patrick.net

40 minutes in and both my heads are still intact and functioning perfectly.

Dan8267 says

40 minutes in and both my heads are still intact and functioning perfectly.

OK, not related to the bullshit that Quigley posted, I'm going to address the content in the video itself.

Basically, the topic of conversation is could it be possible that our universe is a simulation running in another universe, which is a common SciFi pot (Matrix, 13th Floor, Simpson's did it). A few points.

Point 1: Although an interesting conversation piece, the answer does not really change anything regarding the pursuit of fundamental questions.

For example, even if we are a simulation, this does not even address the question of why does anything exist instead of nothing. It just moves the question one step back to why does the universe in which our simulation runs exist in the first place? Of course one can make an infinite regress of simulations or a circular regress. Both are meaningless.

The same goes for all other "big" questions like
1. How did life start?
2. What is the purpose of life?
3. Why is existence the way it is?

Point 2: Any conversation regarding the property of a host universe is impossible without making assumptions already considered invalid.

I do software for a living, and I'm damn good at it. Pretty much all software development I've done in the past 20 years has been related to the Internet in one way or another. The Internet is largely built on virtualization, the running of virtual machines inside physical machines, something I deal with on a daily basis like you guys deal with traffic lights.

In virtualization, a host operating systems runs one or more guest operating systems. For example, you could have a physical server running Windows Server 2012 and have two VMs on it running Windows Server 2012 and Windows 10 respectively. The conversation in the video postulates that our universe may be a guest universe running on a host universe.

A lot of the conversation regarding the host universe assumes that it behaves according to the same laws of logic as our universe. For example, one speaker says that you need the computational power of the universe to simulate our universe. However, everyone one the panel also accepts that the laws of physics in the host universe can be completely different from the laws of physics in our universe. If this is so, we cannot count on the laws of logic of our universe to apply in the host universe either. All bets on the laws governing the host universe are off. Therefore, we cannot conclude that the host universe needs the computational power of our universe to simulate our universe.

Point 3: It is unknown whether or not we could tell whether or not we are in a simulation.

The philosopher on the panel argued that any proof that we are or are not in a simulation could be simulated, therefore we could never answer this question. This statement is based on the assumption that all proofs are simulable, which may feel like a right answer, but is a baseless assumption. In fact, it is an incorrect assumption. Simulations, by definition, are the executions of rules, and rules limit what the simulation can do. Therefore, the simulation cannot do anything. It would not even be a simulation if it could.

A concrete example... We cannot create a logically consistent simulation in which the square root of two is a rational number. If we tried to write this invalid mathematical law into a simulation, we'd inevitability generate contradictions that would be detectable. Any sentient beings in our simulation could, in principle, detect these contradictions and demonstrate that they are a simulation.

Strategist says

60 minutes in. That's half-way through. Any bets on my head exploding or me even being shocked at Tyson claiming "I found god!"? Come on, I'll give you ten to one odds.

Dan8267 says

2. What is the purpose of life?

I know the purpose of life:
The purpose of life is to play your part in the chain of evolution. i.e. to survive, and ensure the survival of your offspring.
There is no other purpose.

1:09:00 in. Finally someone mentioned Gödel's incompleteness theorems.

I've always disagreed with Gödel on this point. I fall into the camp that truth and provability are the same thing for all a prior statements. That is, there is no logical statement that is true but unprovable. Unprovable means not true either by being false or opinion or meaningless. This is a fun subject to discuss.

Strategist says

I know the purpose of life:

The purpose of life is to play your part in the chain of evolution. i.e. to survive, and ensure the survival of your offspring.

There is no other purpose.

That is demonstrated by evolution, not the god hypothesis or the simulation hypothesis.

Of course, eventually human life is going to stop. If we don't kill ourselves off from conservative policies that lead to ecological collapse or nuclear war, our descendants will gladly get rid of their living bodies for virtualized minds in order to gain immortality of sentience. The purpose of life and the purpose of sentience are not the same thing, and sentience is far more important than life. Who gives a shit about biological mechanisms and chemical reactions when what's really important is your mind, your thoughts, your feelings, your experiences, your choices. All of these things are better served by a system that can back up and restore your mind rather than by meatware.

Eventually, technology will enable our descendants to replace their organic bodies with inorganic ones. People will choose immortality over death and decay. So your genes aren't going to mean shit anyway as they are marching towards oblivion already. All paths before us lead to genetic oblivion. One path leads to immortal sentience. [At least immortal for most practical purposes.]

