Don't forget randomness is still just a hypothesis (2006)
nature.com
nature.com
First, I think it's important to define what randomness even is, for which I'll use the most universal definition, i.e., Kolmogorov randomness. A string of data is Kolmogorov random if, for a given universal Turing machine, there is no program shorter than the string that produces the string (yes, you can arbitrarily choose which universal Turing machine, but the invariance theorem makes this fact inconsequential for the most part).
So if we repeatedly set up and measure a quantum system that's not in an eigenstate and then apply the probability integral transform to the individual measurement values, we should expect to find a sequence of values drawn from a uniform distribution, and this sequence should not be compressible by any computer program.
This is where it gets interesting though, because it may very well be the case that this sequence of measurement values is incompressible only because we lack external information, i.e., we are looking at the Kolmogorov complexity of the string from our perspective as experimenters, but from the perspective of a hypothetical observer outside the universe, the conditional Kolmogorov complexity (conditioned on some missing information) could indeed be less than the length of the string.
So where could this missing information be stored? My guess is that it's at the boundary of experimenter/experiment (not referring to a spacetime-local boundary here), since you can't represent the overall quantum state of experimenter + experiment as a separable state. That is, the information necessary to perfectly predict the result of a measurement on a quantum system is inaccessible to us precisely because we — the experimenters — are part of the system itself.
In this way, quantum randomness would be truly random from our perspective in the sense that the future is to some degree fundamentally unpredictable by humans, but just because it's genuinely random to us doesn't imply the universe is indeterministic.
I wonder if there's some way you could design an experiment that distinguishes true indeterminism from the merely unpredictable...
I guess I don't really have a reason for my preference other than thinking, if a universe pops into existence, what would I expect it to act like? Deterministic, or indeterministic? It just seems simpler to assume indeterministic because I can't think of a reason why it should be deterministic.
Edit: I know that chaos theory is built on determinism :-) I'm thinking of things like strange attractors. My ignorance leaves me with no better word for what I'm talking about, unfortunately.
For a statistics perspective, given the youthful nature of Quantum Mechanics as a field it is more likely than not that we're observing a deterministic phenomenon from the wrong angle, so it appears random. This is because that is how first contact played out with pretty much everything else.
That’s not a definition of unbiased randomness. A true unbiased random number could be all 0’s. Nothing about an unbiased random number demonstrates it’s random, otherwise whatever that distinction is would be a bias in it’s generation.
Kolmogorov complexity is it’s own thing, and sequences that seem very complex can have extremely low complexity. Such as long sequences of hashes of hashes.
Anyway, I suggest you reread the end of that paragraph:
“A counting argument is used to show that, for any universal computer, there is at least one algorithmically random string of each length. Whether any particular string is random, however, depends on the specific universal computer that is chosen.”
Kolmogorov complexity is really referring to the fact you can’t have lossless compression of arbitrary bit strings. You can’t encode every possible N+1 bit string using an N bit string. The computer chosen can make an arbitrary 1:1 mapping for any input though. So, it’s got nothing to do with randomness in the context of coin flipping as the mapping is predefined.
Just remember, you’re choosing the computer and at that point any input can be mapped to any output. But, after that point limits show up.
Kolmogorov complexity works only if we have a big sample of random numbers, but we do not know if we have such in this universe or not.
Without access to the architecture of the machine the universe runs on you can’t tell if the initial random string would be one bit or nigh infinite bits.
The citation for that paragraph is a peer-reviewed journal article covering Kolmogorov complexity and randomness. It’s actually a really good article, by someone pretty famous named Per Martin-Löf. Which is all great, except that paper is from 1966, and in 2019 a more studied concept is something called “Martin-Löf randomness” :)
In the same line, from my perspective, randomness appears where the "limit of our measurement resolution" is. In other words, when your only way to measure something is to sample it, then the maximum resolution of your measurements is going to depend on the maximum speed/frequency at which you can sample the thing you are measuring. So in the end all our measurements will always be limited by whatever thing is the fastest that we can handle/operate/understand. Anything faster than that will appear to be random to us. And this is also pretty much the same as what the Nyquist-Shannon sampling theorem says about any wave/information.
