Wednesday, April 11, 2007

Can something be bigger than everything?

Ah, fractals... the video eventually zooms in so far that the entire expanse of the image you started with would now be larger than the known universe. (Kind of like powers of ten, but more purely mathematical, and so even "bigger"!) Pretty cool.

There's some fun discussion at Metafilter [hat-tip: Dillon] about how it is that anything could be larger than the entire universe. In response, one comment offered a nice analogy to highlight the merely virtual nature of the fractal space: "If you play Doom 3, the environment is larger than your house. How can that be when the computer is INSIDE your house?" Clearly, the universe can contain representations as of something bigger than the universe actually is, but the representations themselves -- bits in a computer -- have a more modest reality.

Is it our natural tendency to confuse the ontological status of representing and represented things that makes the fractal video so awe-inspiring? After all, in itself the sequence of images seems nothing all that special. But if you interpret them in such a way as to feel almost drawn into contradiction, and led to ponder the mysteries of the universe, that's something else entirely. Then again, perhaps the provided hint of incoherence plays no crucial role here -- it may be enough simply to vividly represent astronomical scales, and let the dwarfing effect run its own course. What do you think?

6 comments:

  1. I think that if you start from two axioms:

    1- Any size plus any other size equals a greater or equal size.
    2- The size of any part is added to the whole

    It's impossible to avoid the conclusion that the whole cannot be bigger than the parts. If one were false because there were somehow negative sizes it's easy to see how parts could be bigger than the whole but I cannot even begin to conceive of "negative size". I suppose one might also be false if there were incomensurable size quantities like imaginary size but I don't think that's got any bearing on the current puzzle. 2- Seems to me to be a matter of definition.

    So I'd say it's pretty clear that any part or parts can never be bigger than the whole.

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  2. "Clearly, the universe can contain representations as of something bigger than the universe actually is..."

    Unless, of course, the universe is actually infinitely large. Right?

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  3. Paintist,
    I don't think so. Cantor's theorem states that some infinite sets are larger than others. In particular, the power set of everything in an infinite universe would be larger than the universe itself.

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  4. Not sure the DOOM 3 example necessarily works without a very restricted definition of what you mean by size. In terms of "information" for example, (which is a "size" in some sense), the house clearly has more information than the game.

    It seems to me that any discussion about the "size" of non-discrete objects will necessarily come to a discussion of the cardinality of infinite sets, as Alex demonstrated above.

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  5. Hmm ..quote " Cantor's theorem states that some infinite sets are larger than others. In particular, the power set of everything in an infinite universe would be larger than the universe itself."

    then the infinite set is the new 'infinite' .. of which there can be a new power etc ..etc.

    If there CAN be a 'infinite' it is ... well ... infinite .

    end of story ... by definition.

    unless of course, one is a philosopher it seems!

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  6. A - your point about information is a good one. I'd been thinking about that too, but I'm not sure I have a solid enough understanding of what "information" is, to apply it to the original case. Do fractals contain infinite information? Or, since it can be condensed down to a single equation, barely any info at all?

    (Though, via an analogue of Timothy's argument, I guess it's trivial that the universe as a whole contains no less information than any part of it, including the fractal displayed on my computer screen.)

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