Circular reasoning

I ran into my former student Troy today, and he mentioned the ending to Carl Sagan’s novel Contact in which the main character discovers that at some point the digits of pi, when written out in base 11 and arranged in a square of the right size, form a perfect circle of ones and zeros. And this is taken as a sign from God.

Troy mentioned this because he recalls having felt a sense of awe at this idea. I told him that I vividly remember this plot point (which does not show up on the film), from the time I first read Contact. Troy told me that I am the first other person he’s encountered who remembers the God-in-a-circle ending – and he has talked about this to many people.

But here’s the interesting thing: I recall it as the moment I stopped feeling good about Carl Sagan. Up until then he had been a hero of mine – the guy who got people interested in science, the intellect behind Cosmos, the bridge builder who was able to shine a light on the rarified world of cutting edge research, and show its beauty to the general populace.

But when I read about a God who encodes a circle in the digits of pi, as a kind of shout-out to whatever intelligent race might be listening, my blood ran cold. I think that I can safely say that, on an intellectual level, it was the single most disgusting and repugnant thing I have ever read. Bear with me here…

What Sagan is positing is that a supreme being, creator not merely of the universe but of all possible universes – we know this because the message is encoded in pi, which has the same meaning in all universes – is resorting to a gimmick, a cheap and irrelevant trick, to get our attention.

It’s as though the supreme creator, author of all that is and could ever be, had scrawled the words “Hi mom!” onto the firmament, or maybe held its hands up in Plato’s shadow to make cool shapes like barking dogs and bouncing bunny rabbits.

If there were an intelligent being responsible for the universe, and for the beauty of mathematics, for the sheer loveliness that is logic and symmetry and universal truth, that being would not be resorting to cheap vaudeville tricks to signal its existence.

What it felt like to me, reading the novel’s lame conclusion, was that Sagan had sketched out evidence for God’s existence by arranging for Him to show up on stage in a porkpie hat, pull His pants down and fart.

Would you want to live in such a universe?

22 Responses to “Circular reasoning”

  1. Doug says:

    My response when I read Contact was to try to imagine how it would be possible for even an omnipotent god to mess with pi in that way. I could imagine messing with the cosmological constant or other things that we know by measurement, but pi is the limit to certain simple infinite sums. I don’t think there’s any way you can go about changing it, without changing all of logic. Pi is simple to express as a program, Pi with a hidden circle is not.
    I think even most theologians would agree that god couldn’t create a logical impossibility.

  2. AFAIK natural scientists are concentrated on “real-world” statistics (to proof their scientific claims they use simple data fitting and correlations etc..). But actually I never heard concepts like set theory (the root of modern mathematics) and number theory were useful to them (I may be plain wrong in some cases here, but as far as I watch my friends making PhD in that kind of sciences I didn’t observe their interest in those things).
    So, what I try to say: Sagan’s mathematical fallacies didn’t surprise me.

    However, I was inspired today by your post and came into idea of a problem (similar to Lobachevsky geometry concept) – maybe it is actually possible to create some weird metric space to actually put a short message into every circle (or a sphere) ? Maybe it could be fun to waste some time thinking about it ?

  3. Ok, I wasted my time a bit.
    Assume newPI = circumference/(2*radius)
    Take for example 2d metric space with maximum norm:
    ||p-q|| = max(|p.x-q.x|,|p.y-q.y|)
    And “circle” with radius=1 will look-like 2×2 square.
    Length of one square side = max(|2|,|0|) = 2
    (measured with the same norm)
    Length of the whole “circle” curve = 4*max(|2|,|0|) = 8
    newPI = 8/2 = 4
    Interestingly if norm is:
    ||p-q|| = sqrt(p.x^2 + p.y^2) + sqrt(q.x^2 + q.y^2)
    newPI = infinity

    Anyway, how about linear interpolation between maximum and euclidean norm ? ; )
    ||p-q|| = max(|p.x-q.x|,|p.y-q.y|)*(1-f) + sqrt((p.x-q.x)^2 + (p.y-q.y)^2)*f
    for f = 1, newPI = 3.14159265…
    for f = 0, newPI = 4
    for 0<f<1: anything in between! :)

  4. Now… just pick appropriate “f” for which newPI fraction starts from “HELLO WORLD” (two digits per letter).

  5. Ok, found it by very rough approximation technique…
    For newPi = 3.7269767679…
    ‘H’=72 ‘E’=69 ‘L’=76 ‘L’=76 ‘O’=79 (ASCII codes)
    “f” is about 0.19
    (note! for 0.19 only ‘H’ will appear, but you got the point,
    with more precise technique it should be relatively easy to
    say if it is 0.19123… or maybe 0.189593.. )
    Pure pseudoscientific sickness ! : )

  6. troy says:

    My God would have no problem pulling down his pants and farting. Heck, if he’s smart enough to encode messages in pi, he has no reason to be embarrassed about a little potty humor…

  7. Troy says:

    It’s been many many years since I read the story, but, I didn’t really take it as a gimmick. Well, obviously a gimmick from Sagan, but, let’s forget about that for a second.

