Discussion Board

Topic: Universal Library & Fermat's last theorem

From: Robert Danforth
Location: Orlando Florida
Date: 09/02/2008

As I bumped into your Google talk you were discussing the inaccessibility of Pi except by a description of how to get there. I found myself wanting to crawl through the screen and ask about a minor obsession of mine that no real number is accessible except as a manipulation of integers, that could only be discovered in the direct reversal of that manipulation.

This means that 1. all mathematics is just a tide pools next to a vast ocean of numbers that cannot be accessed 2 was the insight Fermat had when he wrote his famous theorem.

So while your library would not contain Pi it would also not be able to contain any number beyond integers and the manipulation of integers, just as it would to describe Pi.

There is a problem here in that actual reality DOES contain all real numbers (and more?) and therefore no measurement can acquire more than an approximation of that reality which of necessity leads to "error" and the butterfly effect (an issue similar looking but quite different than the quantum issues)

I think this also shines a light on the disconnect between reality and any description of reality that must always be an approximation. This is what I think of as the non boring part of fractals, particularly as random and scalar issues are thrown in.

Re: Universal Library & Fermat's last theorem

From: Greg Bear
Date: 09/02/2008

Good to hear from you, Robert! Very interesting questions here. Sufficiently large numbers--and infinitely long numbers--might be compared to infinitely large continuous texts, which are also not contained in the Universal Library, but might be reconstructed by rearranging strings or "volumes" of text. The point well taken here is that any sufficiently complicated reality cannot be encompassed by (or reconstructed from) any description or theorem--or, as Wittgenstein said, (and I quote in the novel) "The map is not the territory." So--was Polybiblios ultimately deluded in his quest...?

Re: Universal Library & Fermat's last theorem

From: Robert Danforth
Location: Orlando Florida
Date: 09/02/2008

My deeper point was that no finite arithmetic can handle more than a scattering of infinite real numbers. You solve an equation and discover a real number that has the first five digits equal to a multiple of Pi (or the root of 2,3, etc) and you know immediately that it will continue to match at the 8th digit and thousandth digit and ten thousandth.

Why not a match at all but one in five hundred digits? or even one in 20? The reason can only be that those other numbers are inaccessible to arithmetic, except by adding another real number. Once that is done the number is again insoluble except by direct back pedal.

Since that is what happens in quadratic equations above the level of the square you have arrived directly at Fermat's theorem.

So without the arithmetic (much less the computer) capable of accurately manipulating any set of real numbers there can only be rough approximations and no matter how many decimals the number, the butterfly effect will await in the next.

Re: Universal Library & Fermat's last theorem

From: patrick
Location:
Date: 09/04/2008

I dunnooo. Simultaneously trying to discourage any idea of anthropic principle, I'm wondering if there is some anthropo-centric bias in the conceptualism(s) above.

Re: Universal Library & Fermat's last theorem

From: Greg Bear
Date: 09/05/2008

That's a very big question. Are numbers human? Is there another way to quantify, find relations, and simulate precise actions based on formally abstracted data?


Is counting metaphysically fundamental, and universal to all problem-solving systems?

In ANVIL OF STARS, the Brothers' idea of mathematics involved "smears" and not individual numbers. Calculus without the inhibition of integers. More of an analog system, less digital.

Re: Universal Library & Fermat's last theorem

From: patrick
Location:
Date: 09/05/2008

Well, that is one way to think of it. There is the phenomenon of analog - yet our modern technology is largely, if not entirely, digital in function. The particle level is quantized, so one might assume that operating at such a level would ultimately mean digital operation....but would it?

There is a wave nature to energy. Something comes to mind - that of the contrast between Hertzian and transverse (scalar) waves. Hmmmm.

In any case, perhaps at least a synthesis of analytic and metaphorical?

And, lastly, to your first paragraph:

- numbers are a convention. It's not (at last, yet) relevant to wonder if they're uniquely human in conception.

- quantification is abstracted. In a context, however, it connotes qualification -- as the identity of a thing is its function/operation - vs what we might call it/it's social function.

- simulation isn't always necessary. There are programs that create output at runtime, given certain parametres and values. A recent one I came across is a digital animation program. I'm thinking this idea will proliferate.

Re: Universal Library & Fermat's last theorem

From: Bill Goodwin
Location: Los Angeles
Date: 09/06/2008

Is this a good time to mention Reuben Hirsch and his humanist perspective on mathematics?

