Category: math

Infinitely fine

In 1910, Henry James published an essay titled “Is There Life After Death?”

Probably not, is the answer he starts with. For one thing, a lot of people don’t even seem to care whether they have immortal souls. “How can there be a personal and a differentiated life ‘after,’ ” he sniffs, “for those for whom there has been so little of one before?” The consciousnesses of people who don’t care about their souls are almost certainly too boring to need to be preserved. In James’s opinion, in fact, it’s not clear why such people exist even once, unless they do so for the same purpose “slugs and jellyfish” do, to provide “amusement” and to represent “wealth and variety” for others in the universe who are more spiritually inclined.

This is divinely snobbish, but as a logical argument, hard to take very seriously. James goes on, however, to raise a more substantive objection to the afterlife: materialism. Science teaches that “we are abjectly and inveterately shut up in our material organs,” he writes, and most people’s experience confirms this. As we get older, we become more and more aware of the limits and compromises that our bodies inflict on our minds, and in the end, we are forced to acknowledge that “even at our highest flights of personality,” we are “the very stuff of the abject actual.” Our hopes and passions turn out to be “but flowers sprouting in that eminently and infinitely diggable soil.” Perhaps the most dismaying confirmation comes when we observe that people “die piecemeal.” After a certain age, most of us become aware of the encroachment of partial death even in ourselves. And if it’s possible for our personality to decline “by inches”—in my own case, the ability to recall people’s names is fast becoming a lost cause—it’s hard to sustain a belief that the mind is ineffable, and hard to resist the conclusion that mind is just an effect produced by “the poor palpable, ponderable, probeable, laboratory-brain.”

Then there’s the paucity of ghosts. This sounds pretty eccentric, as arguments against the afterlife go, but hey, we’re talking about Henry James. What he cares about is personality, and what is a ghost but the essence of the personal? In his nonfiction, it turns out, James doesn’t believe in ghosts, however persuasively he may have written about them in his fiction. (He admits he finds mediums and trances interesting, but only as evidence of the personalities of the mediums.) In James’s opinion, no one has ever come back from death for a visit. Which to James’s mind, more or less proves that no one continues to exist as an individual person after death. Because: No one came back? Really? No one? Could the afterlife be so overwhelmingly interesting that absolutely no one who gets there is willing to spare a moment to check in on the people they used to care so much about? If another world does exist, could the border between it and our world be so impenetrable that none of the greatest and bravest souls who ever lived can figure out a way to cross it, even briefly? And even if we the living are relatively speaking very boring, and even if the obstacles to returning are very high, isn’t a universal failure to come back just incredibly rude?

We think of the particular cases of those who could have been backed, as we call it, not to fail, on occasion, of somehow reaching us. We recall the forces of passion, of reason, of personality, that lived in them, and what such forces had made them, to our sight, capable of; and then we say, conclusively, “Talk of triumphant identity if they, wanting to triumph, haven’t done it!”

If you’re dead to the appeal of society to that extent, then, in Henry James’s opinion, you must really be dead.

In the second part of his essay, however, James changes his mind. He writes that the change began for him with a suspicion that he wasn’t quite sure about the afterlife, after all. He felt he needed to investigate by “trying to take the measure of my consciousness”—only to discover that it wasn’t at all clear that consciousness could be measured, that it had a beginning or an end. Mind seemed to him to be at least as large as the world that contained it, any angle of which it could observe and reflect, at will. “The more and the more one asked of it,” James writes, “the more and the more it appeared to give.” It seemed capable of giving more, in fact, than James thought he could come to the end of in the course of even the longest lifetime. His work as a novelist, in particular, brought this home to him:

. . . it is above all as an artist that I appreciate this beautiful and enjoyable independence of thought and more especially this assault of the boundlessly multiplied personal relation (my own), which carries me beyond even any “profoundest” observation of this world whatever, and any mortal adventure, and refers me to realizations I am condemned as yet but to dream of.

If consciousness is infinite, how can it come to an end? Once we’ve been given a taste of eternity, to yank it away would be, James writes, “a practical joke of the lowest description.” Fate couldn’t possibly be so “vulgar.”

I wish it were true that the universe couldn’t possibly be vulgar, but I’m not sure. I have to admit, however, that writing has sometimes brought me to a similar sense of consciousness as limitless. Writing about life is a strange activity. You take a portion of life that you have lived, and you spend a second portion remembering and re-experiencing the first, in order to create a representation and share it. Life goes on while you are doing the re-experiencing; life runs away, in fact. And the ratio of the second portion to the first, it soon becomes clear, may be any number. To write my first novel, I drew on a year of my life; the writing itself took more than five years. There are hours that it would be easy for me to spend months writing about. It has often happened that I’ve spent a day writing in my journal about the day before. Mathematically, this incommensurability is suggestive. Any set that can be put into one-to-one correspondence with one of its subsets can be shown to be infinite. As a corollary, therefore, since you can spend as much of your life as you want thinking about as brief a stretch of it as you’re interested in, life must be infinite. It can always be described more thoughtfully, more carefully. There is no limit to how much attention you can pay. Which doesn’t mean, mathematically speaking, alas, that it doesn’t or can’t have a beginning and an end. The moments in a life could be like the points in a line segment, which begins at A and ends at B but can be subdivided to any fineness.