    In the story, ET first made contact by bouncing back a video image of Hitler at the opening ceremony of the Olympics that was imbedded in a radio frequency that was some multiple of either helium or hydrogen (can’t remember which, sorry… )

    This was not the message, but, just a marker to get someone’s attention. After looking more closely at the signal, they found something geometric interlaced with the video. Then after more head scratching, found another message within that one that ended up being a blueprint for a machine that allowed them to communicate (or, create a singularity that allows one to traverse space time through a wormhole and talk directly to ET).

    So, with the end of the book, and the discovery of the circle hidden somewhere deep in Pi, I took this as a marker. Some starting point to find other hidden messages, and perhaps a way of communicating with something powerful enough to create the possibility of the existence of Pi.

    Assuming that Sagan meant this to be a sign from God, let’s assume for a second, that this God made humanity in his image. That would mean God likes a good puzzle the same as the rest of us. If so, wouldn’t it be so much more fun to have a Dan Brown style decyphering field trip that starts hidden in the makeup of how we describe the fundamental laws of the Universe? If God made me in his image, that’s exactly how He’d like to start the journey of discovery. Maybe it was a silly gimmick, like the farting clown that you mention earlier, but, maybe it’s just a marker to something much more profound that we find in a yet-to-be discovered Universe “DNA” that we haven’t found the correct constant for. Maybe the “message” is hidden in all things physical.

    The idea of something being capable of such architecture is really what created the “awe” that you described earlier. You gotta admit, Sagan or no Sagan, it would be a pretty awesome trick… :)

    Perhaps the path to heaven can be found hidden in science… So much for buying that stairway…

  8. Gee! Troy, you seem to not understand what PI really is.
    You can measure any curve out there and it is highly probably that if it has a “curvy appearance”, it length will be irrational number.
    Why? To calculate the first fractional digit you can just approximate that curve with finite number of small straight segments. To get more digits you can approximate that curve with smaller segments etc…
    Now you see, the process is iterative ad infinitum, thus after every measure you will get more and more digits…
    Universe contains all the finite messages possible, it’s a fact,
    but we are the one responsible to give meaning to them!
    The point is: that kind of meaning is totally irrational (amazingly, irrational meaning from irrational numbers).
    Since every possible message is already here, which one you consider the one from God ?
    The one that is written down on your forehead or something ?

  9. Oops, forget about my curve measuring and irrational numbers (in general, it’s bullshit), but everything that I said from “Universe contains all the finite messages” is ok.

  10. troy says:

    it’s written on my forehead? clearly you have spent too much time contemplating Uranus…

  11. Excuse me ? ; )
    With all the respect, I’m not the one who was suggesting that inside Uranus might be some hidden DNA ; P

  12. troy says:

    A wise man once said, “never argue with an idiot. Eventually, nobody can tell the difference.”

  13. Do you feel offended by me ? I didn’t mean it, really.
    The point of my “forehead” example was to show that we don’t know any criteria what information is a “marker” and what is not.
    Which subsequence of PI digits you considered as a “marker” to something more profound ? I think it is just us that puts any meaning to those subsequences / digits.
    You can find the same meaning in skin-tone variations on your forehead
    (and here goes my illustrative example that offended you) or those famous life-lines on your hand (fortune tellers used to give meaning to them).
    Anyway, I’m not a native english speaker, I don’t have good intuition what is offensive in USA and what is not (and my grammar may also look ‘handicapped’ to you).
    And you were surely “contemplating Uranus”, by fantasising about physics.. I actually find nothing wrong with that – it’s on par with pseudoscience but (as I said) you were just suggesting that it might be the case, hypothesising, it’s ok. But it can be called “contemplating Uranus” – am I right ? Or I don’t understand this metaphor good enough ?

  14. admin says:

    Wow. I go away from the computer for a few hours and come back to a flame war. I think if you both read back to the earlier emails you’ll see that both of you had really interesting things to say here (before this all degenerated).

