I've heard that too. Truly random, in my book, means the absence of a pattern.

Ramsey Theory

According to Ramsey theory, pure randomness is impossible, especially for large structures. — Wikipedia

Now according to Ramsely theory there's a pattern in these numbers: 2, 3, 5, 7, 11, 13, 17, 18,... [prime numbers]

I hope the pattern in prime numbers isn't just the fact they have only 2 factors (1 and themselves)!

2, 3, 5, 7, 11, 13, 17, 19, 23, 29,...

First difference: 1, 2, 2, 4, 2, 4, 2, 4, 6,... [no pattern?]

Second difference: 1, 0, 2, -2, 2, -2, 2, 2,...[no pattern?]

Third difference: -1, 2, -4, 4, -4, 4, 0,...[no pattern?]

Fourth difference: I'm too lazy. Imagine if there's a pattern in the googolplex^th difference! :scream:

What we need is someone who suffers from severe pareidolia or apophenia.

The Pattern-Patternless paradox.

Assume 3 numerical series: S₁, S₂, S₃.

S₁ and S₂ exhibit (unique) patterns.

S₃ is patternless i.e. it's random.

Suppose now you have another series S₄ that's also patternless.

The part where it gets interesting is S₃ and S₄ have something in common i.e. there's a "pattern" viz. randomness/patternlessness.

Patternlessness is a pattern!

Every attempt to formulate randomness leads to a repetition of a previous sequence. Now this has to be the case in every random infinite sequence of numbers, but then the repetition starts somewhere, eeeh, randomly.

Perhaps part of the problem is a lack of clarity over the meaning of 'random'.

Traditionally, it is supposed to mean that an outcome or next in sequence, occurs without a direct prior cause. Whether the same number appears twice in succession or the spaces' between numbers is consistent or not, the question is whether numbers in the sequence are generated by a prior cause.

This is normally taken to mean a mathematical/formulaic cause, or generated by a machine which operates from formulaic coded instructions, but why couldn't a random sequence be generated by the human mind?

I don't think it is accurate for to say that randomness disappears because two random sequences have something in common (ie. that they are random). If nothing else, the comment being made about the categorization of the set, not the contents of each set/sequence.

I don't think it is accurate for ↪Agent Smith to say that randomness disappears because two random sequences have something in common (ie. that they are random). — Gary Enfield

Yep, one difference between random and nonrandom is algorithms.

The fact that 100 randomicals have their randomic aspects in common, doesn't make them less random. Patterned randomity is still random.

The fact that 100 randomicals have their randomic aspects in common, doesn't make them less random. Patterned randomity is still random. — Raymond

Of course: if I toss a coin 10 times, and within those 10 there's a sequence of 3 heads, that doesnt make it non-random. Then if I repeat the 10 tosses and again get a sequence of 3 heads, that doesnt make either of those sequences of 10 non-random. For coin tossing it is probably harder to prove the result is not random. For that to be so my tossing action would need to be somehow biassed and/or the coin would have to be unsymmetrical enough to affect the result. Let's face it, both are pretty unlikely.
So all I need is a 10-sided dice...

Thinking of national lotteries: they take huge pains to ensure there is no bias in the balls the machine selects. If after each selection that ball was replaced with one the same, then the selected sequence would effectively be random. Chaos theory would make any analysis of the balls' motions within the circulating drum and thus prediction impossible..

"Random" can mean different things. Rigorously defining randomness can be problematic - see this SEP review for starters: Chance versus Randomness.

One important distinction is process vs. product randomness. Very roughly (the above linked article goes into details), process randomness is produced by a random process. What that means is... complicated. Product randomness is something that just presents itself as random. What that means is... no less complicated. Product randomness is often cached out in terms of frequencies, such as the normality criterion, to which I will return in a moment.

I've read that it's impossible to produce a truly random series of numbers. — Tim3003

This likely refers to process randomness, with the assumption being that no process is truly random. This is true at least for ordinary digital computers not equipped with a quantum random number generator (QRNG).

I've also read that the sequence of digits of irrational numbers like the square root of 2 are totally random. — Tim3003

This may refer to the product, or frequency sense of randomness. There are some intuitive criteria of randomness as applied to number sequences. On their own, none of them is perfect, i.e. no single criterion guarantees that every sequence that satisfies the criterion will be perceived as random.

For example, the criterion of normality requires that none of the digits occur more often than any other in the long run. But a number like this, while obviously non-random, would satisfy this criterion:

0.1234567890123456789...

