn to look rather like Robert Goulet.) We see them chatting in the artists’ studio or strolling around contemporary Athens, accompanied by an adorable dog called Manga (Greek slang for “cool dude,” not a reference to Japanese comics). They argue about the developing ­logic-and-­madness theme and fret over whether there’s too much or too little technical stuff for the average reader. It’s almost as if they want to pre-empt the stern judgment of the reviewer. Fat chance.

The story proper opens on Sept. 4, 1939, three days after the Nazi invasion of Poland. Bertrand Russell is giving a public lecture at an American university on the role of logic in human affairs. Angry isolationists in the audience challenge Russell to explain how logic could justify participating in a world war. Ah, he responds, but what is logic?

In a series of flashbacks, Russell recounts his epic struggle with that question. We see him first as a little boy, in the 1870s, being brought up by his grandparents after the mysterious — to him, at least — disappearance of his mother and father. (Before succumbing to disease, Russell’s parents lived in a scandalous ménage-a-trois with a rather sinister amateur scientist.) Russell’s grandfather, Lord John Russell, a Whig aristocrat and reformer, had twice been prime minister, but it was his dour and pious grandmother who dominated his childhood. Not only did he suffer from crushing loneliness, but it was borne in upon him that his Uncle Willy had to be shut away as a violent lunatic. (His Aunt Agatha was none too sane either.) This was the beginning of his lifelong terror of hereditary madness, and the impetus for many a nightmare, which the cartoonists depict with lurid relish.

The adolescent Russell sought refuge in the abstractions of mathematics. (In his autobiography, he claimed it was his love of mathematics that saved him from suicide.) His vision of an enchanted logical world was jarred, however, when he reached Cambridge and found that mathematics as practiced there was little more than a bag of calculating tricks, sloppily based on physical intuition rather than rigorous proof. If certain knowledge was to be achieved, he grew convinced, the house of mathematics had to be rebuilt from scratch on firm logical foundations.

Russell’s quest for certainty coincided with a busy erotic career. We see him courting Alys, the pretty American Quaker girl who would become the first of his four wives. (The cartoonists inexplicably neglect to depict what Russell later described as “the happiest morning of my life,” when Alys allowed him to kiss her breasts). The young couple set off on a tour of the Continent, where Russell seeks out Gottlob Frege, the greatest logician since Aristotle, and Georg Cantor, the creator of the mathematical theory of infinity. Both men, to Russell’s consternation, prove to be slightly daft. In Paris, at the 1900 International Congress of Mathematicians, he witnesses a titanic clash between Henri Poincaré and David Hilbert, the two greatest mathematicians of the day, over the importance of intuition versus proof. Returning to England, Russell spends the next decade laboring with Alfred North Whitehead to complete the epic “Principia Mathematica” — all the while doing his best to seduce Whitehead’s comely wife, Evelyn. Their (stillborn) masterpiece runs many thousands of pages, a mere 362 of which are required to prove the interesting proposition “1 + 1 = 2.”

All of this is presented with real graphic verve. (Even though I’m a text guy, I couldn’t keep my eyes off the witty drawings.) To ginger up the story, the authors often deviate from the actual facts. As they admit in an afterword, Russell never met Frege or Cantor in the flesh. Nor, I am fairly certain, did he ever say to Whitehead, “I’m tired, man.” (You expect Whitehead to reply, “Me too, bro!”) We are assured, however, that no liberties have been taken with “the great adventure of ideas.” And for the most part the ideas are conveyed accurately, and with delightful simplicity. If you don’t know much about infinity, for instance, you are invited to check in to “Hilbert’s Hotel” — which, with its infinite number of rooms, can miraculously accommodate additional guests even when it’s completely full.