    Tomasz, your idea to create an alternate, if slightly warped, universe (literally!) in which pi indeed encodes any desired statistically improbable message is just awesomely cool. I think this may be the first time anybody has ever had this particular idea. :-) But I think you were unfair in suggesting that Troy doesn’t know what pi is. Troy knows perfectly well what pi is. He was just having an entirely different conversation, grooving on Sagan’s concept of divinity as a nested sequence of intellectual challenges.

    Troy, I completely get your enthusiasm for Sagan’s ideas about a kind of recursive “uplift the race”, with the path to divinity embodied by a progression of intellectual puzzles and solutions. I agree that it’s a nifty idea. My problem with it is exactly what I think you like about it, and we probably just need to disagree here: The human-centrism of it all – the implicit assumption that the universe was made for and around us humans (ie: an entire nested stack of progressively higher beings working to help us, in a way that it would never occur to us to help other species). I think there’s no right or wrong in this – it comes down to what first principles one follows.

    I also liked your first comment a lot. It made me laugh out loud, and it had quite a bit of Buddhist wisdom in it – which is impressive, because I’m pretty sure you’re not Buddhist!

  15. Thanks Ken! I tried my best on wasting my time yestarday :)
    I even added a blog entry about this idea – you can click on my name to get there (its called “Message From The Alien God”).
    And yes, I might be unfair to Troy, actually I was also thinking about Universe ‘DNA’ idea, but I thought about ‘DNA’ in every particle, also shared between particles so physics laws can evolve nicely… but you know.. even that idea might be catchy, I am not an expert in physics so it’s more or less “contemplating Uranus” :)

  16. Ken –

    Long time listener, first time caller. :)

    I think there’s something really important that’s lost in translation here. I’m a HUGE Sagan fan as well and I hope that I can shed a little light on what I got from the “pi reveal” moment at the end of the book, it’s fairly straightforward.

    In a nutshell, it’s this: the race that made the “subway” system was capable of modifying the structure of the universe at a very deep level. In order to leave a mark for emerging races to find, they hacked into the mathematical subsystem, if you will, of the universe itself. This concept is what inspired the current ETs (the ones that sent the message that Earth rec’d) to try to communicate in the same sort of way with new emerging races.

    When Ellie encounters the ETs they explain this, in a way, but leave her (and the reader) to discover it on her (their) own, and close out her story.

    I just wanted to take a second to try to explain how I took it, it makes me sad that Sagan’s brilliance as an atheist and as a thinker would be snuffed out for you by a small misunderstanding. :)

    Thanks for listening!

    – John

  17. admin says:

    Hey John. It’s good to get a comment on this topic from pi studios. 😉

    OK, I get it now. If we back off from “God” concepts and look at this as races communicating with each other, then the act makes sense from an anthropological perspective.

    I still have serious issues with this particular way of “hacking into the mathematical subsystem”. As Doug pointed out in the first comment above, what could it possibly mean to hack into pi – a fundamental mathematical constant? To me that’s a little like hacking into “1 + 1 = 2” so that it becomes “1 + 1 = 3”: Essentially a contradiction on the face of it.

    On the other hand, if we forget that inconvenient point, I certainly see that the plot point works as a parable of vastly different races communicating – through the universal language of mathematics. I just wish the particular mechanism Sagan was proposing for this purpose had not been so downright dumb, from a mathematical perspective.

  18. it's-just-fiction-novel says:

    wow! i remember seeing this movie about this guy who was told that pro wrestling is not real! he shouted back IT IS REAL!! c’mon, it’s a fiction novel.

  19. admin says:

    That’s a valid point to raise, but I don’t agree. Sagan was clearly writing a “novel of ideas”, and his ideas deserve to be taken seriously, even if we disagree with them. Voltaire’s “Candide” was fiction, as was almost everything written by Ayn Rand. Abraham Lincoln credited Harriet Beacher Stowe’s “Uncle Tom’s Cabin” – a fiction novel – as one of the prime forces that ignited the American Civil War, by shifting public perception of slavery.

  20. barking dogs says:

    barking dogs…

    I didn’t realise there were so many articles online about Circular reasoning ” Ken’s Blog, it’s great to see that barking dogs is recognised as serious subject….

  21. barking dogs says:

    barking dogs…

    We all talk far too much, especially now we can like this, virtually anonymous ranting! I don’t think Circular reasoning ” Ken’s Blog fits in to that category but I’m coming across more and more that are while I’m looking for info on barking dogs…

  22. barking dogs says:

    barking dogs…

    On Friday I met up with some friends who knew very little about barking dogs, I was quite surprised by this as its a widely discussed topic on the net. I challenged them to go home, do a search and then tell me they don’t know anything!…

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