Any sequence of numbers can be described as a sequence of a polynomial function. Not only by one precise, exact and fitting polynomial function, but actually an infinite number of them. — god must be atheist

This isn't prediction though, only retrospectively the numbers in the series so far can fit an infinite number of polynomial functions. If the numbers are generated by a random oracle, then picking the right polynomial function that predicts all next number in the series is simply 1 out of the infinite available; i.e. impossible to predict.

Though, otherwise, I don't really know what this thread is about.

If you want "pure" random numbers (as far as we know) you use radioactive decay. Radioactive elements have some probability of decaying in some span of time, but exactly when is completely random as far as we know and this (and a bit of math) can be used to create random numbers of reasonable certainty.

To give a simplified example, if we have a series of atoms and convert each one in turn to some radioactive element, it will have some half life time. For a single radioactive element the half life is simply when it has 50% chance of decaying already. For each atom, if it decays before the half-life we can mark a 0 and if survives half-life we can mark a 1. You then get a random binary string by repeating the process. Of course, there are weakness in this simple process as maybe the experiment isn't setup perfectly and there's slightly more 1's than 0's or vice-versa, but gets the basic point across. How atoms actually play half-life is a complicated quantum process I can't explain here.

I guess nothing is completely random. — Gregory

Randomness can be simulated by a mind capable of understanding its mathematical interpretation. Imagine X is such a mind and Y is us. Suppose there's a machine which displays a number from 1 to 6 (a die). X operates the machine but Y doesn't know that. Y checks the machine and X makes it display 1. The next time Y checks the machine, X makes it display 6. Every time Y looks at the display X manipulates it in such a way that the frequency of each digit is 1/6. Y concludes the machine is random when in fact X was merely simulating it.

There seems to me something infinite about randomness. — Gregory

Think about it the other way around. What is the opposite of randomness? Something that repeats itself somehow, that has a pattern. How can you know that something doesn't have a pattern? One hypothetical way looking at it would be to go to infinity and see that there is no pattern and the mathematical object is truly unique, which one obviously cannot do. Hence I think the Algorithmic Information Theory is a very good way to understand the complexity of the issue.

So the next successive number can always be predicted. Or else explained. — god must be atheist

Yes, after the fact. The point is, not before.

This test you propose is not the one that's going to work in establishing randomness of a sequence of numbers. — god must be atheist

Can't establish it if you don't know what it is. What (do you say) it is?

Can't establish it if you don't know what it is. What (do you say) it is? — tim wood

It.

Yes, after the fact. The point is, not before. — tim wood

It's impossible with any sequence of numbers what comes next. The psychological / IQ tests that rely on this are all flawed.

What comes after 1, 2, 3, 4, 5??? Not six. I mean, it could be six, 8, 194, 539430. Any number. There are fuctions in the magnitude of infinity, that will make the next number not six, but any desired number. To know what number the next number must be, you need to know the function that the test writer has in mind, and that involves actual mind-reading. That is not math, and it can't be cited in the proof.

I read your objection re: prediction. Please see the post I made just before this one.

Which means that any number series could be random. That leaves the question, what is random, and what passes for random?

The only definition I know of for a random number is when the number itself is the shortest description of the number. As to what can pass for a random number, as when someone needs one, I have no idea.

The official and ARBITRARY definition for random numbers given in texts of probability math is the numbers that come up on subsequent rolls of a fair die. This definition recognizes that the numbers may not be random, recognizes also that its randomness can't be tested, so it gives the operational categorical definition while recognizing those features. Few things work this way in math, but this one does.

It's impossible with any sequence of numbers what comes next. The psychological / IQ tests that rely on this are all flawed. — god must be atheist

Maybe you could provide a citation for this assessment. :chin:

I see that

Necessity= order

Order= non-random

Contingency= randomness

Randomness= disorder

So unless order is in the eye of the beholder, we should be able to tell as humans what disorder is when we find it. For me, it is the irrational

Yet I can admit there is a randomness of sorts in lots of games that is contained within the field of rationality

Maybe you could provide a citation for this assessment. :chin: — jgill

Reason. Stands to reason.

