Logicizing randomness
So here is the thought experiment:
You are given a machine made by scientists that they say is perfectly random. All the machine does is display ten numbers every year on the same day. But the machine has just been made. So on its very first display of numbers, it lists out ten 5's. Do we assume it's random or assume the scientists made a mistake, in which case the machine is deterministic?
You are given a machine made by scientists that they say is perfectly random. All the machine does is display ten numbers every year on the same day. But the machine has just been made. So on its very first display of numbers, it lists out ten 5's. Do we assume it's random or assume the scientists made a mistake, in which case the machine is deterministic?
Comments (67)
This is too easy. A "machine made by scientists" is a physical device. It therefore has unavoidable imprecisions in its manufacture; and therefore it's not perfectly random, refuting your assumption hence your argument.
The only way to salvage this is to say we have a "machine made by God." Which, by way of example, might as well be taken as the universe. Then I flip a fair coin and it comes up heads a billion times in a row. Is that evidence of the failure of randomness?
Suppose the coins come up in any pattern whatsoever. Isn't the probability that they came up in that exact pattern also [math]\frac{1}{2^{10^9}}[/math]? Wouldn't you take THAT as a failure of randomness?
We are forced to conclude that anything that happens at all is so unlikely that it counts as proof that there is no randomness. The big bang happens. Stars form. Our solar system forms. Eons pass. Life appears on earth. Millions more years pass. Humans crawl out of caves and build civilization. Your parents meet and produce you; and here you sit at your computer or phone and read my words. What is the prior probability of THAT?
Your argument can't be salvaged at all, even by assuming God made the machine. If the sun rose in the east this morning, that is an accident of such unimaginably low probability that it might as well be taken as randomness. Or determinism. Your argument can't tell the difference.
tl;dr: Unimaginably low probability events happen all the time. Look around. What is the prior probability that you are you, and sitting in this particular room with its particular arrangement of stuff? What are the odds that there's any stuff at all? The (prior) odds are virtually zero. Yet here you are. Does that make the universe random, or deterministic? Your thought experiment gives no clue.
Quoting fishfry
It seems everything that happens in human life is highly unlikely - what are the chances of you and all the people in the cinema being there? But, as you say, you are all there!
Lottery paradox. It's rational to conclude that you won't win, therefore you shouldn't play. But somebody must win.
https://en.wikipedia.org/wiki/Lottery_paradox
Randomness seems like a type of freedom while determinism is Newtonian. I appreciate that you saw no difference between between them in my first post. That was the purpose. If any thing can happen with randomness then it seems any result must be accepted. So my conclusion is that pure randomness cannot be understood by us (that is, randomness has parameters)
I honestly don't see what is paradoxical about that.
It is rational to conclude that it is very unlikely for my ticket to win, but it is not rational to conclude that it is certain that it won't win.
Something may happen despite being improbable. Probability is all we need in practice.
EDIT:
At first I applied a GOF-test, but I ignored that I need more to data to justify using that test. My informal response is that there are only 10 strings as concentrated as 5555555555, and that is stronger intuitive evidence against randomness than 2343549094.
That's true, and that's a good point. Each string of ten digits is equally likely, but it's strange that we got such a monotonous sample, all in the same category of '5.' What was really needed in my first post was a goodness-of-fit test, though we don't have enough data to justify. Probably a simulation approach to getting a p-value would be appropriate.
Perhaps what threw you off and made you ask this question was that all the ten numbers are 5, perhaps reminding you of slot machines whose jackpot win sequence is usually a series of identical digits. This creates an illusion that from, say, a list of numbers from 000 to 999, to get numbers like 111, 222, 333, 444, you get the idea, are rarer or special or require skill or for the laws of chance to be violated i.e. we believe, erroneously, that we can't get numbers like these by chance. Wrong! getting 5555555555 or 6666666666 or 1111111111 or similar repeating digits is as likely as getting numbers like 9478321564 with non-repeating digits. The bottom line - there's nothing fishy going on with 5555555555.