Dan8267 says

Subpoints, A simulation

1. is natural, not super-natural

2. is not omnipotent as simulations, by definition and necessity, have limitations in the form of rules including laws of logic and in the form of finite resources

3. is not omniscient as the simulation does not know what is going to happen in the future. Simulations have to crank out results and therefore are learning, and learning is mutually exclusive with omniscient by definition.

I'd call bull on at least these three points. If the universe is all happening in a vast simulation, then it by definition is supernatural not natural, since its creation and perpetuation is entirely dependent on an artificial source that vastly supersedes even what man can accomplish. Is the simulation able to do anything? I'd argue that whoever or whatever controls the simulation could do anything within it as simply as a programmer could make a program conform to his wishes. And since a simulation like the one we are discussing absolutely REQUIRES a creator, this second point seems obvious.
Is it omniscient? Does it keep track of everything that happens within it? I'd say yes, and DUH! Can it predict the future? If the simulation observer exists outside of time then of course they could observe any point in that timeline with 100% accuracy, all the while preserving the choice of individuals.

Sounds an awful lot like a God and his creation. Whether or not the God is benevolent is a matter of opinion, but it could be argued that too much interference would subsume free will, and thus obviate the entire purpose of the experiment. Also if the simulation was entirely safe for its inhabitants, there would be no challenge and the whole thing would be a boring, flavorless waste of time. However if you consider that the participants would eventually "wake the dream(to quote Robert Jordan's Aiel)" then their simulated fate would mean little except as a reflection on their characters. This is exactly in line with many forms of religious thought, and would imply that religion has reached the final solution to the biggest question long before science was able to even understand the problem.

Quigley says

I'd call bull on at least these three points.

I call bullshit on your calling of bullshit.

Natural means obeying the laws of nature. Supernatural means not obeying the laws of nature. It's that simple. If you want to use a different nomenclature, fuck you, that's the nomenclature I'm using for my statements. You don't get to change the meaning of my statements by picking a different nomenclature, and I'm not going to get into a nomenclature argument with you because such arguments are about nothing.

Any simulator that is simulating our universe is, by definition, a natural phenomenon obey some laws of nature in its universe. It's not even remotely like what people think of supernatural beings like their gods and demons.

Furthermore, the simulator was created either by biological beings or other machines, each possibility being a creation of nature either directly or indirectly. None of these things would be gods any more than a mother, who is the creator of a baby, is a god.

Quigley says

Is it omniscient? Does it keep track of everything that happens within it?

Omniscience is knowing everything, not just keeping track of state. The simulation would not even know all possible mathematical proofs as our universe has not, will not, and cannot generate all infinite number of possible mathematical proofs.

Nor would the simulation know anything outside of its programming constraints and data. No one in the host universe would consider the simulation a god. And even if you did considered the simulation a god, it's a non-sentient, amoral god that itself was created by non-gods. That's a pretty pitiful god indeed.

Oh, and by that criteria, every weather simulator used to make weather predictions is a god. So why aren't you worshiping the Accuweather 5000?

Quigley says

Sounds an awful lot like a God and his creation.

Ha ha. Accuweather 5000 sounds like your god and its creation. I suppose the Doopler 200 radar system is his virgin-born son.

I'm an hour and a half into the video, and nothing in it even remotely supports your bullshit. [Christ, someone just mentioned Occam's Razor. It's like obligatory in such conversations.] So far my head hasn't exploded. Ready to admit defeat?

OK, 1:42:00 into the video. Tyson asked each speaker what is the probability, rated as a definitive number from 0% to 100%, that our universe is a simulation. The first speaker gave the most reasonable answer: it's impossible to answer that question based on our current knowledge. One speaker, the bullshitting philosopher, gave a joke answer. Two said basically low, but significant. One said essentially zero, but possible, which is the second most reasonable answer.

Finally, Tyson, breaking his own rule, did not give a percentage probability but simply said "high". Of course, high does not mean more than 50%. One in a million chance could be considered high given the significance of the results. It is unclear whether or not Tyson thinks it's more likely than not that we are part of a simulation. The only thing clear from his statement is that he's open to the possibility, as am I, although I have no reason to believe in this one possibility out of an infinite set of possibilities and no reason to prefer it.