There is one serious misstep, though. It has to do with the notorious paradox that Russell discovered in the spring of 1901: the paradox of the set of all sets that don’t contain themselves as members. (Think of the barber of Seville, who shaves all men, and only those men, who do not shave themselves. Does this barber shave himself or not? Either possibility yields a contradiction.) The authors have fun unpacking Russell’s paradox, but they exaggerate its fallout. The paradox did ultimately doom Russell’s (and Frege’s) project of reducing mathematics to pure logic. However — and this is something that Russell himself failed to realize, along with the authors — it left mathematics pretty much undisturbed. When Cantor heard of Russell’s paradox, he did not react like a madman, the way ­“Logicomix” caricatures him. He calmly observed that it did not apply to his own theory of sets, which evolved into the present-day foundation of mathematics.

It is true that Cantor did suffer fits of madness (the magus of infinity died in a mental asylum), as did many other figures in this story. Frege, the consummate logician, ended up a foaming anti-Semite. Kurt Gödel, who proved that no logical system could capture all of mathematics, starved himself to death out of a paranoid fear that people were poisoning his food. Russell maintained his own grip on sanity, but his fear of hereditary madness was borne out when his elder son became schizophrenic and his granddaughter, also schizophrenic, committed suicide by setting herself afire. Russell’s philosophical confidence, however, was shattered by his onetime pupil Ludwig Wittgenstein, who made him realize that he had never really understood what logic was.

Is it madness to be driven by a passion for something as inhuman as abstract certainty? This is a question the four creators of “Logicomix” ponder as, in a beguiling coda, they make their way through nighttime Athens to an open-air performance of the “Oresteia.” Oddly enough, Aeschylus’ trilogy furnishes the concluding wisdom, which, at the risk of triteness, I’ll condense into a mathematical inequality:

Life > logic.

Jim Holt is the author of “Stop Me if You’ve Heard This: A History and Philosophy of Jokes.” He is at work on a book about the puzzle of existence.

Omar Khayyam...birthday

Omar Khayyam
1048-1131

Quatrain XI in his 1st edition:

Here with a Loaf of Bread beneath the Bough,
A Flask of Wine, a Book of Verse - and Thou
Beside me singing in the Wilderness -
And Wilderness is Paradise enow.

Omar Khayyam was a Persian astronomer, mathematician, and poet in the 11th Century, famous today for Edward Fitzgerald's 1859 translations of his works into English.

From Treatise on Demonstration of Problems of Algebra [1070]:

By the help of God and with His precious assistance, I say that Algebra is a scientific art. The objects with which it deals are absolute numbers and measurable quantities which, though themselves unknown, are related to "things" which are known, whereby the determination of the unknown quantities is possible. Such a thing is either a quantity or a unique rlation, which is only determined by careful examination. What one seaches for in the algebraic art are the relations which lead from the known to the unknown, to discover which is the object of Algebra as stated above. The perfection of this art consists in knowledge of the scientific method by which one determines numerical and geometric unknowns.

I was unable to devote myself to the learning of this algebra and the continued concentration upon it, because of obstacles in the vagaries of time which hindered me; for we have been deprived of all the people of knowledge save for a group, small in number, with many troubles, whose concern in life is to snatch the opportunity, when time is asleep, to devote themselves meanwhile to the investigation and perfection of a science; for the majority of people who imitate philosophers confuse the true with the false, and they do nothing but deceive and pretend knowledge, and they do not use what they know of the sciences except for base and material purposes; and if they see a certain person seeking for the right and preferring the truth, doing his best to refute the false and untrue and leaving aside hypocrisy and deceit, they make a fool of him and mock him.

Omar Khayyam

The Rubaiyat

Rubaiyat of Omar Khayyam

Mathematics and computers 1959 style...Yudell L. Luke

[Click to enlarge and read.]

Yudell L. Luke

June 26th, 1918 to May 6th, 1983

Midwest Research Institute

Mathematician Has the "Lowdown" on Computers

by

Roger Swanson

The Kansas City Times

Tuesday

November 10th, 1959

Not the personal PC but state of the arts UNIVAC was a critical tool for Yudell L. Luke at the fledgling Midwest Research Institute in Kansas City, Missouri in the late 1950s. After leaving Midwest Research Institute, Luke moved less than a mile to the University of Missouri at Kansas City mathematics department.

