Hrvoje1

What is more common in nature, regularities or irregularities?

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I am sorry, but I don't get it again. If that sequence would reach a very large number n, and continue consistently on for as long as we can follow, according to the same rule, as an output of a random number generator, I wouldn't call it perfect, but extremely poorly designed random number generator. In fact I don't know what could be worse than that. Maybe to produce a dice for a sole reason of using it for random number generation, and whenever you throw it, you get the same number? Or a random number generator that produces such output: {1,2,3,4,5, ...}? If I am not missing something here, that would be the same class of failure. Right?

I mean, if you construct something that is not supposed to output something that can be described by such a simple rule, and you still get it, then you didn't do a good job.

OK, I got it you are right, there is no reason why should we expect that random sequences cannot turn out to be easily described by simple rules.

Especially if we believe that nature is regular in its essence. 

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So, the question is, why are random sequences in reality, in nature, not like that, describable by simple rules? With a perfectly balanced dice, getting every time the same number would be in conflict with the law of large numbers. I think I am at the end of my understanding am I confronted here with some oddness or not. My intuition tells me that random number sequence cannot be generated by a simple rule, because, it is not generated by any rule, by definition, and that actually tells me my logic too. Maybe you were seeking to much for some mystery here, when there is none, studiot?

Although, that is also some kind of rule (no rule).

And it's a simple one.

And if you draw a random line in a coordinate system, with a free hand, that is, without a ruler and compass, I bet it would be asymmetric, for the same reason.

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Posted (edited)
On ‎5‎/‎18‎/‎2019 at 12:39 PM, studiot said:

Consider the sequence

{1, 2, 4, 7, 11, 16 ...}

This can be generated in a variety of ways,

Actually, the sequence that you mentioned can be described only in these two ways, using a recurrence relation [math]x_n=x_{n-1}+n , \forall{n>0}[/math], which is an example of a first order linear difference equation, with initial condition [math]x_0=1[/math], and by its closed-form solution [math]x_n=1+\frac{n(n+1)}{2}[/math]. That this formula is a solution of that difference equation can be shown like this:

[math]x_1=x_0+1[/math]
[math]x_2=x_1+2=x_0+1+2[/math]
[math]x_3=x_2+3=x_0+1+2+3[/math]
...
[math]x_n=x_0+1+2+3+...+n=x_0+\sum_{k=1}^{n} k=1+\frac{n(n+1)}{2}[/math]


[math]x_{n-1}=1+\frac{(n-1)n}{2}[/math]

[math]x_n-x_{n-1}=1-1+\frac{n}{2} (n+1-(n-1))=\frac{n}{2}(1+1)=n [/math]

The fact that this sequence cannot be output of a true random generator is a fine example of regularity of nature, which can be expressed by a rule that when certain outcome is not enforced (by some rule, algorithm or physical constraint), it does not happen, because there is a multitude of other, equally probable possibilities. So, finite sequence {1, 2, 4, 7, 11, 16} can be easily a result of random choosing of 6 numbers from a certain range of numbers, for example from the first 16 natural numbers, but the regular infinite sequence {1, 2, 4, 7, 11, 16 ...} cannot be a result of random choosing among all natural numbers.

On ‎5‎/‎16‎/‎2019 at 1:21 PM, studiot said:

The whole beauty of the English language is that it has so many words with similar meanings and even substantial overlap of meaning.
This is because there is so many subtle variations of meaning available.

While the "random" sequence was mildly interesting contribution to the discussion, this was a very pedestrian observation. In fact this is even worse than that, this could have been pedestrian if you mentioned instead the beauty of any natural language, which, for your information, share the same trait.

Edited by Hrvoje1

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Posted (edited)
19 hours ago, Hrvoje1 said:

when certain outcome is not enforced (by some rule, algorithm or physical constraint), it does not happen,

I mean, when certain outcome is not deterministically caused, it happens according to its probability, which is in this case zero. At least in an experiment that is repeated an infinite number of times, such as here. Choosing one infinite sequence among an infinite number of such sequences, means choosing one number among an infinite number of natural numbers, infinitely many times, to form that infinite sequence. Right?

