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Pi and the Movie Mind
A number of recent films deal with mathematics and mathematicians. Can we learn something from them or are they misleading? Peter Stone investigates.
Consider the number pi (π), 3.1415... This number, which equals the ratio of the circumference of a circle to its diameter, has interested philosophers since the days of Pythagoras. He and many other ancient philosophers thought of numbers as having a higher reality above and beyond the sensual world. This idea was plausible for nice neat integers like 2, 6, and 372, but not nearly as plausible for a messy number like pi. Pi is an irrational number, meaning that it cannot be written as the ratio of two whole numbers. It struck the ‘number’ philosophers as disastrous that the pristine higher reality of numbers should contain such a gawky fellow as pi.
In the days of Pythagoras, mathematical concepts such as pi would have been the concern only of a small and discriminating philosophical elite. But times have changed and mathematics has entered popular culture. The results are often not very pretty – much less pretty, even, than the number pi itself.
Consider for example the 1998 movie Pi, directed by Darren Aronofsky. One might expect a movie named after a geometric concept to treat mathematics with the care and sophistication that Pythagoras brought to it. One might expect this…but then again, one might also believe in the Easter Bunny. The two beliefs have more in common than one might think.
Pi tells the story of a mathematician working in number theory. His work attracts the attention of a sect of Orthodox Jews strongly interested in numerology. The mathematician’s computer churns out a 216-digit number that apparently has some special power or significance. (The film never explains what he was trying to do when the number burst forth.) The Orthodox sect believes that one particular 216-digit number, properly translated, is the true name of God, and assumes that the mathematician has found it. The struggle for control of the number coincides with the mathematician’s own descent into madness.
I suppose Pythagoras could not have objected to the film’s efforts to link number theory with the occult. He was, after all, prone to mysticism himself. But Pythagoras would never have made such a fundamental and obvious goof-up when working with numbers as the film does. But not only does Pi make a goof-up regarding its mysterious 216-digit number – it puts this goof-up at the center of one of the film’s climactic scenes.
In this scene, the sect captures the mathematician, and its leader demands that he reveal the number. His response?
“It’s just a number. I’m sure you’ve written down every two hundred sixteen [digit] number. You’ve translated all of them. You’ve intoned them all. Haven’t you? But what’s it gotten you? The number is nothing. It’s the meaning. It’s the syntax. It’s what’s between the numbers. If you could understand, you would. But it’s not for you! I’ve got it. I understand it. I’m going to see it.”
Surely, the mathematician argues, the sect has already tried every 216-digit number, without successfully learning which one encodes the name of God. Therefore, something more is needed – something he has and that he refuses to share with them.
It all sounds very exciting, until one stops and applies even a few grey cells to the situation. Is this mathematician supposed to have memorized the 216-digit number? I once had pi memorized to about 50 decimal places, but it took quite a bit of work. Doing the same with a number four times as large could not be done on the fly by anyone except Dustin Hoffman’s character in Rain Man. And yet the mathematician is never depicted as studying or memorizing the number. Or is the fact that the mathematician saw the number on a computer printout enough? If so, God is setting a pretty low threshold for what He demands from his followers. Regardless of whether the number has been memorized or not, what does it mean to speak of the ‘meaning’ or the ‘syntax’ of a number?
But there is an even more basic problem with this scene, and it’s a mathematical one. The scene depends on the idea that a small sect could write down every 216-digit number, translate them into possible names of God and then speak them so as to get God’s attention. Unfortunately, there is simply no way on God’s green Earth that the sect would ever have been able to do this, as we see if we apply some rudimentary mathematical skills – skills Pythagoras would have had over 2000 years ago – to this claim.
A 216-digit number is big. Really big. And the total number of 216-digit numbers is even bigger: it’s 10216, or a 1 followed by two hundred and sixteen 0s. This is a really really really big number. (This total includes numbers that are 216 digits long only by virtue of having zeroes at the front: eg, the number 1 can be written as a 216-digit number simply by sticking 215 zeroes in front of it. Such numbers are only 216 digits long in a trivial sense. But the point of the exercise is to find the name of God, and the numerical depiction of God’s name might conceivably begin with zeroes. In addition, 90% of all 216-digit numbers are not trivial in this sense, and so I shall stick with this figure.)