It’s suggestive, too, though in a darker way, that the practice of writing, once capable of convincing James that the soul was eternal, looks likely to become an activity that very few humans will still do at any length, a decade or so from now, thanks to the advent of generative artificial intelligence.

When Gödel did the political math

I spent much of the past week coughing when I should have been sleeping, the only boon of which is that I managed to read Jordan Ellenberg’s lively and instructive How Not to Be Wrong: The Power of Mathematical Thinking. I came across an alarming politics-related anecdote about Kurt Gödel, the mathematician who famously demonstrated that in any formal axiomatic system of arithmetic, there will be some true theorems that can’t be proved to be true.

Apparently, when Gödel was studying for the U.S. citizenship test in 1948, he found what seemed to him a fatal flaw. “The document,” Ellenberg writes, “contained a contradiction that could allow a Fascist dictatorship to take over the country in a perfectly constitutional manner.” For better or worse, the exact nature of this flaw has been lost to posterity, but Gödel was apparently so upset that he couldn’t help but talk about his concern with the judge who examined him on behalf of the Immigration and Naturalization Service, despite the advice of colleagues Albert Einstein and Oskar Morgenstern, who thought he should keep his worry to himself. Years later, in 1971, Morgenstern wrote down his memory of the exchange:

The examiner turned to Gödel and said, Now, Mr. Gödel, where do you come from?

Gödel: Where I come from? Austria.

The examiner: What kind of government did you have in Austria?

Gödel: It was a republic, but the constitution was such that it finally was changed into a dictatorship.

The examiner: Oh! This is very bad. This could not happen in this country.

Gödel: Oh, yes, I can prove it.

The examiner, Morgenstern remembered, “was intelligent enough to quickly quieten Gödel and broke off the examination at this point, greatly to our relief.”

Ellenberg’s source is a webpage at Princeton’s Institute for Advanced Study, a page that unfortunately no longer exists, but there’s an account of Gödel’s immigration exam on page 7 of the spring 2006 issue of the institute’s newsletter, and the writer Jeffrey Kegler has put together a synopsis of the documentary evidence and has shared a scan of Morgenstern’s memorandum.

Fractionated

One of the great challenges of swimming, for me, is remembering how many laps I’ve swum. Mostly I just repeat the number in my head as I swim: “This is lap twelve, this is lap twelve, this is lap twelve,” for example, which is a little boring. I have a few mnemonics. Lap number fifteen, for example, is the lap especially devoted to daydreaming about the snack that I will eat when I get home. To most numbers, however, I don’t have any particular association, and my mind strays.

Yesterday, on lap twenty-five, I strayed into thinking about why the fraction one quarter is written in decimals as 0.25. I had always assumed that the way fractions appear in decimal form is more or less arbitrary, but yesterday, maybe because I was swimming, and therefore thinking about my hands at the same time that I was thinking about decimals, it started to seem a little suspicious to me that one quarter, when written in decimals, should take the form of five times five—and especially suspicious given that humans ordinarily write their numbers in the decimal system (a ten-based system—that is, with ten numerals, 0 through 9) in part because they have five fingers on each hand, and two times five equals ten. Is one quarter a special fraction? Is it a coincidence that one quarter, in decimals, looks like five squared?

I couldn’t figure it out while swimming, but this morning, while walking the dog, I tried again.

Suppose you’re writing numerals in a b-based system. (In the usual, decimal system, b is 10, but b could just as well be 8 or 12 or 47.) And suppose that b is the product of the two smaller numbers, the number 2 and a number that we’ll call f, for fingers. (In the world we live in, b = 10 and f = 5.)

Then here’s another way of asking my question: what does the fraction 1/4 look like when expressed in “decimals” in a b-based system?

Suppose we’re going to need at least two “decimal” places to express the fraction 1/4 in the b-based system. (They shouldn’t really be called “decimal” places, of course, in a non-decimal system, and I’m guessing when I say that two is the number of “decimal” places that we’ll need to shift, but I’m pretty sure my guess is kosher.) In order to “see” the two numerals to the right of the “decimal” place, we need to move the “decimal” place over two notches—in other words, we need to multiply by whatever 100 is in the b-based system. In our usual decimal system, 100 is 100—more properly written, 10010 = 10010. In an 12-based system, 10012 is 14410. In general, 100b = b2.

