k2aggie07 said:

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Don't forget the second theorem. If it is complete it also won't be consistent. So it's not just mute on its internal consistency, it can't be both complete and consistent. That's a bit daunting at least for philosophy.

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Nevertheless, there is a good deal of mathematics that is not subject to either of Godel's incompleteness theorems. For Godel's incompleteness theorems to apply your theory must be recursively defined, and sufficient to develop arithmetic. Tarski axiomatized Euclidean geometry in such a way where these two postulates do not hold, and therefore Euclidean geometry can be shown to be both complete and consistent. Similarly, the theory of real numbers can be axiomatized in a system known as the theory of real closed fields, which is both complete and consistent.

Neat I did not know that. Time to read read read.

So, darg, I feel this is a little less strong than you intended. As I'm sure you know Tarski's system is complete but can only define points... not even lines as strings of points.

Tarski's system is complete only as far as the semantic definition of truth relative to that system maps 1-to-1 with theorems of that system...It can't really express a "mathematical truth".

Basically "it's all turtles models all the way down" combined with the observation that every single model is either inconsistent, incomplete, or both is my view of the universe. Any truth is only true relative to some other system.

Some read food
http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Greenberg2011.pdf

>Logic is an indication that the universe operates according to natural law, not supernatural law.
>I agree with that. My question still remains. A "non-created" thing always existing sounds like God.

Being non-created and eternal are 2 attributes of God.

The universe operates according to natural law not super natural law. If there is no God and this entire system is matter plus energy and change > the appearance of logic would have to be supernatural.

For me it's more logical to believe that rationality comes from a rational being (God) than ration comes from matter, time, plus energy.

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Who invented logic?

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If you define logic as the systematic study of the form of arguments then this was formulated in China, India, and Greece. I have a hard time understanding what a god has to do with formulating valid arguments.

I have a hard time understanding what a god has to do with formulating valid arguments.

If there is no God, then our existence is reduced to matter and energy. In our experience matter and energy does not produce the following: Logic, Justice, Love, envy, hatred, morality. In our experience a rational mind capable of logical arguments only come from other beings with rational minds. For me it's more logical to believe that a powerful, creative God, capable of rational thought, created humanity (beings capable of justice, logic, creativity, etc)

k2aggie07 said:

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So, darg, I feel this is a little less strong than you intended. As I'm sure you know Tarski's system is complete but can only define points... not even lines as strings of points.

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I feel like this is a bit misleading. Tarski treats points as primitive objects in his theory and explicitly shuns the use of set theory, therefore we can't do something like say L={(x,y)|y=mx+b} and define L as a line this way. But the set-theoretic way is not the only way of defining a geometric line. In fact, lines, circles, triangles, angles, and polygons in general can all be specified by a finite collection of points, along with the notion of "betweenness" and "equidistant" which are also primitive notions in his formulation. His axiomatization has most of the stuff you think of as Euclidean geometry pop out, and you can do quite a bit of geometry this way, although it is cumbersome. You can talk about things like parallelity, longer, bigger, and all sorts of geometric notions. What's missing from his formulation is a good definition of numbers, but you don't need that for a lot of Euclidean geometry. All that said, the theory is both complete and decidable. Earlier in the thread you stated that "So, as a philosopher you can't consider that its practically irrelevant in a grand sense..." I was merely trying to give an example of a theory both complete and decidable, without giving much regard to it's application. But I think it encompasses quite a bit more than you're giving it credit for.

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Tarski's system is complete only as far as the semantic definition of truth relative to that system maps 1-to-1 with theorems of that system...It can't really express a "mathematical truth".

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How would you define a mathematical truth, then?

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Basically "it's all turtles models all the way down" combined with the observation that every single model is either inconsistent, incomplete, or both is my view of the universe. Any truth is only true relative to some other system.

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I'm still not sure what you're looking for when it comes to a theory of everything. Are you familiar with Conway's game of life? It's basically a set of cells that can either be in the states on or off that evolve through discrete time, where a cell turns on or off in each step based on four simple rules about the cells adjacent to it. If you know what the four rules are (I think it's four -- it might be a few more or less) I would say you have a theory of everything about that system. The fact that there are questions that have a true or false answer, but may be incalculable has no effect on this. An example of such a question might be given an initial configuration of some trillion cells and an infinite amount of time to run does some specific cell ever turn on? I find this analogous to a differential equation that doesn't have an analytical solution or the fact that pi can't be computed exactly as a decimal number. Because we can't find an analytical solution doesn't mean that the differential equation doesn't correctly describe the system.

