What is random?
[s]All irrational numbers are not truly random (theyre psudorandom). https://thephilosophyforum.com/discussion/comment/557638
There are different interpretations of QM, some random, some determenistic.
https://thephilosophyforum.com/discussion/comment/557565
Where can you find true randomness?
https://thephilosophyforum.com/discussion/comment/557612
And what is psudorandom?
https://thephilosophyforum.com/discussion/comment/557747[/s]
Real OP:
Quoting fishfry
There are different interpretations of QM, some random, some determenistic.
https://thephilosophyforum.com/discussion/comment/557565
Where can you find true randomness?
https://thephilosophyforum.com/discussion/comment/557612
And what is psudorandom?
https://thephilosophyforum.com/discussion/comment/557747[/s]
Real OP:
Quoting fishfry
The story actually goes back to Leibniz, who "dreamt of building a machine that could manipulate symbols in order to determine the truth values of mathematical statements."
The problem got its modern form as the Entscheidungsproblem, which is German for "decision problem." According to Wiki,
The problem asks for an algorithm that considers, as input, a statement and answers "Yes" or "No" according to whether the statement is universally valid, i.e., valid in every structure satisfying the axioms.
By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the Entscheidungsproblem can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic.
— Wikipedia
In 1936 Alan Turing published his famous paper, On Computable Numbers, With an Application to the Entscheidungsproblem. Turing was played by Benedict Cumberbatch in the highly fictionalized but definitely worthwhile movie, The Imitation Game.
In this paper Turing showed that there could be no mechanical procedure to decide the truth of mathematical propositions.
As part of his proof, he invented a formalized model of computation called the Turing machine (TM). A TM can be programmed to solve a given mathematical problem. He showed that there are many real numbers whose digits can be cranked out by a TM. He defined such a number as computable. For example any rational number is computable, since given two integers n and m we can just apply the grade school division algorithm to compute as many digits as we like of the quotient n/m.
Surprisingly, even many irrational numbers are computable. For example pi can be computed via the famous Leibniz series:
pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ...
which can easily be turned into a computer program or TM.
However there are many noncomputable real numbers, whose decimal representation can never be computed by any TM or any computer program.
Now, how many real numbers are computable? Since each TM program consists of finitely many finite-length instructions, there are only countably many such programs: finitely many of length 1, finitely many of length 2, and so forth.
On the other hand we know from Cantor's diagonal argument and also from Cantor's theorem, that there are uncountably many real numbers. So there are "too many" real numbers to compute them all with TMs. These are the noncomputable real numbers. They are deserving of the name random, because there is no mechanical or deterministic process to generate their decimal representations.
Now when we say that "almost all" real numbers are noncomputable, we mean that a countable set is vastly smaller than an uncountable one. So "most" real numbers are noncomputable.
We can formalize this idea using measure theory. Any countable set has measure zero. So for example in the unit interval consisting of all the real numbers between 0 and 1, the set of computable reals has measure zero, and the noncomputable reals has measure 1.
Intuitively this means that if you put all the real numbers in a paper bag, closed your eyes, and reached into the bag and pulled out a real number, the probability is zero that it's computable, and probability one that it's not.
Another way to think of it is that if you flip a fair coin infinitely many times, regard heads as 1 and tails as 0, and put a binary point in front of the sequence, you have the binary representation of some real number. The probability that a computer program could crank out that exact sequence is zero; and the probability that no possible computer program or mechanized procedure could do so is one.
This is actually plausible. If we randomly flip infinitely many coins to generate a sequence of 0's and 1's, it's very unlikely that there will be any kind of pattern or mechanical procedure that predicts the sequence. It's far more likely that the sequence will be patternless, or random.
The noncomputable real numbers are sort of the "dark matter" of the real numbers. They don't have names. They don't have individual qualities or characterizations that uniquely describe any of them. They are simply there, holding the real line together.
So my thesis is that even beyond the mysteries of quantum mechanics, if anything in the world can legitimately be called random, it's the noncomputable real numbers.
Constructivist mathematicians don't believe in noncomputable numbers. They try to do all of math using only numbers that can be specified via algorithms or procedures. They're gaining some mindshare these days due to the influence of computers in math.
Well I wrote a lot but I left out a lot, so please ask if anything's not clear.