It would be just as easy for you to research this on Google as for me. I know that I have made the claim, so it should be backing it up, but then again, I am here for fun, nobody pays me what to do and what to say. So I only do those things that are fun for me here.

It's impossible with any sequence of numbers what comes next. The psychological / IQ tests that rely on this are all flawed.
— god must be atheist

Maybe you could provide a citation for this assessment. :chin: — jgill

Given any finite sequence whatever, it can be continued with absolutely any next number and fitted to a polynomial.

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

Also, if an infinite number of universes exist, there are an infinite number of universes where incredible fantastical coincidences are the norm, not the exception. And there would be infinitely many such worlds, so the set of "worlds with extreme amounts of fantastical coincidences" would equal the set of "worlds without extreme amounts of fantastical coincidence". If you didn't know which kind of world you're in (and how would you?), there's a 50/50 chance you're in the world of crazy coincidences.

You assume human error. I always use the example of joining a poker game, and dealer deals himself a royal flush. Nobody would quit the table. After the second one, some people would leave. After the third one, everyone would be gone. As the odds become longer, outside agency starts to look more and more likely.

All sequences are likely, but some are more explainable by chance than others. For example 3478907834617856 is explainable by chance. And what I mean by that is there's no competing theory that does better than "chance" for that string of numbers. However, if the numbers were 12345678901234567890, there is a competing hypothesis that beats the chance hypothesis: human intervention.

in this case I do not believe it is random, something that is created by man and computerised has a set of computed outcomes already and not entirely random.

Also human error could be at play as it was created by scientists.

Also, if an infinite number of universes exist, there are an infinite number of universes where incredible fantastical coincidences are the norm, not the exception. And there would be infinitely many such worlds, so the set of "worlds with extreme amounts of fantastical coincidences" would equal the set of "worlds without extreme amounts of fantastical coincidence". — RogueAI

When physicists use the word infinity they must mean something quite different than what mathematicians mean, else they'd immediately have to ask themselves what is the transfinite cardinality of the set of universes, and whether the universes can be well-ordered, and so forth, or at the very least they'd have to simultaneously note that standard set theory does not apply to their use of the word infinity.

Since you are speculating that there might be infinitely many universes, why don't you suggest answers to those questions, if only to challenge your own thinking.

And what is your chain of logic that, " if an infinite number of universes exist, there are an infinite number of universes where incredible fantastical coincidences are the norm ..." What's the argument that this is so? After all there are infinitely many positive integers 1, 2, 3, ... yet none of them is a purple flying elephant, at least as far as we know. Every positive integer is subject to the Peano axioms. So we already have evidence that your claim is (pending some kind of argument) false.

If you didn't know which kind of world you're in (and how would you?), there's a 50/50 chance you're in the world of crazy coincidences. — RogueAI

Really? Have you got an argument for this? But I have already pointed out earlier that we ARE in a world of crazy coincidences. From the big bang to your being here reading this requires a chain of the most unlikely coincidences and accidents. So your statement here is unsupported and vacuous.

You know I've seen famous physicist Leonard Susskind talk and write about infinity (two separate instances that I have in mind) where he clearly has no idea what he's talking about. Physicists, even some very eminent and famous ones are very imprecise in their notions of infinity.

For example 3478907834617856 is explainable by chance. And what I mean by that is there's no competing theory that does better than "chance" for that string of numbers. — RogueAI

There's a polynomial that inputs 1 and outputs 3; inputs 2 and outputs 4; inputs 3 and outputs 7; and so forth. Polynomials are particularly simple examples of functions. As a "competing theory" as you put it, scientists will take a polynomial every day of the week. In fact when computer scientists can reduce the growth rate of a problem to a polynomial, they are ecstatic.

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

But for that matter there is a constant function f(x) = 3478907834617856 that gives this output for every real number input. If polynomials make computer scientists ecstatic, constant functions drive them absolutely delirious with delight.

Informally I agree with you. But more formally it's not clear that no pattern exists in 3478907834617856 simply because one is not obvious.

I like the standard test of significance approach better. Assume that the numbers are generated by a discrete uniform distribution and then calculate the chance of a sample that extreme. More exactly we could let f(sample) = #_of_different_digits and see that f(sample) = 1 is a rare, extreme value: only 1/10^9 samples give f(sample) = 1. That makes our assumption (intuitively) less likely, though admittedly still possible.

