Blogs / Cosmic Variance

As I reached the end of what would be called high school in the US, I was certain that I wanted more than anything to become a mathematician. Soon afterwards, as a beginning maths undergraduate at Cambridge, I had become even more committed to the subject, after having spent some time reading about the history of the subject, and becoming enthralled by the lives and contributions of some of the great mathematicians. Among these, I found myself personally drawn to Bertrand Russell, partly because I was more interested in philosophy then than I am these days, but mostly because of the sheer grandness of the vision embodied in the Principia Mathematica - the opus co-authored by Russell with Alfred North Whitehead - and it’s later challenge from Gödel.

Nevertheless, as one gets older, reads more, and hopefully gains a more sophisticated knowledge of the subject, one’s tastes tend to change somewhat. In my case, a gradual shift in my interests from pure to applied mathematics, and finally to theoretical physics opened up an increasing range of giants to understand and respect. Somehow though, I have always retained a soft spot for Russell; perhaps because of his atheism, perhaps because of his breadth, but more so I think, because when I think of him I can quite viscerally recall the way reading about him made me feel about mathematics.

Because of this, while I have never become an avid reader of graphic novels, I’m hoping to get hold of a copy of Apostolos Doxiadis’ Logicomix, which I learned about via The Guardian, and which the web site describes as

Covering a span of sixty years, the graphic novel Logicomix was inspired by the epic story of the quest for the Foundations of Mathematics.

This was a heroic intellectual adventure most of whose protagonists paid the price of knowledge with extreme personal suffering and even insanity. The book tells its tale in an engaging way, at the same time complex and accessible. It grounds the philosophical struggles on the undercurrent of personal emotional turmoil, as well as the momentous historical events and ideological battles which gave rise to them.

The role of narrator is given to the most eloquent and spirited of the story’s protagonists, the great logician, philosopher and pacifist Bertrand Russell. It is through his eyes that the plights of such great thinkers as Frege, Hilbert, Poincaré, Wittgenstein and Gödel come to life, and through his own passionate involvement in the quest that the various narrative strands come together.

The web site contains a few samples of what to expect, plus a nice summary of the cast of characters. To a graphic novel newbie like myself, it isn’t obvious what to expect from a telling of this kind of sweeping academic story in such a format. But the team involved looks promising, and I’m sufficiently fascinated by the subject matter that I’m really looking forward to getting a look at Logicomix.

Physicists and mathematicians who think about higher-dimensional spaces are, if they allow their interest to somehow become public knowledge, inevitably asked: “How can you visualize more than three dimensions of space?” There are at least three correct answers: (1) You can’t. (2) You don’t have to; manipulating abstract symbols is enough to help you figure things out. (3) There are tricks to help you pseudo-visualize higher-dimensional objects by cleverly projecting them into three dimensions; see here and here.

But really, why can’t we visualize things in more than three dimensions of space? Could a Flatlander, living in a world with only two spatial dimensions, learn to visualize our three-dimensional world? Could we somehow, through practice or direct intervention in the brain, train ourselves to truly visualize more dimensions?

I can think of a couple of explanations why it’s so hard, with different ramifications. One would be simply that our imaginations aren’t good enough to project our consciousness into a constructed world so very different from our own. Could you, for example, really imagine what it’s like to live in two dimensions? Sure, you can visualize Flatland from the outside, but what about asking what it’s like to really be a Flatlander? The best I can do is to imagine a line, flickering with colors, surrounded by darkness on either side. But the darkness is still there, in my imagination.

The other possible explanation is that the process of visualization takes up a three-dimensional space in our actual brain, preventing us from “tuning a dimensionality knob” on our imaginations. The truth is certainly more complicated than that (and I’m not experts, so anyone who is should chime in); the visual cortex itself is effectively two-dimensional, but somehow our brain reconstructs a three-dimensional image of the space around us.

Maybe this could be a new tantric discipline: visualization in higher dimensions. Or maybe the Maharishi already offers a course?

Randall Monroe (of xkcd fame) has taken on Verizon:

(via failblog)

I sometimes forget that we don’t all read the same blogs, and that it’s good to recommend some of the fun stuff out there on the internets. So let me give a shout-out to Matt Springer at Built on Facts, who had the brilliant idea of discussing a different function every Sunday. Functions are one of those things that are as necessary to math and science as breathing, but which don’t necessarily percolate into the wider world. And he (quite correctly, I think) interprets his self-imposed mandate fairly liberally, taking the time to talk about various issues in middle-level mathematics. Here are some selections from Matt’s series:

• Exponential
• Arctangent
• Witch of Agnesi
• Stirling’s approximation
• Continuous but almost-nowhere differentiable

Consider this an open thread to recommend other stuff we should all be reading. Or your favorite functions.