18 minutes ago
Comments (29)
But to answer your question: Maybe, we can’t tell.
Take a fair, six-sided die [1, 2, 3, 4, 5, 6]. Randomness, in my book, simply means all six possibilities are equiprobable [1/6]. The probability of 1/6 is what we call theoretical probability. If one rolls this die a very large number of times, what we can do is look at the relative frequency of each outcome aka experimental probability. If the experimental probability = the theoretical probability = 1/6 for each possibility, that's as random as random can get, for a die that is.
Thanks for asking this question because, right/wrong, I've managed develop a system of argumentation/justification vis-à-vis ontology. The basic idea is to assume as an axiom that the universe is symmetrical - cold/hot, hard/soft, good/bad, you get the idea. We know that determinism exists in the universe and since the universe is symmetrical, (true) randomness must also exist.
Almost all reals are noncomputable. How is some noncomputable number pseudorandom but not random in your view?
Quoting Wheatley
Among the noncomputable numbers. Pick any one you like. Its binary digits are as random as anything can be.
Give me a minute to edit my OP.
A computable number is a real number whose decimal digits can be generated by a computer program.
Most real numbers are not computable. The digits of a noncomputable number are completely unpredictable. No program or recipe or algorithm or mechanical procedure of any kind can generate them. Noncomputable numbers are random if anything is.
You don't have to know much about this, only that most real numbers are truly random in any sense of the word you could name.
:up:
You have a Minute To Win It! :lol:
A sequence is pseudorandom when it's the output of a deterministic process, but the sequence satisfies all known statistical tests for randomness. When you call the random() function in any computer language, you're not getting a random number, but rather a pseudorandom one.
Pseudorandomness measures the extent to which a sequence of numbers, though produced by a completely deterministic and repeatable process, appear to be patternless.
https://en.wikipedia.org/wiki/Pseudorandomness
Now, this thread is yours! :rofl: :party:
So you were only trolling? I don't get it.
Can you unpack that a bit for the benefit of those without a background in mathematics? Say a bit about why it is the case, and what it means? Thanks.
:clap:
The story actually goes back to Leibniz, who "dreamt of building a machine that could manipulate symbols in order to determine the truth values of mathematical statements."
The problem got its modern form as the Entscheidungsproblem, which is German for "decision problem." According to Wiki,
[quote=Wikipedia]
The problem asks for an algorithm that considers, as input, a statement and answers "Yes" or "No" according to whether the statement is universally valid, i.e., valid in every structure satisfying the axioms.
By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the Entscheidungsproblem can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic.
[/quote]
In 1936 Alan Turing published his famous paper, On Computable Numbers, With an Application to the Entscheidungsproblem. Turing was played by Benedict Cumberbatch in the highly fictionalized but definitely worthwhile movie, The Imitation Game.
In this paper Turing showed that there could be no mechanical procedure to decide the truth of mathematical propositions.
As part of his proof, he invented a formalized model of computation called the Turing machine (TM). A TM can be programmed to solve a given mathematical problem. He showed that there are many real numbers whose digits can be cranked out by a TM. He defined such a number as computable. For example any rational number is computable, since given two integers n and m we can just apply the grade school division algorithm to compute as many digits as we like of the quotient n/m.
Surprisingly, even many irrational numbers are computable. For example pi can be computed via the famous Leibniz series:
pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ...
which can easily be turned into a computer program or TM.
However there are many noncomputable real numbers, whose decimal representation can never be computed by any TM or any computer program.
Now, how many real numbers are computable? Since each TM program consists of finitely many finite-length instructions, there are only countably many such programs: finitely many of length 1, finitely many of length 2, and so forth.
On the other hand we know from Cantor's diagonal argument and also from Cantor's theorem, that there are uncountably many real numbers. So there are "too many" real numbers to compute them all with TMs. These are the noncomputable real numbers. They are deserving of the name random, because there is no mechanical or deterministic process to generate their decimal representations.
Now when we say that "almost all" real numbers are noncomputable, we mean that a countable set is vastly smaller than an uncountable one. So "most" real numbers are noncomputable.
We can formalize this idea using measure theory. Any countable set has measure zero. So for example in the unit interval consisting of all the real numbers between 0 and 1, the set of computable reals has measure zero, and the noncomputable reals has measure 1.