The bottom line is that Tyson has not expressed any belief in
- a god
- the supernatural
- Christianity
- intelligent design
- or anything remotely like religion or faith

Put simply, Quigley took one line of dialog -- and one not precisely crafted to avoid misinterpretation -- ignoring the entire context of the dialog to make a blatantly false statement about Tyson's claim.

My head has not exploded. I know that because I'm shaking it in disappointment at the dishonesty and foolishness of Quigley, who clearly does not get that in the Information Age any bullshit you say will be exposed as the bullshit it is. You cannot get away with incorrect factual statements. Verification is easy. I am literally in my underwear right now disproving Quigley's statement in the original post. Yes, it's Saturday. Why do I need to wear pants? It takes no effort to verify facts today. You literally do not even have to put on pants to disprove false statements like you used to have to when libraries were essential.

I'll answer Tyson's question differently than any of the speakers. Tyson asked, what is the probability that our universe is a simulation.

The question is meaningless. When I say the probability of a coin flip landing on heads is 50%, that statement is meaningful only because coin flips are repeatable events. The actual chance that a given coin flip is heads is either 0% or 100%; i.e., the coin actually lands on one or the other. To say that there is a 50% chance the coin will land on heads is simply short-hand for 50% of coin flips do land on heads.

We only have one universe. That one universe either is or is not a simulation. There are no series of repeatable events from which to extrapolate a probability of our universe being a simulation. The assignment of a probability to the two possible answers to this question is therefore meaningless.

1:52:16 Lisa Randall says "And if you have an explanation to why there's nothing, then there's something there that allowed you to have the rules that [unintelligible, but on the lines of proving that]."

Dan8267 says

You mean, why is there anything instead of nothing? Of course, why really means "how" in science. Asking why an object falls is asking how it behaves, what mechanism causes it to fall. "Why" doesn't mean what is the purpose, only what is the causality.

My philosophical take is that a thing is held together by its opposite. Top is meaningless without bottom. Left is meaningless except in contrast to right. So in order for nothing to exist, it's opposite, something, must also exist. You can only have nothing if there is a thing not to have. So for nothing to exist, something must also exist somewhere or somewhen else. The universe is simply a manifestation of this principle.

Great minds truly do think alike, and there is a reason for this. Correct thought has few faces, maybe even only one.

Wait wait... you have to quote yourself to get a backup? That is too pathetic for words!

It seems that you never understood my point, let alone refuted it. If the universe is a simulation, its creator/controller is indistinguishable from God. After all, if you control a simulation, you are the god of that universe. Therefore by presupposing the likelihood of such is "high," NDT professes belief in such a deity. This doesn't mean that he subscribes to a religion, just that he is a secret deist.

Dan8267 says

We only have one universe. That one universe either is or is not a simulation. There are no series of repeatable events from which to extrapolate a probability of our universe being a simulation. The assignment of a probability to the two possible answers to this question is therefore meaningless.

Quantum theory, an extension of the scientific process, would strongly disagree with you.

I will explode your head, Dan! It will nova!

Dan8267 says

The unmoved mover explains nothing.

Sactly. This is a who is on 1st type deal

As to the other i have no idea or interest in what you are talking about.

Quigley says

Wait wait... you have to quote yourself to get a backup?

I don't know what nonsense you are saying, but you sound like a fool, a childish imp whose throwing a tantrum because he was demonstrated to be a lying sack of shit.

Quigley says

Quantum theory, an extension of the scientific process, would strongly disagree with you.

Actually, no it doesn't. By universe, I mean everything, and by definition, there is only one everything. The multiverse conjecture has nothing to do with our universe being a simulation.

You are simply grasping at straws and muddying the conversation to convince anyone that I made a mistake, however inconsequential, to cover up the fact that I just thoroughly kicked your ass in this thread.

A real man would simply say, "sorry, I was wrong about everything". You simply don't warrant any respect.

Quigley says

I will explode your head, Dan! It will nova!

A childish and impotent threat yapped by a petulant toddler. Your words mean nothing.

I like how you ran away from my central points, choosing instead to adopt insults as your theorem veritas. Weak sauce Dan the impotent.

I guess I must yield to the ancient maxim: "if you wrestle with a pig you both get muddy and only the pig enjoys it."

You are so full of shit. I precisely addressed everything you said. Simply lying about that does not change anything.

You need to learn the difference between a real argument and mere contradiction. Here's an educational video for you.

www.youtube.com/embed/kQFKtI6gn9Y

So you finally admit there is no global warming....

