Prime numbers

[Unity+]

Your implication was that Primes are not randomly distributed. That's drivel of the meaningless kind.

That depends on the definition of random. Seemingly random is different than actually random. And, no it is not meaningless drivel as case in point.

 

And, no I did not repeat the same statement in different ways. It was not redundant for they were similar in subject, but different things were being stated.

 

 

 

I see someguy1 is banned so there certainly will be no citation that I asked him for on the Twin-Prime breakthrough he invoked. But as John mentioned Terrence Tao and Terrence was a part of the team working on that breakthrough I will provide my own citation. NOTE!: That work neither proves the Twin-Prime Conjecture nor does it show Primes are anything but randomly distributed. I'm a little pressed for time so I will just give a link for now and save quoting the pertinent parts for later.

 

Together and Alone, Closing the Prime Gap

Thank you for the article. Also, what does the banning of Someguy1 have anything to do with this?

 

 

That work neither proves the Twin-Prime Conjecture nor does it show Primes are anything but randomly distributed. I'm a little pressed for time so I will just give a link for now and save quoting the pertinent parts for later.

I wasn't claiming that the paper did prove anything at all. My rebuttal was merely that not proving one conclusion does not have other to be correct, which is why, again, unsolved problems within mathematics exist.

Edited June 29, 2014 by Unity+

[imatfaal]

OK Guys - I am involved so I cannot moderate this thread; but please cut back on the rhetoric and concentrate on the subject please

[Acme]

That depends on the definition of random. Seemingly random is different than actually random. And, no it is not meaningless drivel as case in point.

More meaningless drivel.

 

And, no I did not repeat the same statement in different ways. It was not redundant for they were similar in subject, but different things were being stated.

You are mistaken.

 

Thank you for the article. Also, what does the banning of Someguy1 have anything to do with this?

Your welcome. If you had read the thread you would know.

 

... My rebuttal was merely that not proving one conclusion does not have other to be correct, which is why, again, unsolved problems within mathematics exist.

No duh.

[Unity+]

OK Guys - I am involved so I cannot moderate this thread; but please cut back on the rhetoric and concentrate on the subject please

Sorry about that, I will focus. But, aren't I right that there is no absolute proof that primes are randomly distributed in the sense that there is no function, yet discovered, that can predict position of primes or produce all of them?

More meaningless drivel.

 

 

There are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no known useful formula that sets apart all of the prime numbers from composites. However, the distribution of primes, that is to say, the statistical behaviour of primes in the large, can be modelled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability that a given, randomly chosen number n is prime is inversely proportional to its number of digits, or to the logarithm ofn.

http://en.wikipedia.org/wiki/Prime_number

 

If you think Wikipedia is not a meaningful source, I will get another article.

 

EDIT: I put this in bold for everyone to see, and just to make sure I made it red, italisized, and in 24pt font.

Edited June 29, 2014 by Unity+

[Acme]

Sorry about that, I will focus.

Very good. Focus & critical thinking go together like forums and arguments.

 

But, aren't I right that there is no absolute proof that primes are randomly distributed in the sense that there is no function, yet discovered, that can predict position of primes or produce all of them?

The random distribution of Primes is self-evident. Adding the caveat 'no function, yet discovered' only allows us to continue -logically- looking for patterns without giving much thought to the very real possibility of our being on a fool's errand. As Gödel showed, if our math system is consistent it is incomplete and so the question of the distribution of Primes may -logically- be unanswerable.

 

 

http://en.wikipedia.org/wiki/Prime_number

 

If you think Wikipedia is not a meaningful source, I will get another article.

 

EDIT: I put this in bold for everyone to see, and just to make sure I made it red, italisized, and in 24pt font.

The editing is an impediment to the reader, not an aid.

 

Knowing about primes 'in the large' is not a solution to the problem of knowing in the small. This is the same mistake Peter is making when he challenged the quote I gave from Wolfram by focusing on the military precision' idea -the large- while ignoring growing 'like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout', -the small'. It is the small that is the big problem.

[Unity+]

The random distribution of Primes is self-evident. Adding the caveat 'no function, yet discovered' only allows us to continue -logically- looking for patterns without giving much thought to the very real possibility of our being on a fool's errand. As Gödel showed, if our math system is consistent it is incomplete and so the question of the distribution of Primes may -logically- be unanswerable.

