Does Distance exist without time?

[Prince_James]

CptBork:

In a case like or , we first have to say that these things exist, and define them, before we can prove or disprove anything regarding these objects. We know that from the basic axioms of math, these types of terms are undefinable, so in order to define them we'd have to add extra axioms. I don't see how Godel's theorem talks about this anymore than how it talks about whether math can be used to prove or disprove that Hollywood stars wear nice clothes.

A classic example of GIT is the Liar's Paradox-like statements such as:

The following statement is true: The prior statement is false.

Let's treat 0 * N = 0 and x ^ 0 = 1 as axioms.

If you now combine them as 0 ^ 0 you have two conflicting axioms. You have the same sort of problem you have in the pseudo-Liar's Paradox, where the statement is wrong under one rule no matter what. This is a classic example of GITS. There is no truth value in the system of mathematics for this statement. It is neither true nor false, because the system is incapable of coherently adressing the answer.

 

[Prince_James]

BenTheMan:

Prince James---

The concept of infinity is ill-definied, I think. Someone please correct me, but if infinity was well defined, then one could make statements such as for some . But this statement is clearly false, i.e. infinity is never obtainable by adding a finite number of integers together. In other words, infinity is NOT a number. If it were, then it must obey the following algebraic equation:

.

So we MUST define infinity in terms of a limit.

I believe infinity is treated as a hyperreal in mathematics.

But that being said, you are correct: The number one away from infinity is a meaningless concept. No finite number is every anything but infinitely away from infinity. That being said, even if infinity is not a number in the same sense as the whole numbers are, it is still a meaningful concept, and has been used in mathematics in hyper reals, set theory, et cetera, et cetera.

That is to say, it is not hogwash. It is not gibberish. It is not incoherent or absurd. But it is not something that can be reached finitely. That being said, I would say that the ideal value of infinitely counting would produce infinity. THat is to say, given infinite time, you would reach infinity.

 

[Quantum Quack]

What I donot understand is how you can divide a 12 inch brick infinitely and come up with a physically thin slice of infinitesimal thickness and then attempt to reverse teh equation by multiplying that physical size and expect not to exceed the length of the brick by infinity.

if you maintian an infinitismal has dimension thickness then multiplying this dimension by infinity must give you something infinitely big and certainly not a 12 inch brick.
So I fail to see how the use of a minimum size is possible nor logical.

say in very poor maths:
what is proposed:

12" / infinity = infinitesimal value a

reversed
value a * infinity = must equal infinity [ not 12" ]
the same would apply to time I would think.

any one?

 

[CptBork]

What I donot understand is how you can divide a 12 inch brick infinitely and come up with a physically thin slice of infinitesimal thickness and then attempt to reverse teh equation by multiplying that physical size and expect not to exceed the length of the brick by infinity.

if you maintian an infinitismal has dimension thickness then multiplying this dimension by infinity must give you something infinitely big and certainly not a 12 inch brick.
So I fail to see how the use of a minimum size is possible nor logical.

say in very poor maths:
what is proposed:

12" / infinity = infinitesimal value a

reversed
value a * infinity = must equal infinity [ not 12" ]
the same would apply to time I would think.

any one?

There's no problem with it at all; first you consider what would physically happen if you treated the brick as, say, a dozen finite slices. Then you would consider what would happen if it was sliced up 100 times, then 10 000 times, and so on. From the methods of analysis, you can then show that this process reaches a limit, i.e. there is a physical value which you can get arbitrarily close to by taking a sufficient number of slices (or more), and this limit can then be determined from the infinitesimal calculus.

 

[Reiku]

What do you know of Solitons cpt?

 

[Prince_James]

Solitons cpt?

 

[Prince_James]

Quantum Heraclitus:

value a * infinity = must equal infinity [ not 12" ]
the same would apply to time I would think.

Try zero. It is even worse. Zero * n = Zero.

I once reasoned about this more in depth than that. If I can find my notes, I'll talk about it.

 

[AlphaNumeric]

Regarding Godel's Incompleteness Theorem (GIT) and division by zero and other undefines. As you are aware, GIT postulates that every formal system with a complexity that can encompass mathematics will either be complete and inconsistent, or consistent and complete. THat is, if it is truly powerful enough to encapsulate all theorems from the axioms necessary for mathematics, there will be statements which are improvable in the system. As such, we come upon things in mathematics where the application of the axioms cease to make sense and we create theorems which can neither be proven or disproven either which way without a sacrifice somewhere in the system, an inconsistency. 0/0 is traditionally held to be undefined. Infinity/infinity, too (although as seen I prefer the answer "1" for that). These are examples where the theorems cease to be descriptive of a coherent and complete system, and instead represent the Godelian breach that lead to the GIT. 0^0 is another one of these cases. 0 * n = zero; n^0 1. Any answer requires throwing out one of these principles, or inconsistency. As such, it is an example of GIT's hydric heads raising up and ruining logistic programmes of mathematics.

I don't think that's right.

GIT is about statements which can be entirely consistent with a set of axioms wether it's true or not. You have to define it's true or negation but once you do that you're left with an entirely consistent theory.

infinity-infinity or infinity/infinity doesn't fall into that category, because if you say "Right, I define my algebra to be such that $$\frac{\infty}{\infty}=1$$ then you can immediately reach inconsistencies.

$$\frac{\infty}{\infty}=1$$ implies $$1 = \frac{\infty}{\infty} = 2*\frac{\infty}{\infty}=2*1 = 2$$

Infinity acts like a catch off buffer. You can add to or multiply it by anything you like, other than infinity, and get infinity again. But since you've equated it to something which doesn't obey that rule in general you're going to find inconsistencies everywhere. Whatever you define $$\frac{\infty}{\infty} $$ to be equal to it too must not obey the standard rules of algebra on the Reals or else you'd get inconsistencies. And then you end up with another quantity you cannot relate to the reals. Hence you end up developing a seperate algebra, where you cannot cross back and fore to the Reals in a consistent manner, they must remain seperate or travel is 'one way'.

http://en.wikipedia.org/wiki/Non-standard_analysis#Pedagogical

Hence the infinitesimals. Even in them you don't get back from the infinesimals.