Following Scott Aaronson’s advice, I instructed the good folks at Amazon to send me a copy of The Princeton Companion to Mathematics. (In exchange for money, of course.) It’s sprinkled with gems like this, in the article on “Differential Topology” by my former professor Clifford Taubes:

If you are with me so far, suppose now that an advanced alien en route from Arcturus to the galactic center kidnaps you and drops you into some unknown, 2n-dimensional manifold. You suspect that it is Sⁿ x Sⁿ, but you are not sure.

Come on, the screenplay practically writes itself! I’m seeing Ewan McGregor, maybe Natalie Portman. Russell Crowe as the alien. SEEx could help with some of the mathy stuff. If any studio executives are reading this, call me, I’d be happy to bang out a treatment.

Seriously, the book is great fun, and as Scott says it’s surprisingly readable. Not really a popularization; neither equations nor high-level abstractions are shied away from. (After months of jousting with the “grammar checker” in Microsoft Word, I now deploy sentence fragments and the passive voice out of sheer spite.) But put into the hands of the right ambitious high-school student, it could be life-changing.

p.s. You haven’t really lived until you’ve seen Cliff Taubes do his little dance to illustrate the concept of “quantum fluctuations.”

Via Slashdot, I came across the following question

Troy writes:

“I’m a high school math teacher who is trying to assemble an extra-credit reading list. I want to give my students (ages 16-18) the opportunity/motivation to learn about stimulating mathematical ideas that fall outside of the curriculum I’m bound to teach. I already do this somewhat with special lessons given throughout the year, but I would like my students to explore a particular concept in depth. I am looking for books that are well-written, engaging, and accessible to someone who doesn’t have a lot of college-level mathematical training. I already have a handful of books on my list, but I want my students to be able to choose from a variety of topics. Many thanks for all suggestions!”

There are some good suggestions in the comments, and some not so good ones. Surely our wise and mathematically sophisticated readers will be able to help. Add what you can there, and in the comments here if you like.

If you’re like me, all too often while relaxing and watching a good procedural drama on TV you find yourself wondering, “How did they solve that differential equation so quickly?” That’s why we need more hit prime-time TV shows with web pages that explain the mathematical content underpinning each episode.

As far as I know, the only show that rises to this challenge is NUMB3RS, the CBS drama featuring Charlie Epps, a math professor at a suspiciously Caltech-esque university who teams up with his FBI-agent brother to solve crimes. The shows creators, Nicolas Falacci and Cheryl Heuton, had a goal from the beginning of creating an entertaining hour of television that would involve science in an intimate way. (I suppose math is almost as good.) As part of the effort, they’ve partnered with Wolfram Research to follow each episode with a web page delving into the various mathematical concepts that were discussed, including Mathematica notebooks to illustrate the various ideas:

The Math Behind NUMB3RS

Episode 11 this year was entitled “The Arrow of Time.” Here’s the opening:

You can see the full episode here; the math page is here. This stuff would make a great topic for a book.

Having recently slogged through grading an enormous pile of graduate-level problem sets, I am compelled to share one of the most useful tricks I learned in graduate school.

Make your integrals dimensionless.

This probably seems silly to the theoretical physicists in the audience, who have a habit of changing variables and units to the point where everything is dimensionless and equals one. However, in astrophysics, you frequently are integrating over real physical quantities (numbers of photons, masses of stars, luminosities of galaxies, etc) that still have units attached. While students typically do an admirable job of setting up the necessary integrals, they frequently go off the rails when actually evaluating the integrals, as they valiantly try to propagate all those extra factors.

Here’s an example of what I mean. Suppose you want to calculate some sort of rate constant for photoionization, that when multiplied by the density of atoms, will give you the rate of photo-ionizations per volume. These sorts of rates are always density times velocity times cross section:

For a Planck spectrum of photons and a typical energy-dependent cross section above some threshold

which becomes

This integral looks like a rough customer. You can pull some factors out front, but you’re still left with that unpleasant business in the exponential. You’re also using an integrating variable that has units, making it a bit tougher to check the dimensions of your answer to make sure it’s sensible.