For the sake of argument, let's say an omniscient being comes down and gives us a rule or some small set of rules analogous to the one's for Conway's game of life that describe the time-evolution of everything in the universe, maybe as a set of differential equations since that's what we understand. Would you really let the fact that we couldn't answer a question like, "does this particle ever cross the point (x,y,z) in the history of the universe" stop you from calling this a theory of everything? If so, I would disagree with you, and say we're at an impasse. We're looking for the set of rules that govern the universe, not whether or not everything is calculable within that formulation, when we talk about a theory of everything.

Sorry I lost track of this.

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Tarski's system is complete only as far as the semantic definition of truth relative to that system maps 1-to-1 with theorems of that system...It can't really express a "mathematical truth".

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How would you define a mathematical truth, then?

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This gets into a different discussion, because the whole point of this is that truth is only true -- can only be true -- given a definition of true that exists inside that system. To say "what is truth" you have to get outside of the system. Much like Pilate, this is the bedrock of basically any discussion..."What is truth?" I imagine he said it with more contempt, or perhaps disappointment or bitterness.

I don't believe Tarski's geometry can map something like "this statement is false". It can only say "this exists" or "that exists". But I'm not an expert.

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I'm still not sure what you're looking for when it comes to a theory of everything. Are you familiar with Conway's game of life? It's basically a set of cells that can either be in the states on or off that evolve through discrete time, where a cell turns on or off in each step based on four simple rules about the cells adjacent to it. If you know what the four rules are (I think it's four -- it might be a few more or less) I would say you have a theory of everything about that system. The fact that there are questions that have a true or false answer, but may be incalculable has no effect on this. An example of such a question might be given an initial configuration of some trillion cells and an infinite amount of time to run does some specific cell ever turn on? I find this analogous to a differential equation that doesn't have an analytical solution or the fact that pi can't be computed exactly as a decimal number. Because we can't find an analytical solution doesn't mean that the differential equation doesn't correctly describe the system.

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I'm looking philsophically. By your vantage, we can basically say Godel offered no insight to the universe, or math, or philosophy. Of course this is not true... his work was a blow to an entire field, because he told them what they were trying to do was impossible.

Godel doesn't apply to a particular system except as far as the truth-reference in that system must be decided from without. So, your examples are not precisely relevant. We can solve or correctly describe a system, but never all systems.

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For the sake of argument, let's say an omniscient being comes down and gives us a rule or some small set of rules analogous to the one's for Conway's game of life that describe the time-evolution of everything in the universe, maybe as a set of differential equations since that's what we understand. Would you really let the fact that we couldn't answer a question like, "does this particle ever cross the point (x,y,z) in the history of the universe" stop you from calling this a theory of everything? If so, I would disagree with you, and say we're at an impasse. We're looking for the set of rules that govern the universe, not whether or not everything is calculable within that formulation, when we talk about a theory of everything.

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I would say it's a theory of (almost) everything. Because, for example, that small set of rules can't validate themselves. Why is Conway's game life true? Conway says so.

I think perhaps we're getting at different things. What you're describing would be what I would call a theory of the characterization of everything, but this isn't a theory of everything. We can characterize things without being able to predict their formation or why they form.

I'm not sure I can speak coherently on this without someone to bounce ideas off of (and some beers).

Side thoughts that came up though - what if the rule is so simple as to be useless? I mean, look at the complexity that can arise from Conway's rules. On the other hand, look at how little predictive value they have. If you were one of Conway's cellular formations, and you knew the rules -- would you consider those rules useful or even worth knowing?

And I'm still not sure that if the Conway Solution to the Universe and Everything was imposed, that Godel wouldn't apply.

My head kind of hurts.

AstroAg17 said:

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Hey darg, how good are you at chess? I was reading about the tcec, clicked a random link and found you critiquing stockfish. The internet is a small place.

https://www.reddit.com/r/chess/comments/3x8imx/stockfish_vs_komodo_where_did_white_go_wrong/

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Small world indeed.

I'm....okay. Chess has so many different tiers, it all depends on who you ask. A good child/beginner adult will beat a beginner child 9/10 times. A bright child/social adult player will beat that person 9/10 times. An average tournament adult player will beat that guy 9/10 times. A good tournament player will beat that guy 9/10 times. A master will beat that guy 9/10 games. A grandmaster will beat that master 9/10 games. A PC running 2008 software will beat the grandmaster 9/10 games, and a PC running Stockfish will beat that software 7 or 8 out of 10 times. I am probably at the level that most might call a good tournament player, but below that of a master. If you ask a random guy off the street he'd probably say I was pretty good, if you asked a master I'm a patzer (but experienced enough to know the word patzer!).