Logicism is the breaking down of mathematics into its simplest components. I don't know if this is possible with randomness, because since any outcome is as likely as another, we can't really say what the probability of rolling a 5 is, yet we know how it works in the real (i.e. practical) world
There is no test for randomness.
That's why they say god thorws dice. Nobody can generate random numbers, not because it's impossible, but because it is not possible to test for it being truly random.
Back in my meteorology days one approach was to measure small changes in atmospheric pressure. But not a guarantee of randomness.
I've been stumped by brain teasers that have you try to determine the next number in a sequence, so, no, not necessarily random. This is another math topic that not too many professionals give much thought to - unless you work in that area. If I need to generate a random sequence for a computer program I just go to that command and not worry about it. :cool:
We should keep in mind that pRNGs are a big part of modern technology and that there are lots of test. http://www-users.math.umn.edu/~garrett/students/reu/pRNGs.pdf
One can argue that no test is 'perfect,' but then we have to figure out what is meant by 'perfect.' In the same way there are some good definitions of randomness.
I don't think it's that clear. While statistical hypothesis tests are never conclusive (because haunted by the possibility of type-I and type-II error, I think it's fair to say that lower p-values (if understood) are experienced as stronger evidence against the null hypothesis (in this case randomness.)
While each string of digits is equally likely, we can categorize strings by how spread out over the 10 categories they are. The more spread-out strings are more common and therefore more likely. The string is all 5s has only 9 other strings of similar extremity. That's a 1 in 10^9 chance of such a concentration, which is strong evidence against randomness.
That's only because you're lumping all the "evenly spread" events together. There are far more of them. Whatever outcome you got was incredibly unlikely. The fact that it's a member of an arbitrarily large class of outcomes doesn't make any difference except psychologically.
Let's consider a simpler example, so you can see where I am coming from.
Let's flip a coin ten times to test it for fairness and get HHHHHHHHHH.
If H_0: p= 0.5 and H_A: p != 0.5, then the p-value is 2/2^10 = 1/2^9 ~ 0.002.
Is this evidence against the fairness of the coin? If the coin is fair, the chance of such an extreme value is about 0.2%. Let's make this concrete: if you were thinking of using the coin for practical purposes, would you trust it?
IMO, the problem at hand is more mathematically complicated (with 10 categories) but subject to the same principle.
[quote=wiki]
In null hypothesis significance testing, the p-value[note 1] is the probability of obtaining test results at least as extreme as the results actually observed, under the assumption that the null hypothesis is correct.[2][3] A very small p-value means that such an extreme observed outcome would be very unlikely under the null hypothesis. Reporting p-values of statistical tests is common practice in academic publications of many quantitative fields. Since the precise meaning of p-value is hard to grasp, misuse is widespread and has been a major topic in metascience.[4][5]
...
Null hypothesis testing is a reductio ad absurdum argument adapted to statistics. In essence, a claim is assumed valid if its counterclaim is highly implausible.
...
Thus, the only hypothesis that needs to be specified in this test and which embodies the counterclaim is referred to as the null hypothesis; that is, the hypothesis to be nullified. A result is said to be statistically significant if it allows us to reject the null hypothesis. The result, being statistically significant, was highly improbable if the null hypothesis is assumed to be true. A rejection of the null hypothesis implies that the correct hypothesis lies in the logical complement of the null hypothesis.
[/quote]
https://en.wikipedia.org/wiki/P-value
emphasis added
But what then do you make of testing the coin for fairness as in my reply to tim?
It's commonly accepted that the coin is fair if a long sequence of flips meets the usual tests for statistical randomness, which is your point and several other people's point. But that's also the definition of pseudorandom. We can generate completely deterministic sequences of numbers that satisfy every known statistical test for randomness. So if we see a statistically random sequence of numbers or coin flips, we actually can't conclude anything at all for certain. All we can say is that all heads doesn't look random, and a statistically random-looking sequence does look random. But we can never be sure. We can say how things appear; but we can't say for sure how things are.
