Posted by: Alexandre Borovik | April 1, 2012

Felix Klein, who did for the bottle …

A bizarre phrase in an otherwise quite sensible article in the NYT:

Felix Klein, who did for the bottle what August Ferdinand Möbius had done for the strip …

[With thanks to muriel]

Posted by: Alexandre Borovik | March 22, 2012

Liar’s paradox

From a letter from a colleague:

I am reminded of a commentary on logic puzzles of a certain kind; it
was perhaps in a letter to Martin Gardner, reprinted in one of his
books. The puzzles are those about getting about on an island where
each native either always tells the truth or always lies. You reach a
fork in the road, for example, and a native is standing there, and you
want to learn from him, with one question, which way leads to the
village. The “correct” question is “If I asked you if the left way
led to the village, would you say yes?” But why should the native’s
concept of lying conform to our own logical ideas? If the native is a
liar, it means he wants to fool you, and your logical trickery will
not work. The best you can do is say something like “Did you hear
they are giving away free beer in the village today?” and see which
way the native runs. You follow him, even if he says something like
“Ugh, I hate beer!” since then he probably really is lying.

Posted by: Alexandre Borovik | March 19, 2012

Previously unpublished manuscript by Boris Weisfeiler

Boris WeisfeilerOn the size and structure of finite linear groups,

This is a nearly complete, previously unpublished manuscript by Boris Weisfeiler. The results were announced by him in August 1984. Soon after, in early January 1985, he disappeared during a hiking trip in Chile.

The investigation into Boris Weisfeiler disappearance is still ongoing in Chile, see

Posted by: Alexandre Borovik | February 25, 2012

Girls’ Verbal Skills Make Them Better At Arithmetic

Another finding, which appears to contradict a previous post:

While boys generally do better than girls in science and math, some studies have found that girls do better in arithmetic. A new study published in Psychological Science, a journal of the Association for Psychological Science, finds that the advantage comes from girls’ superior verbal skills.
“People have always thought that males’ advantage is in math and spatial skills, and girls’ advantage is in language,” says Xinlin Zhou of Beijing Normal University, who cowrote the study with Wei Wei, Hao Lu, Hui Zhao, and Qi Dong of Beijing Normal University and Chuansheng Chen of the University of California-Irvine. “However, some parents and teachers in China say girls do arithmetic better than boys in primary school.”
Zhou and his colleagues did a series of tests with children ages 8 to 11 at 12 primary schools in and around Beijing. Indeed, girls outperformed boys in many math skills. They were better at arithmetic, including tasks like simple subtraction and complex multiplication. Girls were also better at numerosity comparison—making a quick estimate of which of two arrays had more dots in it. Girls outperformed boys at quickly recognizing the larger of two numbers and at completing a series of numbers (like “2 4 6 8”). Boys performed better at mentally rotating three-dimensional images.
Girls were also better at judging whether two words rhymed, and Zhou and his colleagues think this is the key to their better math performance. “Arithmetic and even advanced math needs verbal processing,” Zhou says. Counting is verbal; the multiplication table is memorized verbally, and when people are doing multiple-digit calculations, they hold the intermediate results in their memory as words.
“Better language skills could lead to more efficient verbal processing in arithmetic,” Zhou says. He thinks it might be possible to use these results to help both boys and girls learn math better. Boys could use more help with verbal strategies for learning math terms, while girls might benefit from more practice with spatial skills.

[with thanks to muriel]

Posted by: Alexandre Borovik | February 25, 2012

Alan Turing and Linear Algebra

2012 is Alan Turing Year but perhaps I have missed a chance to attract attention of my colleagues who, like me, teach undergraduate linear algebra to a significant fact in history of linear algebra which is worth mentioning to students:

LU decomposition of matrices (and, within the routine, systematic use of elementary matrices) was introduced in Alan Turing‘s paper [1948] which was motivated, in Alan Turing’s own words, by

“the advent of electronic computers“.

I told the story to my students in my lecture on Wednesday. Since the idea that

“The process of replacing the rows of a matrix by a linear combination of other rows may be regarded as left-multiplication of the matrix by another matrix, this second matrix having coefficients which describe the linear combinations required” [1948, p. 290]

comes forth at early stages of modern expositions of linear algebra, this semester’s courses are likely to pass the point when history of LU decomposition could be usefully mentioned. But maybe it is not too late to do that in linear algebra courses taught in the Autumn.

[1948] A. M. Turing, Rounding-off errors in matrix processes. Quart. J Mech. Appl. Math. 1 (1948), 287–308.

Posted by: Alexandre Borovik | February 25, 2012

Gender Biases in Early Number Exposure to Preschool-Aged Children

A paper by Alicia Chang, Catherine M. Sandhofer, and Christia S. Brown. Journal of Language and Social Psychology, December 2011 vol. 30 no. 4 440-450. Published online before print August 25, 2011, doi: 10.1177/0261927X11416207.


Despite dramatically narrowing gender gaps, women remain underrepresented in mathematics and math-related fields. Parents can shape expectations and interests, which may predict later differences in achievement and occupational choices. This study examines children’s early mathematical environments by observing the amount that mothers talk to their sons and daughters (mean age 22 months) about cardinal number, a basic precursor to mathematics. In analyses of naturalistic mother–child interactions from the Child Language Data Exchange System (CHILDES) database, boys received significantly more number-specific language input than girls. Greater amounts of early number-related talk may promote familiarity and liking for mathematical concepts, which may influence later preferences and career choices. Additionally, the stereotype of male dominance in math may be so pervasive that culturally prescribed gender roles may be unintentionally reinforced to very young children.