Intuitively this means that if you put all the real numbers in a paper bag, closed your eyes, and reached into the bag and pulled out a real number, the probability is zero that it's computable, and probability one that it's not.
Another way to think of it is that if you flip a fair coin infinitely many times, regard heads as 1 and tails as 0, and put a binary point in front of the sequence, you have the binary representation of some real number. The probability that a computer program could crank out that exact sequence is zero; and the probability that no possible computer program or mechanized procedure could do so is one.
This is actually plausible. If we randomly flip infinitely many coins to generate a sequence of 0's and 1's, it's very unlikely that there will be any kind of pattern or mechanical procedure that predicts the sequence. It's far more likely that the sequence will be patternless, or random.
The noncomputable real numbers are sort of the "dark matter" of the real numbers. They don't have names. They don't have individual qualities or characterizations that uniquely describe any of them. They are simply there, holding the real line together.
So my thesis is that even beyond the mysteries of quantum mechanics, if anything in the world can legitimately be called random, it's the noncomputable real numbers.
Constructivist mathematicians don't believe in noncomputable numbers. They try to do all of math using only numbers that can be specified via algorithms or procedures. They're gaining some mindshare these days due to the influence of computers in math.
Well I wrote a lot but I left out a lot, so please ask if anything's not clear.
//ps hey are you familiar with this website https://www.cantorsparadise.com/
You would find it of interest, I think.
Wouldn’t you actually have to roll it an infinite number of times for any of the six sides to land facing up exactly one-sixth of the time?
We are, of course, speaking here not only of the ideal die, but also the ideal toss. The mass of the former must be equally distributed throughout its entire body, and its edges must be perfectly sharp everywhere (and remain that way during an infinite number of tosses), and the toss itself must be such that any of the infinite positions and velocities of the die upon impact with the playing surface be equally possible.
Heard of it, I'll have a look around. Thanks.
Randomness, patternlessness, can be generated with an algorithm.
Say, I'm the Cartesian deus deceptor (the evil demon) and I see someone about to play a coin-tossing game of chance.
Algorithm for randomness:
1. n = 1
2. If n is odd, make the outcome heads else n is even, make the outcome tails
3. n = n + 1
4 Goto 2
The person playing this game will see no pattern in the coin tosses. P(heads) = 50% = P(tails). In other words this person will think the outcomes (heads/tails) is random and for all intents and purposes if all possible outcomes (heads/tails) are equiprobable, it is considered random. However the deus deceptor used a simple algorithm to generate this randomness.
I've seen theoretical probability matching experimental probability in less than ideal conditions. I guess what comes into play are errors (+/-) instead of biases (++/- or +/--). You need a large number of throws, ideal size of the experiment = infinity, for the errors to cancel each other out. That's my reading though. I maybe wrong!
That's indeterminism.
Well, guess I was wrong then
That's a married bachelor. Algorithms are deterministic. They cannot create randomness.
Quoting TheMadFool
This is precisely what pseudorandomness is: A deterministic sequence that passes all known statistical tests for randomness. It "looks random."
You don't need an evil demon, just a mathematical gadget called a linear congruential generator. That's how random() functions in programming languages work.
Hold on a minute!
Quoting fishfry
That's precisely the point. You can't tell the difference between randomness and determinism (sorry, I can't find a better word). If so, of what use is the distinction? I phrase I learned in an introduction to philosophy book seems apt: A distinction without a difference.
Ok ...
Quoting TheMadFool
We can know for certain that a pseudorandom sequence that we generated from an algorithm is not actually random.
But given an apparently random sequence, we can't know for sure if it's "truly" random (if that concept is even meaningful or possible), or merely pseudorandom.
So the point of pseudorandomness is that it works one way. We can know for sure that a p-random sequence is not actually random. But given an apparently random sequence, we can't tell if it's truly random or only p-random.
But I don't follow your point. Anything that comes out of a deterministic process is not random, by definition. So a p-random sequence is not random. That's a meaningful statement, I don't know why you don't think so.
When the typical person says something is random, they mean it's "truly random," ie NOT the output of a deterministic process. If it's deterministic, it's not random, even if it looks random.