Are you saying that it is an axiom? You would have to give a reasonable argument that it should be considered an axiom of Mathematics.

 

 

 

The editing is an impediment to the reader, not an aid.

 

Knowing about primes 'in the large' is not a solution to the problem of knowing in the small. This is the same mistake Peter is making when he challenged the quote I gave from Wolfram by focusing on the military precision' idea -the large- while ignoring growing 'like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout', -the small'. It is the small that is the big problem.

So, we are going to give a proof by providing a muse of human thinking about numbers...sounds more like personal incredulity.

[PeterJ]

I feel you not doing your opponents justice, Acme. The prime number theorem states the prime distribution is not random, but the result of strict mathematical laws. We can use these laws to predict the next prime, and it is not difficult to do. But it takes forever for large primes, and it's probably just as easy test for them as predict them. Still, we do not have to just stick a pin in the board. Because they are not random we can predict with certainty that they will not appear at 6n+/-1, at 30n+/-5 and so on.

 

For all practical purposes the distribution may as well be random. It is as if it is random. But it is not random. It is as non-random as any sequence of numbers could be. 'Pseudorandom' seems a happy compromise.

 

If the distribution of the primes is truly random then the Riemann hypothesis is false. It would be dead easy to build a simple machine that produced the endless sequence of primes given a non-halting Turing machine, or some different sized wheels and a long enough road.

 

To call the primes random is a mathematical shorthand, not a rigorous description. It is this shorthand that I'm complaining about. I think it is profoundly misleading outside of mathematics. What happens inside mathematics is none of my business.

Edited June 29, 2014 by PeterJ

[Acme]

Are you saying that it is an axiom? You would have to give a reasonable argument that it should be considered an axiom of Mathematics.

I am saying the random distribution of Primes is referred to as a 'random distribution' because no one in a couple thousand years has shown a way to give the n^th Prime even if given the n-1^th Prime. It is an hypothesis that Primes are randomly distributed.

 

So, we are going to give a proof by providing a muse of human thinking about numbers...sounds more like personal incredulity.

I gave no proof; I simply characterized the problems.

 

I feel you not doing your opponents justice, Acme.

You said you didn't understand after explanations from multiple sources. You have received your due.

 

To call the primes random is a mathematical shorthand, not a rigorous description. It is this shorthand that I'm complaining about. I think it is profoundly misleading outside of mathematics. What happens inside mathematics is none of my business.

Again, mathematicians don't care what laypeople think. This is true whether they complain or laude.

Edited June 29, 2014 by Acme

[Unity+]

I am saying the random distribution of Primes is referred to as a 'random distribution' because no one in a couple thousand years has shown a way to give the n^th Prime given the n-1^th Prime.

Just because I haven't shown that the Universe came from nothingness doesn't mean it didn't. Lacking such understanding doesn't make personal suggestion any more correct that lacks proof.

 

 

 

I gave no proof; I simply characterized the problems.

If no proof is given then you are spouting nothing other than speculation(there is a section for that).

 

 

 

Again, mathematicians don't care what laypeople think. This is true whether they complain or laude.

And mathematics doesn't care what mathematicians claim. A mathematician can claim 1+1 = 3 when mathematics itself says otherwise.

 

EDIT: It becomes a useless debate because mathematical proof is the final determination of such conclusions. At this point, the debate would and should remain as 'inconclusive' until otherwise.

Edited June 29, 2014 by Unity+

[Acme]

Just because I haven't shown that the Universe came from nothingness doesn't mean it didn't.

 

If no proof is given then you are spouting nothing other than speculation.

 

And mathematics doesn't care what mathematician claim. A mathematician can claim 1+1 = 3 when mathematics itself says otherwise.

You are babbling. Maybe instead of arguing with me you can quote Bignose or imatfall in their statements that support what I have asserted and then argue with them.

[Bignose]

Our knowledge is unfinished, of course.

Sure, sure. But what best describes our current state of knowledge?

 

To call the primes random is a mathematical shorthand, not a rigorous description. It is this shorthand that I'm complaining about. I think it is profoundly misleading outside of mathematics. What happens inside mathematics is none of my business.

