A Thread for scratch123's Random Math Questions

For the discussion of math. Duh.

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Re: Why do e, pi, and sqrt(2) look less random when multipli

Postby eSOANEM » Mon Feb 11, 2013 12:58 am UTC

scratch123 wrote:From what I have read randomness actually doesn't have a well defined definition. In fact the first sentence here: http://en.wikipedia.org/wiki/Randomness is "Randomness means different things in various fields". Like I said before people are excellent at pattern recognition so if they see something as random chances are you can find mathematical justification for it.


Having multiple definitions in different fields is still well-defined and completely different from not having a definition. Furthermore, by introducing your own definition you're just making the problem [urlhttp://xkcd.com/927/]worse[/url].

And again, you claim that humans are good at pattern recognition but so far have only asserted this fact with no actual basis. Until you provided evidence for this, there is no reason for us to accept it.
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Re: A Thread for scratch123's Random Math Questions

Postby Carlington » Mon Feb 11, 2013 11:06 am UTC

I'm sure he's seen quite a lot of cases of people being good at pattern recognition.
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Re: A Thread for scratch123's Random Math Questions

Postby brenok » Mon Feb 11, 2013 12:55 pm UTC

I would agree that humand are good at pattern recognition. If by "good" you mean "can find them everywhere", of course.

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Re: Why do e, pi, and sqrt(2) look less random when multipli

Postby scratch123 » Tue Feb 12, 2013 4:16 pm UTC

eSOANEM wrote:
scratch123 wrote:From what I have read randomness actually doesn't have a well defined definition. In fact the first sentence here: http://en.wikipedia.org/wiki/Randomness is "Randomness means different things in various fields". Like I said before people are excellent at pattern recognition so if they see something as random chances are you can find mathematical justification for it.


Having multiple definitions in different fields is still well-defined and completely different from not having a definition. Furthermore, by introducing your own definition you're just making the problem [urlhttp://xkcd.com/927/]worse[/url].

And again, you claim that humans are good at pattern recognition but so far have only asserted this fact with no actual basis. Until you provided evidence for this, there is no reason for us to accept it.


No having different definitions in various fields means people can't decide what randomness really means. Like the wikipedia article said sometimes it means lack of pattern and sometimes it means lack of predictably. I am starting to get an idea from the people in this topic that people lean toward the "lack of predictability" definition while I prefer the "lack of pattern" definition. I am also not introducing my own definition of randomness either. It happens to be a definition that I have read about and happened to agree with. It is well known that humans are good at pattern recognition which is why they can beat computers at many tasks. It has to do with the fact that humans process stuff in parallel while computers have to do tasks one at a time.

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Re: Why do e, pi, and sqrt(2) look less random when multipli

Postby mike-l » Tue Feb 12, 2013 4:33 pm UTC

Ok, here's 2 numbers. One is from one of your 'less random' versions of pi and e, and one is a random number I just made in excel. Which is which?

7797204405
7737121582
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The map is not the territory

Postby Schrollini » Tue Feb 12, 2013 4:45 pm UTC

scratch123 wrote:No having different definitions in various fields means people can't decide what randomness really means.

Biologists think that a nucleus is an organelle that contains the cell's DNA. Physicists thing a nucleus is the part of an atom containing the protons and neutrons. Oh no! -- people in different fields can't decide what "nucleus" really means.

Or, maybe, there's not a one-to-one mapping between words and concepts.
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Re: Why do e, pi, and sqrt(2) look less random when multipli

Postby doogly » Tue Feb 12, 2013 5:21 pm UTC

mike-l wrote:Ok, here's 2 numbers. One is from one of your 'less random' versions of pi and e, and one is a random number I just made in excel. Which is which?

