You know, I look at the title I just wrote and it makes it seem as though this will be a monster post of amazing depth and insight. Don’t get your hopes up. 🙂 I am busy working on a couple of Tomorrow’s World scripts for taping next week (one on the biblical “Man of Sin” and one about the observance of Easter), but it helps to unknot the brain every few moments by doing something else. This post will be one of those “something elses” and a nice brain stretch — a standing up, stretching the legs, and walking around before sitting down to hammer away at the task at hand some more.
The Alan Turin biopic coming out — “The Imitation Game” — got my attention when I first saw the initial trailer. It comes across like a WWII intellectual thriller and character study, where viewers will watch Benedict Cumberbatch play the role of Alan Turning as he and others crack the Enigma code. Turing was an interesting figure, one which current movie tastes certainly makes attractive for cinema (more on that in a moment), and the idea of seeing a movie depicting the man who laid the foundation for modern computing and, truly, so much more (as I will also get to in a moment) is appealing.
And yet it isn’t. Alan Turing was also an unrepentant and publicly professed homosexual in a time when such activity was illegal. The talk I hear concerning the movie, which premiers November 28, is that it is an “important story for our times” (or something like that), and such language, given its subject matter, suggests to me that much will be made of Turing’s sexuality and how someone so crucial to victory in the war–someone so gifted, etc.–was persecuted for simply “loving differently.” Whatever. Hollywood is very good at crafting stories influencing us to admire the heroes it presents us with in such a manner that our actively cultivated admiration may cause us to overlook whatever chosen element of immorality they are trying to change our minds on. It would be as if the Bible were rewritten by an author hoping to use David’s virtues to get us to think less critically of adultery, as opposed to the Bible’s actual approach, which is to present its flawed heroes as just that: flawed heroes. And recognizing someone as a flawed hero requires one to recognize flaws for what they are: flaws.
Alan Turing’s immoral sexual behavior was a flaw of character. It does not diminish the greatness of his intellect or insight. However, neither does the greatness of his intellect or insight diminish the immorality of his sexual behavior. I suspect that Hollywood is hoping that we will ignore the second of these facts.
So, I’m not as interested in seeing the movie as I otherwise would be. And why would I be interested in seeing the movie? Because I think Turing’s work, like Kurt Gödel’s and, more recently, Gregory Chaitan’s, has played a significant role in changing not only how mathematics is seen, but how reality is seen.
I’ve mentioned Gödel before — actually, not on the blog, I think, but in a telecast: “What Is Truth?” Don’t have 28 minutes to watch? We also made a TW Short version of “What Is Truth?” that is only 3 minutes long and an article of the same title. Actually, Kurt Gödel is mentioned in all three incarnations of that work, and including a mention of his “logical nuclear bomb” is one of my all time “feel good moments” concerning my work in our media. (And kudos to our video editors, who also allowed me to get images of the Fundamental Theorem of Calculus in there.)
That “logical nuclear bomb” was his work on the incompleteness of mathematics, demonstrating mathematically that not all mathematical statements can be proven, nor can the consistency of mathematics be proven mathematically. (That’s my own summary–forgive me for washing over details for those who are nitpickers.) I’ve long thought it fascinating since I first saw the result mentioned in a PBS program as a child (probably a NOVA episode, but I am not sure). However, an old OMNI magazine article (anyone remember OMNI?) pressed me to consider what the result might be saying about reality, given the intimate connection between reality and mathematics — an intimacy deeper than that of the physical sciences, since it is the relationship on which the sciences depend.
I’m currently enjoying Chaitan’s popular book Meta Maths: The Quest for Omega, which talks about the author’s own fascination with Gödel’s work and his extension of Turing’s discoveries concerning the halting problem (i.e., the question of whether or not there would ever be a way to predict which programs, out of all possible programs, a computer could run that would come to a stopping point versus running forever without stopping; turns out there is no way to do this for all programs). The point of the book is to discuss the implications of the number omega, defined to be the probability of a randomly constructed program of halting. It is a well-defined number that surely exists, and yet no computer will ever be able to calculate it — not because of limits to memory, computing power, programming language, etc., but because it is actually, in its existence, impossible to compute. Its every digit, in a sense, represents a mathematical truth that mathematics cannot determine, though the existence of the number is well established.
Among the things I am enjoying about the book is that Chaitan discusses thoughts that have been rattling around in my noggin for a few years, now (though he does so with intelligence, experience, and insight, where as my thoughts have been characterized by more of a dull hum…). For instance, do what we call the “real numbers” exist in the world? I don’t mean that in the sense of Platonism (i.e., is there some sort of abstract “reality” where these things exist in a non-physical sense), but, rather, is there any real physical representation of them anywhere? We can take a square that is exactly 1 meter by 1 meter in dimension, and its diagonal would be the square root of 2 meters. And we can take a circle that is exactly 1 meter in diameter and its circumference would be π (pi) meters. So, surely, the square root of 2 and π are things that exist in reality… except that there is no square in existence where the sides are exactly 1 meter each, nor is there a circle in existence with a diameter of exactly 1 meter. Can such numbers still be said to exist?
OK, just imagine the points invisibly in space, without a physical object assigned. Can’t we define such numbers by these invisible distances between these infinitely small points, which, though conceptual, surely do exist as locations in real space? No, not necessarily. Evidence, by some accounts, continues to mount that our physical reality is not continuous like what we imagine the line of a perfect circle to be but discrete and made up of “bits” like a circle on a computer screen looks when you look close enough to notice the pixels making up the image. Life (and reality) would not be a continuous flow, but a passing from frame to frame, like a strip of film showing in a movie theater–an illusion of continuous movement but actually a series of stills shown in rapid succession.
If there is no true continuity to existence, if all is discrete, then there is no room for infinite strings of digits in reality. And without infinite strings of digits, the vast majority of real numbers on the number line disappear into nothingness — including our favorites, like π and e. They remain only as “useful fictions” that allow us to use an imaginary continuity to model a very discrete reality.
Leaving earth behind, might there be a perfect circle in heaven, or a perfect square? Since the word “perfect” here reflects not an actual perfection of morality or existence but, rather, conformity to an idea than man has defined, I’m not so sure there are.
My old Platonism is looking pretty tattered these days. 🙂 As much as I love Cantor’s work and believe there is real value to it, more and more I am beginning to think of Leopold Kronecker’s famous statement “God made the natural numbers; all else is the work of man” as being possibly true at more levels than I had ever given it credit for.
Anyway, that sort of stuff, among other things, is what comes to mind when I hear of Alan Turing. And I look forward to the general resurrection when, if I can be so indulged at that time, maybe I can see Gödel, Turing, Cantor, and Kronecker at tea together not only discussing such insights but comparing and contrasting their thoughts with a revealed view of reality, all while having the privilege of helping them to get to truly know the Author of all they had been studying.
Wow — I really wandered around in this one, huh? Probably fell into a few ditches, too. Looking back, I read some of what I wrote and think, “Just what was in this salted caramel mocha, anyway?” If you read this far without falling asleep or getting a headache, congratulations! This probably hasn’t been the best expression of my thoughts on these things, but it still feels good to get it into writing in some way. I might try and write about these things again in the future — I’ve been wanting to write about how I first began to lean toward the belief that zero might not actually be a number in a very real sense, and if some of you out there have been having a hard time falling asleep, let me know and I will go there for you. 🙂 And the break has been nice — now, back to the scripts!