Some numbers have been found to satisfy all popular empirical randomness tests, and this is perhaps what you have heard. Not all irrational numbers would fit the bill though. For example, this number is irrational but clearly non-random:

0.1001000100001...

I've also read that the sequence of digits of irrational numbers like the square root of 2 are totally random. — Tim3003

As with many things in math, and other topics as well, you sometimes have to be careful how you express your thoughts. The sequence of the digits (in the decimal representation) of the square root of two is hardly random. What is meant - what you should be saying - is that it is not a repeating pattern or sequence of numbers. It is a non-repeating decimal. Which is another way of saying that it is not a rational number, because every rational fraction corresponds either to a decimal fraction that terminates, e.g., 1/2 = .5, or that repeats.

As to what a random number is, that is not-so-simple, and best to look on line and to spend some time thinking about what you find in the way of technical definitions, along with accompanying discussions and commentary.

after each selection that ball was replaced with one the same, — Tim3003

You mean replacing balls every time? There are few perfectly symmetric objects to throw. The throwing is mostly random, but the falling on a side depends on the shape. I think if I pick a coin from a heap, the side with which I throw it up is random. Unless all coins in the heap face up. If I make the same movements every time, I will throw it with the same side up every time. Then also my hand makes non-random motions, though I let the coin fly quite arbitrarily. So the coin shape is important. The motion of atoms in gas is random. This can be used in random number generation.

Random choices are impossible to make.

This is a mathematical can of worms, with definitions and counter-definitions. In the physical world certain phenomena may exhibit what we normally think of as randomness. What comes to mind is recording very slight changes of atmospheric pressure in some sort of controllable context.

Since there are algorithms predicting successive digits of the square root of two, from one perspective those digits - the sequence - would not be considered random. But for practical purposes the sequence could be considered random.

Put a tiny mirror on a Brownian particle, shine a light on it, and record the reflected light. The record of randomness.

As to what a random number is, that is not-so-simple, — tim wood

Isn't randomness in fact the most simple?

Isn't randomness in fact the most simple? — Raymond

Go for it.

Randomness and total order are seemingly opposite but two sides of the same medal. Total order displays one globally and locally uniform pattern, while randomness displays none at all, nor locally, nor globally. The interesting patterns seem to arise in between. Inside the medal, so to speak.

A random sequence of 1's and 0's shows nor locally, nor globally a pattern. The 1's and 0's are independent of each other, non connected by a formula, while the 1's and 0's in a maximally ordered sequence are locally and globally connected by the same formula.

Both are simple.

A random sequence of 1's and 0's shows nor locally, nor a pattern. — Raymond

Hmm. How about the decimal sequence 11111111111111111111111111111111? What is the next number? Ordered? Not ordered? Random? Not random?

It depends. If the 1's are independent, it's random. That can only be decided by writing extra 1's and 0's. It's more likely it has total order. If the next sequence would be the same number of 1's then 0's again, the 1, etc. There would be total order. Though all 1's or 0's would be even more orderly. In a totally random sequence, your row occurs once in a while. But how can we tell by looking? We can't. Only by knowing if there is a dependency between the 1's and 0's. That might be the case for this row of 1's. It can be seen in a casino though. I was sure red had to appear after 13 times black. And it did.

A lot of information security is based on cryptographically-secure randomness. Such random number generators allow you to produce sequences that others cannot reliably predict.

Although you can deterministically predict them because you know the initial configuration, it remains fully random for others as long as they don't know the configuration and don't have access to the generator. Not truly random, but effectively unpredictable. Like the square root of 2.

But how can we tell by looking? We can't. — Raymond

Part of the challenge in defining a random number. We all kind-of know informally what one is, but what it actually is, in the sense of this or that number being random, is more difficult.

Here is an interesting video on "All of the numbers."
https://www.youtube.com/watch?v=5TkIe60y2GI

Random is more a algorithmic collection of shapes, such as a marked circle in a square. If the circle rotates and the square ticks synonymously in 'algorithmic' manner, if you pick a corner and twist the whole collection, it's too complex to be a matter of skill as to which corner the mark is closest to.