How do you support this? How is it profoundly misleading? If you know of a definitive pattern, then prove it and publish it! None has come out yet. If there is no pattern, then the word 'random' is applicable.

 

I'll repeat what I wrote above -- that just because we don't know a pattern today doesn't mean one doesn't exist. Nor does it mean it is useless to look for one even if one is never found. But these statements made by your guys need to accept the current state of knowledge. That there is no deterministic function or pattern that is 100% accurate. That the adjective random, in both its colloquial and mathematic sense, is appropriate. Again, if you disagree, just publish something demonstrating otherwise and again, start collecting awards as it will be a major advancement of our knowledge. Otherwise, can you please drop this objection to what is an accurate description of our knowledge today?

Edited June 29, 2014 by Bignose

[Acme]

...

I'll repeat what I wrote above -- that just because we don't know a pattern today doesn't mean one doesn't exist. Nor does it mean it is useless to look for one even if one is never found. But these statements made by your guys need to accept the current state of knowledge. That there is no deterministic function or pattern that is 100% accurate. That the adjective random, in both its colloquial and mathematic sense, is appropriate. Again, if you disagree, just publish something demonstrating otherwise and again, start collecting awards as it will be a major advancement of our knowledge. Otherwise, can you please drop this objection to what is an accurate description of our knowledge today?

+1 +50 if it were allowed.

 

Formula for Primes@ Wiki

In number theory, a formula for primes is a formula generating the prime numbers, exactly and without exception. No such formula which is efficiently computable is known. A number of constraints are known, showing what such a "formula" can and cannot be. ...

Failing to read the entire article and its supporting links, but continuing to argue against the randomness of Primes is ...erhm...uhhhh... ill advised.

[Unity+]

Sure, sure. But what best describes our current state of knowledge?

Why must I repeat myself? Our current state of knowledge has nothing to do with the question at hand, besides whether we will find a proof of the dilemma at hand or not.

 

 

How do you support this? How is it profoundly misleading? If you know of a definitive pattern, then prove it and publish it! None has come out yet. If there is no pattern, then the word 'random' is applicable.

Unless there is a proof that it is randomly distributed with no ability to produce a formula then the logic here is misleading.

 

 

I'll repeat what I wrote above -- that just because we don't know a pattern today doesn't mean one doesn't exist. Nor does it mean it is useless to look for one even if one is never found. But these statements made by your guys need to accept the current state of knowledge. That there is no deterministic function or pattern that is 100% accurate. That the adjective random, in both its colloquial and mathematic sense, is appropriate. Again, if you disagree, just publish something demonstrating otherwise and again, start collecting awards as it will be a major advancement of our knowledge. Otherwise, can you please drop this objection to what is an accurate description of our knowledge today?

This paragraph is a logical fallacy all together. As I have stated before, Mathematics does not give you the ability to state "since this has not been proven, the other conclusion is correct." It is an inconclusive result until proof is presented.

 

Our knowledge of today does not grant the ability to use the informal definition of randomness within mathematics. "Seemingly random" would be a much more accurate description rather than simply "random" until proof is presented. Until then, speculate away.

 

EDIT: One thing I think that should be made clear from here on out is the difference between science and mathematics. In science, it can be declared that current knowledge of reality is what should be accepted. However, in Mathematics what is accepted is what is proven mathematically with 100% certainty.

 

 

 

In number theory, a formula for primes is a formula generating the prime numbers, exactly and without exception. No such formula which is efficiently computable is known. A number of constraints are known, showing what such a "formula" can and cannot be.

I'm assuming this statement means that certain constraints prohibit some forms of method from being developed.

Edited June 29, 2014 by Unity+

[Acme]

...

Formula for Primes@ Wiki

 

Failing to read the entire article and its supporting links, but continuing to argue against the randomness of Primes is ...erhm...uhhhh... ill advised.

Knowing full well that my advisement will go unheeded, I will quote some from the article and add some bolding and commentary.

 

...

Prime formulas and polynomial functions

It is known that no non-constant polynomial function P(n) with integer coefficients exists that evaluates to a prime number for all integers n. The proof is as follows: Suppose such a polynomial existed. Then P(1) would evaluate to a prime p, so P(1) \equiv 0 \pmod p. But for any k, P(1+kp) \equiv 0 \pmod p also, so P(1+kp) cannot also be prime (as it would be divisible by p) unless it were p itself, but the only way P(1+kp) = P(1) for all k is if the polynomial function is constant.