Open Library

ScientificCommons


Linda Hall Library

Midwest Research Institute

I remember as a young person over 45 years ago taking a school field trip there and being amazed when a scientist carrying a fluorescent tube walked into a special room and the tube glowed just as if it were connected to an AC circuit. Oddly enough the same circumstance happened once before when my fluorescent desk lamp that was turned off began to illuminate when a strong thunderstorm was in the area. Physics rules.

Thanks to Bill Ashworth, Cindy Rogers, and Bruce Bradley...Linda Hall Library, Kansas City, Missouri.

S. Hawking...better

"Stephen Hawking expected to make full recovery"

Doctors say condition of scientist, 67, is improving after he was taken to hospital 'very ill'

by

Ian Sample and Robert Booth

April 21st, 2009

guardian.co.uk

The family of physicist Stephen Hawking said today they were looking forward to him making a full recovery after he fell ill and was admitted to hospital yesterday.

Hawking, 67, was taken by ambulance to Addenbrooke's hospital, Cambridge, for tests after he fell "very ill", but his condition appears to have improved and he was said to be in a "comfortable" condition today.

"Professor Hawking is being kept in for observation at Addenbrooke's hospital this morning," a spokesman for Cambridge University said. "He is comfortable and his family is looking forward to him making a full recovery."

Hawking has been unwell for a couple of weeks, and earlier this month pulled out of a headline appearance at a science conference in Arizona to recover from a chest infection.

A Cambridge University spokesman said Hawking was still having tests for a condition that was not related to his respiratory infection, and was not life threatening.

The scientist, who rose to wider public prominence in 1988 with the publication of his bestselling A Brief History of Time, began to develop the symptoms of incurable motor neurone disease in the 1960s, gradually losing the use of his limbs and voice.

He has worked at the university's department of applied mathematics and theoretical physics for over 30 years, but is due to step down as Lucasian professor of mathematics, a post once held by Sir Isaac Newton, at the end of the academic year. It is customary to retire from the post at 67, though Hawking intends to continue as professor emeritus.

In a career spanning almost 50 years, Hawking has wrestled with some of the most puzzling questions in cosmology. With Sir Roger Penrose, at Oxford University, he used the physics of collapsing stars to argue that space and time could begin at points in the universe called "singularities".

In a lecture he gave in 2007 in honour of Nasa's 50th anniversary at George Washington University in Washington DC, Hawking suggested primitive alien life might be common.

In 2002 Hawking, one of the most recognisable figures on the streets of Cambridge, drove his high-powered wheelchair into a wall while in a rush to get into town. He broke his hip and almost missed his 60th birthday celebrations.

In 2007 he became the first disabled person to experience weightlessness aboard a Boeing 727 that replicates the freefall conditions of being in orbit. The plane, which flew from Nasa's Cape Canaveral site in Florida, performed eight steep dives over the Atlantic, allowing the physicist to float freely for 25-second spells.

Hawking has since signed up to fly to the edge of space next year as one of Sir Richard Branson's first space tourists aboard the Virgin Galactic spacecraft.

His progressive disease has left Hawking reliant upon a computer screen and a voice synthesiser to communicate. His cultural reach has led to appearances in The Simpsons, Futurama and Star Trek: The Next Generation.

Motor neurone diseases steadily destroy the nerves that control muscles. Doctors usually give patients three years to live after their first symptoms appear. Hawking, who is thought to have amyotrophic lateral sclerosis (ALS), is one of the world's longest-surviving MND patients and has round-the-clock care from a team of nurses.

Brian Dickie, director of research at the Motor Neurone Disease Association, said only 5% of people diagnosed with ALS survive for 10 years or longer. Hawking "is at the extreme end of the scale when it comes to survival", Dickie said.

Peter Haynes, head of the department of applied mathematics and theoretical physics, said: "Professor Hawking is a remarkable colleague. We all hope he will be amongst us again soon."

Hawking was born in Oxford and grew up in St Albans, Hertfordshire. He studied at Oxford University before moving to Cambridge to carry out research in cosmology. He was awarded the CBE in 1982, made a Companion of Honour in 1989 and is a fellow of the Royal Society.


