The Platonic Mathematical World

[Vraith]

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.

I don't think that follows, though I agree mostly with the conclusion: It's extremely unlikely that human intelligence will understand everything.
However [and partly repeating what I said earlier] what he showed was only [heh...only in its hugest sense] that no single formal system will work. So there are still at least 2 ways to get around it: use an array of different formal systems, [so they cover each others "blind spots"] or the inconceivable-so-far non-formal but consistent system(s).

[SerScot]

Vraith,

But isn't an "array of formal systems" just another "formal system"?

[Vraith]

Vraith,

But isn't an "array of formal systems" just another "formal system"?

I don't think so...or at least they don't have to be.
Analogically, there's a reason for having both eyes and ears. They gather fundamentally different data. Understanding of the world is completely altered by having just one, or the other, or both.
The analogy falls short, of course.
You'd need a layering, or nesting [that might be infinite...which is why I agree the chance of knowing all is slim to none] of systems in which each of them proves assumptions and/or fills gaps of another system [and also provides unique, provable information of its own].
The selection of systems can't be entirely systematic/formal, though: formal selection would be incomplete itself. It would have to be, at least in part, pragmatic/experimental. [or nested systems as well].

[Zarathustra]

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.

I don't think that follows, though I agree mostly with the conclusion: It's extremely unlikely that human intelligence will understand everything.

I agree, well said. Godel didn't show anything about the universe at all, much less our ability to understand it. But even if we restrict SerScot's claim to the subject matter which Godel addressed--formal systems--Godel still didn't show that they are beyond our understanding. In fact, we can understand something about them that is beyond the limits of those formal systems themselves, simply by understanding Godel's Theorems.

But I do agree that it's likely the universe is too complex for us to ever understand it completely.

However [and partly repeating what I said earlier] what he showed was only [heh...only in its hugest sense] that no single formal system will work.

What do you mean by "work?" We seem to get by just fine. I think his theorems say more about how we think of formal systems, philosophically, than their applicability and usefulness.

[Vraith]

What do you mean by "work?" We seem to get by just fine. I think his theorems say more about how we think of formal systems, philosophically, than their applicability and usefulness.

Yah, what I meant by "work" in this particular case is that unless we're "outside the box" of a single system, we're permanently stuck. If I'm reading you correctly, we agree here, I just spoke sloppily.
It is obvious to me that when we "think," we are performing an act fundamentally different than when we "think about thinking."
A squared + B squared=C squared is the first, what Godel did is the second.

[SerScot]

Time for thread necro. Zarathustra, interesting point. Are you saying Godel says we can understand the Universe but that we cannot prove our how we understand the Universe? Isn't the key to Godel's theorems (and the limit of their applicability) that they limit their assertions to the ability of formal systems to be proven with their own axioms. My mistake is equating proof with understanding?

[Avatar]

I think he's saying that there will always be a limit to our current understanding. That isn't to say we won't understand more, but rather that each time we learn something new, it simply changes that particular boundary.

--A

[SerScot]

Avatar,

I think he's saying that there will always be a limit to our current understanding. That isn't to say we won't understand more, but rather that each time we learn something new, it simply changes that particular boundary.

--A

The more I read the more I don't believe the incompleteness theorems are about "Knowing" at all. They are about the ability to rigorously "prove" what we know. Not being about to prove something doesn't mean we can't know it. It also doesn't mean it does not have an existence independent of the human mind.

[Avatar]

No, but it does mean that we can't prove it. While we might "know" something unprovable, it must be proved in order to have that independant existance.

--A

[SerScot]

Avatar,

No, but it does mean that we can't prove it. While we might "know" something unprovable, it must be proved in order to have that independant existance.

--A

I doubt Godel, who was a Platonist, would agree with your assessment. He, and I, believe that existence is distinct from proof. Just because something may not be provable with our logic does not mean it cannot exist in a way that is unprovable.

Godel's proof showed that there are mathematical concepts that can be true but that cannot be proven using the axioms of a given formal system. Therefore, the same may hold true for the idea of mathematics being independent from human conception of mathematics despite the fact that we cannot within the axioms we can create for mathematic prove its independence from the human mind.

[Avatar]

*shakes head* Existence is different from knowing. If something exists objectively, there must be proof for it. Subjective existence can be iin your head. Concrete, physical existence requires proof. It requires that everybody can see and experience whatever it is.

The idea of something is not the same as its physical presence. Plato's ideal forms do not exist in any realistic way. Or at least, not in any physical way. They exist in our heads maybe, (our knowledge of what a certain thing is), but that knowledge comes from our perception and memory of what is real. What we have already experienced, or learned from the experience of others.

--A

[SerScot]

Avatar,

*shakes head* Existence is different from knowing. If something exists objectively, there must be proof for it. Subjective existence can be iin your head. Concrete, physical existence requires proof. It requires that everybody can see and experience whatever it is.

The idea of something is not the same as its physical presence. Plato's ideal forms do not exist in any realistic way. Or at least, not in any physical way. They exist in our heads maybe, (our knowledge of what a certain thing is), but that knowledge comes from our perception and memory of what is real. What we have already experienced, or learned from the experience of others.

