Proof Infinity is Impossible

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Thomyum2
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Re: Proof Infinity is Impossible

Post by Thomyum2 » June 27th, 2020, 2:16 pm

Hello devans99, a very lively and interesting discussion you’ve started! I’m late to join, but will give you my take, for what it’s worth.

As I see it, you are trying to make a proof by contradiction here by postulating an infinite object and trying to show that its existence must results in an impossibility. But the contradictions you describe don’t follow from your argument, in my mind. So…
devans99 wrote:
June 26th, 2020, 7:11 am
A finite brick with left and right end is the type of brick we are used to. An actually infinite length brick would have a left end but continue forever to the right - IE it would have no right end - if it had a right end it would of course be a finite length brick.

If we reduce the length of our actually infinite brick to finite length, then we have a finite length brick with a left end but no right end. But such a brick cannot possibly exist - if it has no right end, it has no middle (because the middle would count as the right end) and if it has no right end, it cannot have a left end either (as the left end would also count as the right end) - so the brick cannot exist.

Reenforcing this, a brick with a left end but no right end obviously has an UNDEFINED length and all objects must have a non-zero, positive length to exist.
I think the first problem with the logic is here – I’d agree that an object must have a non-zero, positive length to exist (physically), but it doesn’t follow that if the length of an object is undefined that it cannot exist. In other words, to exist, the object must have length, but it is not a requirement for existence that the length be measurable or defined.

And as you said a little further along that:
devans99 wrote:
June 27th, 2020, 11:59 am
…length implies the ability to measure something. You can't measure something if it has no start or end or both.
I would again agree that an infinite object cannot be measured, but that an object has a length has nothing to do with whether or not you have ability to measure it. So again, a brick with no end cannot be measured, which means only that the length cannot be determined, not that it cannot exist.
devans99 wrote:
June 26th, 2020, 7:11 am
Returning to the ‘set’ of natural numbers:

{1, 2, 3, 4, 5, 6, 7, 8, … }

If something existed in reality with the entire structure of the natural numbers, then like the brick, it would have no right end. If it has no right end, then it has no end-1 (because that would count as the end). If it has no end-1, it cannot have an end-2. If it has no end-n, it has no end-(n+1). We can then use mathematical induction to work backwards through all the natural numbers to show the sequence has no start either. So anything with the structure of the entire natural numbers (IE actual infinity) cannot possibly exist in reality, or even logically - it would have no end; therefore no start and so it would not exist at all.

So (actual) infinity appears logically impossible?
I’m not clear what you mean about a physical object having the ‘entire structure of the natural numbers’, but I think you’re continuing the same error here. Natural numbers are counts, not measures. In other words, there’s a difference between infinite in quantity and infinite in size. You’re applying the logic of numbers to measurements but they’re not the same thing. One is analytic and one is synthetic/empirical.

Lastly, you’ve said a couple of times that:
devans99 wrote:
June 27th, 2020, 9:44 am
The proof I am talking about is in the OP - infinite objects are impossible. Maths claims that infinite objects are possible (axiom of infinity), hence raising this post.
...and I’m not a mathematician by any means, but I'd just point out that I don’t think the proofs say that infinite objects are possible - they say that infinite sets are possible. So similarly, sets have a quantity but do not have a dimension that can be measured, so the analogy cannot work to disprove the possibility of an object of infinite length. As I see it, that's the essential confusion here, and the problem with the concept of infinity in general, that it's problematic to use mathematical concepts that are analytical to draw conclusions that rely on empirical observations for verification since these are two different realms of knowledge.

A proof by contradiction succeeds if it is one that results in a logical impossibility, but you are trying to show proof by describing a situation that is an empirical impossibility. But empirical observations can't be said to be either possible or impossible - we just observe what we are able to observe. So if you postulate that an infinite brick exists as a premise, then you can't rely on the limitation of our ability or to observe it, or to imagine that we could potentially observe it, to prove the impossibility of its existence.

Thoughts?

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Terrapin Station
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Re: Proof Infinity is Impossible

Post by Terrapin Station » June 27th, 2020, 2:29 pm

devans99 wrote:
June 27th, 2020, 11:53 am
Philosophy includes the philosophy of science - I don't think we can completely avoid physics whilst discussing this issue.
Philosophy of science isn't at all the same thing as physics, is it?
A hydrogen bomb is quite small - maybe a couple of meters long - that's not enough matter to destroy a city. I don't know the physics that well, but nuclear fusion converts hydrogen atoms to helium atoms - but the conversion is not 1:1 - a huge amount of additional energy is released in the process - from a very small amount of matter.
So it turns out that re: "Can you cite the observational data you'd like to use to support the claim (that we're observing energy sans matter when it comes to any phenomena involving hydrogen bombs)?"

The answer is actually, "No."

If you're going to do philosophy, you need to be critical of claims being made. It sounds like you don't even know how a claim is being made re hydrogen bombs that we're ever talking about energy sans matter.