Dan8267 says

Oh, and by that criteria, every weather simulator used to make weather predictions is a god. So why aren't you worshiping the Accuweather 5000?

Dungeness says

So you finally admit there is no global warming....

Much like your mother saying "I love you son" that never happened anywhere except your fantasies.

Wow. Jumping from global warming denial to mother jokes in one post.
I love it when I hit a nerve...

Dungeness says
So you finally admit there is no global warming....

Dan says
Much like your mother saying "I love you son" that never happened anywhere except your fantasies.

Quit cumming all over yourself, Shrek. Your trolling isn't as effective as you think. Like most things in your mind, it's just a delusion.

The objective of trolling is to distract from the thread subject and lead the trollee off topic down some inane path.
Currently You are not talking about Tyson, instead blathering mother jokes and vomiting inept responses to my troll. From where I sit I'd say it is quite effective...
He he he...

Dan8267 says

. Your trolling isn't as effective as you think.

Oh honey, I've already said everything there is to say in this thread, so no, you aren't being an effective troll. But feel free to cum all over yourself in trollish rejoicement of your imaginary accomplishments. You and CIC are truly pathetic losers.

www.youtube.com/embed/HNTxr2NJHa0

And you've spent your night fucking goats again, I see.

Dan8267 says

I've always disagreed with Gödel on this point. I fall into the camp that truth and provability are the same thing for all a prior statements. That is, there is no logical statement that is true but unprovable. Unprovable means not true either by being false or opinion or meaningless. This is a fun subject to discuss.

You could say that Godel's theorem is wrong in that it assumes consistency. i.e. the assertion G brought about by Godel can be proven true (and was by us) so is a theorem, and so an inconsistency. That would still be a much weaker statement than to say all true mathematical statements must be provable.

Dan8267 says

I've always disagreed with Gödel on this point. I fall into the camp that truth and provability are the same thing for all a prior statements. That is, there is no logical statement that is true but unprovable.

There is no such camp, because nobody who actually understands the definitions Gödel used would argue about this.

If you dislike the definition of "true" which logicians use, then pick another word for it. But there is no dispute as to the correctness of Gödel's result: there are well-formed statements which hold in all models (definitions of the symbols in the statements) of systems derived from consistent axioms, that nonetheless cannot be proven from those axioms. "True" means "holds in all models" and doesn't say much, because the statements are not earth-shattering proclamations such as the existence or non-existence of god.

The camp which quarrels with Gödel's definition of "true" is not a camp which Gödel would have bothered to argue with, because "true" is a term defined so that the theorems can be stated. The theorems are correct, so there's nothing to argue.

You actually have no disagreement with Gödel. You have a disagreement with an entire profession (logicians and mathematicians) who have no interest in your points, because you're talking about the ideas of "truth" in another sense. Your gentlemanly disagreement which you imagine you're having with Gödel is really just a disagreement between a contractor and a customer who doesn't understand what drywall is.

Heraclitusstudent says

You could say that Godel's theorem is wrong in that it assumes consistency. i.e. the assertion G brought about by Godel can be proven true (and was by us) so is a theorem, and so an inconsistency

“Consistency" only means that the axioms don't clash with one another. If said axioms are the foundation of an incomplete system, in the sense that true statements exist which are unprovable from those axioms, there's still no inconsistency whatsoever.

Perhaps there's an inconsistency with common sense and the world we want to believe we live in, but that may arise from a failure to comprehend the term "true" in the sense that logicians used it: holding in all possible interpretations of the symbols in the axioms.

Gödel's results are profound, but they say far less about the universe(s) and the meaning of life than they do about our feeble human attempts to axiomatize arithmetic so that we can sleep better at night.

HydroCabron says

there's still no inconsistency whatsoever.

I'm not sure what you mean. If it is inconsistent it is inconsistent. A system can be inconsistent whether it is incomplete or not.

HydroCabron says

“Consistency" only means that the axioms don't clash with one another.

But the point about this theorem is that it applies to *all* systems once they reach a sufficient content. So it's not a question of a specific set of axioms - once this content is reached.
The entire Godel's reasoning is that either G is a theorem, in which case we have to admit that our axiomatization of arithmetic - relying on seemingly irrefutable axioms - is inconsistent, or it is not, and in that case these axiomatizations are incomplete. And from that Godel jumps to the conclusion: they are incomplete. And this applies to any extended axiomatization which is fairly damning.

