The Platonic Mathematical World

[SerScot]

Avatar,

But it would exist. I hope you don't subscribe to the Heisenberg belly gaze that claims if we were not here the Universe would not "really" exist.

[Zarathustra]

Avatar,

But it would exist. I hope you don't subscribe to the Heisenberg belly gaze that claims if we were not here the Universe would not "really" exist.

But even if Heisenberg and/or Avatar were correct, that wouldn't affect the truth of logic and math in any way. 2+2 would still equal 4 even if we didn't exist to witness it. The logical necessity of formal relations don't depend upon subjectivity. They are not relative (like physical quantities such as space, time, and mass), nor are they dependent upon context or reference frame (such as measured velocity).

I'm open to the position that observation plays a large role in collapsing quantum proxy waves of possiblity to "create" the observable world of actuality. That doesn't mean that the universe doesn't exist until someone looks at it--it only means that it has a different kind of existence prior to observation. Or in other words, what we observe isn't the whole picture. Maybe we're seeing what an infinite, timeless mathematical reality looks like from the "inside," from the perspective of a self.

[Avatar]

Avatar,

But it would exist. I hope you don't subscribe to the Heisenberg belly gaze that claims if we were not here the Universe would not "really" exist.

Bah, as far as I'm concerned, the universe will stop existing when I die.

--A

[Fist and Faith]

Good think I'm a bit older than you, and will likely die first. Otherwise, I might be sitting here one day, minding my own business, when suddenly *poof*!

[SerScot]

Avatar,

Avatar,

But it would exist. I hope you don't subscribe to the Heisenberg belly gaze that claims if we were not here the Universe would not "really" exist.

Bah, as far as I'm concerned, the universe will stop existing when I die.

--A

I sincerely hope there will never come a day when I hear very bad news and post in this thread "I refute Avatar, thus."

[SerScot]

Zarathustra,

Do you think Godel's theorems prove there is a larger world in Mathematics that we cannot capture in our formal systems? Do they really reenforce his Platonism? Are the Post-modernist correct that they mean everything is socially based and a construct of our minds, simple games that don't mean anything (as Wittgenstein attempt to say about Godel's theorems). Or is there impact something else?

[Vraith]

Zarathustra,

Do you think Godel's theorems prove there is a larger world in Mathematics that we cannot capture in our formal systems? Do they really reenforce his Platonism? Are the Post-modernist correct that they mean everything is socially based and a construct of our minds, simple games that don't mean anything (as Wittgenstein attempt to say about Godel's theorems). Or is there impact something else?

I don't think he's saying quite that about Godel. What he's saying is the fact that we can understand Godel, and think beyond that implies there is more. [I think...if I'm wrong, he'll surely correct me.. ]
I'd love to know if someone has a real idea of what such a system would look like/how it would work, what kind of frame/shape it has, if any.
And...that's not really all there is to Post-modernist. And Wittgenstein didn't say quite that: he said all we can possibly claim as knowledge, what we mean by knowledge, is language games. He didn't say nothing else existed, only that we must "remain silent" about it.
[heh...depending on who you listen to...people still argue about what a lot of his work really meant.]

[SerScot]

Vraith,

This is what Wittgenstein said about Godel's Theorems:

Mathematics cannot be incomplete; any more than sense can be incomplete. What ever I can understand, I must completely understand. this ties up with the fact that my language is in order just as it stands, and that logical analysis does not have to add anything to the sense present in my propostion in order to arive at complete clarity.

...

No calculus can decide a philosophical problem. A calculus cannot give us information about the foundations of mathematics.

...

My task is not to talk about Godel's proof, for example. but to by-pass it.

Wittgenstein was a self important blowhard who didn't like results that contradicted his own ideas.

[Vraith]

Vraith,

This is what Wittgenstein said about Godel's Theorems:

Mathematics cannot be incomplete; any more than sense can be incomplete. What ever I can understand, I must completely understand. this ties up with the fact that my language is in order just as it stands, and that logical analysis does not have to add anything to the sense present in my propostion in order to arive at complete clarity.

...

No calculus can decide a philosophical problem. A calculus cannot give us information about the foundations of mathematics.

...

My task is not to talk about Godel's proof, for example. but to by-pass it.

Wittgenstein was a self important blowhard who didn't like results that contradicted his own ideas.

Heh...well he was self-important and a blowhard by most accounts. Nevertheless, what he said here is essentially correct as a critique. The incompleteness isn't a result/property of math-in-total, it's a result/property of how we do math. If/when we can bypass/transcend the axiomatic and formal, yet retain coherence/consistency, the incompleteness will vanish. [to some extent, we can do this: fundamentals of Euclidean geometry that can't be shown within that geometry have been proven utilizing other forms of mathematics...these other systems have their own parallel problems though, requiring still other forms.]