When physicists use the word infinity they must mean something quite different than what mathematicians mean, else they'd immediately have to ask themselves what is the transfinite cardinality of the set of universes, and whether the universes can be well-ordered, and so forth, or at the very least they'd have to simultaneously note that standard set theory does not apply to their use of the word infinity.

I don't know about any of that. But many cosmologists advocate for a multiverse with infinitely many universes where the values of the physical constants are different.

Since you are speculating that there might be infinitely many universes, why don't you suggest answers to those questions, if only to challenge your own thinking.

What's with the snark? My reply to you in this thread didn't even have a question in it. I was making a bunch of points about infinite universes.

And what is your chain of logic that, " if an infinite number of universes exist, there are an infinite number of universes where incredible fantastical coincidences are the norm ..." What's the argument that this is so?

The values of the physical constants are different. I'm not talking about a set of identical infinite universes. For example, there would be universes (an infinitely many of them) consisting of nothing but Boltzmann Brains constantly popping into and out of existence.

After all there are infinitely many positive integers 1, 2, 3, ... yet none of them is a purple flying elephant, at least as far as we know. Every positive integer is subject to the Peano axioms. So we already have evidence that your claim is (at least arguably, pending some kind of argument) false.

Purple flying elephants are physically impossible. Picture worlds where people win the lottery 20 times in a row, and people always go into spontaneous cancer remission after they drink from a certain fountain. Erosion patterns constantly spelling out the truths of the natural world, E=MC2. Stuff like that.

Really? Have you got an argument for this?

Yes. Countable infinite sets are equal and there are infinitely many worlds where the laws of nature are real, and where the laws of nature are nothing but descriptions of fantastical coincidences happening over and over again. If you don't know what set you're in, and both sets are equal, it's a 50/50 chance if you're guessing.

But I have already pointed out earlier that we ARE in a world of crazy coincidences. From the big bang to your being here reading this requires a chain of the most unlikely coincidences and accidents. So your statement here is unsupported and vacuous.

Aren't you just the pleasure to talk to.

You know I've seen famous physicist Leonard Susskind talk and write about infinity (two separate instances that I have in mind) where he clearly has no idea what he's talking about. Physicists are very imprecise in their notions of infinity.

Possibly.

But that just limits you to sets of numbers. Suppose we take the poker example I gave earlier. How many royal flushes does the dealer have to deal himself before you leave the table? Which is to say, how many royal flushes until the probability "dealer cheating" > "chance". That's a tough one to nail down because it's so subjective. Bayes Theorem works great in those kinds of situations.

Fish, if a lottery was being run for the first time, and you were the manager, and the winning ticket's numbers were 314159265359, what would you conclude?

I don't know about any of that. But many cosmologists advocate for a multiverse with infinitely many universes where the values of the physical constants are different. — RogueAI

Yes, and I'm saying they haven't thought through the consequences of that claim. I've heard the number $10^{500}$ "types" of universes, which is still a finite number and avoids the question of how many universes there are.

I
What's with the snark? My reply to you in this thread didn't even have a question in it. I was making a bunch of points about infinite universes. — RogueAI

No snark. I'm asking you to ask yourself the same questions I ask everyone who claims the universe instantiates infinity in any way. It's the same question I'd love to put directly to Leonard Susskind, who also doesn't understand the point. I am inviting you to challenge yourself to think about what it means to claim there are infinitely many of anything physical. That's not snark, it's a question designed to get people to contemplate the weakness of their own thinking along these lines.

I
The values of the physical constants are different. I'm not talking about a set of identical infinite universes. For example, there would be universes (an infinitely many of them) consisting of nothing but Boltzmann Brains constantly popping into and out of existence. — RogueAI

Yes, so I've heard. But that's no argument for infinity of anything.

I
Purple flying elephants are physically impossible. Picture worlds where people win the lottery 20 times in a row, and people always go into spontaneous cancer remission after they drink from a certain fountain. Erosion patterns constantly spelling out the truths of the natural world, E=MC2. Stuff like that. — RogueAI

Not an argument for infinity.