In fact some argued that this means AI is impossible since humans are apparently able of something any formal system couldn't do. (proving their G assertion is true).

I tend to agree with Dan this is BS. (whether for the same reasons or not). There is simply no step in this demonstration a computer wouldn't do for any axiomatic system, including one that describes the computer itself.

To me the right interpretation is that Godel pointed to a paradox (which is not structurally different from other self referential paradoxes): A proposition that is both true and false. But this interpretation leaves us with 2 problems:
- one is just an idiosyncrasy of mathematical logic: the entire structure of formal logic breaks down in presence of such an inconsistency. F => T This is easily remedied, though it would probably take a lot for logicians to in fact do it. It probably involves changing the definition of what is true, and also admitting that A => B is not a function of A and B.
- the second is deeper: what does it mean that arithmetic includes such paradoxes? and if they exist, how do we avoid them? To me the simple explanation is to separate the universe of discourse (here arithmetic) from the knowledge structures describing them. Knowledge applies arithmetic, and to itself. But only self referential knowledge structures have this consistency issue. However they say nothing about the domain itself. So the meaning of Godel, as I see it, is that there are well formed sentences about knowledge itself that are inconsistent but in fact they say nothing semantically about the universe considered.

Heraclitusstudent says

But the point about this theorem is that it applies to *all* systems once they reach a sufficient content. So it's not a question of a specific set of axioms - once this content is reached.

The incompleteness theorem states that any set of axioms which is a consistent axiomatization of number theory is incomplete, in the sense that there are true propositions within the system which are not provable using those axioms. The definition of "true" in this case is as I describe above.

This applies to ALL consistent axiom sets - nowhere do I say I'm fixing the axioms.

As for axioms "reaching a certain content", I'm not sure what you mean. Either a particular set of axioms is consistent, or it isn't. They don't grow like children or trees. (I have never thought of consistency itself being an undecidable question, but I don't think that's what you're referring to.)

Heraclitusstudent says

The entire Godel's reasoning is that either G is a theorem, in which case we have to admit that our axiomatization of arithmetic - relying on seemingly irrefutable axioms - is inconsistent, or it is not, and in that case these axiomatizations are incomplete.

What's G here? Is it a well-formed statement? If so, and it's a theorem, then this says nothing about the consistency of the axioms. Only in the case that all true statements were theorems would we know that our axioms were inconsistent, because the system would be complete (therefore inconsistent). The provability of any single formula tells us nothing.

Heraclitusstudent says

A system can be inconsistent whether it is incomplete or not.

Sure, no argument from me here: there are inconsistent systems that are incomplete. All the incompleteness theorem says is that no consistent system (strong enough to axiomatize arithmetic) can be complete.

Heraclitusstudent says

And from that Godel jumps to the conclusion: they are incomplete. And this applies to any extended axiomatization which is fairly damning.

He doesn't "jump" anywhere. The incompleteness theorem has a mathematical proof: he shows that any consistent axiomatization satisfying the hypothesis (consistency and strong enough to include arithmetic) will be incomplete. I have read a proof; I don't know how close it was to his original proof, but it's at the advanced undergraduate level, which means it's totally non-controversial, settled mathematics.

Godel wasn't writing a on op-ed, or an article in a philosophy journal.

I believe people make far too much of the incompleteness theorem. It merely lays out a fairly profound limitation of any project to axiomatize mathematics so that all true statements are theorems. There are still plenty of interesting, profound theorems, and if some fundamental question (e.g., the Riemann Hypothesis - "cough") were found to be undecidable, we could either rethink the standard axioms, and perhaps come up with a different set which still seems consistent (we'll never know for sure whether the existing Zermelo-Fraenkel axioms, plus the axiom of choice, collectively called ZFC, constitute a consistent system, anyway - the whole thing could be a house of cards).

I love this stuff, particularly the issue of the Continuum Hypothesis being independent of the ZFC axioms, but I don't see deep philosophical implications, beyond a basic warning that you shouldn't take mathematicians too seriously on questions beyond the realm of their little sandbox, because Godel showed that mathematics has clear limitations. At least mathematics can claim the honor of having been more upfront than any other discipline as to its weaknesses.

But that's all it is. No discussion of the existence of god should rest on mathematics. I think we don't need the incompleteness theorem to see that.