 

The same reasoning shows an even stronger result: no non-constant polynomial function P(n) exists that evaluates to a prime number for almost all integers n.

 

Euler first noticed (in 1772) that the quadratic polynomial

P(n) = n2 − n + 41

is prime for all positive integers less than 41. The primes for n = 1, 2, 3... are 41, 43, 47, 53, 61, 71... The differences between the terms are 2, 4, 6, 8, 10... For n = 41, it produces a square number, 1681, which is equal to 41×41, the smallest composite number for this formula. If 41 divides n, it divides P(n) too. The phenomenon is related to the Ulam spiral, which is also implicitly quadratic, and the class number; this polynomial is related to the Heegner number 163=4\cdot 41-1, and there are analogous polynomials for p=2, 3, 5, 11, \text{ and } 17, corresponding to other Heegner numbers. ...

Note I have bolded the reference to Ulam's spiral and that I early on here gave it as an example that showed Primes are randomly distributed. See post #5

[Unity+]

Knowing full well that my advisement will go unheeded, I will quote some from the article and add some bolding and commentary.

 

 

Note I have bolded the reference to Ulam's spiral and that I early on here gave it as an example that showed Primes are randomly distributed. See post #5

I know about Ulam's spiral, and it is not proof of the random distribution, defined as "with no comprehensible sense of a pattern", of prime numbers. Therefore, there is a reason the advisement went unheeded.

 

 

 

It is known that no non-constant polynomial function P(n) with integer coefficients exists that evaluates to a prime number for all integers n. The proof is as follows: Suppose such a polynomial existed. Then P(1) would evaluate to a prime p, so P(1) \equiv 0 \pmod p. But for any k, P(1+kp) \equiv 0 \pmod p also, so P(1+kp) cannot also be prime (as it would be divisible by p) unless it were p itself, but the only way P(1+kp) = P(1) for all k is if the polynomial function is constant.

Merely a constraint.

 

EDIT: Huh, someone down voted me... Interesting.

Edited June 29, 2014 by Unity+

[Acme]

I know about Ulam's spiral, and it is not proof of the random distribution, defined as "with no comprehensible sense of a pattern", of prime numbers. Therefore, there is a reason the advisement went unheeded.

 

Merely a constraint.

 

EDIT: Huh, someone down voted me... Interesting.

Down vote? I can't imagine. Your continued arguing for argument's sake in the face of contrary facts is little more than trolling. Whether you agree about the use of terms is irrelevant to what those terms describe. I dare say nothing you have presented would be taken seriously at any gathering of mathematicians or in any mathematical journal.

[John]

Did you even read the slides you cite here?

Dial it back a bit, thank you. I have no dog in this fight, and the slides were meant simply as interesting reading, not as a support for my limited understanding (which I've been reading this thread in an effort to improve). Perhaps I should have said my understanding "has been" rather than "is."

 

 

He repeats several times that there are no deterministic methods to generate primes.

 

This is exactly what has been said the last 40 posts or so. If someone thinks there is a deterministic way of generating primes, then present it. And then go and collect your Fields Medal or other similar prizes in mathematics. Because no one has found a deterministic method to date.

 

Then, the next slides in your link there go on to talk about some of the random and pseudomethod random methods that have shown some success. But none of them are anywhere near 100% accurate. Some show some promising leads into the idea, but again, if anyone actually could demonstrate something definitive, it would be a major advancement.

 

I get the appeal of looking for a pattern in the primes. The human mind craves patterns. But we can't claim a pattern unless it is demonstrated. This doesn't mean that we shouldn't keep looking or that even if the looking is never successful, that it wouldn't be worth it.

 

And so, with no definite pattern been proven, and no deterministic method available... what other words describes our current state of knowledge about the distribution of primes other than 'random'?

 

Keep in mind that the distribution of primes does have some interesting structure. Is "random" exactly the right word in this case?

 

And as for reading the slides themselves, towards the end there is a slide which says, and I quote, "Of course, the primes are a deterministic set of integers, not a random one, so the predictions given by random models are not rigorous."