Stephen Hawking ill

Deceased--Jacob T. Schwartz

Jacob T. Schwartz
January 9th, 1930 to March 2nd, 2009

"Jacob T. Schwartz, 79, Restless Scientist, Dies"

by

John Markoff

March 4th, 2009

The New York Times

Jacob T. Schwartz, a mathematician and computer scientist who did seminal research in fields as diverse as molecular biology and robotics, died Monday at his home in Manhattan. He was 79.

He died in his sleep of liver cancer, his wife, Diana, said. He was chairman of the computer science department at New York University, which he founded, from 1964 to 1980.

During a career that also included 42 years as a professor at the Courant Institute for Mathematical Sciences at the university, Dr. Schwartz wrote more than a dozen books and more than 100 scientific papers and research reports. At his death Dr. Schwartz was actively working on research in both molecular biology and logic.

Throughout his life, Dr. Schwartz, who was known as Jack, moved from one scientific field to the next. He was not a dilettante, but mastered each field in turn and then made significant contributions.

"He didn't dabble, he plunged," said W. Daniel Hillis, a computer scientist and founder of the Thinking Machines Corporation, an early maker of massively parallel thinking machines.

Invited to spend a summer consulting at the start-up firm in the early 1980s, Dr. Schwartz arrived and asked to see the documentation of the company's CM-1 supercomputer, Dr. Hillis recalled. When he learned that the computer manual did not exist yet, he sat down and wrote it single-handedly.

His most influential publication was one of his first. As a mathematics graduate student at Yale, he worked with Nelson Dunford. The two men cooperated on "Linear Operators," which was published in three volumes in 1958 and remains in print today as a standard in the field, referred to by mathematicians simply as "Dunford and Schwartz."

During the 1960s, Dr. Schwartz turned his attention to computing and became involved in the nascent field of computer science.

He spent time as a visiting scientist at I.B.M., which led to a collaboration with two I.B.M. researchers, John Cocke and Frances E. Allen. That led to pioneering work in optimizing compilers, software tools that are used by programmers to increase the performance of application programs. Dr. Allen would later become his second wife.

His background in mathematical algorithms led Dr. Schwartz to develop an early programming language called SETL, based on the mathematical theory of sets. The language was intended to make it possible for programmers to efficiently express algorithms. The language would later influence the designer of the Python programming language, widely used by programmers today.

The Courant Institute had financial support both from the National Science Foundation and from the Atomic Energy Commission and housed an early version of the Control Data Corporation 6600 supercomputer, designed by Seymour Cray.

Dr. Schwartz went out of his way to open the supercomputer to a generation of scientifically inclined high school students who were given access to the machine, which initially could be used by only one user at a time.

In the late 1970s, Dr. Schwartz's interests shifted toward parallel computing, and he designed an innovative early parallel computer called the Ultracomputer.

His later interests ranged as far afield as robotics, to which he made important theoretical contributions on how robots move around obstacles.

In the mid 1980s, he spent a year as the head of the Defense Advanced Research Projects Agency's Information Processing and Techniques Office.

While there he shifted the research focus of the organization away from artificial intelligence.

In 1999, Dr. Schwartz became interested in molecular biology and began a multiyear collaboration with Michael Wigler, a professor at Cold Spring Harbor Laboratory.

The son of Ignatz and Hedwig Schwartz, Dr. Schwartz was born on Jan. 9, 1930, in the Bronx. He received his bachelor of science degree from City College of New York in 1949 and his master’s degree and Ph.D. from Yale.

His marriages to Dr. Allen and to Sandra Weiner ended in divorce. Besides his third wife, Diana, survivors include his daughters, Abby Schwartz of Manhattan and Rachel Fainman of Winnipeg, Manitoba; and a sister, Judith Dunford, the widow of the literary critic Alfred Kazin.

So fearsome was Dr. Schwartz's early reputation as a mathematician that when John Forbes Nash Jr., the Nobel Prize winning mathematician and economist, learned that he was attempting to solve an extremely challenging mathematical problem known as the "embedding problem," he became agitated, apparently fearing Dr. Schwartz might beat him to a solution, said Sylvia Nasar, author of "A Beautiful Mind," a biography of Nash.