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Re: Proof Infinity is Impossible

Post by Sculptor1 » June 27th, 2020, 2:35 pm

Atla wrote:
June 27th, 2020, 7:28 am
Sculptor1 wrote:
June 27th, 2020, 6:37 am

There is a computer on my lap.
There is not an infinite number of computers because I can see something other than computers, because if computers were infinite they would have to take up all available space.
They can be infinite and not take up all available space.
FFS how?

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Sculptor1
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Re: Proof Infinity is Impossible

Post by Sculptor1 » June 27th, 2020, 2:37 pm

devans99 wrote:
June 27th, 2020, 8:04 am
Steve3007 wrote:
June 27th, 2020, 7:59 am


Maths does not claim that.
Maths does claim that an infinite, physical thing exist:

https://en.wikipedia.org/wiki/Axiom_of_infinity
Maths is ONLY a model.
Its abstract.

Atla
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Re: Proof Infinity is Impossible

Post by Atla » June 27th, 2020, 2:43 pm

Sculptor1 wrote:
June 27th, 2020, 2:35 pm
Atla wrote:
June 27th, 2020, 7:28 am

They can be infinite and not take up all available space.
FFS how?
Space can be a bigger infinite ffs.

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Re: Proof Infinity is Impossible

Post by devans99 » June 27th, 2020, 2:45 pm

Thomyum2 wrote:
June 27th, 2020, 2:16 pm
Hello devans99, a very lively and interesting discussion you’ve started! I’m late to join, but will give you my take, for what it’s worth.

I think the first problem with the logic is here – I’d agree that an object must have a non-zero, positive length to exist (physically), but it doesn’t follow that if the length of an object is undefined that it cannot exist. In other words, to exist, the object must have length, but it is not a requirement for existence that the length be measurable or defined.
I would again agree that an infinite object cannot be measured, but that an object has a length has nothing to do with whether or not you have ability to measure it. So again, a brick with no end cannot be measured, which means only that the length cannot be determined, not that it cannot exist.
I’m not clear what you mean about a physical object having the ‘entire structure of the natural numbers’, but I think you’re continuing the same error here. Natural numbers are counts, not measures. In other words, there’s a difference between infinite in quantity and infinite in size. You’re applying the logic of numbers to measurements but they’re not the same thing. One is analytic and one is synthetic/empirical.
...and I’m not a mathematician by any means, but I'd just point out that I don’t think the proofs say that infinite objects are possible - they say that infinite sets are possible. So similarly, sets have a quantity but do not have a dimension that can be measured, so the analogy cannot work to disprove the possibility of an object of infinite length. As I see it, that's the essential confusion here, and the problem with the concept of infinity in general, that it's problematic to use mathematical concepts that are analytical to draw conclusions that rely on empirical observations for verification since these are two different realms of knowledge.

A proof by contradiction succeeds if it is one that results in a logical impossibility, but you are trying to show proof by describing a situation that is an empirical impossibility. But empirical observations can't be said to be either possible or impossible - we just observe what we are able to observe. So if you postulate that an infinite brick exists as a premise, then you can't rely on the limitation of our ability or to observe it, or to imagine that we could potentially observe it, to prove the impossibility of its existence.

Thoughts?
Hello Thomyum2!

I think that the requirement for an object to have a non-zero, positive, length does rule out any/all objects from having a UNDEFINED length:

An UNDEFINED length means the object is missing a start or an end or both. Such objects are invalid topologically. For example, an infinite brick with a left-end but no right-end, if it has no right-end, it cannot have a middle (because the middle would count as the right-end). And if it has no right-end, it cannot have a left-end, because the right-end would count as the left-end.

We can start measuring a brick with no end, but we won't get very far - none such brick can exist.

By ‘entire structure of the natural numbers’ - imagine a ruler with every natural number on it. Its topologically very similar to the brick analogy I'm using.

Natural numbers are measures by themselves and their quantity is equal to their size.

There is no proof in maths that actual infinity exists. They just assume it does (axiom of infinity).

Sets can be arrange linearly - then they have a dimension - so they are measurable with the natural numbers .

I hope you can see that set of natural numbers, arranged linearly, is a measure and I hope you see that it is analogous to an infinite ruler or infinite brick. But none of these things can possibly exist in reality. Its a finite world.

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Re: Proof Infinity is Impossible

Post by Gertie » June 27th, 2020, 2:48 pm

devans
With your infinite rock in space and laser beam, you still have an object with a start but no end - topologically invalid - cannot exist. In maths terms, you are effectively doing:

∞-1=∞

The above equation says we changed something (∞) and it did not change. That's a contradiction. That contradiction proves that infinity is possible.
Well as I said my basic point is that maths is descriptive, not prescriptive. The territory dictates the nature of the maps we create, not the other way round.

So far it works well. But it doesn't mean we can't ever come across such an object, and if we do, then the maths should change to reflect reality.