I agree. I don't see any way around type-I and type-II error. This is one reason I like to frame things practically. Let's say we are doing quality control on some product. At what point do we decide to reject the object? We know that an error is always possible, but we also don't want to ignore p-values and just guess.
Quoting fishfry
I agree here too. As we discussed earlier, we seem to have some intuition of perfect randomness (the ideal fair coin.) And our tests seem to be built on this intuition. But as you say, we have PRNGS (which I understand to be defined in terms of being actually deterministic), that pass such tests.
There's a recent thread on the nature of probability, a notoriously tricky philosophical subject. Perhaps there are some clues there. Also I enjoyed Nassim Taleb's book, Fooled by Randomness, in which he argues that we often confuse random events with meaningful ones, as in survivor bias.
I've read much of Fooled by Randomness. Taleb is great. I like that he programs simulations (and I'm quite fascinated by PRNGs and all that they can be used for.)
If you look into algorithmic information theory, you'll see randomness as irreducible. It has no useless space in it. It's thick. And a truly random (infinite) sequence contains an infinite amount of information (which can be thought of as an unpredictable string of yes/no answers to a countably infinite number of questions.)
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.
So the next successive number can always be predicted. Or else explained.
This test you propose is not the one that's going to work in establishing randomness of a sequence of numbers.
I guess nothing is completely random. When we see things floating in space, they move randomnly. But they still make sense; they are not irrational in their movements
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.
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.
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.
It.
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.
Maybe you could provide a citation for this assessment. :chin:
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
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.
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 human error could be at play as it was created by scientists.
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.
Quoting 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.
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.
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.
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.
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.
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.
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.
Aren't you just the pleasure to talk to.
Possibly.
Yes, and I'm saying they haven't thought through the consequences of that claim. I've heard the number [math]10^{500}[/math] "types" of universes, which is still a finite number and avoids the question of how many universes there are.
Quoting RogueAI
No snark. I'm asking you to [i]ask yourself[i] 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.
Quoting RogueAI
Yes, so I've heard. But that's no argument for infinity of anything.
Quoting RogueAI
Not an argument for infinity.
Quoting RogueAI
Not an argument for infinitely many of anything. Even the physicists only allow for [math]10^{500}[/math] variations in the parameters. That's a finite number.
Quoting 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.
Quoting 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.
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.
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
ETA:
"[i]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.[/i]"
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.
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.
Quoting 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.
Quoting 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.
Quoting 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?
Quoting 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.
Thank you, Fishfry. Now we need another quote that it can be continued with absolutely any next number and fitted to any of an infinite number of non-congruent polynomials. That's also true.
If we consider the all too human component of this thought experiment, us, we must not fail to consider the fact that our brains are pattern detectors. This somewhat negates the classical notion of randomness in my mind. Ultimately it is nothing more than a lack of information on our part, what we consider randomness essentially is somewhat synonymous with inexplicable. Even when considering this through the lens of probabilistic outcomes, that something with a vanishingly small likelihood of occurring happens, seems more aptly attributed to desirability bias - this shouldn’t have happened! It had a one in all but infinite probability of being the case! - which I think also treats the issue of the original post and addresses what responses I did get through.
That said, another plausible interpretation of randomness (though only subtly different, it is philosophically distinct) is unpredictability. In this sense, the case of 10 5’s appearing and being an unsatisfactory result from the randomness machine would entail you A) had an expectation of what the randomness should look like or B) any result with too much of a “common” pattern would be unsatisfactory and make the machine seem faulty.
My background isn't math, so I can't contribute too much along these lines. The other day, I was reading about proposals to take the infinitely large set of worlds and partition it in some non-arbitrary way so that probabilities can be assigned, but I can't find it now.
Sure.
I concede the point. There might be some fundamental aspect of things that makes a universe of nothing but Boltzmann Brains physically impossible. But that doesn't seem to be the case currently. There doesn't seem to be anything preventing, say, "casino worlds" in Hitchhiker's Guide to the Galaxy (if you haven't read the book, it's a world where random erosion patterns just happened to have created glittering casinos everywhere).