 

Perhaps Tao is using multiple meanings of the word "deterministic" here, or something.

 

In any case, given the tone of this thread, I'm out. Take care.

 

Edit: I thought I'd add, there is a cleaned up copy of Tao's presentation available as a PDF from Springer here: http://www.springer.com/cda/content/document/cda_downloaddocument/9783642195327-c1.pdf?SGWID=0-0-45-1140839-p174105361

 

The general idea does seem to be that the primes are probably pseudorandom, though no proof has been found.

Edited June 29, 2014 by John

[Unity+]

Dial it back a bit, thank you. I have no dog in this fight, and the slides were meant simply as interesting reading, not as a support for my limited understanding (which I've been reading this thread in an effort to improve). Perhaps I should have said my understanding "has been" rather than "is."

 

 

 

 

Keep in mind that the distribution of primes does have some structure. Is "random" exactly the right word in this case?

 

And as for reading the slides themselves, towards the end there is a slide which says, and I quote, "Of course, the primes are a deterministic set of integers, not a random one, so the predictions given by random models are not rigorous."

 

Perhaps Tao is using multiple meanings of the word "deterministic" here, or something.

 

In any case, given the tone of this thread, I'm out. Take care.

This is the last post I am making in this thread because it simply is getting no where.

 

Anyways, my last words are I hope there is a proof developed that ends the debate.

[Acme]

Phhhh.

 

The Beauty of Bounded Gaps: A huge discovery about prime numbers and what it means for the future of math

 

...

Building on the work of many predecessors, Zhang is able to show in a rather precise sense that the prime numbers look random in the first way we mentioned, concerning the remainders obtained after division by many different integers. From this (following a path laid out by Goldston, Pintz, and Yıldırım, the last people to make any progress on prime gaps) he can show that the prime numbers look random in a totally different sense, having to do with the sizes of the gaps between them. Random is random!

 

Zhangs success (along with the work of Green and Tao) points to a prospect even more exciting than any individual result about primes that we might, in the end, be on our way to developing a richer theory of randomness. How wonderfully paradoxical: What helps us break down the final mysteries about prime numbers may be new mathematical ideas that structure the concept of structurelessness itself.

...

[Bignose]

Unless there is a proof that it is randomly distributed with no ability to produce a formula then the logic here is misleading.

If you wish to state that, then it must be said that your claiming that there IS a pattern has got to be more misleading. I guess I stick to science's conservative nature in which I am not going to just assume that a pattern exists without a proof thereof.

 

Dial it back a bit, thank you. I have no dog in this fight, and the slides were meant simply as interesting reading, not as a support for my limited understanding (which I've been reading this thread in an effort to improve).

I didn't have a tone -- this is the danger in reading more into posts than are really there. All I was trying to do was point out that what you claimed the link was saying was in fact NOT what it was saying. I thought the slides were well measured in the word choices it used, as opposed to what has been used in this thread.

 

Keep in mind that the distribution of primes does have some interesting structure. Is "random" exactly the right word in this case?

Random variables can have plenty of structure. A normal (a.k.a. Gaussian) distributed random variable has plenty of structure. But it is still a random variable. I don't think anyone is claiming that the primes are uniformly randomly distributed. But, the fact remains, that even if there is structure, there still isn't a deterministic way to produce primes.

 

The 'structure' may increase the odds on where to bet to find prime -- just like the 'structure' in the sum of two fair 6 sides dice -- I'll bet on the sum being 7 every time -- but you can roll a lot of non-sevens in a row with two dice -- just ask any avid craps player. And that same structure in the randomness is not the same at deterministic.

 

Maybe this has all been a nomenclature issue, but I do think that it is important to get it right.

Edited June 30, 2014 by Bignose

[John]

Very well. If I read too much into what you said, then I apologize. Disregard my abandonment of the thread.

 

The original point of contention was that Acme declared the primes to be randomly distributed, and Unity+ responded by saying they could very easily appear random without actually being so, i.e. the distribution of the primes could easily be pseudorandom, and we have no proof for or against that idea.

 

What I linked above certainly doesn't assert that the primes are distributed randomly. Really, the best (and only, barring some major advances no one's yet mentioned and we don't know about) answer to this entire "debate" is simply that we don't know for sure, but it's quite possible (and useful in some ways to believe) that there is ultimately some pattern we may uncover.