A classic example of GIT is the Liar's Paradox-like statements such as:

The following statement is true: The prior statement is false.

Let's treat 0 * N = 0 and x ^ 0 = 1 as axioms.

If you now combine them as 0 ^ 0 you have two conflicting axioms. You have the same sort of problem you have in the pseudo-Liar's Paradox, where the statement is wrong under one rule no matter what. This is a classic example of GITS. There is no truth value in the system of mathematics for this statement. It is neither true nor false, because the system is incapable of coherently adressing the answer.

I don't think this is true either.

GIT talks about a system which is consistent irrespective of the truth or negation of a statement which is construtable from the axioms. That is not what your example is an example of. Your example is a paradox. If a system construct what is essentially "A=0" and "A=1" then it's inconsistent (or trivial, like the trivial ring). It's not an example of the GIT in action.

I'll ask you again, have you actually studied set theory? You keep giving the impression you're very confident on this material but you keep missing the mark. By a long way.

What do you know of Solitons cpt?

What relevance does that have?

If you want to know about solitons, Euler is the guy to ask.

 

[Reiku]

Well, it's just that QQ is talking about infinitesimally breaking things down. It just crossed my mind concerning solitons... the pre-proposed theory that there is something smaller than even a photon, or a superstring. Basically useless, in other words. I was just thinking out loud.

 

[AlphaNumeric]

Solitons are more than just "There's something smaller than a photon". They arise in wave systems (infact, it was a concept seen by a mathematician while walking by a canal and it took 30 years before anyone could explain it) and quantum mechanics. Instantons and solitions can be used to explain a number of things in quantum field theory, beyond what you said.

 

[Quantum Quack]

There's no problem with it at all; first you consider what would physically happen if you treated the brick as, say, a dozen finite slices. Then you would consider what would happen if it was sliced up 100 times, then 10 000 times, and so on. From the methods of analysis, you can then show that this process reaches a limit, i.e. there is a physical value which you can get arbitrarily close to by taking a sufficient number of slices (or more), and this limit can then be determined from the infinitesimal calculus.

excuse my ignorance please but all this is saying to me is that you can not slice a 12 inch brick infinitely as there is a finite limitation.
Why are we saying we can when we can't?
a sort of pseudo infinity maybe?

 

[CptBork]

excuse my ignorance please but all this is saying to me is that you can not slice a 12 inch brick infinitely as there is a finite limitation.
Why are we saying we can when we can't?
a sort of pseudo infinity maybe?

Nope, it's just saying that as you consider bricks made out of thinner and thinner slices, the math which describes each of these cases approaches a limit, it converges to a specific value or answer. So say you first do the calculation, with the brick made out of 100 slices. Then you do the calculation with 10 000 slices, and you get a vastly different answer. Then you do 1 000 000 slices, and you again find a different answer, but this time not so different than you got for 10 000 slices. Now you do it for 10 000 000 slices, and you find it makes almost no difference. In cases like these, you can prove that as you take thinner and thinner slices, the answer converges to a fixed value. So you're guaranteed to reach a point where, say, the difference between calculating what happens with 10^27 slices or 10^100 slices is negligible, and making the slices even thinner just brings you even closer to the value obtained by calculus.

This used to be considered a problem with the foundations of calculus, and it had mathematicians stumped for the good part of a century. However, the problem was resolved about 150 years ago thanks to folks like Karl Weierstrass and the techniques of Mathematical Analysis they developed. Unfortunately, the concepts are difficult to fully understand unless you have a solid math background coming into the typical Analysis course.

 

[Reiku]

Solitons are more than just "There's something smaller than a photon". They arise in wave systems (infact, it was a concept seen by a mathematician while walking by a canal and it took 30 years before anyone could explain it) and quantum mechanics. Instantons and solitions can be used to explain a number of things in quantum field theory, beyond what you said.

I admit, i know very little about them, other than they should be loosely coupled sub-subatomic waves/particles?

 

[Quantum Quack]

Nope, it's just saying that as you consider bricks made out of thinner and thinner slices, the math which describes each of these cases approaches a limit, it converges to a specific value or answer. So say you first do the calculation, with the brick made out of 100 slices. Then you do the calculation with 10 000 slices, and you get a vastly different answer. Then you do 1 000 000 slices, and you again find a different answer, but this time not so different than you got for 10 000 slices. Now you do it for 10 000 000 slices, and you find it makes almost no difference. In cases like these, you can prove that as you take thinner and thinner slices, the answer converges to a fixed value. So you're guaranteed to reach a point where, say, the difference between calculating what happens with 10^27 slices or 10^100 slices is negligible, and making the slices even thinner just brings you even closer to the value obtained by calculus.

This used to be considered a problem with the foundations of calculus, and it had mathematicians stumped for the good part of a century. However, the problem was resolved about 150 years ago thanks to folks like Karl Weierstrass and the techniques of Mathematical Analysis they developed. Unfortunately, the concepts are difficult to fully understand unless you have a solid math background coming into the typical Analysis course.

Ok thanks I appreciate your efforts.
I understand that it is more a limitation on the use of mathematics than on the perfect and puristic notion of infinity as used in philosophy or other fields.

hmmmm......If I get a chance I might have a look at your Karl Weierstrass... but as you know I have no math background so I am not sure it would be of any real value at present.