--A

Nope. You should read up more on the Incompleteness theorems. Godel's proofs showed that, in mathematics, there are some true propositions that will remain unprovable for a given formal system. It's pretty interesting.

Try Incompleteness by Rebecca Goldstein. She gives a pretty cogent explanation of the Incompleteness theorems that are accessible even to the non-math geek, which I most assuridly am.

[Avatar]

In the meantime, why not give me an example of something that is true, yet cannot be proved?

--A

[SerScot]

Avatar,

The best I can do for now is quote this old statements:

"This very sentence is a lie." Is the statement true or false? Or the idea of "the set of all sets not contained in a set". Does that set contain itself? If it does how can it be "a set of all sets not contained in a set?

How the anwers to these assertion be proven?

I'm going to quote the passages of Incompleteness we I have more time.

[SerScot]

Rather than quote extensively from Incompleteness let me refer you to the wiki on Godel's Incompleteness theorems. Specifically, to the section discussing "The Liar's Paradox".

en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems

That should explain what I'm trying to illustrate. Godel showed that by replacing "truth" with "provability" it was possible to have a true statement that was fundamentally unprovable.



I hope you have a Merry Christmas!

[Avatar]

"This very sentence is a lie." Is the statement true or false? Or the idea of "the set of all sets not contained in a set". Does that set contain itself? If it does how can it be "a set of all sets not contained in a set?

How the anwers to these assertion be proven?

But it can be a set of all sets contained in a set, can't it? The liar paradox is of course a classic one, but since it's self-referential, there is no one solution, and as always, the solutions are therefore pretty damn subjective.

I've always quite liked the idea that it's not a paradox at all...that the statement is a claim that the given statement is false. In other words, that the claim (A) is that A is false. Therefore A=(A=False). If A is not able to be both true and false, then the equation is logically unsolvable.

But, it isn't an example of something that is true and unprovable I think. Not being able to prove that the sentence is false does not make it true.

--A

[SerScot]

NM

[Hashi Lebwohl]

I really regret being away from here for so long and missing this particular thread. There is no way I could realistically reply to every point made thus far, so I will simply jump in at no particular starting point.

Mathematics, that which we call the system of rules about numbers we have discovered to date, exists independently of human beings. It doesn't matter who you are or where you come from, if I put three rocks on a table and then another four rocks in a different pile, you will understand it when I combine the rocks that 3 + 4 = 7. Bertrand Russell looked into this and noted that numbers aren't really numbers; rather, they are equivalence classes of sets containing a certain number of elements.

Even if you begin with only positive integers, you can quickly develop arithmetic. Positive integers leads to addition, addition leads to subtraction (its opposite), subtraction leads to zero (a number minus itself), and thence to negative numbers ("what is 4 - 7?"). Addition, when repeated, leads to multiplication, thence to division (its opposite), thence to fractions/rational numbers. Multiplication also leads to squaring integers, which leads to square roots, which then leads to irrational numbers.

If you work arithmetic problems but replace one number with a letter then you have algebra. Algebra leads to graphing equations; if you have the concept of the slope of a line then you ultimately get to calculus (derivatives are slopes at a single point), etc.

Sagan was right--if we ever meet any extraterrestrial beings of sufficient sentience and advancement then we will be able to "talk" to each other only via mathematics at first.

Godel is also correct vis-a-vis truth versus provability. I can tell you "I have a roll of paper towels in the lower drawer of my desk at work"--which is true--but none of you can prove it. Any system of axioms that claims to be "complete" cannot prove all true statements in the system.

The numbers we call "prime" are made up, in a certain sense. If I put 29 rocks on a table, there is no way I can arrange the rocks in a rectangular grid pattern to simulate a multiplication problem other than 1x29--29 is prime, so no integers divide it leaving a remainder of 0.

The universe also exists independently of us and our ability to perceive it. The best definition of reality I ever heard is "that which continues to exist (or be true) even when you quit believing in it".

Godel, Escher, Bach is probably the most difficult book I have ever read. I recommend it to anyone and everyone.

[Avatar]

Giving up SerScot? I am open the the possibility of being completely wrong, y'know. Hell, I might even be talking complete nonsense.

Godel is also correct vis-a-vis truth versus provability. I can tell you "I have a roll of paper towels in the lower drawer of my desk at work"--which is true--but none of you can prove it. Any system of axioms that claims to be "complete" cannot prove all true statements in the system.

But whether or not you have those paper towels in your desk is provable, just not immediately and by us. We could contact somebody in your office and have them independantly verify the fact if we wanted.

Mathematics, that which we call the system of rules about numbers we have discovered to date, exists independently of human beings. It doesn't matter who you are or where you come from, if I put three rocks on a table and then another four rocks in a different pile, you will understand it when I combine the rocks that 3 + 4 = 7

That's what I'd call arithmetic, not mathematics. 3+4=7, 2+2=4, clearly and indisputably common and verifiable.

But when you get to mathematics, an assumption of commanlity is just that. An assumption. Those hypothetical aliens might agree that 2+2=4, but will they agree that E=MC2?

(Nice to see you around by the way. )

--A

[SerScot]

Avatar,

The key is whether it is provable a priori without obervation. That's the defintion of "proven" in mathematics. My wife explained why I was having difficulty with this over the weekend.