Axiom of Choice? “Would you rather be deaf or blind?”

Sir Michael Francis Atiyah, OMFRSFRSEFAA, a British mathematician
in Mathematics in the 20th Century


Brief Book Review: James Gleick’s “The Information: a History, a Theory, a Flood”

Overall James Gleick’s book The Information: a History, a Theory, a Flood is an excellent read. Given that it’s an area with which I’m intimately interested, I’m not too surprised that most of it is “review”, but I’d highly recommend it to the general public to know more about some of the excellent history, philosophy, and theory which Gleick so nicely summarizes throughout the book.

Book Cover: The Information

There are one or two references in the back which I’ll have to chase down and read and one or two, which after many years, seem like they may be worth a second revisiting after having completed this.

Even for the specialist, Gleick manages to tie together some disparate thoughts to create an excellent whole which makes it a very worthwhile read. I found towards the last several chapters, Gleick’s style becomes much more flowery and less concrete, but most of it is as a result of covering the “humanities” perspective of information as opposed to the earlier parts of the text which were more specific to history and the scientific theories he covered.

Review originally posted at

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Book Review: Charles Seife’s “Proofiness: The Dark Arts of Mathematical Deception”

Proofiness: The Dark Arts of Mathematical Deception by Charles Seife (Penguin)
Proofiness: The Dark Arts of Mathematical Deception Book Cover Proofiness: The Dark Arts of Mathematical Deception
Charles Seife
Mathematics, Popular Science
September 23, 2010

The bestselling author of Zero shows how mathematical misinformation pervades-and shapes-our daily lives. According to MSNBC, having a child makes you stupid. You actually lose IQ points. Good Morning America has announced that natural blondes will be extinct within two hundred years. Pundits estimated that there were more than a million demonstrators at a tea party rally in Washington, D.C., even though roughly sixty thousand were there. Numbers have peculiar powers-they can disarm skeptics, befuddle journalists, and hoodwink the public into believing almost anything. "Proofiness," as Charles Seife explains in this eye-opening book, is the art of using pure mathematics for impure ends, and he reminds readers that bad mathematics has a dark side. It is used to bring down beloved government officials and to appoint undeserving ones (both Democratic and Republican), to convict the innocent and acquit the guilty, to ruin our economy, and to fix the outcomes of future elections. This penetrating look at the intersection of math and society will appeal to readers of Freakonomics and the books of Malcolm Gladwell.

Charles Seife doesn’t prove that mathematics is essential for a democracy, but he certainly shows how the lack of proper use of mathematics can fray heavily at the edges!

Proofiness was a great book to have read over a long Fourth of July holiday. Though many people may realize some of the broad general concepts in the book, it’s great to have a better structure for talking about concepts like Potemkin numbers, disestimation, fruit packing, cherry picking, apple polishing, comparing apples to oranges, causuistry, randnumbness, regression to the moon, tragedy of the commons, and moral hazard among others. If you didn’t think mathematics was important to daily life or our democratic society, this book will certainly change your mind.

Seife covers everything from polls, voting, politics, economics, marketing, law, and even health to show how numbers are misused in a modern world that can ill-afford to ignore what is really going on around us.

This is a fantastic book for nearly everyone in the general public, but I’d highly recommend it for high school students while taking civics.

Original review posted on on 7/9/12.

Reading Progress
  • 07/07/12 marked as: currently reading
  • 07/07/12 23.0% #
  • 07/09/12 52.0%
  • 07/09/12 Finished book
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You Cannot Learn Too Much Linear Algebra

Benedict Gross, Ph.D., George Vasmer Leverett Professor of Mathematics, Harvard University
in Abstract Algebra, Lecture 2 at 14:25 via Harvard Extension


Benedict Gross standing in front of chalkboard with equations from Abstract Algebra Class
Benedict Gross teaching abstract algebra
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Mathematics in Popular Science Books | The Economist

Big bang (The Economist)
Popular physics has enjoyed a new-found regard. Now comes a brave attempt to inject mathematics into an otherwise fashionable subject

This review of Brian Cox and Jeff Forshaw’s forthcoming book The Quantum Universe: Everything That Can Happen Does Happen sounds intriguing. I’m highly impressed that so much of the review focuses on the author’s decision to include a more mathematical treatment of their subject for what is supposed to be a popular science book. I always wish books like these at least had the temerity to include much more in the way of the mathematical underpinnings of their subjects; I’m glad that the popular press (or at least The Economist in this case) is willing to be asking for the mathematics as well. Hopefully it will mark a broader trend in popular books on scientific topics!