Nash's own reputation in mathematics was cemented by his 1954 solution to the embedding problem, which was perceived by mathematicians as a more formidable challenge than game theory, for which he won his Nobel, she said.

"They were at some level of birds of feather, which was probably why Nash was so concerned," she said.

Jacob T. Schwartz

Deceased--Ilya Piatetski-Shapiro

Ilya Piatetski-Shapiro
March 30th, 1929 to February 21st, 2009

"Ilya Piatetski-Shapiro, Math Theorist Who Clashed With Soviets, Dies at 79"

by

Kenneth Chang

March 5th, 2009

The New York Times

Ilya Piatetski-Shapiro, whose outstanding mathematical contributions stretched over a long career despite hardships as a Jew in the Soviet Union and later the debilitating effects of Parkinson’s disease, died Feb. 21 in Tel Aviv. He was 79.

His death was announced last week by Yale University, where he was a professor of mathematics.

Working with James W. Cogdell, his main collaborator over a quarter-century, starting in the mid-1970s, Dr. Piatetski-Shapiro shaped a proof of what is known as the Converse Theorem, which finds some deep relationships between different fields of mathematics.

The mathematical constructs are "quite mysterious, even to mathematicians," Dr. Cogdell explained in an interview this week. But the theorem has wide applications, including playing a small but important role in the proof of Fermat's Last Theorem that Andrew Wiles, a Princeton mathematician, completed in 1994.

"It's a very powerful tool," said Peter C. Sarnak, a professor of mathematics at Princeton. "He made sure it's in a form everyone can use it."

Born in Moscow, Ilya Piatetski-Shapiro became interested in mathematics when he was 10. In a memoir, he said that when he learned about negative numbers from his father, a chemical engineer, he was struck "by the charm and unusual beauty" of the concept.

After completing his undergraduate degree in 1951 at Moscow University, he wanted to continue there for his graduate studies. Despite the strong recommendation of Alexander O. Gelfond, a prominent mathematician and member of the Communist Party, the application was rejected.

"That was a time of great anti-Semitism," said Dr. Piatetski-Shapiro's son, Gregory.

Through Gelfond's efforts, he was admitted to the Moscow Pedagogical Institute. But in 1952, he received a letter from government officials ordering him to go to Kazakhstan to teach high school. His parents, fearing that he might be sent to Stalin’s labor camps if he refused, told him to go, but he decided not to. After about a year, another letter informed him he did not have to report for the teaching assignment.

During this time, he produced a proof that asserts that certain sequences of integers defined through the power function contain an infinite number of prime numbers. "It was rather unexpected," Dr. Sarnak said.

Dr. Piatetski-Shapiro received his doctorate in 1954.

Dr. Piatetski-Shapiro also attended seminars at the Steklov Mathematical Institute in Moscow, where Prof. Igor Shafarevich influenced his interests toward modern number theory and algebraic geometry.

In 1958, he became a professor of mathematics at the Moscow Institute of Applied Mathematics, and in 1965, he gained an additional professorship at Moscow State University.

Except for a short trip to Hungary, Dr. Piatetski-Shapiro was not allowed to leave the Soviet Union. The authorities told him that he would be free to travel if he joined the Communist Party.

He responded that party membership would distract him from his work.

In the early 1970s, the Soviet Union let more Jews emigrate to Israel, including Dr. Piatetski-Shapiro’s first wife, Inna, whom he had divorced, and his son. As a result, he was fired from his Moscow State University professorship. In 1974, he also applied for an exit visa, but was denied. He lost his job at the Institute of Applied Mathematics, and even his access to mathematical libraries.

In 1976, Dr. Piatetski-Shapiro finally received an exit visa. That broke up his second marriage, as his wife remained in Moscow.

Starting in 1977, he divided his time as a professor between Tel Aviv University and Yale.