Source: the weather

The problem with randomness is that just by looking, like at a long array of 1's, it's impossible to say if there is a connection between the elements of the array. A long array of 1's could be random, or each element of the array could be related to the other ones in a scale-independent way. The atoms in an ideal gas all are independent of one another. The atoms in a perfect crystal are orderly connected. If the local connections between the atoms are globally invariant, the atoms display total order. You just need to look at the connections locally to know the global order. In a gas the situation is the same. If you know the local appearance, you know the whole. The gas might show a fluctuation of orderly structure but only temporarily. The crystal structure is fixed in time.

Now how does this relate to rows of 1 and 0? If you are given a random array of them how we know it's random? In fact, every array of them could be random, even an infinite sequence of 1. If all the 1 and 0 are independent of one another they still could decide all to show 1. Like all particles of gas could find themselves in a corner. All particles of the perfect crystal are static (at zero temperature). But they are as dull a sight as the gas. The 1 and 0 array of sqrt 2 seems random. But they have a connection. To discover randomness, dynamics and initial conditions have to be taken in account. If all gas particles are given the same direction, their motion is not random, like the motion of "weather gasses", the atmosphere is only partially random. The fun lies between complete randomness and complete order. But what is complete randomness?

How can we determine randomness? Chance is invented for that. If there are correlations (interdependencies) found between stochastic variables, like the gas particles, there is a degree of non-randomness. If the particles in a gas showed a continued interdependence, it's not a fluctuation. If they find themselves all in a corner at a given moment (could be), and they stay there, there is a non randomizer (not to be confused with a sodomizer) active, like boundary conditions or internal changes (maybe the particles all hold hands suddenly). Are the initial conditions of the universe random?

Isn't randomness in fact the most simple? — Raymond

One could argue that that which is random is the least simple (as in the least compressible or amenable to abbreviation or summary.)

Part of the challenge in defining a random number. We all kind-of know informally what one is, but what it actually is, in the sense of this or that number being random, is more difficult. — tim wood

Surely there's no such thing as 'a random number'. Randomness is about a process of selecting a series of numbers, such that no single selection of a number affects any subsequent selection of a number. It's the definition of a random series or sequence we need to worry about..

The definition of such a number that I've seen, and like, is whether shortest description of the number (perhaps in terms of a computer program that reproduces it) is just the number itself.

143 is a random number algorithm, where 1 is a constant, 4 is a change and 3 is a choice.

Such as plates, poles and seismic activity, whether you choose to change the constant the outcome is random.

Randomness, as per Wikipedia, is unpredictability. Unpredictability, given n possibile outcomes, is basically equiprobability $(\frac{1}{n})$ of these outcomes . Think of it, any probability other than $(\frac{1}{n})$ would introduce a perfectly serviceable level of predictability to even phenomena that are inherently probabilistic.

Are there any truly random phenomena in the universe? That, my friend, is the million dollar question.

Furthermore, probabilistic predictability is a thing (re quantum physics) and that means just because one's able to make a probabilistic prediction about a given system, it doesn't follow that the system is deterministic. Those interested in free will might find that of some use. Psychological research showing, say, 90% of people think/say/do something (mentalists - there are no green elephants in Denmark), doesn't clinch the argument for no free will.

Are there any truly random phenomena in the universe? — Agent Smith

The outcome of measuring spin up or down is random. It is actually the device measuring the electron spin that should be used in casinos. "I have put all my money on spin up". "That's stupid! Spin up has fallen 7 times already". "I have put my last money on upupupup!"

The true challenge lies in the search of random choice. Can we randomly choose? Totally unpredictable?
If we mentally envision a device measuring spin up or down, and mentally watch the outcome...
Best is to memorize the ups and downs in a real experiment, though even there it remains to be seen if the superposition is really 50-50. And maybe God intervenes.

:ok:

Particle spin, totally random? Good to know.

It's direction. It can exist in a superposition of two equally probable states. But coming to think about it, how can we be sure? Maybe a hidden force plays with the probabilities. Hidden variables. Consider it part of my paranoiatical nature. Something is after the probabilities? Is it a determined force? I guess, as that force will help me win the lottery. It's written...

The first episode of the TV serie NUMB3RS is about a serial rapist-murderer. This criminal attempts to throw the cops off his scent by trying to randomize his victim abduction points in the city.

Unfortunately for the antagonist, his randomization is (too) perfect; in the real world, clusters are a regular feature, none appear in the rapist-murderer's crime zone/hunting ground!

Are there any truly random phenomena in the universe? That, my friend, is the million dollar question. — Agent Smith

The outcome of a quantum collapse is random (not to be confused with uniformly distributed).

Here's a white paper by a company that produces a true random number generator using quantum collapse.