Yes. Countable infinite sets are equal and there are infinitely many worlds where the laws of nature are real, and where the laws of nature are nothing but descriptions of fantastical coincidences happening over and over again. If you don't know what set you're in, and both sets are equal, it's a 50/50 chance if you're guessing. — RogueAI

Not an argument for infinitely many of anything. Even the physicists only allow for $10^{500}$ variations in the parameters. That's a finite number.

Aren't you just the pleasure to talk to. — RogueAI

The point in which this was a response (stretching the definition of response) was that we already live in a fantastically unlikely universe. And if "just the pleasure to talk to" means that I'm asking you to make a coherent argument for infinitely many universes, an argument thateven professional physicists don't make, and that you are unable to do so, well then I guess that's what you mean.

Don't take it personally. I have made this argument many times and have never gotten a good response. And when you dig into the literature, you find that the professional physicists don't seriously make the claim. They use "infinity' as a synonym for a really big finite number.

Fish, if a lottery was being run for the first time, and you were the manager, and the winning ticket's numbers were 314159265359, what would you conclude? — RogueAI

That it's a hell of a coincidence that the digits of pi came up. Or that someone's playing a joke. I don't follow your question at all.

But you are really taking me wrong about this infinity business. I'm not being snarky with you. You claimed that there are or might be an actual infinity of universes in the multiverse. Even the physicists don't make this literal claim, they only use the word metaphorically to mean a big number. I'm challenging you to make actual sense of your own claim. If you can't, have some self-awareness and admit that you only meant a really big finite number; and that if you truly mean an actually infinite number of universes, you have to grapple with the set-theoretic implications.

Given any finite sequence whatever, it can be continued with absolutely any next number and fitted to a polynomial. — fishfry

Of course, but the point was the opinion that the use of such sequences in IQ tests is invalid, which I question. I don't think sequence puzzles are necessarily invalid in these tests. Such sequences are good for pattern recognition. We're not talking about fitting a polynomial to a set of points. I used to take the Denver Post, and each day there would be a puzzle created by, I believe, a Canadian math teacher. Frequently he would present a sequence and ask for the next term. Some were pretty clever and I got stumped occasionally. The idea is to find the simplest mathematical structure generating the next term. And, yes, sometimes there was more than one answer.

You can find them all over the internet, now. Polynomial interpolation is beside the point.

Such sequences are good for pattern recognition. — jgill

Right. They test the ability to get the answer that the examiner expects. Which of course measures a type of IQ but not creativity. It tests for conformity to what "clever people" think is cleverness. It tests for the kind of thinking that caused Einstein to be unable to obtain an academic post after getting his doctorate. He was a genius, but terrible at agreeing with the answers other clever people got.

Classic example. What's the next number: 1, 2, 4, 8, 16, ____?

Well it's 31 of course. It's the number of distinct regions you get by drawing chords between n points on a circle.

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

Fishfry, my point isn't about whether the multiverse is infinite or not. I'm OK assuming we don't know one way or the other and will likely never know. My point was about the ramifications if there are infinitely many universes with different physical constants. IF that is the case, the set of universes "everyone is a Boltzmann Brain" is infinite and the set "everyone is a real person" is infinite, and they're both countably infinite sets, so how would you decide which set you're in if you don't know? It's a coin toss, in that situation. If the multiverse isn't infinite, none of that applies, of course, but philosophy is about speculation, so I'm speculating here.

ETA:
"And even though the pocket universes keep forming, there’s always a volume of exotic repulsive gravity material that can inflate forever, producing an infinite number of these pocket universes in a never-ending procession...

...The problem with having an infinite multiverse is that if you ask a simple question like, ”If you flip a coin, what’s the probability it will come up heads,“ normally you would say 50 percent. But in the context of the multiverse, the answer is that there’s an infinite number of heads and infinite number of tails. Since there’s no unambiguous way of comparing infinities, there’s no clear way of saying that some types of events are common, and other types of events are rare. That leads to fundamental questions about the meaning of probability. And probability is crucial to physicists because our basic theory is quantum theory, which is based on probabilities, so we had better know what they mean."
https://www.scientificamerican.com/custom-media/biggest-questions-in-science/the-founder-of-cosmic-inflation-theory-on-cosmologys-next-big-ideas/

I don't know. Maybe SA is talking out their ***. They usually don't. But I'm just including this to buttress my tangential point that infinite universes is taken seriously in cosmology.