When Dr. Cogdell was a graduate student at Yale, Dr. Piatetski-Shapiro became his thesis adviser. As part of the oral examination, Dr. Piatetski-Shapiro assigned him some assertions to prove.

Dr. Cogdell, now a professor at Ohio State, recalled that he thought he had found the solutions.

"Then I realized there was a mistake," Dr. Cogdell said. "His response was, 'Very good.' Then I said, 'I know how to fix it,' and he said, 'Even better.'"

Dr. Piatetski-Shapiro is survived by his wife, Edith Piatetski-Shapiro, who lives in New Haven and Tel Aviv; his son, Gregory, of Brookline, Mass.; a daughter, Shelly Shapiro of New York City; a stepdaughter, Niki Lipkin of Tel Aviv; and two grandsons.

As the Parkinson's worsened, Dr. Piatetski-Shapiro relied on others to record notes and he had more trouble talking. In 1992, he decided to stop traveling, but a year or two later, he changed his mind. "He missed all of the contact and finally decided he wouldn't feel better if he traveled or not," Dr. Cogdell said.

His third wife, Edith, a mathematician he had met in Israel, and others helped him walk. His speech fell to a whisper and often he could not talk at all.

For a number of years, when Dr. Piatetski-Shapiro was in a good state, he would phone Dr. Cogdell, and it would be a three-way conversation with Edith repeating Dr. Piatetski-Shapiro’s words to Dr. Cogdell and then handing the phone to Dr. Piatetski-Shapiro as Dr. Cogdell replied.

One of the highest honors in mathematics is to be invited to lecture at the International Congress of Mathematics, held only once every four years, and Dr. Piatetski-Shapiro was invited to speak four times, in 1962, 1966, 1978 and 2002, when he was in his 70s and still "knocking off well-known, longstanding problems," said Dr. Sarnak, the professor at Princeton.

Dr. Piatetski-Shapiro was able to deliver the lecture in person only once, in 1978, after he had immigrated to Israel. For the first two invitations, the Soviet Union did not allow him to leave the country and colleagues delivered his speeches for him; the last time, he was too tired to attend, and the address was delivered by Dr. Cogdell.

Ilya Piatetski-Shapiro

March 3rd, 2009--3/3/09--√9 = 3--3² = 3 × 3 = 9

Nearly slipped by...this is "Square Root Day" celebrated by math aficionados. It is a rare mathematical function event that won't happen again until April 4th, 2016. Tuesday is "Square Root Day", a holiday that occurs when the day and the month are both the square root of the last two digits of the current year. Previous dates were: 1/1/01, 2/2/04, 3/3/09, 4/4/16, 5/5/25, 6/6/36, 7/7/49, 8/8/64, and 9/9/81. Ron Gordon, a Redwood City, California high school teacher, created the first "Square Root Day" on September 9th, 1981 [9/9/81].

"Lewis Carroll in Numberland: His Fantastical Mathematical Logical Life"

A new book is out discussing the mathematics of Lewis Carroll...Lewis Carroll in Numberland: His Fantastical Mathematical Logical Life by Robin Wilson.

"How to Measure a Cheshire Grin?"

by

John Allen Paulos

February 1st, 2009

The New York Times

Charles Lutwidge Dodgson, better known as Lewis Carroll, was a mathematician at Oxford University for most of his life. His fanciful "Alice's Adventures in Wonderland" and "Through the Looking Glass" are quite familiar to us, as, to a lesser extent, are his photographs of young children. In "Lewis Carroll in Numberland," the distinguished British mathematician Robin Wilson has filled a perceived gap in the writings about Carroll by describing in a straightforward, jabberwocky-free fashion the author's mathematical accomplishments, both professional and popular.

Wilson begins this fine mathematical biography with an account of Dodgson's idyllic North England childhood. Born in 1832, the eldest son in a large family, Dodgson was mathematically gifted like his clergyman father. He read widely, wrote amusing pamphlets for his siblings and dazzled his teachers. As Wilson documents, some of Dodgson's later concerns with logic, time and puzzles were already apparent in his pamphlets and letters.