[SerScot]

Vraith,

So, are Godel's theorems proven or not? If they are how is anyone going to find a formal system that can be proven by the rules of that system without finding propostions within that system that are true and unprovable within the system?

Godel's theorems, to me, are simply an mathematical way of expressing the saying, "The universe is not only stranger than we imagine, it is stranger than we can imagine."

[Vraith]

Vraith,

So, are Godel's theorems proven or not? If they are how is anyone going to find a formal system that can be proven by the rules of that system without finding propostions within that system that are true and unprovable within the system?

Godel's theorems, to me, are simply an mathematical way of expressing the saying, "The universe is not only stranger than we imagine, it is stranger than we can imagine."

We're going to end up partially agreeing...oh, the HORROR!...
Exactly...how are they going to find it? They're not. Godel is proven, for formal systems. To "get beyond" it...I don't think anyone has even the slightest inkling. Informal, yet rigorous...logically consistent, yet non-axiomatic...the words make perfect sense, the concept is "Huh???" [which bears a resemblance to your second paragraph.]

I have an idea...I could never do it myself, and for all I know someone is already on the job...for a direction to look, though. When Chaos first burst out [fractal stuff happened roughly the same time, and they're related in interesting ways] there was a "moment" of "What the hell?" But almost immediately, they started finding "patterns of chaos." [not to mention instability forming the stable].
Someone should look at the regions of incompleteness in all the maths, and compare them to each other. Huge leaps were made from Chaos and the patterns of it...the same could be true for incompleteness and the patterns of it.

[SerScot]

Vraith,

Harness the incompleteness?

[Vraith]

Vraith,

Harness the incompleteness?

Odd as it sounds, that's as good an analogy as any.

[Avatar]

I sincerely hope there will never come a day when I hear very bad news and post in this thread "I refute Avatar, thus."

Even if you do, it will be a fair bet that my universe has ended.

--A

[SerScot]

Avatar,

I sincerely hope there will never come a day when I hear very bad news and post in this thread "I refute Avatar, thus."

Even if you do, it will be a fair bet that my universe has ended.

--A

I don't know that I'd even say that. I don't hold to a simple materialist position that what we perceive with our five senses is all we can know that there is. As I've said before "the Universe is not only stranger than we imagine, it is stranger than we can imagine". As such I will not say with certianty or even say it's a safe bet to say there is nothing after we die.

[Avatar]

Ah, sorry, forgot you're a Christian. I tend toward the opinion that when we die, that's it.

That isn't to say that there aren't things out there that we're incapable of perceiving with our senses...the infra-red spectrum is just one example of something that's all around us, but we can't see it.

But that equally doesn't mean that there is anything "supernatural" about those things we can't yet perceive.

--A

[SerScot]

Avatar,

I suppose we'll all find out, eventually.

[Zarathustra]

As an atheist who largely agrees with Serscot in this thread, for this subject matter, I don't think it necessarily commits one to the supernatural.

So, are Godel's theorems proven or not?

Yes. But it wasn't proven using the same technique that that individual theorems within a formal system are proven (i.e. by using the rules of that system). It "stepped out of" formal systems, so to speak. It was a proof about formal systems in general.

If they are how is anyone going to find a formal system that can be proven by the rules of that system without finding propostions within that system that are true and unprovable within the system?

A formal system "proven" in what sense? Proven complete? You can't; that's Godel's point. Proven consistent? Again, you can't; that's Godel's point.

I'm not sure why this is a problem. Do you think we should be able to prove formal systems complete and consistent?

[SerScot]

Zarathustra,

As an atheist who largely agrees with Serscot in this thread, for this subject matter, I don't think it necessarily commits one to the supernatural.

So, are Godel's theorems proven or not?

Yes. But it wasn't proven using the same technique that that individual theorems within a formal system are proven (i.e. by using the rules of that system). It "stepped out of" formal systems, so to speak. It was a proof about formal systems in general.

If they are how is anyone going to find a formal system that can be proven by the rules of that system without finding propostions within that system that are true and unprovable within the system?

A formal system "proven" in what sense? Proven complete? You can't; that's Godel's point. Proven consistent? Again, you can't; that's Godel's point.

I'm not sure why this is a problem. Do you think we should be able to prove formal systems complete and consistent?

I'm not saying Godel proved the Supernatural must exist. However, I think he showed that the Universe will always be too complex for us to have true understanding of everything in it.

[Avatar]

Avatar,

I suppose we'll all find out, eventually.

Only if you're right. If you're wrong, you'll never know.

--A