Taking my own crackpot off the back burner:

You can't get away from the anthropic element. Place mathematics in the Platonic realm where it exists independently of "us" and you come up against dualism in another of its guises, which (in my opinion) is worse. How are number (generality) and matter (specificity) related? The only answer seems to be to locate consciousness (yours, mine) at The End Of Time, where these ostensibly separate systems converge. Matter is directed forwards (hence our divergent selves), mind rearwards (hence the "backpedaling," hence we are able to comprehend each other) and the only duality is an epistemological one.

All very mystical but there it is. Intelligence digital, consciousness analog (concerned with meaning, which always slips through terms). I actually find myself alarmed at the runaway digitalization of our culture. Are kids as conscious as we are (were)? The hobby shops are closing. Action figures have become avatars. Is there a historicity to consciousness? Did it mean the same thing to a medieval monk to be conscious as it does to me or will to my neurally augmented descendant? Did the (pick one) republican/democratic convention sound like a beehive to you? Will we even know it when the eidolons nooify?

Synthesis of analytical and metaphorical--yes! 20th-century philosophy was a (perhaps necessary) wasteland. Wittgenstein et al. overreacting to mechanical marvels. Time to go back to Goethe and his subjectivism. End of sermon.

Re: Universal Library & Fermat's last theorem

From: Greg Bear
Date: 09/09/2008

Leibnitz was quite charmed by his binary number system, and believed that it accurately described God's method in creating the universe: 1's (God) and 0's (nothing) coming together to make number, which underlies the universe. Pretty idea... Who'd have thought that God and nothing would team up to make the computer work?

And who supplied the venture capital for that project?

Re: Universal Library & Fermat's last theorem

From: Bill Goodwin
Location: Los Angeles
Date: 09/10/2008

Sounds similar to Goethe's theory that colors are created by the tension when darkness and white light meet.

And conversely, if God is NOTHING, then nothing is SOMETHING. And so secular science must yield up the quantum vacuum. What fun! Supress the demiurge on one side, it pops up on the other.

Leibniz was also interested in the I Ching, which posits a dynamic balance of opposites. His Characteristica Universalis (a sort of 17th-century XML) was concieved as inseperable from a universal compendium of all knowledge-- much like Polybiblios' Babel in City at the End of Time. You've got artificial languages in Moving Mars, I recall--a thread here?

Hey! Nothingness is the perfect mirror. God teamed up with himself to make the universe work.

As for the venture capital...Cthulu? (Last words before the Big Bang: "Wait, don't touch that--")

Speaking of which, they're reving up the Large Hadron Collider for the first time tonight.

I hope you get this...

Re: Universal Library & Fermat's last theorem

From: Greg Bear
Date: 09/15/2008

Chthulhu lies waiting! The devil is in the details, you know...

Re: Universal Library & Fermat's last theorem

From: patrick
Location:
Date: 09/15/2008

Okay, boys. No, Bill. Anthropism is a by-product of an identity based in culture and, more fundamentally, gender. There's your answer. Most of Western (and particualarly modern and contemporary) philosophy, though entertaining, is essentially the indignant cry of biological self-importance.

As for 0s and 1s, well: 1) 'nothing' is determined by context, hence it's merely a convention; and, 2) zero is the same entity, albeit in the digital realm (one might say it isn't even an integer), and it has a twin - infinity.

Re: Universal Library & Fermat's last theorem

From: Greg Bear
Date: 09/25/2008

Might be worthwhile to explore the idea of non-biological "observers," or systems which can track and respond to minute physical processes, and thus lock them into reality. I believe Heinz Pagels explored one possible mechanism for reality to continue to exist without human or sentient observers. Hmm... is zero infinity turned inside out? I have NO idea what that means.

Re: Universal Library & Fermat's last theorem

From: Robert Danforth
Location: florida
Date: 09/25/2008

I seem to recall a story, not sure all the details are correct. Zeno was sailing to Alexandria, and proposed that motion was impossible because to reach any point the halfway point would have to be reached first, and of course there was an infinite regression of halfway points. Supposedly they threw him overboard.(I may be mixing stories, not sure)

However while passing all those halfway points is trivial, measuring to an infinite level of precision and calculating a change based on each one is not. That is the nature of the butterfly effect, and why such a perfect prediction system is as impossible as perpetual motion etc.

I believe that was also what Fermat was thinking about.

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