To find the numerals to the right of the “decimal” point that express the fraction 1/4 in a b-based system, in other words, you need to multiply 1/4 by 100b, or 1/4 x b2.

But b = 2 x f. So the numerals to the right of the “decimal” point that express 1/4 in a b-based system also equal 1/4 x (2f)2, or 1/4 x 4 x f2, or f2.

Not a coincidence, in other words. It’s because humans have five fingers, and because they write their numbers in a system based on ten, which is five times two, that the fraction 1/4 is expressed by the square of five when written in decimals.

What’s more, it’s possible to generalize. If humans had six fingers on each of their two hands and counted in a 12-based system, then the fraction 1/4 would be expressed to the right of the “decimal” point by the same numerals that express 6 squared—3610, or 3012. In other words, 1/4 = 0.3012, and it wouldn’t be a coincidence that 3012 is the square of 612. And if humans had four fingers on each hand and counted in an 8-based system, the fraction 1/4 would be expressed to the right of the “decimal” point by the same numerals that express 4 squared—1610, or 208. In other words, 1/4 = 0.208, and it isn’t a coincidence that 208 is the square of 48.

You can generalize in another direction, too. If you look at the fraction 1/8, and notice that 8 is the cube of 2, you’ll see that its expression in decimals is 0.125, or 5 cubed—not a coincidence, either.

What if humans had h hands, instead of just 2, and counted in a system based on b = h x f? It would still be the case that the fraction 1/h2 will be expressed as f2 in b-decimals. For example, if humans had three hands and four fingers on each hand, and therefore chose to count in twelves, the fraction 1/9 would take the form 0.1412, and it wouldn’t be a coincidence that 9 is the square of 3, and 1412 (a.k.a. 1610) is the square of 4.

I don’t know how a three-handed human would swim, however.

Who’s wonking who?

About a third of the way into Ezra Klein’s new essay “How Politics Makes Us Stupid,” I met a stumbling block. Klein begins his essay by describing a 2013 study that tested whether political affiliation could compromise people’s ability to solve a simple statistical problem. In an experiment, researchers gave some subjects a stats problem about the efficacy of a skin-rash lotion, and others a structurally identical problem about the efficacy of a gun-control law. Here’s Klein’s summary of the results:

Being better at math didn’t just fail to help partisans converge on the right answer. It actually drove them further apart. Partisans with weak math skills were 25 percentage points likelier to get the answer right when it fit their ideology. Partisans with strong math skills were 45 percentage points likelier to get the answer right when it fit their ideology. The smarter the person is, the dumber politics can make them.

Consider how utterly insane that is: being better at math made partisans less likely to solve the problem correctly when solving the problem correctly meant betraying their political instincts. People weren’t reasoning to get the right answer; they were reasoning to get the answer that they wanted to be right.

Something’s not quite right with Klein’s inferences here, I’m pretty sure. Here’s a link to the research paper that Klein is describing: “Motivated Numeracy and Enlightened Self-Government” by Dan Kahan, Ellen Peters, Erica Dawson, and Paul Slovic. And here’s how the original authors phrase the results that have caught Klein’s eye:

On average, the high Numeracy partisan whose political outlooks were affirmed by the data, properly interpreted, was 45 percentage points more likely (± 14, LC = 0.95) to identify the conclusion actually supported by the gun-ban experiment than was the high Numeracy partisan whose political outlooks were affirmed by selecting the incorrect response. The average difference in the case of low Numeracy partisans was 25 percentage points (± 10)—a difference of 20 percentage points (± 16).

Klein has reported the numbers accurately, but his interpretation of them is fallacious. As you can see by comparing Kahan et al.’s words with Klein’s, Klein is correct when he writes that “Partisans with weak math skills were 25 percentage points likelier to get the answer right when it fit their ideology. Partisans with strong math skills were 45 percentage points likelier to get the answer right when it fit their ideology.” But Klein is in error when he adds, “The smarter the person is, the dumber politics can make them.” If higher-numeracy subjects are 45 percent more likely to identify the correct answer when they find it congenial, and lower-numeracy subjects are only 25 percent more likely to do the same under the same conditions, then math skills improve the ability to solve the problem under those conditions by 20 percentage points, as Kahan, Peters, Dawson, and Slovic note. Smarter people are in fact smarter (the trouble is that they only bother to use their smarts to confirm their political bias—more on that in a moment).