 

Edit: And again, all I said about the material linked in my first post was that Terence Tao had given a presentation on the subject some of us might find interesting to read. I made no claims as to the content of the presentation, though admittedly the wording of my post may have implied otherwise.

Edited June 30, 2014 by John

[PeterJ]

We do know for sure. We know that the primes are not random. It is obvious that they are a completely determined and predictable sequence of numbers.

 

The facts are there for anyone to see and we need not argue. We're talking about the definition of 'random', that's all. That and Acme's intransigence.

 

But I'm away for a few days so I'll have to stop here.

Edited June 30, 2014 by PeterJ

[Bignose]

We do know for sure. We know that the primes are not random. It is obvious that they are a completely determined and predictable sequence of numbers.

But this ISN'T a fact, and it IS arguable. Because if this bald assertion can be proven, then let's see it. Show me 'completely determined'. If it is so determinable, show us right here what the next largest prime will be. Heck, I'd even accept a prediction of exactly how many digits it has to be at least somewhat meaningful.

 

If you are making this claim, then you need to provide definitive evidence of it. Because it sure doesn't represent what is being published as our current best knowledge today.

 

The original point of contention was that Acme declared the primes to be randomly distributed,

... and I think people took this to mean uniformly randomly distributed, which I think we all agree is not right.

Edited June 30, 2014 by Bignose

[Acme]

...

 

 

The original point of contention was that Acme declared the primes to be randomly distributed, [and Unity+ responded by saying they could very easily appear random without actually being so, i.e. the distribution of the primes could easily be pseudorandom, and we have no proof for or against that idea.

... and I think people took this to mean uniformly randomly distributed, which I think we all agree is not right.

 

I added some of what you left out of John's quote Bignose, as it is germane to some of the misunderstanding. And yes, I agree I did not suggest uniformly randomly distributed.

 

Above John says 'could easily be pseudorandom' and earlier Peter liked pseudorandom as well. Let's explore that term.

 

Pseudorandomness

A pseudorandom process is a process that appears to be random but is not. Pseudorandom sequences typically exhibit statistical randomness while being generated by an entirely deterministic causal process. ...

Note that counting is an entirely deterministic process. Let's explore 'statistical randomness'.

Statistical randomness @ Wiki

A numeric sequence is said to be statistically random when it contains no recognizable patterns or regularities; sequences such as the results of an ideal dice roll, or the digits of π exhibit statistical randomness.[1]

 

Statistical randomness does not necessarily imply "true" randomness, i.e., objective unpredictability. Pseudorandomness is sufficient for many uses, such as statistics, hence the name statistical randomness.

 

Global randomness and local randomness are different. Most philosophical conceptions of randomness are globalbecause they are based on the idea that "in the long run" a sequence looks truly random, even if certain sub-sequences would not look random. In a "truly" random sequence of numbers of sufficient length, for example, it is probable there would be long sequences of nothing but repeating numbers, though on the whole the sequence might be random. Local randomness refers to the idea that there can be minimum sequence lengths in which random distributions are approximated. Long stretches of the same numbers, even those generated by "truly" random processes, would diminish the "local randomness" of a sample (it might only be locally random for sequences of 10,000 numbers; taking sequences of less than 1,000 might not appear random at all, for example).

A sequence exhibiting a pattern is not thereby proved not statistically random. According to principles of Ramsey theory, sufficiently large objects must necessarily contain a given substructure ("complete disorder is impossible"). ...

So we can see patterns within larger otherwise patternless structures. Let's visit the Wiki on 'randomness'.

 

randomness @ Wiki

...Randomness versus unpredictability

 

Randomness, as opposed to unpredictability, is an objective property. Determinists believe it is an objective fact that randomness does not in fact exist. Also, what appears random to one observer may not appear random to another. Consider two observers of a sequence of bits, when only one of whom has the cryptographic key needed to turn the sequence of bits into a readable message. For that observer the message is not random, but it is unpredictable for the other.

 

One of the intriguing aspects of random processes is that it is hard to know whether a process is truly random. An observer may suspect that there is some "key" that unlocks the message. This is one of the foundations of superstition, but also a motivation for discovery in science and mathematics.