Fair enough.
This is an assumption, but I think it a fair one. If there are infinite universes, why wouldn't they be countable? But maybe they're not.
Maybe. I don't know much about the Continuum hypothesis.
That's fine. Your speculations are interesting. I'm going to have to read more about Continuum hypothesis. Infinity is interesting.
No, I'm not assuming they're equally likely or distributed uniformly. That's not required to generate the dilemma of have to choose between two infinite sets to figure out which one you're in, but like you said, the true odds may be different. For example, if you're jumping off a tall building, there are two sets to consider: the set of universes where you survive and the set where you don't, and obviously your odds of surviving aren't 50/50, so there's something going on there, and yet, at a fundamental level, reality either is as it appears to be (actual laws of nature, not just fantastic coincidences over and over, we're not Boltzmann brains, etc.) or reality isn't as it appears to be. If there are an infinity of universes of each type, and you don't know what kind of universe you're in, how is it anything other than 50/50? You would have to assert some limiting principle where the multiverse just doesn't produce universes where fantastic coincidence isn't the norm, but what on Earth would that mechanism be?
After the first exchange, I thought you were making some errors, and I don't have much of a math background, so I asked a probability question about Pi. Do you know Bayes Theorem well?
Let me know if you find it, I'd be interested. There is no uniform probability distribution on a countable set. That is, there is no way to assign probabilities to, say, the positive integers, in such a way that each one has an equal chance of being picked. There is no conceivable way to do this, and the proof is straightforward.
Quoting RogueAI
I'm afraid I'm the only one who read that book and thought it was silly. So your point is lost on me, although it would resonate deeply with pretty much everyone else.
Quoting RogueAI
Well, there's only one countably infinite cardinality; but there are so many uncountable cardinalities that they're too big to be corralled into a set. So if the probabilities are uniformly distributed -- like your nonsense worlds -- then the odds are unimaginably small that the number of worlds is countable.
But this isn't good reasoning. Mine, meant to be facetious to prove a point; or yours, meant to be serious.
Quoting RogueAI
Here's all you need to know in order for me to explain my point. Cantor defined infinite cardinalities in such a way that they proceed one after another: [math]\aleph_0, \aleph_1, \aleph_2, \aleph_3, \dots[/math].
Now Cantor's famous diagonal argument and also his more general and beautifully simple Cantor's theorem shows that there are [math]2^{\aleph_0}[/math] real numbers. And the question is, which Aleph is that??. Perhaps [math]2^{\aleph_0} = \aleph_5[/math], or [math]2^{\aleph_0} = \aleph_{\text{googolplex}}[/math].
The Continuum hypothesis is the claim that
[math]2^{\aleph_0} = \aleph_1[/math]
That is, that the cardinality of the real numbers is the very next cardinality after that of the natural numbers; or equivalently, that there is no infinite cardinality strictly between that of the naturals and the reals.
The question of whether this is true vexed Cantor and vexed everyone till Cohen proved as recently as 1963 that the question is independent of the standard axioms of set theory. In other words there are models of set theory in which it's true, as Gödel showed in 1940; and models in which it's false, as discovered by Cohen. In fact Cohen earned the only Fields medal every granted for mathematical logic for his pioneering work in showing how we can cook up arbitrarily weird models of set theory in order to investigate such independence questions.
Now. My point is this. Every time a physicist casually says, "The number of universes might be infinite," or. "The size of the universe might be infinite"; or some breathless pop science writer who knows less than you or I do about the topic makes the same type of claim; they should immediately exclaim: "This is very exciting! It means that mathematical problems like the Continuum hypothesis, which were formerly relevant ONLY to the realm of pure, abstract, non-physical mathematics, are now potentially amenable to study by physicists!"
But they never say that. Nobody has EVER said that. And in my own opinion, the reason that they don't, is that they do not take their own suggestion seriously enough to have spent five minutes considering the profound mathematical and physical implications of what they're saying.