Instead, if you force the variable you’re integrating over to be dimensionless:

the integral reduces to something that you can start to wrap your brain around:

Now you have the business end of the integral out front, where you can check the units and the scaling of the answer to see if it makes sense. The integral is also something that is far simpler to evaluate (although in this case, it’s actually not a trivial integration, but at least you can recognize that early and plan on how to deal with it). If you’re in a situation where you have to integrate by parts, the dimensionless integral will save you a world of pain. Even if you make a mistake in evaluating the integral, you’re usually only off by a simple multiplicative factor like pi, or 2. All these things are good.

David Foster Wallace died last night. He was found at home by his wife — apparently he hanged himself. It’s a terrible tragedy for American literature.

Wallace’s big, famous book was of course Infinite Jest, but among his other words was a quirky history of the concept of infinity, Everything and More. Like everything he wrote, it was sprawling and inventive and chock full of discursive footnotes. One such footnote now seems especially poignant:

In modern medical terms, it’s fairly clear that G. F. L. P. Cantor suffered from manic-depressive illness at a time when nobody knew what this was, and that his polar cycles were aggravated by professional stresses and disappointments, of which Cantor had more than his share. Of course, this makes for less interesting flap copy than Genius Driven Mad By Attempts To Grapple With ∞. The truth, though, is that Cantor’s work and its context are so totally interesting and beautiful that there’s no need for breathless Prometheusizing of the poor guy’s life. The real irony is that the view of ∞ as some forbidden zone or road to insanity — which view was very old and powerful and haunted math for 2000+ years — is precisely what Cantor’s own work overturned. Saying that ∞ drove Cantor mad is sort of like mourning St. George’s loss to the dragon; it’s not only wrong but insulting.

Chad laments that we don’t hear that much about the decathlon any more, because Americans aren’t really competitive. I also think it’s a shame, because any sport in which your score can be a complex number deserves more attention.

Yes, it’s true. The decathlon combines ten different track and field events, so to come up with a final score we need some way to tally up all of the individual scores so that each event is of approximately equal importance. You know what that means: an equation. Let’s imagine that you finish the 100 meter dash in 9.9 seconds. Then your score in that event, call it x, is x = 9.9. This corresponds to a number of points, calculated according to the following formulas:

points = α(x₀-x)^β   for track events,

points = α(x-x₀)^β   for field events.

That’s right — power laws! With rather finely-tuned coefficients, although it’s unclear whether they occur naturally in any compactification of string theory. The values of the parameters α, x₀ and β are different for each of the ten events, as this helpful table lifted from Wikipedia shows:

Event	α	x0	β	Units
100 m	25.437	18	1.81	seconds
Long Jump	0.14354	220	1.4	centimeters
Shot Put	51.39	1.5	1.05	meters
High Jump	0.8465	75	1.42	centimeters
400 m	1.53775	82	1.81	seconds
110 m Hurdles	5.74352	28.5	1.92	seconds
Discus Throw	12.91	4	1.1	meters
Pole Vault	0.2797	100	1.35	centimeters
Javelin Throw	10.14	7	1.08	meters
1500 m	0.03768	480	1.85	seconds

The goal, of course, is to get the most points. Note that for track events, your goal is to get a low score x (running fast), so the formula involves (x₀-x); in field events you want a high score (throwing far), so the formula is reversed, (x-x₀). Don’t ask me how they came up with those exponents β.

You might think the mathematics consultants at the International Olympic Committee could tidy things up by just using an absolute value, |x-x₀|^β. But those athletes are no dummies. If you did that, you could start getting great scores by doing really badly! Running the 100 meter dash in 100 seconds would give you 74,000 points, which is kind of unfair. (The world record is 8847.)

However, there remains a lurking danger. What if I did run a 100-second 100 meter dash? Under the current system, my score would be an imaginary number! 61237.4 - 41616.9i, to be precise. I could then argue with perfect justification that the magnitude of my score, |61237.4 - 41616.9i |, is 74,000, and I should win. Even if we just took the real part, I come out ahead. And if those arguments didn’t fly, I could fall back on the perfectly true claim that the complex plane is not uniquely ordered, and I at least deserve a tie.

Don’t be surprised if you see this strategy deployed, if not now, then certainly in 2012.