So if random is a philosophical judgment, then my detractors are no less intransigent then I.

 

Ah well, so it goes. I would like to sum up with a quote from Terrence Tao. It comes from some of his lecture notes from 2007 that were linked to from the last article I quoted from on the Yitan Zhang affair.

 

Simons Lecture I: Structure and randomness in Fourier analysis and number theory

The dichotomy between structure and randomness seems to apply in circumstances in which one is considering a high-dimensional class of objects (e.g. sets of integers, functions on a space, dynamical systems, graphs, solutions to PDE, etc.). For sake of concreteness, let us focus today on sets of integers (later lectures will focus on other classes of objects). There are many different types of objects in these classes, however one can broadly classify them into three categories:

 

●Structured objects objects with a high degree of predictability and algebraic structure. A typical example are the odd integers A = {,-3,-1,1,3,5,}. Note that if some large number n is known to lie in A, this reveals a lot of information about whether n+1, n+2, etc. will also lie in A. Structured objects are best studied using the tools of algebra and geometry.

 

●Pseudorandom objects the opposite of structured; these are highly unpredictable and totally lack any algebraic structure. A good example is a randomly chosen set B of integers, in which each element n lies in B with an independent probability of 1/2. (One can imagine flipping a coin for each integer n, and defining B to be the set of n for which the coin flip resulted in heads.) Note that if some integer n is known to lie in B, this conveys no information whatsoever about the relationship of n+1, n+2, etc. with respect to B. Pseudorandom objects are best studied using the tools of analysis and probability.

 

●Hybrid sets sets which exhibit some features of structure and some features of pseudorandomness. A good example is the primes P = {2,3,5,7,}. The primes have some obvious structure in them: for instance, the prime numbers are all positive, they are all odd (with one exception), they are all adjacent to a multiple of six (with two exceptions), and their last digit is always 1, 3, 7, or 9 (with two exceptions). On the other hand, there is evidence that the primes, despite being a deterministic set, behave in a very pseudorandom or uniformly distributed manner. For instance, from the Siegel-Walfisz theorem we know that the last digit of large prime numbers is uniformly distributed in the set {1,3,7,9}; thus, if N is a large integer, the number of primes less than N ending in (say) 3, divided by the total number of primes less than N, is known to converge to 1/4 in the limit as N goes to infinity. In order to study hybrid objects, one needs a large variety of tools: one needs tools such as algebra and geometry to understand the structured component, one needs tools such as analysis and probability to understand the pseudorandom component, and one needs tools such as decompositions, algorithms, and evolution equations to separate the structure from the pseudorandomness. ...

Random, pseudorandom, hybrid, unpredictable, structured, pseudo-randomly-hybrid-structured, yada, yada, yada; in any case I dare say no one can give me the 9347593714679763402706530846308460346034760^th Prime before next Thursday.

[Acme]

I finished reading the above piece and while I don't pretend to understand every argument or equation I do find that such reading over time allows a goodly amount to seep in. Following the body of the paper, Tao accepts and answers questions from readers. Anyway, this particular response from Tao is telling.

 

First, the question by a reader:

It is always interesting to consider the subject of generating random

numbers but given that EVERYTHING around us in the physical

sense is computing and given that ALL computations are at least

POTENTIALLY reversible (if you could live long enough) what exactly

is RANDOM?

Terry's response:

Dear Mario,

 

It may well be that the universe itself is completely deterministic (though this depends on what the true laws of physics are, and also to some extent on certain ontological assumptions about reality), in which case randomness is simply a mathematical concept, modeled using such abstract mathematical objects as probability spaces. Nevertheless, the concept of pseudorandomness - objects which behave randomly in various statistical senses - still makes sense in a purely deterministic setting. A typical example are the digits of pi = 3.14159...; this is a deterministic sequence of digits, but is widely believed to behave pseudorandomly in various precise senses (e.g. each digit should asymptotically appear 10% of the time). If a deterministic system exhibits a sufficient amount of pseudorandomness, then random mathematical models (e.g. statistical mechanics) can yield accurate predictions of reality, even if the underlying physics of that reality has no randomness in it.

Edits for typos and formatting.

Edited June 30, 2014 by Acme