And this is from  a post in the NYT Blog, under the title Mothers Talk Less to Young Daughters About Math:

Even [when their children are] as young as 22 months, American parents draw boys’ attention to numerical concepts far more often than girls’. Indeed, parents speak to boys about number concepts twice as often as they do girls. For cardinal-numbers speech, in which a number is attached to an obvious noun reference — “Here are five raisins” or “Look at those two beds” — the difference was even larger. Mothers were three times more likely to use such formulations while talking to boys.

And this is from my collection of testimonies made by professional research mathematicians about their earliest exposure to mathematics (I collect such stories for my forthcoming book Shadows of the Truth):

My Mother told me the following story.  When I was about two and a half  a small flock of birds flew overhead.  I said: “Look, there are two and three birds”.  I didn’t yet know the number five but I understood simple counting.

What mattered was that Mother found this conversation significant.  And yes, of course, she was talking to a boy …

[with thanks to muriel]

Posted by: Alexandre Borovik | February 21, 2012

Mathematics Genealogy

I have just learned  from the Mathematics Genealogy Project that I am one of 92526 descendants of Friedrich Leibniz.

Posted by: Alexandre Borovik | February 16, 2012

Ian Stewart: The mathematical equation that caused the banks to crash

Ian Stewart in the The Observer Sunday February 12 2012:

The Black-Scholes equation was the mathematical justification for the trading that plunged the world’s banks into catastrophe

It was the holy grail of investors. The Black-Scholes equation, brainchild of economists Fischer Black and Myron Scholes, provided a rational way to price a financial contract when it still had time to run. It was like buying or selling a bet on a horse, halfway through the race. It opened up a new world of ever more complex investments, blossoming into a gigantic global industry. But when the sub-prime mortgage market turned sour, the darling of the financial markets became the Black Hole equation, sucking money out of the universe in an unending stream.

Read more in The Guardian.

[with thanks to muriel]

Posted by: Alexandre Borovik | February 16, 2012

Aaron Sloman on vision.

A recent talk by Aaron Sloman,

What is vision for, and how does it work? Some considerations for philosophy of perception [600 kB PDF].

With thanks to Seb Schmoller.

Posted by: Alexandre Borovik | February 11, 2012

Cinderella and GeoGebra

[My ancient post (December 17, 2006) from a now defunct blog on Blogger: ]

In his comment on one of my previous posts, Euclid’s Elements, Interactive, Rick Booth brought my attention to GeoGebra, a free interactive geometry package. GeoGebra is nice, simple, intuitive to use, and could be a great help at school.

For some years, I was an occasional user of a commercial package, Cinderella. I decided to run a quick comparision of the two packages on a problem about the radical axis of two circles — I briefly mention the problem in my book. The problem is easy to solve with some help from Pythagoras:

Prove that the set of points P in the plane such that the tangents from P to two given non-intersecting circles are equal, is a straight line (it is called the radical axis of the two circles).

The problem is even easier when the two circles intersect – in that case, the radical axis is exactly the line through the two points of intersection. And in the case of tangent circles it is even easier — in that case the radical axis is the common tangent line.

It is interesting to compare the behaviour, in Cinderella and GeoGebra, of a simple interactive diagram: two interesecting circles of varying radii and the straight line determined by their points of intersection. In GeoGebra, when you vary the radii or move the centers of the circles and make the circles non-intersecting, the line through the points of intersection disappears — exactly as one should expect. In Cinderella, the line does not disappear, it moves following the movements of the circles, always separating them; when circles touch each other and start to intersect again, the line happens to be, again, the tangent line or the line through the points of intersection. And it is exactly the radical axis of the two circles!

I have not read documentation for GeoGebra, but the behaviour of this diagram suggests that, in GeoGebra (at least in the default mode), the underlying mathematical structure is the honest real Euclidean plane. In Cinderella, the underlying structure is the complex projective plane; what we see on the screen is just its tiny fragment, a real affine part. The radical axis of two non-intersecting circles is the real part of the complex line through two complex points of intersection; since the intersection points of two real circles are complex conjugate, the line is invariant under complex conjugation and therefore is real and shows up on the real Euclidean plane.

The Help Index of GeoGebra says about complex numbers:

GeoGebra does not support complex numbers directly, but you may use points to simulate operations with complex numbers.

However, even if one cannot do in GeoGebra the complex tricks of Cinderella, it is not a serious drawback — one should work hard to find a problem, of a secondary school level, where the difference in the nature of real and complex mathematical engines becomes visible. Such a problemis likely to involve circles or other conics, that is, quadratic equations — but remember, even the distance function is already involving a quadratic form, therefore complex numbers can pop up in rather unexpected places.

[I can add that another problem:

find the points of kiss of three mutually tangential cirles with centers at three given points

is linear, not quadratic, and can be solved by affine tools, say, with the help of a ruler with two parallel edges. ]

But still, when assessing software for mathematics teaching, it is useful to check the power, both computational and theoretical, of the mathematical engine. Unfortunately, most reviews of software for mathematics teaching never discuss this point.

To finish on a note more suitable for the festive season, I wish to mention that, in my book, radical axes appear in a solution of the following problem:

The Holes in the Cheese Problem. A big cubic piece of cheese has some spherical holes inside (like Swiss Emmental cheese, say). Prove that you can cut it into convex polytopes in such way that every polytope contains exactly one hole.

The previous discussion is the strongest hint for its solution.

And finally, a quote from

“Emmental […] is considered to be one of the most difficult cheeses to be produced because of its complicated hole-forming fermentation process.”

Is it realy surprising that compex numbers naturally appear in the cutting the wheel of cheese?

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