20 hours ago, Hrvoje1 said:

there is a multitude of other, equally probable possibilities

Infinite random sequences are all equally possible outputs of true random number generator, while deterministic infinite sequences are all equally impossible.

The only problem with randomness is that the Bourbaki school considered the statement "let us consider a random sequence" an abuse of language.

Edited by Hrvoje1

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Posted (edited)

Actually, I was solving first order difference equation [math]\Delta(x_{n-1})=n[/math] while you proposed second order difference equation [math]\Delta^2(x_n)=1[/math], which are equivalent and have the same closed-form solution, which can be shown like this:

[math] \Delta(x_{n-1})=x_n-x_{n-1}=n \implies \Delta(x_n)=n+1 \implies \Delta(x_{n+1})=n+2[/math]

[math]\Delta^2(x_n)=\Delta(x_{n+1})-\Delta(x_n)=n+2-(n+1)=1[/math]

So, this adds a little bit of variety to defining the same sequence.

Edited by Hrvoje1

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Posted (edited)

Of course that I should have said that you proposed [math]\Delta^2(x_{n-1})=1[/math], because that also holds true if [math]\Delta^2(x_n)=1[/math], it only would be less clumsy to show the connection with my proposal:

[math]\Delta^2(x_{n-1})=\Delta(x_n)-\Delta(x_{n-1})=n+1-n=1[/math]

But the truth is that these two proposals are not equivalent, I dropped one summation constant in the process. Whoever has a first clue about math, knows the link between difference equation [math]\Delta^2(x_n)=1[/math] that has solution [math]x_n=x_0+(x_1-x_0-1)n+\frac{n(n+1)}{2}[/math] and differential equation [math]x''(t)=1[/math] which has solution [math]x(t)=x(0)+x'(0)t+\frac{t^2}{2}[/math]

So, basically, as there are two integration constants, for a second order differential equation, here are two summation constants for a second order difference equation, which represent initial conditions.

On ‎5‎/‎23‎/‎2019 at 10:35 PM, Hrvoje1 said:

Choosing one infinite sequence among an infinite number of such sequences, means choosing one number among an infinite number of natural numbers, infinitely many times, to form that infinite sequence.

Of course that this is only true for random sequences, for deterministic sequences only initial conditions can be chosen, and all other values are bound to each other by a rule, or bound to a rule, by that rule. So, after sorting out that confusion, and understanding from the start that certain sets or sequences, such as prime number sequence, cannot be expressed by efficiently computable formula, I soon became curious about a classification of deterministic algorithms that generate sequences, and study of random sequences, and this led me to fascinating subjects I was clueless about, such as:

https://en.wikipedia.org/wiki/Arithmetical_hierarchy

https://en.wikipedia.org/wiki/Seven_states_of_randomness

https://en.wikipedia.org/wiki/Recursively_enumerable_set

https://en.wikipedia.org/wiki/Computable_function

https://en.wikipedia.org/wiki/Primitive_recursive_function

... and so on.

Edited by Hrvoje1

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What first struck my mind is the analogy of hyper sieve: to find a random sequence, you have to filter out all possible deterministic sequences, just as you have to filter out multiples of all previous primes to find a next one. Although, it is not a great analogy, more like a bizarre idea.

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On ‎5‎/‎17‎/‎2019 at 4:33 AM, wtf said:

That's the best you can do? As moderator setting the example for the tone around here?

I resent that tone too, and I noticed that wtf's presence on this forum ceased from that moment on. I regret that, because that person is by far the best discussant I have encountered around here, and I hope that this absence is not for good.

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19 hours ago, Hrvoje1 said:

I resent that tone too, and I noticed that wtf's presence on this forum ceased from that moment on. I regret that, because that person is by far the best discussant I have encountered around here, and I hope that this absence is not for good.

!

Moderator Note

First, if swansont is involved in a mainstream thread's discussion, he doesn't moderate that thread. He's always been extremely good about this in all the years he's been doing this. In this thread, he's a professional physicist discussing the science.

