Chesskingloop wrote:
Im so sad im an f in idiot
It's university level mathematics. Don't feel bad unless you've have a maths degree, in which case you have no excuse. 
Does True Randomness Actually Exist? ( ^&*#^%$&#% )
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Nov 9, 2022
0
#4162
Elroch wrote:
Optimissed wrote:
Just to explain for the benefit of he O.P., this discussion regarding the true nature of infinity is fundamental to the general discussion regarding randomness, since the incorrect concept of infinity certainly gives a very different answer from that given by the correct one, regarding the behaviour of randomness in an infinitely long, random sequence.
With all due respect you don't have the understanding of such things that those who work with them have. It obstructs your understanding not to respect all of what is already understood. You will find a summary of the basics of stochastic processes in the Wikipedia article, including a mention of the Bernoulli Process, perhaps the simplest example, which also has its own article. Go to the latter link to see the theory of random sequences of two-valued variables (as we are discussing).
The key foundations are the notions of probability spaces, random variables, sequences of random variables and measure spaces.
Regarding the infinite, it is interesting to track the history of this concept in mathematics. Implicitly, Euclid was dealing both with the infinite algebraic object we call the natural numbers and with infinite 2 and 3 dimensional vector spaces, though he saw the latter in a very elegant axiomatic way rather than constructing them after constructing the real numbers as it common today. Indeed Euclid had not have a model of the real numbers, but did understand that not all geometrically constructible ratios are rational (there is a nice little proof that the square root of 2 is not rational).
Newton got by with a less rigorous approach than Euclid - typical of today's mathematical physicists as well! - dealing with infinitessimals (the inverses of infinite quantities) in a way which was not fully justifiable mathematically at the time but which, guided by his intuition, led him to correct conclusions about calculus.
He had no formal construction of the real numbers (although decimal numbers were effectively introduced in the 17th century, the formal construction of real numbers had to wait for Cantor in 1871!) so relied informally on obvious facts about magnitudes that were the extrapolation of intuition from explicit example calculations. Later his intuition was justified using the mathematics of hyperreal numbers, a consistent and powerful extension of real numbers that adds infinitessimals and infinities (yes, plural).
It would be unfair to say that mathematicians before the modern era didn't know they were not being rigorous. Archimedes (one of the best mathematicians of the classical era) already had an understanding that using infinitessimals was difficult to justify and preferred to get round this by using other techniques.
With all due respect I probably have 20 to 30 IQ points on you and at that end of the spectrum it makes a massive difference, if it's used right.
At best, it should be accepted as a genuine difference of opinion. I know I'm right. You know you're right.
Nov 9, 2022
0
#4164
Optimissed wrote:
The numeric difference should also do so, one would imagine.
I didn't have to imagine. My first version of the program looked at every possible sequence and simply added the difference of each then divided by number of sequences. This leaves nothing to the imagination. Nothing at all.
This gave me an additional way to verify the final version for small numbers of flips. (Not for larger numbers since for 1000 flips you'd have to check 2^1000 individual sequences... I guess it would be possible, but the run time would be... years? Something like that).
So even if for some bizarre reason you don't trust fundamental mathematics, you can literally count.
Nov 9, 2022
0
#4165
Hilarious, @Optimissed.
It's not me you are arguing with, it's the entire body of mathematical knowledge that deals with the infinite, including that about the Bernoulli Process.
The problem is that it seems all you have to stack up against that is a claim about your ability to do IQ tests. If you have some result you want to share - as I have on several occasions: some remembered, and some independently discovered (without any claim of precedence) - do so!
llama36 wrote:
Optimissed wrote:
The numeric difference should also do so, one would imagine.
I didn't have to imagine. My first version of the program looked at every possible sequence and simply added the difference of each then divided by number of sequences. This leaves nothing to the imagination. Nothing at all.
This gave me an additional way to verify the final version for small numbers of flips. (Not for larger numbers since for 1000 flips you'd have to check 2^1000 individual sequences... I guess it would be possible, but the run time would be... years? Something like that).
You should have taken part in the "solving chess" threads. The computing time is astronomical. I wonder where IS tygxc. But it was a real eye-opener for me when we did the calculations as to how much longer than five years it would take. I got a number that I didn't like to admit to so I just called it millions and didn't think even that would be believed. tygxc claimed 5 years.
Nov 9, 2022
0
#4167
llama36 wrote:
Optimissed wrote:
The numeric difference should also do so, one would imagine.
I didn't have to imagine. My first version of the program looked at every possible sequence and simply added the difference of each then divided by number of sequences. This leaves nothing to the imagination. Nothing at all.
This gave me an additional way to verify the final version for small numbers of flips. (Not for larger numbers since for 1000 flips you'd have to check 2^1000 individual sequences... I guess it would be possible, but the run time would be... years? Something like that).
So even if for some bizarre reason you don't trust fundamental mathematics, you can literally count.
A proof of a slightly more general version of the fact that the expectation of the square of the differences goes up linearly should be pretty easy. One version would use induction.
Actually, the clearest way is to prove that the variance of the sum of two independent random variables is the sum of their variances: it is a very simple result from that. Anyone want to prove that? (Looking up the proof will suffice, if you like).