That is my point. And I admit that I've probably stated it so many times on this forum, going all the way back many years to the predecessor of this forum, that by now I often state it quickly without providing sufficient context for people seeing it for the first time. For which I take responsibility.
Quoting RogueAI
Thank you. I should say that I am often snarky, but was not being snarky (at least intentionally) with you. I'm actually trying to be less snarky these days, and your remark reminded me that I was unsuccessful in this instance.
Quoting RogueAI
Yes it is! And its profound implications are never considered, even momentarily, by all the people, from ignorant pop-sci writers to famous world-class physicists, who casually claim that some aspect of the world might be infinite. Because if infinity is instantiated in the world, then all the set-theoretic questions of infinity immediately become matters of physics; just as the bizarre mathematics of non-Euclidean geometry suddenly became relevant to physics when Einstein developed general relativity.
Quoting RogueAI
The jumping off building analogy applies exactly here. Even if you don't know the true odds, it seems (to me) perfectly obvious that it's RARELY the case that the odds of two mutually exclusive events are 50-50. In fact Boltzmann brains are extremely statistically unlikely.
Quoting RogueAI
Well, one point I made was that we DO happen to live in a world of fantastic coincidence leading directly to our existence at this moment. And on the other hand is the building analogy and the Boltzman brain analogy. Boltzman brains are statistically highly unlikely. But then again, which is less likely? A Boltzman brain? Or a fully formed human being? Both are statistically unlikely. In fact it's one of the arguments against Darwinian evolution (among scientifically-minded neo-anti-Darwinists) that there literally hasn't been enough time for pure chance to have produced humans on earth.
Quoting RogueAI
What was the question? I honestly don't get it. First you said 123456789 and contrasted that to a random-looking string in order to get me to admit that one string looks random and one doesn't. [Good point actually]. But then the pi example confused me, because then you have two non-random looking strings of digits. So I didn't understand the point being made.
Quoting RogueAI
I know Bayes' theorem but not well. I get the idea of priors but I've never been able to get very worked up over the apparent dispute between Baysians and frequentists. I know that people can use Bayes' theorem to show that if you test positive for some awful disease, it may still be much more likely that you have a false positive than that you actually have the disease. That's pretty much all I know.
For years, the inability to calculate ratios of infinite quantities has prevented the multiverse hypothesis from making testable predictions about the properties of this universe. For the hypothesis to mature into a full-fledged theory of physics, the two-headed-cow question demands an answer."
https://www.quantamagazine.org/the-multiverses-measure-problem-20141103/
I take this as a datapoint in favor of my thesis. Guth is a heavy hitter, a physics superstar in both the physics and the popular communities. He's a Big Cheese. And he hasn't spend five minutes -- five seconds -- considering the implications of what he says. Nor is he aware that there's no uniform probability measure on a countable set. This is exactly the kind of thing I'm talking about. You know there are infinitely many multiples of 1000000000 and infinitely many positive integers that aren't multiples of 1000000000, but we can still calculate their respective asymptotic densities (which isn't the same thing as a probability measure). This is not a deep point, it's very trivial. In aa countably infinite multiverse, two-headed cows are like multiples of 1000000000 and one-headed cows are like all the other numbers. What of it? It's shallow masquerading as deep. (I'm not yelling at you, I'm yelling at the world. "Old man yells at cloud.")
https://en.wikipedia.org/wiki/Natural_density
By the way the asymptotic density of the primes (of which there are infinitely many) in the positive integers is zero. Mathematicians have thought about these things. Only the physicists pretend that their ignorance is deep thought.
Superstar physicists who don't know sh*t about the mathematics of infinity and haven't thought about what they're saying. I was shocked when I saw Susskind do it, now I'm no longer surprised. It is in fact a common pattern. And pop-sci reporters write this stuff down and then the general public absorbs these confusions.
Thanks for that link, I will definitely read the article and probably get my blood pressure raised.