Second, anyone who reads this thread can tell you're being emotional about a rational stance. swansont made a comment about personal opinion not being meaningful when applied to science, and he was right. Your objections are therefore more about the fact that he disagreed with you, or that he wouldn't let you strawman him or move the goalposts (as both you and wtf were obviously doing). 

Third, if you have a problem with another poster, use the Report Post function. We have a whole procedure for that. No need to take a thread off-topic to voice your frustrations.

 

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I didn’t notice that fact that he disagreed with me.

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And the rest of that post is equally questionable, especially the mentioned moving of goalposts. That could make sense if debating is considered an adversarial competitive sport discipline in which the primary goal is to defeat your "opponent" (that should have been your fellow participant in discussion), by proving him wrong whatever he claims. Even better, prove him wrong even when he does not claim anything, just asks. Such as for example here, I learned from Phi for All's post that swansont disagreed with me even though I only asked >>What is more common in nature, regularities or irregularities?<< and left open for discussion what is exactly regularity in nature. Although I felt I should mention at least some examples, I never thought it can become an immediate source of disagreement.
However, I consider debating an activity in which one of main virtues is to come up with a question that is worth debating in the first place, in which such sports-like metaphors are meaningless. If during the discussion one realizes there is a better question, such as >>What is more fundamental principle in nature, regularity or irregularity?<<, it is not "cheating" to pose such question in the same thread. There is no "goal moving", the goal is always the same: to find the most meaningful question for discussion, only the question may be changed, ie it may be improved. Only a narrow minded person whose whole idea of participating in a forum is to humiliate other discussants could consider that as goalposts moving. Because, like, someone invested some time and effort to defeat me, by proving my claim wrong, or my question meaningless, and now I am cheating by finding better question. We are obviously not playing the same game here, although engaged in same activity, if you can draw such conclusions from what I wrote.
Let me be clear, I don't consider neither of these questions a great achievement of neither originality nor importance, as I already said, I'm afraid I didn't do that good job as I wanted, but I certainly did not "strawman" anybody (what a strange verb), or "move goalposts".

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13 minutes ago, Hrvoje1 said:

And the rest of that post is equally questionable, especially the mentioned moving of goalposts.

!

Moderator Note

Then you need to report it instead of posting about it, off-topic, in this thread! A third moderator or an administrator will look into it. Complaining in the thread is against the rules.

 

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15 hours ago, Hrvoje1 said:

That could make sense if debating is considered an adversarial competitive sport discipline in which the primary goal is to defeat your "opponent" (that should have been your fellow participant in discussion), by proving him wrong whatever he claims.

How do you deal with claims that are false, or sufficiently supported? Do you not post with a contrary position?

15 hours ago, Hrvoje1 said:

Even better, prove him wrong even when he does not claim anything, just asks. Such as for example here, I learned from Phi for All's post that swansont disagreed with me even though I only asked >>What is more common in nature, regularities or irregularities?<< and left open for discussion what is exactly regularity in nature. Although I felt I should mention at least some examples, I never thought it can become an immediate source of disagreement.

You went further than that, though, in defining regularity as perfect symmetry. You didn't "just" ask a question. And I disagreed with this assessment, and later on you seemed as well, with the mention of things having axial symmetry. 

Symmetry is a continuum in its degree, not a binary condition, and even then, there is the notion of identical particles, as was mentioned. There are also conservation laws in physics, which reflect underlying symmetries in nature.

People also pointed out that "regular" has multiple meanings, and within math, is used in a particular way. 

 

 

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Posted (edited)

So, the notion of identical particles is a clear indicator of regularity of nature, but perfect symmetry at that very small scale, that is a direct consequence of that identicalness, and wouldn't be possible without it, is not? How very logical. You obviously oppose not because of what I say, but just because you don't want to agree with me. Which is fine by me, because I have a few points of disagreement with you too. I don't think it is necessary to insult intentionally your fellow participant in discussion, instead of saluting at the end of discussion, and I don't think it is wise to underestimate the wisdom of common people, I met a lot of them that are smarter then some registered, licensed scientists.

Edited by Hrvoje1

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