Klein also writes, “Being better at math made partisans less likely to solve the problem correctly when solving the problem correctly meant betraying their political instincts.” That’s not an accurate report of Kahan et al.’s results. In their study, being better at math did make partisans a tiny bit more likely to solve the stats problem correctly even when the correct answer contradicted their partisan druthers. (For the evidence, see the dotted blue and solid red curves in the lower graph of figure 6 in Kahan et al.’s paper; the drift is upward in both cases, though it’s an exceptionally modest upward; that is, when solving a puzzle that declares that gun control increases crime, a liberal’s odds go up very slightly as his math skills improve, and so do a conservative’s odds when solving a puzzle that declares that gun control lowers crime.) Kahan et al. didn’t discover that math hurt problem solving. They discovered that math skills helped disproportionately more when the correct answer confirmed the subject’s political biases.

Klein writes that “People weren’t reasoning to get the right answer; they were reasoning to get the answer that they wanted to be right.” In fact, the original researchers’ explanation was a bit more subtle. They noted that an easy wrong answer tempts anyone who first glances at the type of statistics puzzle they chose, and they suggested that when the easy wrong answer confirmed a partisan’s bias, he was more likely to fall for it. Partisans resorted to brain-taxing math skills only when the easy wrong answer contradicted what they hoped to hear.

Kahan et al. did discover that math skills increased polarization. Not polarization in political bias, though: within the experiment’s sample of subjects, polarity in political bias was a given. The polarization that worsened was between likelihood of solving the problem correctly when it confirmed biases and likelihood of solving it correctly when it contradicted biases. Intriguingly, that polarization was not only higher when math skills were higher. It was also higher among conservatives than among liberals. (The evidence is in the lower two graphs in figure 7 of Kahan et al.’s paper. In both graphs, the red bumps are much further from one another than the blue bumps are, which suggests that conservatives’ ability to solve the problem diverges more according to bias than liberals’ ability does.)

Audio files of my 2003 interview with David Foster Wallace

On 17 October 2003, I interviewed David Foster Wallace at New York’s Park South Hotel about his book Everything and More: A Compact History of ∞, which was then just being published. A week later, a condensed and edited version of the interview was published in the Boston Globe, a version that has since been reprinted in Stephen J. Burn’s Conversations with David Foster Wallace.

After Wallace died in September 2008, I went back to my transcript of the original interview and posted on this blog a few passages that I hadn’t been able to shoehorn in to the published version, including some to-and-fro about God and infinity that verged on the mystical. I intended even then to make audio files of the interview available some day, but at the time, I was a bit shy about the fact that during the interview, Wallace briefly turned the tables and spent a few minutes interviewing me (fortunately, he let me turn the tape recorder off for most of those minutes). Also, it turned out to be trickier than I expected to connect an old-fashioned cassette player to a newfangled laptop. In fact I didn’t figure out the proper Radio Shack doohickey until a few days ago.

Here, then, at last, are MP3 files of my interview with Wallace. The first side of the cassette is about 48 minutes long; the second, 33 minutes. The sound quality isn’t great. In both segments, the tape recorder is turned off and on several times, which may be confusing to a listener. If you hear sudden non sequiturs, you’ve probably just passed a silent lacuna of this kind. From time to time, Wallace plays with the conventions of the audio interview by making a gesture that contradicts the words he’s saying aloud; if you hear me laughing even though Wallace doesn’t seem to have said anything funny, that’s probably why. As I explained when I published excerpts of the transcript in 2008, when the first side of the tape ran out, it took me a few minutes to notice, and so I lost what may have been the best part of the exchange, about Godel’s Incompleteness Theorem and the Infinite Jest character Pemulis (I posted my notes about the content of the missing minutes in the 2008 blog post).

These files probably won’t be riveting to listen to unless you’re both a David Foster Wallace fan and a math nerd, and even if you are, the first twenty minutes or so are fairly slow. I confess that I myself can’t any longer follow all the math talk in this interview; even at the time, I was a little out of my depth. But despite these limitations to its appeal, the audio does offer an unedited sample of what Wallace sounded like in conversation, and I hope it will be of interest to some.

Interview with David Foster Wallace, 17 October 2003, side 1 (mp3, 48:08)

Interview with David Foster Wallace, 17 October 2003, side 2 (mp3, 33:20)

UPDATE, 21 February 2017: A reader/listener named Peter Demers volunteered to excise some of the tape hiss from my files and has shared with me new versions. I’m leaving the original MP3s above, but his versions, which are in the M4A (AAC) format, do sound a little sharper:

Interview with David Foster Wallace, 17 October 2003, side 1 (m4a, 47:58)

Interview with David Foster Wallace, 17 October 2003, side 2 (m4a, 33:05)

NOTE, 5 June 2019: I’ve chosen to share these interviews here on my own website, but they remain under copyright. Please feel free to listen to the audio files and to download them for personal use or personal archiving, but please don’t post them on Youtube or anywhere else online. Thanks!