Relating to the Kolgomorov randomness under the above, something would be random when we can't sample it fast enough to rebuild its waveform with perfect fidelity within the time frame that it appears to be random.
With some creativity I believe Nyquist-Shannon can be applied to all measurements. For example you could think of a single measurement as the equivalent of a sampling rate of 1 in the time period in which the measurement was made.
Interesting to note that the Bible implies something like this paradigm, since it describes God as having total control of the universe but also says we have free will.
When you first encounter that pair of ideas in the text they seem contradictory, but further reflection eventually leads many to some variation on this idea.
I don't believe it's the full picture of free will and predestination in Christian theology - just think it's interesting that it fits with this perspective that would have been quite non-obvious to its authors (at least in regards its relationship to modern physics).
>We try to understand countless things without have a use case in mind at the time of study.
You're making a pretty clear reference to mathematics & science here, but in those disciplines we study things with well-defined structures. We don't study flighty nonsense because it's not ever going to be useful. You shouldn't invoke this phrase to excuse a lack of precision and clarity.
...dunno why but this is one of those things that seem to me so incredibly intuitively obvious down to the bones of your mind, like in "how could even the thought of it being any other way" be possible :) I'd say it's just that modern physics doesn't want/need to have anything to do with such hypothetical "outer observers", so that's why we accept the convention of "true randomness" and work with it. And, it makes sense, otherwise you'd end up with science being polluted with useless metaphysics blabberings.
"The overall Kolmogorov complexity of a string is thus defined as K(x)=|p| where p is the shortest program string for language L such that L(p)=x and we consider all programming languages."
This is false. No one considers "all programming language", we consider one fixed language (by fixed you can/should mean: independently of the input, or in a different way: you define the language for all possible inputs).
(But it can be ANY universal language chosen from all languages (due to the provably constant overhead) - that's true, but according what follows in your link, your definition not meant it this way.)
It is very difficult to imagine root source for truly randomness? If there is truly randomness in the universe, source of it would be perhaps the most important discovery of science ever.
True randomness and infinity are horrible potential features of the universe - especially if both are true.
I know nothing about this field, but it strikes me as wrong intuitively. Say that I have a certain amout of data. I can find specific patterns on it (for example, a chain of 10 ceros) that are compressible (0*10). For a large enough amount of data, that can save me enough space to include a program that can print the decompressed string in less space than the original string, thus implying my original string wasn't random - but we've then reached the absurd, because it is perfectly understandable that randomness could create locally compressible substrings.
What am I missing?
If you are familiar with programming I suggest doing the following experiment:
1. make an image of one solid color
2. make another image of the same size that with each pixel being a random RGB value
3. losslessly compress both images any way you can
4. compare the compressed file sizes to the uncompressed bitmap file size
Make sure to make a hypothesis before the experiment!
To dig a little deeper on this: all subsequences are equally probable in a random sequence. The full implications of that requires a bit more playing around and reading. I think if you explore it you’ll find that there is an intuition that can be built up.
It can't. You'll find that, in a truly random sequence, the "compressible substrings" will be infrequent enough that you will use up your data budget just specifying where they go.
Let's take your run-length example. Let's work in bits to make it simple. Your chain of 10 zeros - 10 bits of information - happens on average every 2^10 bits. Let's say we magically compress this sequence down to 0 bits - we just assume that statistically it's in there somewhere, so we don't need to store it. All that's left is to specify where it goes! How many bits do we need for that? Well... the sequence occurs on average every 2^10 bits. We need all 10 of the saved bits just to say where the sequence goes! We haven't saved anything!
The more compressible the substring, the less frequent it is, and the more information is required to specify its location. This is also why we can't compress files by specifying their offset in the digits of pi, incidentally.
