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November 9, 2006

Mathematics Under the Microscope

Posted by David Corfield

Alexandre Borovik has published the first chapter of his book, Mathematics Under the Microscope, at his website. It has been his experience as a blogger which persuaded him to opt for free access. I read the whole book a while ago and thoroughly recommend it. You even have the opportunity to send comments to the author.

Concerning a book for which there is not free access, in December you will at least be able to purchase my book more cheaply in paperback form for 25 pounds or 45 dollars.

Posted at November 9, 2006 9:35 AM UTC

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12 Comments & 0 Trackbacks

Re: Mathematics Under the Microscope

Nice one, Alexandre! And congratulations, David!

Posted by: Tom Leinster on November 9, 2006 5:52 PM | Permalink | Reply to this

Re: Mathematics Under the Microscope

Chapters 1-3 are now available.

Posted by: David Corfield on November 14, 2006 3:27 PM | Permalink | Reply to this

Re: Mathematics Under the Microscope

I have not finished the first 3 chapters, but I enjoy this very much. Having taught elementary algebra in college to students who’d failed it two or even three times before, I had to become very attentive to what I (as a mathematician) do as opposed to debugging what these boys and girls did, who had adopted very wrong axioms or algorithms very early in life. Teaching is hard, unteaching the mistaught is very hard. But, over 5 semester, with the exception of those who gave up (didn’t do homework, skipped exams) I saved all but one student.

This book seems, in a few places, partly based on several 2nd hand notions. “Meme” was not coined by Dawkins, for instance. When I took Psycholinguistics at Caltech in 1972 or 1973, I was already able to quote the word “MEMEME” from existing specialized literature.

Similarly, the evolutionary computing here was something that I was exposed to earlier, when I “beta-tested” the then unpublished manuscript of John Holland’s seminal “Adaptation in Natural and Artificial Systems.” I was in grad achool then, and taking Category Theory, but not really getting it.

More comments when I read further.

Oh, and “Math Disability” or “Dycalculia” when diagnosed is roughly 1/3 of the time, impossible to overcome due to mysterious aspects of the how those people’s brains are wired. 2/3 of the time, the proper very careful unteaching and proper re-teaching DOES let the person finally “get” mathematics at some practical and conceptual level.

This is a VERY important subject, as the USA currently throws away the careers of that 2/3 who are mistaught.

My wife gave a popular paper at an American Society for Engineering Education conference on unteaching Physics students who’d gotten it wrong in high school, from wrong textbooks and clueless teachers. My wife’s university praised this paer too, anhd suggested that her road to promotion is to follow up on it, with an expert menttor and more publications.

Posted by: Jonathan Vos Post on November 17, 2006 2:21 AM | Permalink | Reply to this

Ecstasy and Me; Re: Mathematics Under the Microscope

Oh, and my father, the late Samuel Herbert Post, solicited, edited, and published “Ecstasy and Me,” the autobiography of Hedy Lamarr, whom the author promises to get to in Chapter 9.

Posted by: Jonathan Vos Post on November 17, 2006 2:40 AM | Permalink | Reply to this

Re: Ecstasy and Me; Re: Mathematics Under the Microscope

Can you suggest a source of public domain photographs of Hedy Lamarr?

Posted by: Alexandre Borovik on November 19, 2006 12:49 PM | Permalink | Reply to this

Learned dimensionality; Re: Mathematics Under the Microscope

I’m intrigued by the footnote 6 on page 41 of the PDF.

I think that one CAN learn 4-D geometry early in life. The proof was Alicia Boole Stott (yes, that Boole). She had specially made colored cubes of wood to play with, and rules imposed on color adjacency. She published 4-D and 5-D gemotry papers when adult. This also led to the classic science fiction story “Mimsy were the Borogoves.”

I have often discussed 4-D visualization both with those who can’t do, and with those who do it much better than I (who could do it well at age 12, and only in flashes since then).

I’ve insisted for many years (having beta-tested the first arcade computer games “Pong” and “Computer Space” – which became +Asteroids” – for their inventor Nolan Bushnell.

Really good “flight simulators” as the author says (I wrote a textbook on flight suimulators for the Air Force) or adventure computer games could almost surely raise a generation of children with great intuitive 4-D geometry. The USAF noticed a generation ago that students in flight school were much better, due to playing computer games beforee enlisting. My father was a flight instructor in World War II, when the Link Trainer was electromechanical, rather than digital.

Imagine a million or more people in a “Second Life” online, but in 4-D. What might they do? Invent the warp Drive? Solve Quantum Gravity? Or does our genetic programming limit this? I’ll assume that brain plasticity in youth is sufficient for 4-D, and more exotic spaces and manifolds.

Mimsy Were the Borogoves (1943)
Lewis Padgett (aka Henry Kuttner and Catherine L. Moore)

Far in the future, humans have not only improved their digestive tracts (eliminating the appendix and shortening their large intestine), they have not only invented a time machine, but they have also invented educational toys which guide their children to learn abstract mathematics (non-Euclidean geometries and algebras in which 2+2 is not four). Two boxes of such toys, in fact, are sent back in time and have a dramatic effect on the kids that find them.

The math here is certainly important to the story, though it is not discussed very convincingly. The psychologist who figures out what is going on confuses logic, arithmetic, and geometry – for instance – when he suggests a “non-Euclidean geometry” in which “two and two needn’t equal four”. But this story is fun to read and the idea that learning alternative mathematical structures (other than the arithmetic of the real numbers and Euclidean geometry) could expand one’s mind and abilities is probably true and a nice advertisement for mathematics.

This story originally appeared in Astounding Science-Fiction in 1943. It has been reprinted in The Ascent of Wonder and you can read the comments that accompany it in that volume here.

Spoiler: Stop reading now if you don’t want to know how it ends! The title of the story, of course, is taken from Lewis Carroll’s “Jabberwocky”. According to this story, it was actually written by Alice herself and is not just a poem but a mathematical formula with which one can travel to bizarre alternate realities. How does she know this? Well, of course, Alice discovered one of the boxes of toys from the future!

Posted by: Jonathan Vos Post on November 17, 2006 3:28 AM | Permalink | Reply to this

Re: Learned dimensionality; Re: Mathematics Under the Microscope

I took liberty of copying your comment to my blog.

Posted by: Alexandre Borovik on November 19, 2006 12:52 PM | Permalink | Reply to this

Re: Learned dimensionality; Re: Mathematics Under the Microscope

We may do some cross-referencing, but in general Alexandre’s blog is the place to discuss his book.

Posted by: David Corfield on November 19, 2006 3:09 PM | Permalink | Reply to this

Re: Mathematics Under the Microscope

Thanks for corrections. But I do refer to Holland in the main text. Introduction cannot mention everything.

Posted by: Alexandre Borovik on November 19, 2006 1:10 PM | Permalink | Reply to this

Re: Mathematics Under the Microscope

Chapters 7, 9, 10, 11 are now available.

Posted by: David Corfield on November 21, 2006 5:53 PM | Permalink | Reply to this

Re: Mathematics Under the Microscope

Now it’s all there. Send comments to the author here.

Posted by: David Corfield on November 24, 2006 9:07 AM | Permalink | Reply to this

Re: Mathematics Under the Microscope

The paperback of my book appears to be out.

Posted by: David Corfield on December 14, 2006 10:21 PM | Permalink | Reply to this

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