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Infinity and the Mind - Queue
January 11th, 2006
02:51 am

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Infinity and the Mind
Rudy Rucker was mentioned in a meeting at work last week. It was also mentioned that my bosses know him, seeing as they published his book Infinity and the Mind. That title seemed familiar to me. I checked when I got home, and, lo, it was indeed the book that hrafn got me years ago that I never read. After pulling it off of the shelf, I set it on the table (I'm noticing lots of typos, which is understandable since it's almost 3 fucking o'clock in the morning and I'm still up, so all you Queue-Point types should pay attention). Tonight after dinner, I happened to mention in conversation a discussion I had with a college professor who was teaching a statistics and probability course that I was taking. This discussion had to do with his assertion that it was impossible for something to be exactly two inches long and my refusal to accept that assertion. Part of that discussion had to do with the claim that infinitely many zeros could actually add up to something. Anyway, I made a comment that maybe I should go home and read Infinity and the Mind to get some insight into this question. So when I got home, I started reading the book (after a phone conversation). I got sucked into reading the whole first chapter. It's a really interesting book, and absolutely accessible to lots of people, with no complicated numbers or calculations whatsoever. This is all a roundabout way (and I could make it more roundabout, since I remembered exactly how the conversation got around to me talking about my college class) of posing the following query. Let me just quote from the book (each chapter ends with some questions, and the answers are at the back of the book. What follows is the last question from the first chapter, followed by its answer).

Question:
Here is an example of an infinite regress. Suppose that some person wishes to prepare a text in which every appearance of the letters "man" is replaced by the letters "woman." If this is rigidly adhered to, then "man and woman" becomes "woman and wowoman," then "wowoman and wowowoman," and so on. What do you reach in the limit?
Answer:
Full equality: "wowowowowo . . . and wowowowowo . . . ," with an infinite number of wo's in each case.
I understand the bit about how the number of wo's is going to be equal in each of them, how one more than infinity is really the same thing as infinity. What I don't understand is what the fuck happened to the man at the end. I mean, each instance of "man" is replaced with "woman," so it's always still there. I understand the difficulty, though. You can't say that it just appears after the last wo, since there is no last wo. Infinity is not a number that you can count to, so you will never reach the end to be able to say, "This is where we add the man." But, Jesus, it can't just disappear, can it? I mean, is it still there but we just have no way to express it? Or does it really disappear when you perform the infinite regression? And if it disappears, where the fuck does it go? This is really bothering me. Anyone have any insight?

For extra credit, rewrite my entry as if I were not only sleep-deprived but also high.

P.S. Seriously, why doesn't the LJ spellchecker recognize "blockquote" as a legitimate word? Every fucking time I quote something, it complains about that.

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From:prog
Date:January 11th, 2006 09:29 am (UTC)
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I have no issues with the answer but I do with the notation of the answer.

The way I see it (thinking as a programmer, anyway), you basically have two lists that start ("wo", "man") and ("man") and then pitch them at an infinite loop that prepends another "wo" element to each list on every iteration. In my eye, it's not so much the end (and that poor "man") that vanishes into undefinition as the start of the list, which shifts backwards forever.

I'd express the answer as "...wowowowoman and ...wowowowoman" except abusing the mathematical ellipse to refer to the element after it, for lack of knowing a more appropriate symbol.
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From:treacle_well
Date:January 11th, 2006 02:05 pm (UTC)
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Well it was kind of weird when I first hit Reply and got an error message that included the phrase "Infinite loop in style or layer". But that seemed to be temporary.

I don't have any insight really. My thought was that you always have two man's and lots and lots of wo's. If you ended up with just wo's and no man's then you end up in a situation where you can no longer replace man with woman and the cycle would end--hence not infinite.

I don't even understand the question "What do you reach in the limit". If it's infinite then there is no limit, right? But this is probably some mathy thing and I have hardly any training in that area.
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From:pinkfish
Date:January 11th, 2006 02:56 pm (UTC)
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Thanks for asking the right question,treacle_well. You are exactly right to ask, "what does 'in the limit' mean here?", and I think you are right that it is a mathy thing.

I don't have time to go into all the details; but the notion of a limit requires a metric on the space of the things in the series, which in this case are strings. And you need that metric to be defined on the entity that you want to prove is the limit. Interestingly, each entity in the series is a string (i.e., a bunch of characters, one following another). The limiting entity, however, is not (how do you ever get to the "and", if there are an infinite number of "wo"'s before it?

Until the author defines this metric, so that it can be consistently applied to these new objects ("infinite strings"), the notion "in the limit" has no meaning. Perhaps the author did this legwork in the book, but he hasn't in the quote.

So you are perfectly right to be confused; you don't have enough info not to be.
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From:treacle_well
Date:January 11th, 2006 06:26 pm (UTC)
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Thanks.

how do you ever get to the "and", if there are an infinite number of "wo"'s before it?

Ah, of course. I hadn't thought of that. That does make things interesting.
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From:jaq
Date:January 11th, 2006 03:08 pm (UTC)
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The Livejournal style engine uses a different definition of "infinite" than the usual one, which is based on some processing exceeding a time limit - which is why you occasionally see a "Infinite loop in style or layer" for no apparent reason.
From:coorr
Date:January 11th, 2006 05:34 pm (UTC)
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whew... for a moment there I thought they might have solved the halting problem!
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From:queue
Date:January 11th, 2006 05:50 pm (UTC)
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OK, that's just freaky. I went to reply to this comment, and I got the infinite whatever message, too.
From:coorr
Date:January 11th, 2006 05:31 pm (UTC)

like... wow dude.... just... wow...

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"For extra credit, rewrite my entry as if I were not only sleep-deprived but also high."

im sorry... but isn't extra credit supposed to be challenging? or are you going to be testing for "performance enhancing" drugs?

as for where the man goes, its kind of like a constant in a limit... if you think of of this as (wo^n + man) for as n approaches infinity the constant basically drops out of the equation.

maybe its the CS major in me that is just all to willing to drop constants like this.

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From:hrafn
Date:January 11th, 2006 06:10 pm (UTC)
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Rudy Rucker was mentioned in a meeting at work last week. Which was totally freaky, cause, like, I just dreamed about him! Only in the dream, he wasn't mentioned in a meeting, we were, like, at a bus stop or something but I can't really remember what happened cause all I can think about now is the way theese littl specks of light keep flashing out of all of the o's I'm typing which is really distracting
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