How big is 10216? you ask. Well, this is 10216, the number of 216-digit numbers, written out without exponents –
How big is that? It’s one thousand thousand million billion trillion quadrillion quintillion septillion octillion nonillion decillion undecillion. (An undecillion, in case you didn’t know, is a thousand decillions.) That’s big. Really big.
How big? Well, the number of cells in a human brain is estimated to be 1011, at most. The number of stars in our Milky Way galaxy is of the same order of magnitude. The number of stars in the universe may be about 1023. The number of atoms in a human body is somewhere between 1028 and 1029. The estimated number of atoms in the universe is in the range of 1078. All of these numbers are paltry beside 10216, which is millions of billions of trillions times bigger than any of them. In other words, the number of 216-digit numbers is fantastically bigger than any number the human mind ever practically encounters. So big, in fact, that there’s no possible way the sect could have tested every single one of them, as Mr Number Theory so blithely assumes in the scene.
Suppose for example that the sect consists of 1000 zealous Orthodox Jews. Assume that all of these men worked non-stop, 16 hours a day, 365 days a year, testing 216-digit numbers. (Presumably God will forgive them for not resting on the Sabbath.) Also assume that it takes one of these men 10 minutes to write the number down, translate it into a possible name of God, double-check the translation, speak the name, and wait for God’s reaction or lack thereof. In a single year, the sect would test 35,040,000 names – a respectable figure, but only in the millions. Assume now that the sect has been working at it for 6,000 years (ie from the Biblical date of Creation to the present), with each dying member being instantaneously replaced by another: then the total goes up to 210 billion – a paltry 12-digit number. We’ve got undecillions and undecillions of names left to test, boys and girls.
Let’s try relaxing our assumptions beyond any point of sanity. Assume that six billion people (roughly the population of the Earth today) start testing names. Assume that each person can test a thousand names a second, and tests non-stop all day, every day. Also assume that this work has carried on from the beginning of the universe until today – about 15 billion years. Then the total number of names tested by the human race would be about 3 thousand billion billion billion, or 3 nonillion. That’s a 31-digit number, and it’s very impressive – indeed, it is quite a bit larger than the estimated number of stars in the universe. But it’s less than a ten thousandth of a nonillionth of a nonillionth of a nonillionth of a nonillionth of a nonillionth of a nonillionth of one percent of the total number of 216-digit numbers.
One does not require cutting-edge mathematics to recognize what a transparent howler Pi has committed here. Indeed, it doesn’t really even require high school level algebra. One need not even do the math to understand the intuition as to why the idea of counting every single 216-digit number is utterly absurd. All that is needed is a sufficient level of comfort with numbers to know the difference between a thousand, a million, a billion, a billion billion, etc. The failure to have some idea of how large or small a number is is a basic sign of mathematical illiteracy, or what mathematician John Allen Paulos dubbed innumeracy. To borrow a line from Paulos, what the movie Pi has done is “analogous to a sign at the entrance to the Lincoln Tunnel stating: New York, population more than 6; or McDonald’s proudly announcing that they’ve sold more than 120 hamburgers” – only worse. Much worse.
Pi was an artsy film intended for an intellectual audience. By the standard of artsy films, there is no question that it must be judged a success, garnering numerous awards and favorable reviews, and currently enjoying a 7.5 rating out of 10 at the Internet Movie Database, imdb.com. But there is no question that Pi makes a serious mathematical blunder at a crucial point in the development of its plot. Either the makers of Pi were completely oblivious to the level of their innumeracy, or else they were aware of the problem but decided that their (allegedly sophisticated) audience would not be, by and large. If the latter is the case, the sad fact is that judging by the mathematically uncritical reception Pi received, they were right.