Incidentally, there's a physics professor at A&M (or at least was when I was around) named Ralf Rapp who was for all intents and purposes a master.

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Sorry I lost track of this.

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No worries, so did I for a while.

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This gets into a different discussion, because the whole point of this is that truth is only true -- can only be true -- given a definition of true that exists inside that system. To say "what is truth" you have to get outside of the system. Much like Pilate, this is the bedrock of basically any discussion..."What is truth?" I imagine he said it with more contempt, or perhaps disappointment or bitterness.

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Fair enough, but unless I'm misunderstanding what you said earlier, mathematicians typically define mathematical truth in a way you seem to be shunning which is why I asked.

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I don't believe Tarski's geometry can map something like "this statement is false". It can only say "this exists" or "that exists". But I'm not an expert.

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If we're talking about Tarski geometry as a formal system then meta-mathematical statements are clearly outside of its domain. But its strength is that it allow you to derive many properties about Euclidean space in a complete, consistent, and decidable manner. It can only prove statements true or false within that system, but that's true for any axiomatic system.

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Godel doesn't apply to a particular system except as far as the truth-reference in that system must be decided from without. So, your examples are not precisely relevant. We can solve or correctly describe a system, but never all systems.

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My only thought on this is that the universe needn't conform to all logical systems. We only need to find the one(s?) where it does. What you're pointing out might be a problem for philosophy but I don't see it as a problem in terms of what is physically real.

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I think perhaps we're getting at different things. What you're describing would be what I would call a theory of the characterization of everything, but this isn't a theory of everything. We can characterize things without being able to predict their formation or why they form.

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Yes, I think we are perhaps talking about different things. In such discussions having definitions straight is probably the most important thing. It sounds to me like you want the axioms to essentially justify themselves for something to truly be called a theory of everything. Is this correct?

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Side thoughts that came up though - what if the rule is so simple as to be useless? I mean, look at the complexity that can arise from Conway's rules. On the other hand, look at how little predictive value they have. If you were one of Conway's cellular formations, and you knew the rules -- would you consider those rules useful or even worth knowing?

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I'm not sure what the relevance of the complexity of the rules is. This is wild speculation, but physical theories are generally rather simple I think, and if we do find a theory of everything I don't see why it would deviate from that trend.

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And I'm still not sure that if the Conway Solution to the Universe and Everything was imposed, that Godel wouldn't apply.

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I'm perfectly content admitting that it does. My point was that it's still what I would call a theory of everything. Of course, going back to one of your previous points, I do think we're talking about different things here though.

What do you like to play?

Play something that allows you to play more on opening principles and less on having to know lines so that's not a problem.

If you would like to post a game or two I can give you some commentary on them. I've generally found that kind of thing helpful.

What concepts are you familiar with at this point?

It'll probably be useful already. Sure the computer will point out tactical errors, but there are ideas that a human can point out in a position that you've never thought of that you might not see by just studying computer lines. For instance, I was playing a hustler in Vancouver once who pointed out to me just how strong an h3, Nh2 manuever can be at the right time(with white castled kingside and knight on f3.) But your call of course, no pressure.

By concepts I mean are you familiar with basic tactics (forks, pins, skewers, x-rays, etc), rudimentary positional concepts (knights vs bishops, pawn strucure, good bishop vs bad bishop etc), endgame ideas, opening principles and so on. There's a lot of really good material out there.

I personally find the Sicilian quite difficult to play, but obviously it's very sound. Still, it seems to me to be an opening that you have to know the lines quite in depth. Something like the Najdorf just makes no sense to me. I played the dragon quite a bit several years ago, and they were the most fun games I've probably ever played, but my rating took a hit.

A surprise opening I occasionally like against d4 is the Albin countergambit. It's sound enough that you can play it without too much difficulty, and always has the potential for setting up the Lasker trap (look it up if you haven't heard of it -- coolest opening trap I've seen as it involves an early underpromotion).

(I get excited by Chess-talk!)

DirtDiver said:

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>Logic is an indication that the universe operates according to natural law, not supernatural law.
>I agree with that. My question still remains. A "non-created" thing always existing sounds like God.