You should fully understand the implications of "disagreeing" with Godel, since his work is completely uncontroversial within the standards of what is considered a normal, workaday mathematical proof - believe me, there are research mathematicians whose loose arguments are way beyond anything logicians like Godel would ever pull, because logicians are among the most rigorous of mathematicians.

If you have a disagreement with Godel, then you are really arguing with the entire culture of mainstream mathematics over the past 175 years. I applaud you for your courage, moxie and utter intellectual integrity in choosing that route -seriously, some good could come of such a project - but you must understand what it means: It's equivalent to choosing a completely different path, even further from the existence-of-god thing. You would be beating mathematics itself with a huge pipe wrench, as the constructivists did, bless them. I'd be sad to wave goodbye to some of my favorite theorems (I think that Schroder-Bernstein would be gone, as well as the lovely architecture of the transfinite ordinals, under any common-sense rebuild of mathematics), but maybe mathematics could use a re-think. I'm dead serious. Just don't fool yourself as to exactly what it means to "disagree" with a theorem that is among the most mundane and settled results in modern mathematics.

HydroCabron says

As for axioms "reaching a certain content", I'm not sure what you mean

I simply mean a set of axioms that at least describe the number theory. The theorem shows that any added axiom beyond that will not provide completeness.

HydroCabron says

What's G here?

G is the assertion that is the center of the proof of the theorem. i.e. a well formed assertion that could be read at a meta level as "G is not a theorem".

HydroCabron says

He doesn't "jump" anywhere. The incompleteness theorem has a mathematical proof: he shows that any consistent axiomatization satisfying the hypothesis (consistency and strong enough to include arithmetic) will be incomplete.

I'm well aware it's a mathematical proof. Nonetheless the theorem is called "incompleteness" theorem because the assumed conclusion is the fundamental incompleteness of any axiomatization of the number theory. This omits the possibility of inconsistency because inconsistency in arithmetic is considered absurd. The proof reaches the point where the theory is either inconsistent or it is incomplete, and eliminating inconsistency means incompleteness. This is the "jump" I'm talking about. In reality of course, the theorem only states (as you are prudently doing) that "consistent axiomatizations will be incomplete". "Incompleteness" is the standard interpretation, but only interpretation.

HydroCabron says

If you have a disagreement with Godel, then you are really arguing with the entire culture of mainstream mathematics over the past 175 years. I applaud you for your courage, moxie and utter intellectual integrity in choosing that route -seriously, some good could come of such a project - but you must understand what it means: It's equivalent to choosing a completely different path, even further from the existence-of-god thing.

No I'm not. What I'm saying above has been said by many people before me.

In fact to interpret this theorem as incompleteness, you need to assume there is step in the demonstration that is achieved by a human mathematician but cannot be achieved by executing logical rules. This is quite simply BS. There is no such step. All steps of the demonstration are trivial logic. (I particular I think someone made a rule based program that will prove the "G" for any formal system containing the number theory, proving that humans are NOT doing anything that cannot be modeled by rules when showing that "G" is true). But if this is true then it follows that inconsistency is the only option. (because adding such rules in the system would lead to inconsistency). And I don't know why this would be in anyway revolutionary. Paradoxes are common in 2nd order logic where you can represent self-referential assertions. And by using numbers as symbols to represent assertions, Godel is implicitly in the second order.

I would add that I'm not sure it's me that's the wing nut here. Jean Dieudonne said something like "If everything logicians have done since 1931 were to disappear tomorrow, one wouldn't even notice.", referring to the fact not even mathematicians use mathematical logic in practice. Indeed if I were to create an AI system tomorrow, I would implicitly write a system including logic and arithmetic, constituting by itself a complete rewrite of the way logic was modeled by mathematicians since the 19th century. It would necessarily differ radically, starting with the definition of truth. But that would probably not even be noticed as an achievement, considering the part about intelligence itself.

Heraclitusstudent says

axiomatization of the number theory

Heraclitusstudent says

I think someone made a rule based program that will prove the "G" for any formal system containing the number theory

I have a hard time following your use of the term number theory here. Number theory can be used in place of arithmetic (we wouldn't say "the arithmetic"), but it's actually higher level higher arithmetic dealing with topics such as factorization of primes, and many of the theorems that are the basis of all that weird 20th century math, that go back to people such as Euler and Fermat.

I think Godel's first incompleteness theorem refers to arithemetic, the field axioms etc.