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Re: Proof Infinity is Impossible

Post by devans99 » June 27th, 2020, 2:50 pm

Sculptor1 wrote:
June 27th, 2020, 2:37 pm
devans99 wrote:
June 27th, 2020, 8:04 am


Maths does claim that an infinite, physical thing exist:

https://en.wikipedia.org/wiki/Axiom_of_infinity
Maths is ONLY a model.
Its abstract.
Maths explicitly declares there exists a set with a greater than finite number of objects in it.

Exists means either it exists in the mind or in reality. If it exists only in the mind, it does not exist. So maths says that actually infinite things exist in reality.

This has for example inspired cosmology into various infinite time/space theories... it is misleading people.

Maths legitimises infinity (without proof) and that misleads others in the physical sciences.

devans99
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Re: Proof Infinity is Impossible

Post by devans99 » June 27th, 2020, 2:55 pm

Terrapin Station wrote:
June 27th, 2020, 2:29 pm
A hydrogen bomb is quite small - maybe a couple of meters long - that's not enough matter to destroy a city. I don't know the physics that well, but nuclear fusion converts hydrogen atoms to helium atoms - but the conversion is not 1:1 - a huge amount of additional energy is released in the process - from a very small amount of matter.
So it turns out that re: "Can you cite the observational data you'd like to use to support the claim (that we're observing energy sans matter when it comes to any phenomena involving hydrogen bombs)?"

The answer is actually, "No."

If you're going to do philosophy, you need to be critical of claims being made. It sounds like you don't even know how a claim is being made re hydrogen bombs that we're ever talking about energy sans matter.
Sure, they tested a whole lot of hydrogen bombs back in the 50s. They really do convert matter to energy. I've seen archive video footage of it- counts as 'observational data'.

So the fact that matter and energy are equivalent is supported empirically as well as theoretically.

devans99
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Re: Proof Infinity is Impossible

Post by devans99 » June 27th, 2020, 3:03 pm

Gertie wrote:
June 27th, 2020, 2:48 pm
Well as I said my basic point is that maths is descriptive, not prescriptive. The territory dictates the nature of the maps we create, not the other way round.

So far it works well. But it doesn't mean we can't ever come across such an object, and if we do, then the maths should change to reflect reality.
Interesting point. Maths appears to describe reality (via the various physical theories). Does that imply if its mathematically impossible, it physically impossible? Not deductively, but inductively: we have never found a logical contradiction in reality - so reality can be said to be logical. Maths is built from logic, so I think we can inductively conclude with a high degree of confidence that if its mathematically impossible, it physically impossible?

Then is it possible to (theoretically) construct realities which are not logical? For example, we could abandon one of the 3 laws of logic, like the LEM. My feeling is that we would not be able to construct viable realities that a not logical (hence not mathematical). Not sure quite how you'd go about proving that... sounds tough.

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Re: Proof Infinity is Impossible

Post by Atla » June 27th, 2020, 3:11 pm

devans99 wrote:
June 27th, 2020, 2:50 pm
Exists means either it exists in the mind or in reality. If it exists only in the mind, it does not exist. So maths says that actually infinite things exist in reality.
All of maths only "exists in the mind". Math is concepts / math is abstract, which isn't quite the same as "does not exist".
Maths says nothing about physical reality.

What is so difficult about this?

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Re: Proof Infinity is Impossible

Post by Terrapin Station » June 27th, 2020, 3:18 pm

devans99 wrote:
June 27th, 2020, 2:55 pm
Sure, they tested a whole lot of hydrogen bombs back in the 50s. They really do convert matter to energy. I've seen archive video footage of it- counts as 'observational data'.
You've seen video footage of energy that's not matter in motion? lol

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Re: Proof Infinity is Impossible

Post by devans99 » June 27th, 2020, 3:22 pm

Atla wrote:
June 27th, 2020, 3:11 pm
devans99 wrote:
June 27th, 2020, 2:50 pm
Exists means either it exists in the mind or in reality. If it exists only in the mind, it does not exist. So maths says that actually infinite things exist in reality.
Maths says nothing about physical reality.
I disputed that in the post above.

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Re: Proof Infinity is Impossible

Post by Atla » June 27th, 2020, 3:25 pm

devans99 wrote:
June 27th, 2020, 3:22 pm
Atla wrote:
June 27th, 2020, 3:11 pm

Maths says nothing about physical reality.
I disputed that in the post above.
You didn't. Sets are still part of maths and say nothing about physical reality.

devans99
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Re: Proof Infinity is Impossible

Post by devans99 » June 27th, 2020, 3:25 pm

Terrapin Station wrote:
June 27th, 2020, 3:18 pm
devans99 wrote:
June 27th, 2020, 2:55 pm
Sure, they tested a whole lot of hydrogen bombs back in the 50s. They really do convert matter to energy. I've seen archive video footage of it- counts as 'observational data'.
You've seen video footage of energy that's not matter in motion? lol
You can see atomic bomb blasts - with your eyes - that's got to be energy - photons.

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