Does True Randomness Actually Exist? ( ^&*#^%$&#% )

This advanced topic has no consequence to the simple fact that a sequence of an infinite number of heads is one of the uncountable set of possible sequences (all of which individually have probability zero). That is a triviality.>>

I'm afraid I disagree with that triviality, where I think that word is misplaced but don't think that it can alter practical results so if they've all made a misake, which I believe they have, it would indeed be a triviality. However, before I believe me entirely, I'll check what @Caproni also has to say on the matter. I'll explain that there's a difference beween you and me, try to get him to see my side and see where he ends up. I think he'll be interested in the fundamental ideas because his new job, starting next month, is managing a team of data scientists and so statistics is going to be very heavily a part of it.

As an entertaining aside, I found a Wiki article on special relativity, apparently written in Glaswegian. I don't know if you're aware of it so I'll post it as soon as I've copied the link. It's quite good and they don't change all the words, which is the correct way to proceed, so it can be understood. It just gives the flavour.

It's a triviality because it is provable in a couple of lines from the definition.

https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&cad=rja&uact=8&ved=2ahUKEwjQ4v6Hxp_7AhU2QEEAHYlDAJ0QFnoECBAQAQ&url=https%3A%2F%2Fsco.wikipedia.org%2Fwiki%2FSpecial_relativity&usg=AOvVaw0QrIuPkFBqdgdRzM1PjHzu
I think it may be part of a series, including general relativity. I remember, as a boy I had no difficulty in understanding the equations for special relativity but of course, general relativity is more complex and difficult. After a bit of concentration, I understood how the special was derived from the general. I don't think I've looked at those equations more than once or twice since the early 1960s, so I think I need to see if I can still understand general relativity from looking at the equations.

That's the problem, in my view .... that it's seen as a triviality may show slipshod thinking.

I'm keeping on at this, Mike. Elroch's mentioning that it's a two-line proof from a definition is very interesing, because it ought to be the oher way round. That is, a definition cannot or should not proceed to an understanding of reality but from that direction. You can't find the nature of reality from a definition of it, which is why I asked if it actually affects any calculations. If not, then there's no reason that the theorists should have corrected a probable mistake.

Thankyou for putting it in English. I'll comply with that. But Elroch and Llama are taking a mathematician's view of a philosophical discussion. Rather logical-positivisic. Perhaps it's true that mathematicians don't understand the idea of infinity. I've heard that said before and had no idea that it could be true.

Philosophers take a mathematician's view too. There is no other way to deal with abstract things like an idealised coin-flip repeated a countably infinite number of times.

Regarding your mention of infinity, remember that what matters are concepts and truths, not words. Mathematicians understand a great deal about infinite entities. The adjective "infinite" is simple enough - it means "not finite", so it's meaning depends only on a definition of finite. But it is worth noting that even this has two distinct meanings, one relating to counting and the other relating to measuring on a real number scale.

"Infinity", the noun is an English word which has less value. I direct you to all the individual mathematical constructions and definitions, which are of many different (if sometimes related) things.

@4139

Note on a technical point, in mathematics we always have to be careful to express things in a way that is meaningful. To say that half the coin tosses in an infinite sequence are heads is not meaningful as it stands.

Would it be meaningful to say, "As the number of tosses approaches infinity,  the number of heads and the number of tails converge such that the difference approaches zero"?

I think that's incorrect. If you look at various different expressions, the difference diverges even though the proportion goes to 50%.

Imagine some sequence such as this (heads on top, and total flips on bottom) 11/20 , 102/200 , 1003 / 2000 . . . this is getting closer to 50% even though the difference between the number of H and T are diverging.

I assume the law of large numbers formula would be a rigorous way to show the proportion goes to 50% and the expression I gave earlier shows how getting an equal number of H and T becomes less likely as the number of trials increases.

You can also view this simply on desmos. Set p to 0.5 (because coin flip) and notice that as you slide n higher the peak goes lower (for even numbers of n, the peak represents an equal number of H and T).

I understand what you're saying, and you imply that you're not sure. Nor am I. Clearly, the % difference approaches 0, but whether the real difference does as well...... I've been back and forth on that several times since reading your post. 
Where I am right now is in agreement with you. I think that you are correct....the % difference is what is driven down by the increased tosses....not the numerical difference. 

And my trend is toward agreement. Your example sequence is compelling.

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.

That isn't correct. The idea is different.

Llama is correct that the percentage difference becomes closer, the longer we toss the coins. That is because the distribution of uneveness between heads and tails as results, throughout the sequence, is naturally compensated for by probability. If there was a superfluity of heads, sooner or later there will be a superfluity of tails. Not caused by the previous superfluity of heads but simply because the uneveness evens out more, as the sequence is exended, due to overall probability.

The idea that a genuinely infinite series of results is half and half is because there is no way to negate that, since if infinity cannot be measured then neither may half of infinity. Note that we are not even considering an infinite series of heads or tails. That is because the nominal probability of it occuring is still zero but when dealing with an infinitely long sequence, that nominal improbability becomes a nominal impossibility. This is just something that 500 years of mathematicians haven't realised. Maybe some have known that the others are misaken, of course, but as a consensus, it's incorrect.

It's caused by complacency and vanity.

Yes, I see the light. Llama convinced me in post 4150 that while the % difference approaches 0, the actual difference does not. 

Im so sad im an f in idiot

The expectation of the square of the difference between the number of heads and the number of tails goes up linearly with the number of samples.

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.

Ok, I wrote a program that graphs the average deviation of coin flips (in other words, how many more H than T, or how many more T than H).

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Of course only afterwards did I think to google for a formula for average difference... surprisingly it's very simple. For coin flips it reduces to the number of trials multiplied by the probability of getting the mean.

https://academic.oup.com/biomet/article-abstract/45/3-4/556/234469

So in one sense my time was wasted, but in another sense it was fun, and it allowed me to verify my graph is correct.

Another way to verify I did it right

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Elroch probably has a few decades of experience over you...

It's the same way a 1000 rated player isn't an idiot, they're just less experienced.