Heraclitusstudent says

constituting by itself a complete rewrite of the way logic was modeled by mathematicians since the 19th century

20th century math might not be applicable to AI, but that has nothing to do with whether it's valid. A lot of it was in areas such as group theory, representation theory. graph theory and so on. I wouldn't be surprised if some of it were useful to modeling AI.

marcus says

I have a hard time following your use of the term number theory here.

With regard to the Godel theorem we are talking of the number theory as a logical theory. It's about logic more than arithmetic.
Most of mathematics exist without having to care about formal logic. Mathematicians use logic but not in a formal way.

marcus says

I wouldn't be surprised if some of it were useful to modeling AI.

Of course computer science is based on a lot of discrete math concepts.
But with AI, presumably we have to build a knowledge system that includes some type of logic. The question of the logical bedrock of math reappears. We have to build a formal framework that is convenient for a lot of things, would be used in a program to deal with maths, and potentially could be used by mathematicians. It would remain formal but would feel far more natural and usable than current mathematical logic. The goal of current mathematical logic was to remove semantic (here the notion of truth), and reduce math operations to tiny steps that are considered obvious, so that precisely we can establish the consistency of maths. This effort mostly failed with Godel proving the incompleteness - or inconsistency - of this edifice, while other mathematicians ignored the framework totally.

Heraclitusstudent says

In fact to interpret this theorem as incompleteness, you need to assume there is step in the demonstration that is achieved by a human mathematician but cannot be achieved by executing logical rules

I don't follow this. There's no human anywhere in the theorem.

There is no assumption in the hypothesis or proof of the incompleteness theorem as to the humanity of anyone considering the truth or provability of a proposition. It simply says that in any consistent axiomatization there will be statements which are true but not provable from the axioms.

There are propositions which are true, such as A OR (~A). This means they hold in all possible models - no human mathematician required - A OR ~A has the added virtue of being provable. So at least some of these are provable (again, a statement not requiring a human to execute the proof of each one); in an incomplete system, not all are provable, in the sense that no proof from the axioms exists (and sending in teams of humans to search for one won't change this, because it won't create something that has been proven not to exist).

I really hate to use the term "paradox" around this stuff, because there is no paradox. All that happened here was that we humans were too optimistic when we assumed that a program to produce a consistent, complete axiomatization of mathematics could be successful. A total bummer, but not a paradox.

HydroCabron says

I don't follow this. There's no human anywhere in the theorem.

Well... it doesn't have to be human BUT:
- on 1 side you have a system mechanically deriving theorems as a way to prove something
- on the other side you have a human mathematician.

Then you exhibit this assertion that is proved true by the mathematician but you claim that the mechanical system will never be able to mechanically derive this assertion (the assertion corresponding to itself). Which is exactly why people refer to it in the context of the existence of God: You are implicitly saying that human mathematicians seem to be doing something that machines cannot do.

This is nonsense. Everything the mathematician has done can be done by a large enough formal system as well.

Which then implies inconsistency.

Heraclitusstudent says

With regard to the Godel theorem we are talking of the number theory as a logical theory. It's about logic more than arithmetic.

Most of mathematics exist without having to care about formal logic. Mathematicians use logic but not in a formal way.

I'm guessing you weren't a Math major.

I see you're still talking about "the number theory." I don't know, maybe it's different in Europe ? (you also referred to "Maths"). Canada maybe ?

I took an undergrad course called number theory. Number theory is hugely important to Abstract algebra which is about topics such as groups, fields, and rings built up off of axioms. But even basic arithmetic (not just doing it, but proving that it's valid) requires a lot of axioms and theorems.

Heraclitusstudent says

Most of mathematics exist without having to care about formal logic. Mathematicians use logic but not in a formal way.

Most of the Math classes I took that were 200 level or higher were 95% proofs. The professor would spend the entire class doing proofs. (somewhat formal) There were exercises, but we were on our own with those. It's not like they did examples of similar problems in class, which is what Math is at the level that I teach.

As for this discussion, I find Godel's Theorem to be fairly easy to understand, but not it's proof, but then I haven't dug that deep in to it. Maybe I'm too simple minded about this, but I think that if anything this theorem propelled Math forward by giving some Mathematicians the ability to be okay with a system being consistent and effective but not complete. Therefore even being okay with sometimes running in to an axiom that might be thought to be needed for some trivial theorem (not required for effectiveness), but which would cause an inconsistency and therefore leaving that axiom out.

