Proof that pattern recognition is unending?

[Feynstein100]

So I've thought of an admittedly crude proof that the process of pattern recognition i.e. finding patterns, be they linguistic, mathematical, artistic, whatever, is a process that can never end.

It goes like this: Imagine we find all patterns, and I mean ALL of them, and we list them on a piece of paper. But then we notice that there are now patterns within the patterns of our list. No problem. We'll just add them to our existing list. However, this now creates new patterns which weren't there before, ad infinitum. Our list will never be complete. There will always be something new in it.

Basically, because patterns are self-interacting, the mere process of collecting them (i.e. getting them to interact spatially), creates new ones which didn't exist before. There will always be a new one that isn't on our list, meaning the process of finding new patterns will never end.

Anyway, can this be considered an actual proof or have I made a mistake somewhere? Has someone discovered this already perhaps?

 

[phinds]

It goes like this: Imagine we find all patterns, and I mean ALL of them, and we list them on a piece of paper.

You're going to need an infinitely long piece of paper.

 

[FactChecker]

So I've thought of an admittedly crude proof that the process of pattern recognition i.e. finding patterns, be they linguistic, mathematical, artistic, whatever, is a process that can never end.

"Can never end" is a vague term. It would be better to say that they can never be listed in a countable, finite, list, if that is what you mean.

It goes like this: Imagine we find all patterns, and I mean ALL of them, and we list them on a piece of paper. But then we notice that there are now patterns within the patterns of our list.

You would need to prove that there are some that are not on the list.

No problem. We'll just add them to our existing list. However, this now creates new patterns which weren't there before, ad infinitum. Our list will never be complete. There will always be something new in it.
Basically, because patterns are self-interacting, the mere process of collecting them (i.e. getting them to interact spatially), creates new ones which didn't exist before. There will always be a new one that isn't on our list,

Can you construct one that you can prove is not on the list?

meaning the process of finding new patterns will never end.
Anyway, can this be considered an actual proof or have I made a mistake somewhere? Has someone discovered this already perhaps?

Sorry, this is not a proof. You should look at the Cantor's diagonal proof that the real numbers in the [0,1] interval can not be listed in a countable (even infinite) list. See this:

 

[Feynstein100]

You're going to need an infinitely long piece of paper.

Well, that kind of assumes that there were already an infinite number of patterns lol. That kind of defeats the purpose of the proof

 

[Feynstein100]

You would need to prove that there are some that are not on the list.

I thought that was implied. The fact that we noticed patterns between the patterns means that they weren't there before, otherwise they'd already be on the list

 

[FactChecker]

I thought that was implied. The fact (emphasis mine) that we noticed patterns between the patterns means that they weren't there before, otherwise they'd already be on the list

You mean "The assumption", not "The fact". Now prove that your assumption is true.

 

[Feynstein100]

You mean "The assumption", not "The fact". Now prove that your assumption is true.

Hmm good point. I guess it doesn't work as a proof, but more of an if....... then......... structure. If new patterns do emerge from the list of patterns, the process will repeat ad infinitum. However, that's not guaranteed to happen. I feel like it should though. Still, that's just a hunch, not actual proof.

 

[Feynstein100]

You should look at the Cantor diagonal proof that the real numbers in the [0,1] interval can not be listed in a countable (even infinite) list. See this:

Yes, I'm already familiar with that, and with Veritasium's video. I don't see how that's relevant, however

 

[FactChecker]

Yes, I'm already familiar with that, and with Veritasium's video. I don't see how that's relevant, however

My point was that Cantor's proof actually constructed a new number that he proves is not already on the list. That is what I think you have to do. You have to construct a specific pattern that you can prove is not already on your list.

 

[jbriggs444]

Imagine we find all patterns

Before we can do that, we should define "pattern".

Edit to add a possible definition:

Definition: A string (possibly infinite) has a "pattern" if it is computable.

Claim: The set of "patterns" is countably infinite but not recursively enumerable.

 

[FactChecker]

Before we can do that, we should define "pattern".

Very good point. And I think that shows how hard it might be to develop a formal proof. Although I suspect that a lot of formal groundwork has already been done on this subject.

 

[FactChecker]

Is this is an uncountable set of patterns?: For each $r \in \mathbb{R}, Pattern_r = {r, r+1, r+2, r+3, ...}$. And since the real numbers are uncountably infinite, this is a set of patterns that could be considered uncountably infinite.
Although this set of patterns can be easily defined, I don't think that you could say it is computable in any way because the initial real number is not defined. Can we lump all of these together to call them one pattern?
This brings up another issue -- not only would we need to define what a pattern is, but we also need to define what makes two patterns different. On a two-dimensional graph, the set of patterns defined above would just be a set of parallel lines. The entire set of patterns could easily be considered one pattern on a graph.
I don't think that I can add anything more constructive to this thread, but I would be interested to know if there has been formal work done on this subject.

 

[Stephen Tashi]

Before we can do that, we should define "pattern".

Yes. Suppose we take the "cheapest" definition: A pattern in a set S is some subset of S.

Let S(n+1) denotes the set who elements are the subsets of the set S(n). If S(n) is finite, then S(n+1) is finite. But the collection (set) of all the S(i) , i = 1,2,3.... is not finite if we consider each S(i) as one distinct element of that collection.