Archive for the ‘math’ Category
A Mathematician’s Apology
August 11th, 2010
I finally read A Mathematician’s Apology, G.H. Hardy’s classic defense of a lifetime dedicated to the study of pure (“impractical”) mathematics. It’s a remarkably sad book, in which Hardy, near the end of his life, famously describes mathematics as a “young man’s pursuit”1 in which the elderly have little to contribute. However, it also contains some really well-composed thoughts:
A man who is always asking, “Is what I do worth while?” and “Am I the right person to do it?” will always be ineffective himself and a discouragement to others. He must shut his eyes a little and think a little more of his subject and himself than they deserve.
The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics. … It may be very hard to define mathematical beauty, but that is just as true of beauty of any kind — we may not know quite what we mean by a beautiful poem, but that does not prevent us from recognizing one when we read it.
1 The usual formulation of Hardy’s rule is that, “if a mathematician’s going to do any significant work, it’ll be done before they’re thirty.” This is true so long as we ignore the later work of Archimedes, Cauchy, Descartes, Euler, Fermat, Frege, Gauss, Hilbert, Newton, Peano, Poincare, Russell, von Neumann, Weierstrass, and most recently Andrew Wiles. I would guess that Hardy’s opinion on the matter was influenced by his relationship with the mathematical prodigy Ramanujan, who died at 33.
Numbers Are New
April 1st, 2010
You’ve probably heard of bands of people in Australia or Amazonia whose concept of number is limited to, “1, 2, 3, 4, many.” Here’s a really good article describing that phenomenon in a lot more detail.
One especially interesting result was the notion that people intuitively distribute numbers on a logarithmic scale rather than a linear one. As it turns out, children do this, too.
It is Pica’s belief that understanding quantities in terms of estimating ratios is a universal human intuition, due to the fact that ratios are much more important for survival in the wild. Historically, faced with a group of adversaries, we needed to know instantly whether there were more of them than us. When we saw two trees, we needed to know instantly which had more fruit hanging from it. In neither case was it necessary to enumerate every enemy or every fruit individually. The crucial thing was to be able to make quick estimates of the relevant amounts and compare them; in other words to make approximations and judge their ratios.
On Finishing Books
March 16th, 2010
I just finished W.G. Sebald’s Austerlitz, which all things considered is a pretty good book, despite my occasional furious claims to the contrary.1 I started it sometime around October, read most of it, and then put it aside. I just finished the last hundred-odd pages a few minutes ago. This seems to be a pattern — I start a book really enthusiastically, then get a little bored or distracted and put it down for awhile. Eventually I get so tired of seeing it sitting in my queue, shamefully reminding me of my miniscule attention span, that I just plant myself down on the couch and force myself to finish it. The satisfaction of finishing the book outweighs the grueling completion process.
This isn’t usually the case, of course. Only for certain books. I’m pretty sure I’d hate reading if this was the normal situation.
Note that the concept of book-graphing has been explored before, which reminds me: I want the book equivalent of a pedometer, so I can settle this graph thing once and for all. Then I want to correlate information about the structure, genre and subject matter of the book with the pace at which I read it.
If I worked at Amazon, I would be spying on customers’ Kindle usage and mining that data so hard.
1 No chapters? Really? Why would you do that, Sebald? I demand discrete chunks.
Logicomix
March 14th, 2010
I’m pretty sure that if you like this blog you’d like a graphic novel biography of Bertrand Russell, which is exactly what Logicomix is.
art+design, books, history, math, old dead white guys | 2 Comments »
Math Ruins Everything
March 12th, 2010
Julia just sent me one of Paul Krugman’s analyses of the causes of the financial meltdown: How Did Economists Get It So Wrong?
The whole article’s good, but I especially love this point:
… economists, as a group, mistook beauty, clad in impressive-looking mathematics, for truth… the central cause of the profession’s failure was the desire for an all-encompassing, intellectually elegant approach that also gave economists a chance to show off their mathematical prowess.
Pointedly ignoring the differences between a beautifully abstract model and the messy underlying reality that it (supposedly) represents? Well I’ve certainly never been guilty of that particular intellectual sin.
Budget Proposal 2011
February 2nd, 2010
The New York Times has a lovely interactive infographic detailing the proposed 2011 budget.
Nomograms
January 10th, 2010
Nomograms (or sometimes nomographs) are graphical single-purpose analog computing devices. They range from the very simple – like the above BMI calculator – to the (often beautifully) complex. Once upon a time they were commonly used for navigation, astronomy, surveying, and countless other things. Now, what with cheap omnipresent digital computers, they’ve fallen into disuse.
Like beautiful math? Need a calendar for 2010? Download a copy of Ron Doerfler’s Graphical Computing Calendar.
We Know You’re Out There, Spiderman
November 22nd, 2009
Abstract
Using absolutely bulletproof science, we demonstrate that 35.3 spidermen are created annually and that hundreds live secretly among us.
Introduction
Prompted by arguments about the possibility of radioactive spidermen living among us, Mr. Harman and I decided to use science to determine how many spidermen (if any) exist on Earth. It’s difficult to extrapolate from the single known instance of a spiderman (hereafter the SKI), but following the example of the Drake equation we’ve developed a predictive formula. Behold the incontrovertible majesty of the Harman-Schwartz equation:
Our equation states that:
N = Pe × fs × fr × fl × fp × fg
where:
Total Population
Note that by using the above formula we’ve only calculated the number of spidermen being generated each year and not the total number of spiderman living on earth at any given time. This can be calculated by the following equation:
Ntot = N(<Ad> – <Ac>)
where:
- Ntot
- is the total number of spidermen living on earth at any given time.
- <Ad>
- is the expected age at which a spiderman dies.
- <Ac>
- is the expected age at which a spiderman is created.
Plugging in the Numbers
- Pe
- The population of the Earth is around 6.67 billion.
- fs
- We estimate that about 1.66 × 10-3% of people are bitten by a spider each year.
- fr
- Between Chernobyl, Hiroshima/Nagasaki, and assorted other tests and accidents, about 2.55% of the land area of the Earth has been irradiated to some degree. We can use this as fr if we assume an evenly distributed spider distribution.
- fl
- The vast majority (about 99.9%) of people survive spider bites, but obviously irradiated spiders are more deadly. Let’s set fl to 50%.
- fp
- Working off the SKI, we’d have to assume that this is 100%. Let’s be conservative, though, and say only a tenth of people bitten by radioactive spiders develop superpowers.
- fg
- We’re totally guessing here and saying that 25% of superpowered radioactive spidermen will dedicate their lives to doing good.
Plugging those figures into the equation, we estimate that on average, 35.3 spidermen are created annually.
- <Ac>
- The median age in the world’s population is 27.5 years, which is what we’re using.
- <Ad>
- This is a controversial term. For the purposes of our study, we’ve made the simplifying assumption that spidermen have an average lifespan equal to the human average (73.1 years). It could be argued for the that spidermen are especially prone to an early violent death, but following the example of the SKI we argue that the rates of violent death and cloning are approximately equal, thereby sidestepping the whole issue.
By plugging these numbers into our final equation, we find that at any given time on Earth, on average there are 1,609 radioactive spidermen living secretly among us. ☐