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January 10th, 2019, 08:09 AM   #11
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$\displaystyle (\sum_{i=1}^{n} x_{i}^{2} )(\sum_{i=1}^{n} a_{i}^{2} )\geq (\sum_{i=1}^{n} x_i a_i )^{2}$
Set $\displaystyle x_k =1$ and it is done.
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January 10th, 2019, 09:04 AM   #12
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Start from assuming all $a_k=\frac{1}{n}$. Then sum $=\frac{1}{n}$. Show that any deviation increases the sum. You might be able to use mathematical induction starting from n=2.
Yes, using cyclic symmetry we know the critical value occurs for:

$\displaystyle a_k=\frac{1}{n}$ where $\displaystyle 1\le k\le n$

And hence:

$\displaystyle \sum_{k=1}^n\frac{1}{n^2}=\frac{1}{n}$

Then we could let all but one term be zero and the one term be 1, and:

$\displaystyle \sum_{k=1}^n a_k=1>\frac{1}{n}$

And so we conclude, given the constraint, that:

$\displaystyle \frac{1}{n}\le\sum_{k=1}^n a_k$
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January 10th, 2019, 06:17 PM   #13
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Yes Erdős, what a strange and interesting character who lead a unique life. Truly made his own path. Wandering mathematician.

Do you think there is a Book? A book that contains, for each theorem, its most beautiful and perfect proof? Can there be such a valuation, a way of measuring beauty that attains a unique maximum? Could even God know such a valuation?

Off the top of my head one could conceptualize how to formalize such a thing. We know from mathematical logic that a theorem is nothing more than a well-formed sentence of some formal language, along with a derivation according to given inference rules, starting from some axioms, and ending in the theorem. So a proof is nothing more than a finite sequence of legal statements, where legal is defined by whatever inference rules you adopt; where each statement in the sequence is either an axiom, or follows from previous statements by the rules of inference. This is a purely syntactic definition of a proof, and makes no reference to the meaning of the symbols.

Given a theorem, we can certainly form the set of all proofs of it. Such a set would simply consist of a collection of proofs, each one a finite sequence of statements. We can partition the set of all proofs by saying that two proofs are equivalent if their last element is the same. That is, they both lead to the same theorem.

Now given a theorem, we just consider the class of all the proofs that lead to it. We want to assign it a beauty score. But that's too philosophically complicated. Let's do the next best thing. Let's assign it any well-defined score at all. One that leaps to mind is each proof's Gödel number. Now we just say that the "most beautiful" is the one with the highest number.

Now that is very unsatisfying. We see that we can rank proofs using arbitrary mathematical functions; but we don't actually know how to syntactically define beauty. So we do not know how to build an algorithm that detects and rates beauty. For that, we require a human mind. Or God. I submit that this is something that only a human being can do, and that a computer can't do. Because there is no syntactic way to define beauty. Only a mind can recognize and rank beauty. Computers can't do that.

Last edited by Maschke; January 10th, 2019 at 06:36 PM.
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January 10th, 2019, 07:20 PM   #14
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Yes they can. You just have to train a mineral network to recognise beauty in a proof and then set it loose on all the proofs you have.

The beauty of that approach is that we may very well never understand from the network how beauty is defined, only agreeing (largely) with its results.
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January 11th, 2019, 09:17 AM   #15
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Now given a theorem, we just consider the class of all the proofs that lead to it. We want to assign it a beauty score. But that's too philosophically complicated. Let's do the next best thing. Let's assign it any well-defined score at all. One that leaps to mind is each proof's Gödel number. Now we just say that the "most beautiful" is the one with the highest number.
Might there not be a “highest” Gödel number for some theorems, implying an infinite number of different proofs? At most, there could only be a countably infinite number of them, but maybe that is a limit on our approach. Might there be a theorem with uncountably infinte ways to prove it? If so, Gödel numbers would be incomplete... hmmmm.???
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January 11th, 2019, 09:23 AM   #16
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Might there be a theorem with uncountably infinte ways to prove it?
No. It is always countable.
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January 11th, 2019, 10:00 AM   #17
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No. It is always countable.

When using a finite formal language and taking into account only a finite string of symbols for any sentence, the list of sentences is countably infinite, yes. I do understand that.

Let’s say I want to prove the existence of the set {1} within a model of ZF. The set {1} is a subset of uncountably many sets within the model, so doesn’t the axiom schema of specification imply an uncountably many different ways to assert the existence of the set {1} though too?

More directly, if each set within the model can be defined using a sentence within the language of ZF, then there would be an uncountably infinite number of sentences. There are plenty of noncomputable sets though...

This is just my initial take without much thought... Will ponder it more.
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January 11th, 2019, 10:51 AM   #18
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Might there not be a “highest” Gödel number for some theorems, implying an infinite number of different proofs?
Yes, sorry, wasn't thinking and got myself bogged down on the wrong point.

My main point is that there's no way to teach a computer what beauty is. v8Archie notes that we can train a neural network to do that, and that's an excellent point. We no longer care about defining things, we only care about mining data. That's a good point.

Regarding the lesser point of numbering proofs, there are only countably many proofs all together (finite strings over a countable alphabet). But there's no reason one theorem can't have infinitely many proofs. In fact given any proof of a theorem, we can prepend extra statements like 1 = 1, 2 = 2, etc. to immediately create infinitely many proofs.

I suppose there must be some formal way to eliminate unnecessary statements from a proof. Even so, I can't think of a reason there could not be infinitely many essentially distinct proofs for a given theorem. So forget my idea of Gödel numbering proofs, that was a bad idea (although it's an interesting question about how to eliminate extraneous statements and whether there can still be infinitely many proofs of a given theorem).

But now to v8archie's point. It's just like the history of automated chess. In the 1980's people tried to teach computers what the game means. This is a good position, that's a bad position. The same way we train human players.

That approach reached a limit. It turned out to be a better idea to simply let the computer play a lot of games and keep track of which moves lead to which outcomes. This approach recently achieved spectacular success with Alpha Zero, which starts with NO knowledge of the game past the basic rules of what's a legal move. Machine intelligence is no longer limited by human knowledge. Instead, we just use advanced datamining techniques and the computer gets really good, really fast.

There are a lot of philosophical issues about what knowledge is. If we train the Chinese room on data, it speaks Chinese but doesn't understand anything. Or does it? That argument's been going on for decades.

So I take both Aplanis's and v8archie's points. I think I raised some questions, but I certainly didn't answer them, and my argument was weak.

Still ... would Erdős be satisfied that God writes the Book using datamining techniques? That's the core philosophical question. If we train a neural net on the contents of all the art museums in the world, does the computer know beauty?

Last edited by Maschke; January 11th, 2019 at 10:55 AM.
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January 11th, 2019, 11:39 AM   #19
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Still ... would Erdős be satisfied that God writes the Book using datamining techniques? That's the core philosophical question. If we train a neural net on the contents of all the art museums in the world, does the computer know beauty?
Well, on the basis that each human has their own personal definition of beauty, I would say that the concept is not well defined. Of course, we can reasonably say that an objet d'art that is considered beautiful by a large majority of people (that have seen it) is beautiful. And if the network correctly categories 99% of the art it evaluates by that same measure, we might reasonably say that it classifies art correctly as beautiful or not. But "knowing beauty" is more that simply recognising it, isn't it? Or is the internal reaction that we feel on seeing a beautiful piece of art simply the output of our own neural networks?
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