Fishfry, my point isn't about whether the multiverse is infinite or not. I'm OK assuming we don't know one way or the other and will likely never know. — RogueAI

Ok that's fair. But if we are speculating, isn't it fair for me to point out some things that need to be considered? If the universe instantiates actual infinity in any way: infinitely many sub-universes, infinitely many distinct times within a finite interval of time like 1/2, 1/4, 1/8, ... infinitely many planets, infinitely many anything ... then we must ask ourselves the question: Does the mathematical theory of infinity apply? If yes, then we must ask if things like the Continuum hypothesis and the axiom of choice have now become amenable to physical experiment; and if not, we must then develop a new physical theory of infinity.

I know you weren't thinking of these things, but (in my opinion) the moment one says that there MIGHT be a physical infinity, these questions immediately come to mind. My mind, in any event.

My point was about the ramifications if there are infinitely many universes with different physical constants. IF that is the case, the set of universes "everyone is a Boltzmann Brain" is infinite and the set "everyone is a real person" is infinite, — RogueAI

This I disagree with. Am I allowed? As Jules played by Samuel L. Jackson says in Pulp Fiction: "Allow me to retort!" The set of positive integers exists. Are there as many numbers equal to 47 as not? No. Are there as many numbers that can be exponents in Fermat's equation? No, 2 is the only one, proven as recently as 1994. Are there infinitely many numbers that are part of a prime pair? Unknown. It is most definitely not the case that every possibility occurs infinitely many times. In the multiverse you have no idea what the actual rules are. Truth is you have no way of knowing that there are infinitely many universes that contain Boltzmann brains. Perhaps there's some as-yet-unknown physical constraint that only allows finitely many such. So your speculation is not fully thought out in my opinion.

Excessive pickiness on my part, maybe. Not snark. I'm making a point. I'm disagreeing with your reasoning.

and they're both countably infinite sets, — RogueAI

Ah! And you know this, how? This is one of my questions. Let us suppose, arguendo, that the number of sub-universes in the universe (or universes in the multiverse) is actually infinite. Is it countably infinite or uncountably infinite? Well, you just made an assumption. So if I got you to state one of your unstated assumptions, my objections have not been in vain. And why should the number be countably infinite? And if it's uncountable, what might its cardinality be? Set theorists have some mighty large cardinals these days. So IMO these are the kinds of questions that come immediately to mind whenever someone speculates on physical instantiations of infinity.

After all, if there are even countably many of anything in the physical world, then we can in principle count its number of subsets; and depending on which cardinal number that happens to be, the Continuum hypothesis is therefore amenable to physical experiment. I take it as proof, or at least meta-proof, that physicists don't take infinite universes seriously; else postdocs would be applying for grants to determine the truth of the Continuum hypothesis. No such grant applications have been applied for; ergo, physicists don't take physical infinity seriously at all.

Why are you allowed to speculate about the consequences of physical infinity, but not me? Can you see that I am actually trying to join in your game, by making my own speculations about the implications of physical infinity.

so how would you decide which set you're in if you don't know? It's a coin toss, in that situation. — RogueAI

Without knowledge of the actual probability distribution, that's like guessing it's 50-50 to land alive after jumping off a tall building. Perhaps some configurations of the multiverse are far more likely than others. You're assuming all configurations are distributed uniformly. Isn't that an assumption?

If the multiverse isn't infinite, none of that applies, of course, but philosophy is about speculation, so I'm speculating here. — RogueAI

Why can't I play too?

So what was the point of the lottery that comes up with the digits of pi? That example went right over my head. 123456789 and a bunch of digits of pi both seem equally contrived.

Well the cards might as well be numbers. We can translate everything about poker into numbers. But I agree that there is something subjective. With p-values, 0.05 is conventionally used as a threshold, but there's nothing magical about 0.5.

I like to thing of these things in terms of automating decisions. If you programming/designing quality control in a factory...or if you were to write a cheating-detection program for chess.com..you seemingly have to decide on some boundary or on many boundaries, despite the ineradicable possibility of error.