Is There In Truth no Beauty?

nitro

nitro

"The well-known theorem of Gödel (1931) shows that every system of logic is in a certain sense incomplete, but at the same time it indicates means whereby from a system L of logic a more complete system L may be obtained. By repeating the process we get a sequence L, L1 = L, L2 = L 1 … each more complete than the preceding. A logic Lω may then be constructed in which the provable theorems are the totality of theorems provable with the help of the logics L, L1, L2 , … Proceeding in this way we can associate a system of logic with any constructive ordinal. It may be asked whether such a sequence of logics of this kind is complete in the sense that to any problem A there corresponds an ordinal α such that A is solvable by means of the logic Lα...."

Thus begins probably the most famous paper in computer science, and imo in this sentence lies a great mystery of rational thought.
  1. Why are systems incomplete apriori?
  2. Since 1 is true as proved by Godel, how is it that some mathematical statements, by the addition of more completeness, allow us to solve problems that are unsolvable in the lower system? if one looks at the new axioms, the new problems that it solves are unexpected.
  3. How is the human mind able to leap to these new languages from the [uncountable ?] number of possible languages?
When you look at modern mathematics, the application of 2 is ubiquitous. It was Grothendieck who said it best:

"...There are, of course, times when it does pay to examine the inner workings of things.Jean-Pierre Serre, another titan of modern mathematics with whom Grothendieck had an intimate working relationship, was often, in Grothendieck’s words “the yang to my yin”. If there was a nut to be opened, Grothendieck suggested, Serre would find just the right spot to insert a chisel, he’d strike hard and deftly, and if necessary, he’d repeat the process until the nut cracked open. Grothendieck, by contrast, preferred to immerse the nut in the ocean and let time pass. “The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough.”

I find it fascinating when a problem is redefined in a more general language and its solution is completely trivial. What seemed impossible, in the new point of view seems almost inevitable. Is abstraction the magic that makes this happen? Is moving up the logic ladder moving towards greater and greater abstraction?

What is the relationship between abstraction and concreteness. Does the brain violate the second law of thermodynamics?
 
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If you're interested in this line of reasoning, I suggest reading 'The Emperors New Mind' by Roger Penrose (if you have not already). It's a fascinating subject, my view is that we are simply missing something about the way brains work, something more than computation as we understand it seems to be going on. The ability of humans to see the truth of statements which cannot be formally proved seems a clincher to me.
 
Yes thank you, I read TENM long ago in a galaxy far far away. I put it in the top 10 favorite books of all time in the non-fiction for the layman category. I remember saving a weeks pay when I was a kid to buy it when it came out.

The problem with what Penrose posits is that there is no known mechanism by which the brain can tap into quantum mechanics since it is wet and hot. Recently though, I saw this:

http://www.technologyreview.com/view/422551/quantum-water-discovered-in-carbon-nanotubes/?ref=rss

I agree with your assessment that we are missing something. I will point out though, that people once thought that computers would never beat a human chess player. That the human brain would rein supreme.

Here is why I think Penrose is correct though. The amount of wattage the brain uses is next to nothing. To me that is the smoking gun that what it does is not computational in the classical sense. You don't need to colocate me in cold air conditioned rooms next to a power utility to think.
 
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I read TEHM when I came out as well, I think I was about 20, long time ago.
You're correct, of course we are not privy to the course of technology in the future, we cannot know future developments. However, brains do not intuitively seem like a complexity limit of current technology. Given the difficulty we have in reproducing even the most basic animal behaviors using current techniques.
 
nitro said:
I agree with your assessment that we are missing something. I will point out though, that people once thought that computers would never beat a human chess player. That the human brain would rein supreme.
More...

This is only because it's a fixed landscape as their failing in thought was considering chess as a game of infinite possibilities. It's a finite game, so whoever can compute depth the fastest wins.

However, the human brain *is* capable of reigning supreme in situations that present infinite depth.
 
i960 said:
This is only because it's a fixed landscape as their failing in thought was considering chess as a game of infinite possibilities. It's a finite game, so whoever can compute depth the fastest wins.

However, the human brain *is* capable of reigning supreme in situations that present infinite depth.
More...
I mostly agree with this.

I have always wondered if say, having a computer proving the Riemann Hypotheses wouldn't be a better "Turing Test" than the Turing Test.

 
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