Fundamental physics

Big bang

Popular physics has enjoyed a new-found regard. Now comes a brave attempt to inject mathematics into an otherwise fashionable subject

Nov 5th 2011 | from the print edition

The Quantum Universe: Everything That Can Happen Does Happen. By Brian Cox and Jeff Forshaw. Allen Lane; 255 pages; £20. To be published in America in January by Da Capo Press; $25.

PREVIOUSLY the preserve of dusty, tweed-jacketed academics, physics has enjoyed a surprising popular renaissance over the past few years. In America Michio Kaku, a string theorist, has penned several successful books and wowed television and radio audiences with his presentations on esoteric subjects such as the existence of wormholes and the possibility of alien life. In Britain Brian Cox, a former pop star whose music helped propel Tony Blair to power, has become the front man for physics, which recently regained its status as a popular subject in British classrooms, an effect many attribute to Mr Cox’s astonishing appeal.

Mr Cox, a particle physicist, is well-known as the presenter of two BBC television series that have attracted millions of viewers (a third series will be aired next year) and as a bestselling author and public speaker. His latest book, “The Quantum Universe”, which he co-wrote with Jeff Forshaw of the University of Manchester, breaks the rules of popular science-writing that were established over two decades ago by Stephen Hawking, who launched the modern genre with his famous book, “A Brief History of Time”.

Mr Hawking’s literary success was ascribed to his eschewing equations. One of his editors warned him that sales of the book would be halved by every equation he included; Mr Hawking inserted just one, E=mc2, and, even then, the volume acquired a sorry reputation for being bought but not read. By contrast, Mr Cox, whose previous book with Mr Forshaw investigated “Why does E=mc2?” (2009), has bravely sloshed a generous slug of mathematics throughout his texts.

The difficulties in explaining physics without using maths are longstanding. Einstein mused, “The eternal mystery of the world is its comprehensibility,” and “the fact that it is comprehensible is a miracle.” Yet the language in which the world is described is that of maths, a relatively sound grasp of which is needed to comprehend the difficulties that physicists are trying to resolve as well as the possible solutions. Mr Cox has secured a large fan base with his boyish good looks, his happy turns of phrase and his knack for presenting complex ideas using simple analogies. He also admirably shies away from dumbing down. “The Quantum Universe” is not a dry undergraduate text book, but nor is it a particularly easy read.

The subject matter is hard. Quantum mechanics, which describes in subatomic detail a shadowy world in which cats can be simultaneously alive and dead, is notoriously difficult to grasp. Its experiments yield bizarre results that can be explained only by embracing the maths that describe them, and its theories make outrageous predictions (such as the existence of antimatter) that have nevertheless later been verified. Messrs Cox and Forshaw say they have included the maths “mainly because it allows us to really explain why things are the way they are. Without it, we should have to resort to the physicist-guru mentality whereby we pluck profundities out of thin air, and neither author would be comfortable with guru status.”

That stance might comfort the authors, but to many readers they will nonetheless seem to pluck equations out of thin air. Yet their decision to include some of the hard stuff leaves open the possibility that some readers might actually engage in the slog that leads to higher pleasures. For non-sloggers alternative routes are offered: Messrs Cox and Forshaw use clockfaces to illustrate how particles interact with one another, a drawing of how guitar strings twang and a photograph of a vibrating drum. A diagram, rather than an equation, is used to explain one promising theory of how matter acquires mass, a question that experiments on the Large Hadron Collider at CERN, the European particle-physics laboratory near Geneva, will hopefully soon answer.

The authors have wisely chosen to leaven their tome with amusing tales of dysfunctional characters among scholars who developed quantum mechanics in the 1920s and beyond, as well as with accounts of the philosophical struggles with which they grappled and the occasional earthy aside. Where the subject matter is a trifle dull, Messrs Cox and Forshaw acknowledge it: of Heinrich Kayser, who a century ago completed a six-volume reference book documenting the spectral lines generated by every known element, they observe, “He must have been great fun at dinner parties.” And they make some sweeping generalisations about their colleagues who pore over equations, “Physicists are very lazy, and they would not go to all this trouble unless it saved time in the long run.”