Either I'm confused (definitely possible) or this is sort of implicitly equivocating between two different senses of "determinism". There are experiments we can perform that appear to demonstrate quantum randomness. Though it may sound superficially plausible that any particular such random outcome is actually the deterministic output of a hidden pseudorandom number generator, that hypothesis is ruled out by Bell's theorem.
What Bell's theorem can't rule out is the hypothesis that not only any individual quantum measurement, but the sum total of everything that happens in the universe including the experimenters' choices of what actions to take during the experiment, is all part of a single deterministic causal path for the whole universe, that just so happens to play out in such a way that we never see anything that visibly contradicts Bell's theorem. This can't really be empirically falsified, but there are various philosophy-of-science reasons to be a priori skeptical of it (depending on which philosophers of science you ask, of course).
But perhaps we do not even need to abandon locality, if we modify it a bit. There is no good reason to believe that space at short distances should be similar to Euclidean space, one interesting hypothesis is that space is more like a graph, and entangled particles in addition to the normal long path through the graph are also connected directly, which allows measurement on one of them to change the state of the other.
Shouldn't this change the wave function though?
As far as I understand Bell's theorem only rules out the hypothesis that the random outcomes are the deterministic output of a hidden pseudorandom number generator that obeys locality. There could be a deterministic, non local process that generates the "quantum randomness", and this could be detectable if it exists.
But there's something going on there that I can't predict, and we need a language to talk about it. That language is the same whether "true physical randomness" exists or not. Calling a coin flip "50-50" is just as valid in a deterministic universe as it is in a random one. Probability is a language more than a theory.
Too many people get hung up on "true" randomness, when that's probably not relevant to the the situation they're describing.
how is that substantively different from what i said?
>99.99% accuracy is probably well above the quantum-weirdness level of accuracy.
what is quantum-weirdness scale and what is well above? hbar is 34 zeros out.
By quantum weirdness scale, I mean that the accuracy (epsilon) of a necessary measurement includes accurate position and momentum measurements. If epsilon is too small, we might not be able to hypothetically measure both to the necessary precision. I'm guessing that the necessary epsilon for essentially perfect prediction is large enough that you can hypothetically measure both to within that precision.
Hint: it’s a free body under rotation on a parabolic trajectory. You won’t end up needing Tony Stark hardware for this one.
Bonus factoid before bed: coins flipped by humans empirically have a small (~51%) bias toward heads. Supposedly folks tend to start with heads up more often, and the coin tends to flip over 0 times more often than you’d think, and that adds up to a bias you can measure in an afternoon.
You’re talking about something different from algorithmic randomness, which would be a property of (infinite...) sequences. When I throw a coin in the air and say “heads or tails”, you have to make a prediction as to where it will land based on the information you have. Not someone else’s information— _your_ information. If you feel like you have as good a chance of winning as losing, you’ll tell us you think it’s a fair — a “random” — coin toss.
You’re talking about betting behavior. It’s not a wild tangent the way some would think — from this we get the structure of probability theory, we get derived preferences, we get the entire mechanics of Bayesian inference that underlies all systems which learn.
From studying this math — the math of the coin flip with a dollar on the table, the math of the universe — we learn that there _cannot be_ “chances” or “odds” out there in the world that we can see if we look hard enough— what we see if we look closely enough will be the structure of our perception, which is a perception itself constituted by subjective probabilities.
We learn that belief states must be subjective, must be local. There can’t be a fact of the matter in any way that makes sense to anyone. And yet we also can prove that we cannot truly reserve judgment, that we cannot be true skeptics— unless we are also not anything recognizable as scientists.
So, yes, absolutely. In the consensus reality we all agree to occupy, how you decide to answer the question “heads or tails” matters — in the deepest possible sense — just as much as anything else. And folks can get _so_ wrapped up in the idea of what an imaginary computer might do, they forget to notice what they are doing with every waking moment.