One other thought that might be part of the confusion. The bizzarre logic Godel used proving that such a statement would exist: "A is true iff it's unprovable" or something to that effect, doesn't mean that therefore every inconsistency that you might run into as you added the possibly infinite number of axioms required for completeness is of a weird self referential nature. That's simply the route he went to prove the theorem.

marcus says

I took an undergrad course called number theory. Number theory is hugely important to Abstract algebra which is about topics such as groups, fields, and rings built up off of axioms. But even basic arithmetic (not just doing it, but proving that it's valid) requires a lot of axioms and theorems.

It looks like what you call Number Theory has nothing to do with logic. In the current context what I call Number Theory is a logical theory . A logical theory is a formal system where AXIOMS are used to mechanically derive THEOREMS using rules. I capitalize the terms because they don't mean exactly the same as in the rest of mathematics. In traditional mathematics is simply an assertion that is proven in given conditions. Here it is an assertion that is formally derived using a (long) sequence of small steps. No mathematicians ever bother with such formal methods.

marcus says

Maybe I'm too simple minded about this, but I think that if anything this theorem propelled Math forward by giving some Mathematicians the ability to be okay with a system being consistent and effective but not complete.

It didn't propel mathematics forward. It represented the failure of Hilbert's program: an attempt to establish sound basis for mathematics. The fact that they are incomplete means these formal logical systems are useless.

marcus says

I find Godel's Theorem to be fairly easy to understand, but not it's proof, but then I haven't dug that deep in to it.

To understand Godel's theorem you need to understand the proof. More specifically the final ad absurdum part I mentioned. You need to understand why it requires semantic, which is the part the human is doing but not the system.

marcus says

doesn't mean that therefore every inconsistency that you might run into as you added the possibly infinite number of axioms required for completeness is of a weird self referential nature. That's simply the route he went to prove the theorem.

Well, arithmetic is consistent if you have a physical interpretation of what numbers are. So you can't simply add any axiom you want: They have to be true within the usual interpretation of numbers. So inconsistencies do not simply arise like this. However there are a number of well known paradoxes, for example Russell's paradox in the set theory, and they all involve self-reference or referential loops. Self-reference within a language is the source of the problem, not arithmetic.

Heraclitusstudent says

You need to understand why it requires semantic, which is the part the human is doing but not the system.

As I said, this is just how he proved it. It doesn't mean that if complete, the inconsistencies will be of this nature.

Heraclitusstudent says

Self-reference within a language is the source of the problem, not arithmetic.

I believe you've got it wrong, and I'm not saying there is a problem with arithmetic. IT's that Godel used arithmetic(the existing axioms of arithmetic) regarding the properties of the natural numbers and logic to prove that a system such as this is either consistent or complete but not both. The self referential statements he came up with to prove this were the result of the logic mathematicians use. That is simply how he proved it. The result is generalizable and true not because of the way he proved it. That is, the result is not a special case as you believe having to do with the way he proved it.(this is logic)

But also, if you're interpretation were correct, then there isn't a problem anyway. In either interpretation, as long as there are no non-trivial inconsistencies we're good. That is, as long as all useful results are consistent, we're good.

By the way, I don't claim a high level of expertise in this topic, but I do have some understanding and experience with how an axiomatic system is built.

marcus says

It doesn't mean that if complete, the inconsistencies will be of this nature.

What other nature is possible? What paradox do you know that is not based on referential loop?
Do you believe arithmetic is in fact inconsistent?

marcus says

The self referential statements he came up with to prove this were the result of the logic mathematicians use. That is simply how he proved it. The result is generalizable and true not because of the way he proved it.

It is true what I said about self-reference is incidental here. It doesn't change what I said before about interpreting this as inconsistency rather than incompleteness.

marcus says

if you're interpretation were correct, then there isn't a problem anyway. In either interpretation, as long as there are no non-trivial inconsistencies we're good.

Well we're good except... as I said...
- First, admitting that there is an inconsistency utterly destroys the way logic itself was formalized in mathematics. Because if an inconsistency exists then every assertion becomes true, because of the way A => B is formalized. So we're good... provided you rewrite all mathematical logic since the 1900's in a different format.
- Second, you don't get to claim there is an inconsistency in this number theory without explaining what it means with regard to arithmetic in general. i.e. you need to explain why there is an inconsistency, and how to circumscribe it so that it doesn't contaminate standard arithmetic. This is what I did by ascribing it to self-reference in the language rather than a consequence of numbers properties.