It would be wrong to beat up solely on Pi. It is intellectually no better, but no worse, than many recent films involving mathematics and mathematicians, such as Good Will Hunting (1997), A Beautiful Mind (2001), and Proof (2005). All of these films treat mathematics in ways that would make Pythagoras cringe. For example, A Beautiful Mind depicts what is supposed to be a major moment for the mathematician John Nash. In this scene, Nash receives an epiphany that makes possible his most famous result, a result that revolutionized the branch of mathematics known as game theory. But the cinematic epiphany gets the heart of Nash’s result pretty much wrong. It might be good cinema to say that Nash proved “Adam Smith was wrong” in what he said about competition, but it’s bad mathematics. (For a critique of A Beautiful Mind’s treatment of Nash’s result by an economist well-versed in game theory, see Steven Landsburg, ‘Mindless’, Wall Street Journal, February 22, 2002.) The film compounds this error by depicting what happens when Nash turns his epiphany into a paper. Nash shows his paper to a professor at Princeton, and the paper he shows is in fact a real paper by the real Nash – but it’s the wrong paper! The paper with a cameo in the film was an earlier paper dealing with a different subfield of game theory.
Mathematics is receiving more attention in popular culture than ever before, thanks to such movies. But ironically, all the contemporary motion pictures depicting mathematicians and their work display a profound innumeracy. This does not add up to a straightforward condemnation of these films; I for one would rather see math depicted badly in films than not at all. Also, the average level of mathematical sophistication among the general public is surely as low as that of films like A Beautiful Mind or Pi, if not lower, and it would be a stretch to blame these films for popular innumeracy. Instead, there is an important reciprocal effect at work here. While filmmakers are no doubt responsible for the films they make, and the effect those films have on the general public, they also have no choice but to respond to public taste. And if the public does not want to watch mathematically literate films, then filmmakers would be foolish to act otherwise.
I once had the chance to ask Sylvia Nasar, who wrote the book upon which the film A Beautiful Mind was based, what she thought of the way the film dealt with mathematics. She replied that she was quite pleased with the film’s treatment of Nash’s work, given the difficulty of getting an audience to pay attention to mathematics. This problem, she pointed out, is acute for authors – she even invoked Stephen Hawking’s famous quip that each equation included in a book halves its sales – but it is even more serious for filmmakers. After all, most people expend far less mental effort watching a movie than they do reading a book. Moviegoers can get quite obstreperous if asked to exert themselves.
Nevertheless, this fact hardly absolves filmmakers of all responsibility for their bad treatment of mathematics. While the proof of John Nash’s most famous result is highly technical, the result itself can be stated without technical notation, and can be understood by any layperson willing to expend a little mental energy. (See Morton D. Davis, Game Theory: A Nontechnical Introduction, revised edn, Basic Books 1983). And even if filmmakers cannot get away with saying a great deal about mathematics to their innumerate audiences, they can at least avoid making mistakes in what they do say about the subject. It is one thing for A Beautiful Mind or Pi to avoid delving deep into game or number theory; it is quite another for them to completely misstate Nash’s result, or to make wildly inaccurate assumptions about the number of 216-digit numbers. Filmmakers can always avoid saying wrong things, even if their audiences do not let them say much. In short, filmmakers can and should at least adhere to a form of Hippocratic Oath: when it comes to public opinion about ideas and theories – first do no harm.
It’s not impossible to imagine a movie dealing with mathematics that is accessible and yet smart enough to get its facts right. Such a movie could be profoundly educative and yet entertaining at the same time. Something like what I have in mind happens in Michael Frayn’s play Copenhagen, which features Neils Bohr and Werner Heisenberg discussing quantum physics at a level that non-physicists can understand. If this can be done for quantum physics, it surely can be done for game theory, number theory, or other branches of mathematics. Movies that dealt with mathematical topics in a more mature way – in a way that demonstrated basic numeracy – could help make such numeracy a widespread reality. How important such a change would be depends upon how serious a problem one takes popular innumeracy to be. I for one would like to think that Pythagoras and his philosophical descendents would favor any change that might help the general public onto the path to truth. An important stop along that path is the study of mathematics.
© Peter Stone 2007
Peter Stone is an Assistant Professor of Political Science at Stanford University. He is active in a number of philosophical organizations, such as the Bertrand Russell Society.