A troll tour guide says, "Over here we have our random number generator." The troll places its hands on a slab of rock and relays the message of "nine nine nine nine nine nine." Dilbert asks, "Are you sure that's random?" The troll responds, "That's the problem with randomness. You can never be sure."
But radioactive decay is supposed to be random. As in, every atom is the the same, but some, randomly, decay. That never made sense to me.
Because there is no known mechanism such as your hypothetical precise robot with perfect situational knowledge to determine the radioactive decay.
Doesn't it also depend on the size of the universe ? We don't have any idea how big it really is. It could be infinite in which case it's not only likely, but inevitable.
All I said is that this is a large state space, which has been largely unexplored. The reason it’s largely unexplored is because most of the state space is useless, inert garbage. The amount of time it takes to create a genome this large is proportional to the size of the genome, not the size of the state space. That’s how evolution by natural selection works. If you hypothesized a world without evolution, where things appeared completely by chance arrangement of molecules, that’s when the size of the state space becomes important.
So I would say that your argument is not an argument against molecular evolution, it is an argument against something else.
* For example some genes are repeated, a bad copy may repeat a gene and the DNA get longer. Virus may cause some duplications too.
* Some genes are almost repeated,each copy has a slightly different function, so each one has a variant that is better for each function. The idea is that an error in the copy made two copies of the original gene and then each copy evolved slowly differently.
* Some parts of the DNA are repetitions of a same short pattern many many many times. IIRC these apear near the center and the extremes of the chromosomes, and are useful for structural reasons, not to encode information. The DNA can extend the extreme easily because it's just a repetition of the pattern.
* Some parte are just junk DNA that is not useful, but there is no mechanism to detect that it is junk and can be eliminated, so it is copy form individual to individual, and from specie to specie with random changes. (Some parts of these junk may be useful.)
So the idea is that the initial length was not 1000000000, but that the length increased with time.
Your calculation does not model the theory of "molecular Darwinism". Your calculation is about the probability that is a "human" miraculously apear out of thin air with a completely random genome, it will get the correct one [3].
[1] Or perhaps RNA, or perhaps a few independent strands of RNA that cooperate. The initial steps are far from settle.
[2] It's not strictly increasing, it may increase and decrease the length many times.
[3] Each person has a different genome, so there is not 1 perfect genome. The correct calculation is not 1/4^1000000000 but some-number/4^1000000000 . It's difficult to calculate the number of different genomes that are good enough to be human, but it's much much much smaller than 4^1000000000. So let's ignore this part.
They, like, evolved, right? As the GP says, there was a short sequence that worked, a little got built on, a little more...
There was never any time that any creature was generated by random choice.
This thread of discussion is about the computationally intractable nature of 4^1,000,000,000
Got math? Maybe post a proof?
Edit: Microsoft Windows 10 is 9GB. It would be impossible to try 8^9000000000 different programs. Yet, Windows exists, and most of us believe it's contained in those 9GB.
You wouldn't code that way and nature doesn't either.
In practice the changes happening in each generation are all sorts of different rearrangements, but that's different from proving the basic and obvious fact that when you have multiple steps you don't have to spontaneously create the entire solution at once.
Bogosort will never ever sort a deck of cards. Yet it takes mere minutes to sort a deck of cards with only the most basic of greater/less comparisons. Even if your comparisons are randomized, and only give you the right answer 60% of the time, you can still end up with a sorted-enough deck quite rapidly.
(Why sorted-enough? Remember that reaching 'human' doesn't require any exact setup of genes, every single person has a different genome. It just has to get into a certain range.)
Again, this thread is about the computationally intractable nature of 4^1,000,000,000.
Got math? A proof maybe to support your statements?
There is no final optimization step that analyze the 4^1,000,000,000 possibilities. We are not the best possible human-like creature with 1,000,000,000 pairs of bases.
> method of gradient descent
Do you know the method of gradient descent? Nice. It is easier to explain the problem if you know it. In the method of gradient descent you don't analyze all the possible configurations and there is no guaranty that it finds the absolute minimum. It usually finds a local minimum and you get trapped there.