Whether or not readers of “The Quantum Universe” will follow all the maths, the authors’ love for their subject shines through the book. “There is no better demonstration of the power of the scientific method than quantum theory,” they write. That may be so, but physicists all over the world, Messrs Cox and Forshaw included, are longing for the next breakthrough that will supersede the claim. Hopes are pinned on experiments currently under way at CERN that may force physicists to rethink their understanding of the universe, and inspire Messrs Cox and Forshaw to write their next book—equations and all.

from the print edition | Books and arts

John McCarthy on Arithmetic

John McCarthy (), an American computer scientist and cognitive scientist who was one of the founders of the discipline of artificial intelligence
in Computer Scientist Coined ‘Artificial Intelligence’ in The Wall Street Journal


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The Response of the Schoolmaster

This must certainly be the quote of the week from English author Alan Bennett’s play Forty Years On:

Foster: I’m still a bit hazy about the Trinity, sir.
Schoolmaster: Three in one, one in three, perfectly straightforward.  Any doubts about that see your maths master.


Paul Halmos on Prerequisites

Definitely the quote of the day:

Paul Halmos (1916 – 2006, Hungarian-born American mathematician
in Measure Theory (1950)


This is essentially the mathematician’s equivalent of the adage “Fake it ’til you make it.”

Riemann’s On the Hypotheses Which Lie at the Foundations of Geometry

One must be truly enamored of the internet that it allows one to find and read a copy of Bernhard Riemann’s doctoral thesis Habilitation Lecture (in English translation) at the University of Göttingen from 1854!

His brief paper has created a tsunami of mathematical work and research in the ensuing 156 years. It has ultimately become one of the seminal works in the development of the algebra and calculus of n-dimensional manifolds.

Terence Tao Teaching Real Analysis at UCLA this Fall

Holy cow! I discovered on Friday that Terence Tao, a Fields Medal winner, will be teaching a graduate level Real Analysis class this fall at UCLA.


Surprisingly, to me, it ony has 4 students currently enrolled!! Having won a Fields Medal in August 2006, this is a true shock, for who wouldn’t want to learn analysis from such a distinguished professor? Are there so few graduate students at UCLA who need a course in advanced analysis? I would imagine that there would be graduate students in engineering and even physics who might take such a course, but perhaps I’m wrong?

Most of his ratings on RateMyProfessors are actually fairly glowing; the one generally negative review was given for a topology class and generally seems to be an outlier.

On his own website in a section about the class and related announcements we seem to find the answer to the mystery about enrollment.  There he says:

 I intend this to be a serious course, focused on teaching the material in the course description.  As such, students who are taking or auditing the course out of idle curiosity or mathematical “sightseeing”, rather than to learn the basics of measure theory and integration theory, may be disappointed.  I would therefore prefer that frivolous enrollments in the class be kept to a minimum.

This is generally sound advice, but would even the most serious mathematical tourists really bother to make an attempt at such an advanced course? Why bother if you’re not going to do the work?!

Fans of the Mathematical Genealogy Project will be interested to notice that Dr. Tao is requiring his Ph.D. advisor’s text Real Analysis: Measure Theory, Integration, and Hilbert Spaces. He’s also recommending Folland‘s often used text as well, though if he really wanted to scare off the lookie-loos he could just say he’ll be using Rudin‘s text.

Baum’s Point Set Topology

You know it's a good start to the weekend when you've just acquired Baum's Point Set Topology for $2.72!

Elements of Point-Set Topology (Dover Books on Advanced Mathematics) Elements of Point-Set Topology by John D. Baum

Commenting only after reading to page 11, but having skimmed some other parts/sections, it’s a nice and condensed volume with most of the standard material on point set topology. It reads somewhat breezily, is well laid out, and isn’t bogged down with all the technicalities which those who haven’t seen any of this material before might have interest in. It seems better for those with some experience in axiomatic mathematics (I’ve always enjoyed Robert Ash’s A Primer of Abstract Mathematics for much of this material), but in my mind isn’t as clear or as thorough as James Munkres’ Topology, which I find in general to be a much better book, particularly for the self-learning crowd. The early problems and exercises are quite easy.

Given it’s 1964 publication date, most of the notation is fairly standard from a modern perspective and it was probably a bit ahead of it’s time from a pedagogical viewpoint.

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