Nov 9, 2022
0
#4168
Elroch wrote:
llama36 wrote:
Optimissed wrote:
The numeric difference should also do so, one would imagine.
I didn't have to imagine. My first version of the program looked at every possible sequence and simply added the difference of each then divided by number of sequences. This leaves nothing to the imagination. Nothing at all.
This gave me an additional way to verify the final version for small numbers of flips. (Not for larger numbers since for 1000 flips you'd have to check 2^1000 individual sequences... I guess it would be possible, but the run time would be... years? Something like that).
So even if for some bizarre reason you don't trust fundamental mathematics, you can literally count.
A proof of a slightly more general version of the fact that the expectation of the square of the differences goes up linearly should be pretty easy. One version would use induction.
Actually, the clearest way might be to prove that the variance of the sum of two independent random variables is the sum of their variances: it is a very simple result from that. Anyone want to prove that? (Looking up the proof will suffice, if you like).
Looks like I'm doing it all backwards, going for the easiest rout last...
https://en.wikipedia.org/wiki/Poisson_distribution
Variance (which is standard deviation squared) is = to the mean (for Poisson).
And mean = number of trials / 2 (for coin flips).
So... a linear relationship.
Not a formal proof, but I'm not a mathematician 
Elroch wrote:
Hilarious, @Optimissed.
It's not me you are arguing with, it's the entire body of mathematical knowledge that deals with the infinite about, including that about the Bernoulli Process.
The problem is that it seems all you have to stack up against that is a claim about your ability to do IQ tests. If you have some result you want to share - as I have on several occasions: some remembered, and some independently discovered (without any claim of precedence) - do so!
I know. There's no problem. Either I'm right, which I believe I am, or they are right. Or neither is right. I'm pretty bright and there have been other occasions where I've thought for myself.
Why are you even trying to argue in an authoritarian manner rather than trying to really think about it? I'd be more interested in what my own peers make of it anyhow. Philosophy versus maths. I would back philosophy every time because maths is process driven and results driven. It deals with real quanties, not imaginary ones. That is, not imaginary in the same way as infinity, of course. It's a difficult concept but if a faulty understanding makes no difference to the processes, why should they question their faulty conceptions?
You've accused me of arrogance but all I'm doing is transmitting what I genuinely think. Where's the arrogance in that?
Tell you what ... give the 2-line trivial proof you mentioned. I'll look at it and tell you if I think it needs clarificattion. That is, it has to be in intelligible English. One of two results will ensue. Either I'll find the mistake or I won't understand it.
Nov 9, 2022
0
#4172
Sure.
The definition of the set of results R is:
R = {all functions f with domain the natural numbers and range the set {H, T}}
The proposition is:
g is a member of R where g is defined by
g(n) = H for all n in the natural numbers [i.e. g is the sequence of all heads]
Proof
1. The domain of g is the natural numbers
2. The values of g always lie in {H, T}
QED
_______________________________
I told you it was trivial.
After all, you can't expect others to believe your assertions, where you claim that they are supported by a trivial, two-line proof, if you aren't prepared to give that proof. I'm always open-minded and enjoy learning new things. If it turns out I'm wrong, I will then try to track down the relevant, cognitive disassociation, to find out how that happened, since it would seem to be quite an unusual occurence and since it seems so obvious (to me) that it's 500 years of mathematicians that are wrong and not me. People have been wrong before, after all. It wouldn't be a big deal. I mean, to me. Maybe to the mathematicians though.
Just teasing in case you didn't post it.
No, it should be in English instead, because that's in code. It's possible you don't understand your own code and you genuinely believe it proves it even if it doesn't. Can't you write it in English?
I'll wait til the morning when my head may be clearer. I just ate some chocolate raisins and it gave me a suger hit. I may be able to understand the mathematical code but at a glance I don't think that's a proof at all. It will be a case of your having defined conditions into existence.
I was, I think, one of the first people to properly analyse ontological arguments, to determine where the deliberate mistakes were. I'm used to smuggled, disguised and variable premises.
Nov 9, 2022
0
#4179
LOL. Tell it to the people who write all the mathematics.
Amusingly, I actually did translate the mathematical symbols I would have written into English. I did leave some high school maths symbols (in full: the "=" symbol; the notation for a function and its argument (parentheses); curly brackets { .... } for a set).
But you do need to understand the English. And a few high school mathematical symbols.
The proposition is like the one "0.77777... (recurring) is a decimal number" and has a similar proof. Normally such things are clear enough without proof.
Nov 9, 2022
0
#4180
Optimissed wrote:
It seems so obvious (to me) that it's 500 years of mathematicians that are wrong and not me.
Intuition alone cannot break down what both intuition and rigor have built up.
And in any case, the stance "everyone else is wrong but me" is actual insanity, which is to say I don't mean that in a pejorative sense, it's a statement of fact: it's insane.
Llama is correct that the percentage difference becomes closer, the longer we toss the coins.
Yes, I see the light. Llama convinced me in post 4150 that while the % difference approaches 0, the actual difference does not.
The numeric difference should also do so, one would imagine. If not, one might be forgiven for imagining that it isn't a fair coin. If there's an infinite series of heads, a double headed coin is going to be the cause. I have some double headed 1880s dollars and 1930s English pennies. They turn up every so often, like bad pennies.