For this method you need to calculate the derivatives, analytically or numerically. And looking at the derivatives at a initial point, you select the direction to move for the next iteration.
An alternative method is to pick e few (10? 100?) random points nearby your initial point, calculate the function in each of them and select the one with the minimum value for the next iteration. It's not as efficient as method of gradient descent, but just by chance half of the random points should get a smaller value (unless you are to close to the minimum, or the function has something strange.) So just this randomized method should find also the "nearest" local minimum.
The problem with the DNA is that it is a discrete problem, and the function is weird, a small change can be fatal of irrelevant. So it has no smooth function where you can apply the method of gradient descent, but you can still try picking random points and selecting one with a smaller value.
There is no simulation that picks the random points and calculate the fitness function. The real process in the offspring, the copies of the DNA have mutations and some mutations made kill the individual, some make nothing and some increase the chance to survive and reproduce.
who is talking about neurons? Beneficial random mutations propagate, negative don't, on average. In this way, the genetic code that survives mutates along the fitness gradient provided by the environment. The first self-propagating structure was tiny.
It's not literally the gradient descent algorithm as used in ml, because individual changes are random rather than chosen according to the extrapolated gradient, but the end result is the same.
>computationally intractable nature of 4^1,000,000,000
which is a completely wrong number, even if only because of codon degeneracy. Human dna only has 20 amino acids + 1 stop codon, which are encoded by 64 different sequences. Different sequences encode the same amino acid.
If you know more than others, it would be great if you'd share some of what you know so the rest of us can learn something. If you don't want to do that or don't have time, that's cool too, but in that case please don't post. Putting others down helps no one.
Your example insinuates that (a) all of the human genome is required to correctly model the human phenotype, i.e. each bit is significant, and, more importantly, (b) the human genome came into existence as-is without a history of stepwise expansion and refinement.
I can't know whether you're a creationist, but I will point out that your attempted argument is on (e.g.) #8 on Scientific American's list of "Answers to Creationist Nonsense" (https://www.scientificamerican.com/article/15-answers-to-cre...). Amusingly, SciAm's rebuttal even explains how the "monkeys typing Hamlet" fails as an analogy to the human genome.
On the other hand, randomness is not the same as nondeterminism. We could have a specific infinitely long incompressible bitstring that is the source of randomness in our universe, but the bitstring is not changing, so it is deterministic. There would be no empirically distinguishable difference between this and an actual undetermined source of randomness.
And finally, what we call random and nonrandom is essentially an arbitrary choice based on a certain mechanism called a Turing machine. As Dilbert says the number 4 is a random number, and the only random number you need. It is unclear what makes compressible bitstrings special, although it is clear that they are rare. But why does observing a compressible bitstring demand an explanation? We could hypothesize reality is fed by an infinitely long bitstring which some parts are compressible and some parts incompressible. In fact, an infinite random bitstring is guaranteed to have arbitrarily long compressible subsequences, so we could explain any empirical observation of compression as being due to an infinite random bitstring and devoid of true explanation beyond itself.
In practice randomness is about lack of knowledge not about actual randomness or processes.
(Edit: Sorry, but this clickbait title redefining randomness as something other than what everybody understands it to be annoyed me.)
Randomness is perhaps a description of correlation? Which obviously relates to probability, prediction of uncertain outcomes, but maybe more general?..
We have not proven that if you repeat the experiment tomorrow, you will get the same result, it is only an hypothesis.
The randomness of the QM is not proven, but for now we don't have a better alternative to predict the results of the experiments. Just like gravity that is has not been proven, but for now we don't have a better alternative to predict the results of the experiments.
Unitary evolution means there's neither information loss nor gain, and if there was anything random you would at the very least expect to see information gain (as new bits of information are created from the "random" result of an observation).
Randomness in quantum mechanics isn't even a hypothesis, it's an interpretation.