Being non-created and eternal are 2 attributes of God.

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Attributes are just that. Attributes. Multiple things can share the same attributes.

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The universe operates according to natural law not super natural law. If there is no God and this entire system is matter plus energy and change > the appearance of logic would have to be supernatural.

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I have argued that logic was discovered, not invented. You're saying logic appeared. When and how?

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For me it's more logical to believe that rationality comes from a rational being (God) than ration comes from matter, time, plus energy.

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Ignoring that by your logic, a rational being could be because it would have to come from a rational being, ration doesn't come from matter, time plus energy. Ration (logic) just is, just as the concept that 1+1=2 just is. The mathematical law of addition isn't specific to this universe, and doesn't even require any universe to be true. It is a concept that we have discovered and applied labels to. God didn't invent "1+1=2". If he proclaimed that "1+1=3" the only way that could be true is if '3' really meant what we know of as '2'. If he invented a universe where one item plus one item were actually three items, or that A!=A then his universe would fail due to unresolvable conflicts. i.e. Even God, if there is a God, has to adhere to the rules of logic.

So much Platonism

Some musings.

One, in order to have a Conway Solution to the Universe and Everything (CSUE) it would be completely deterministic. Maybe that was obvious to you, but it wasn't immediately obvious to me. But that means it really is a theory of everything: given the rules, and the initial condition, everything is knowable. So this means complete, I think.

Two...the attachment to Godel then becomes if it is consistent and if it is a formal system.

But even this doesn't have be direct. Even though our CSUE is formally deterministic doesn't mean it's a mathematical formal system (i.e., axioms that can express addition and so on). However... what if the behavior of things in this deterministic system can be interpreted as patterns in particular ways...ways that might represent mathematical functions? Then we're back where we started - we can't use the observation / interpretation / application of these patterns backwards for a complete understanding of reality.

Which seems paradoxical, somehow.

Hm. Further.

Given:
A consists of (1) classical logic, (2) the mathematical functions of successor, addition, and multiplication, and (3) an infinite supply of natural numbers
B is a "CSUE" that defines, describes, predicts, or circumscribes everything in the universe

And C:
we can describe things in the universe using (1)
we can observe in the universe things that look like (2)
we can fathom or imagine (3)

And D:
our descriptions, observations, or imaginations are part of the universe

And E:
For any formal system S whose axioms are strong enough represent certain mathematical functions (successor, addition, multiplication):
If S is consistent, then S is incomplete (i.e., there are mathematical truths that cannot be derived in S).

Then:
Because of C, A exists
Because of D, A is described by B

Either ~E or ~B. That is to say, either Godel is wrong or there is no possible CSUE.

Sorry K2, I'm not ignoring you, but I do believe we're talking about somewhat different things so I'm not sure I have much more to contribute.

I had a tangentially related thought, though. I'm kind of curious as to how/if Godel would apply when AI becomes a realistic thing, or if a computer would ever encounter Godel when dealing with something like Asimov's laws of robotics. Can something like that even be recast into a formal system?

dargscisyhp said:

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Sorry K2, I'm not ignoring you, but I do believe we're talking about somewhat different things so I'm not sure I have much more to contribute.

I had a tangentially related thought, though. I'm kind of curious as to how/if Godel would apply when AI becomes a realistic thing, or if a computer would ever encounter Godel when dealing with something like Asimov's laws of robotics. Can something like that even be recast into a formal system?

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No worries. I am trying to figure out where the error is in elaborating -- if any formal system is incomplete, then there is also no formal system that describes everything that can be described by a formal system. I.e., not only is any formal system incomplete, but any set of formal systems must be incomplete / inconsistent. I think I'm wrong, but I'm running against the limits of my abilities here.

There are lots of people who argue both sides of whether Godel's two theorems make AI impossible. Chapter 6.3 on the Stanford Encyclopedia of Philosophy entry on Godel's Incompleteness Theorems talks about this. It comes down to consistency. And, if we talk about the laws of robotics, if they are consistent the robot is deterministic (which I think Asimov would agree with... that's the tongue-in-cheek and wink assumption of all of the vingettes, right?). But we run into the same problems above... if it is consistent and deterministic, it must be complete. Which means anything that the robot may do that can be interpreted as a mathematical function, suddenly we're at an impasse.

So, the out is that the AI mind must be consistent while able to entertain the notion of inconsistency. But this, I think, pushes the laws too far. Suddenly they're not iron-clad, right? How do you control inconsistency?