Proceeding linearly through Dodgson's life, Wilson pays particular attention to his early career at Oxford, including the sometimes tedious details of exams, classes and the tutoring of fellow students. But even at the beginning of his career, Dodgson demonstrated a playful approach to mathematics, frequently injecting little puzzles into his lessons. (One of his classics: A cup contains 50 spoonfuls of brandy, and another contains 50 spoonfuls of water. A spoonful of brandy is taken from the first cup and mixed into the second cup. Then a spoonful of the mixture is taken from the second cup and mixed into the first. Is there more or less brandy in the second cup than there is water in the first cup?)

During these early years, Dodgson developed what would become a lifelong fascination with geometry, an interest that led to his many explications and pedagogical enhancements of Euclid’s "Elements." Wilson explains these clearly for those without mathematical background, as well as Dodgson’s later work on algebraic determinants and his idiosyncratic techniques for evaluating them to solve systems of linear equations.

Dodgson's mathematical career — and perhaps even his literary career — would not have been possible without the Rev. Henry Liddell. In 1855, Liddell became dean of Christ Church and the following year appointed Dodgson a lecturer in mathematics. Liddell had four children, including one little girl named Alice. Wilson briskly dismisses the argument that Dodgson's photographs of Alice and other girls, sometimes nude or semi-nude, show he was a pedophile. "In common with many of his generation, he regarded young children as the embodiment of purity and he delighted in their innocence," Wilson writes, adding that Dodgson's vows of celibacy, which he took in 1861 (though he never became a priest), "would have outlawed any inappropriate behavior, and there has never been a shred of evidence of anything untoward."

Despite his voluminous output, Dodgson, who never married, remains inscrutable as a person, at least to me. A religiously, politically and personally conservative man, he revealed no unseemly visceral urges, but he did have an interest in politics. Through it he came to investigate voting and apportionment systems; he pointed out the possible faults of majority rule, runoffs, eliminations and other procedures, and proposed various alternative arrangements, which also had shortcomings. These insights foreshadowed Kenneth Arrow’s 1951 theorem showing that any voting framework satisfying certain minimum conditions sometimes produces unfair results.

Wilson also discusses Dodgson's proposals for ciphers and codes, his suggestions for improvements in sports and tournament scoring, and his logic diagrams, which in some ways better elucidate the conclusions of syllogisms than the more familiar Venn diagrams. But the variety of his endeavors aside, Dodgson's mathematics has not proved influential or enduring. More long lasting have been his geometrical, arithmetical and logical puzzles, well-chosen examples of which Wilson strews throughout the book, along with excerpts from some of his often teasing letters.

Wilson's aim is to concentrate on Dodgson's scholarly work rather than on the whimsical "Alice" books, but when pushed too hard the dichotomy between them breaks down. Even Dodgson’s mathematical work contained wordplay and humorously literal interpretations. Contrariwise, his popular work, always under the pseudo­nym Lewis Carroll, refers obliquely to serious mathematical and philosophical issues.

Wilson doesn't mention it, but this unwarranted bifurcation brings to mind the philosopher George Pitcher's 1965 essay "Wittgenstein, Nonsense, and Lewis Carroll," which highlights telling similarities between the philosophical writings of Wittgenstein and the popular work of Carroll. Both men were concerned with nonsense, logical confusion and language puzzles. But while Wittgenstein was tortured by these things, Carroll was, or at least appeared to be, delighted by them. The relation between the two is similar in this respect to that between, say, Soren Kierkegaard and Woody Allen.

Finally, in case you're wondering, there’s exactly as much brandy in the water as there is water in the brandy. Most frabjous!

[John Allen Paulos is a professor of mathematics at Temple University. His most recent book is "Irreligion: A Mathematician Explains Why the Arguments for God Just Don’t Add Up."]

Lewis Carroll in Numberland: His Fantastical Mathematical Logical Life

by

Robin Wilson

ISBN-10: 0713997575
ISBN-13: 978-0713997576

Lewis Carroll fun

Mr. Klein and Mr. Möbius

Newton's Rings

Sir Arthur Stanley Eddington--poetic tribute