Now, whether the underlying physics is truly random, or whether it's deterministic and the projection only represents a sort of Bayesian update of prior information (a la MWI), that is indeed a matter of interpretation. And completely unfalsifiable by definition, and therefore not even really a question for physicists. It's philosophy at best.
Hypercomputation (halting problem) and Infinite memory/storage
Ascribe either of these to nature, and nature can be deterministic and still, the probability of us discerning its RNG's operations is 0.
This is a good video explaining the different intricacies of how "God's dice" might be constructed:
Many phenomena thought to have been random due to their quantum nature have been found to be based on their initial conditions instead. See spontaneous emission photon phase for example: https://iopscience.iop.org/article/10.1088/1367-2630/9/11/41...
Well unless you use a one time pad but nobody does (hopefully).
Was under the impression intel agencies and militaries use OTP's regularly and the keys are carried in diplomatic pouches around the world.
Leibniz has already proposed essentially the same centuries earlier. And there have probably been people who said the same even earlier.
(Also, mentioning Konrad Zuse made me directly suspect that the author is German. He's Swiss, but close enough.)
"Randomness" in the small can and does appear to be non-random in the large - we make predictions as to what we will see over large numbers of events when we are unable to determine if any single event will fulfill that prediction. Radioactive decay is a good exampe of this. Two-slit refraction patterns are also another example. Much, if not all of our technology, depends on this, whether this be semiconductor design or manufacturing any material products such as steels or concrete.
What does happen is that we have more and more interesting research areas in which we can investigate the underlying principles that govern our universe. But, we must not make the mistake that we will "know" what those principles will be be. We can and do develop workable and useful models and theories to help us get a handle on understanding this universe we live in. We live on one small planet in an isolated region of our galaxy in an extraordinary and immense universe. We do not kave the ability to explore that universe in any detailed way except by proxy observations. So, instead of getting caught up in being "sure", let us have fun in exploring everywhere we can and continue to gather data and discuss what this data means and develop workable theories and models that we can use.
As a disciple of the living God who created all the that we see and do not see, I consider that the universe has a set of specific rules and laws by which it operates and that we can and should try to understand what those laws are. For me that is an act of worship to investigate and understand the what and how.
For those who are of other belief systems, whether that be Hindu, Buddhist, Moslem, Atheists, etc., there is just as much an incentive to study the universe around us and understand what and how it works. There may be additional questions that might be raised from each viewpoint that is not of concern for any of the other viewpoints like "why".
BUt what it all boils down to, is that we live in a wonderful and extremely interesting universe and there is much to learn about it and have fun while learning about it.
I started writing a longer version of this comment but I think that a core part of the question is whether “randomness” is an epistemological convenience, a statement about “order“ or “rules”, or something else.
My takeaway is that this is the reason some physicists claim 'everything is information' because there is some underlying form that gives the statistical quantum physics a consistent pattern instead of devolving into randomness.
In the absence of high enough computational resolution one would perceive this as randomness. This is also related to the quest for determining if we live in a simulation.
If someone know which site I might be alluring to, please post it here.
But it turns out to be a lot harder to design a really good emulator of randomness than one might guess. Certain Monte Carlo simulations using the Mersenne Twister turned out to be oddly but unmistakeably biased.
So, one measure of generators is how many numbers you pull before you get the same sequence again. MT's cycle is very long, so in practice you never see a repeat, even if you see the same number many times. (In many simpler generators, seeing 3 then 8 means next time you see 3, the next number will be 8. A great deal of simulation was done with such generators.) The numbers from an MT satisfy many different measures of apparent randomness.
Monte Carlo investigations consume very many numbers. They might use the numbers in a more or less periodic way, so that any match-up between cycles in the problem and cycles in the generator can skew the results. The main MT cycle is very long, so any skewed results probably point to lesser cycles as the bits stirred are later encountered again. But it's hard to imagine a way to detect such cycles deliberately from the bits you get out. Encountering a process that finds them accidentally is amazing.