7797204405
7737121582

I'm confused. What point are you trying to make with this ridiculous question?
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Re: A Thread for scratch123's Random Math Questions

Postby mike-l » Tue Feb 12, 2013 5:24 pm UTC

It's a ridiculous question in response to Scratch's ridiculous claims.
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Re: A Thread for scratch123's Random Math Questions

Postby doogly » Tue Feb 12, 2013 5:25 pm UTC

Fair enough.

Also, scratch, performing an operation in parallel is something that has a technical meaning. Please do not use it the way you did there, because that is just absolute nonsense.
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Re: A Thread for scratch123's Random Math Questions

Postby Stickman » Tue Feb 12, 2013 7:13 pm UTC

doogly wrote:Fair enough.

Also, scratch, performing an operation in parallel is something that has a technical meaning. Please do not use it the way you did there, because that is just absolute nonsense.


Is that a joke? I can't quite tell!

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Re: A Thread for scratch123's Random Math Questions

Postby doogly » Tue Feb 12, 2013 7:35 pm UTC

It's completely sincere.
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Re: A Thread for scratch123's Random Math Questions

Postby Stickman » Tue Feb 12, 2013 8:19 pm UTC

I am not a neuroscientist, so feel to correct me if I'm mistaken. My impression is that using "parallel" to describe neuron activity in the brain seems to be pretty standard practice in computational neuroscience. Especially in tasks such a visual pattern recognition, there do appear to be processes that run in parallel, in the sense that various features are identified simultaneously by different neurons or regions, and then synthesized. I have found a few essays decrying the use of "massively parallel" to describe brain function, since the interconnectedness of the neural structures mean that the brain generally doesn't consist of independent processing units in the sense of a parallel computer. Even in electronic computation, parallel processing usually includes some form of synthesis (expect in the most extreme cases), so it seems like many aspects of the brain would fit the definition of parallel computing, even if the analogy is not perfect?

Maybe the definitions of "parallel" are just a bit different between neuroscience and CS? What definition of "parallel processing" are you using? That's why I thought you might be joking :)

Edit: I noticed you're posts says "performing an operation in parallel", which seems to imply some form of equivalence of the computational units. This is stronger than what is generally meant by "parallel processing".

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Re: Why do e, pi, and sqrt(2) look less random when multipli

Postby gmalivuk » Tue Feb 12, 2013 8:35 pm UTC

scratch123 wrote:sometimes it means lack of pattern and sometimes it means lack of predictably.
But lack of predictability implies lack of a pattern, because a true pattern must have some sort of predictability to it. So something that's random in the "not predictable" sense, as you say we're advocating, must *also* be random in the "not patterned" sense.

The patterns you see everywhere, if they're real, will allow you to make predictions about what to expect next.

But since you can't make any such predictions (or at least, since you have thus far failed utterly to do anything remotely of the sort, despite the multitude of patterns you claim to have found all over the place), there must not actually be such patterns.
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Re: A Thread for scratch123's Random Math Questions

Postby doogly » Tue Feb 12, 2013 8:46 pm UTC

scratch claimed that humans do parallel but computers don't
there is no sense of parallel in which humans do but a computer can't
(this could be rendered trivially true by realizing that humans are a subset of computers, but even without that little pearl it's still demonstrable)
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Re: Why do e, pi, and sqrt(2) look less random when multipli

Postby arbiteroftruth » Tue Feb 12, 2013 9:28 pm UTC

gmalivuk wrote:The patterns you see everywhere, if they're real, will allow you to make predictions about what to expect next.


Not necessarily, depending on how you define what it means to be a pattern. It seems like scratch is using something like information compressibility as a measure of what constitutes a pattern. So that the Feynman point, though no less likely than any other string of 6 digits, and though it provides no general predictive power for the digits of pi, still constitutes a local pattern.

I also suspect that scratch's notion of "true randomness" of a sequence might be something like "optimally random at any length". A sequence like 01100101 maintains the distribution of digit sequences as uniformally as possible for whatever length of the string you consider. While the example from the wikipedia page he linked to starts with 01110111, which as a substring is not uniformally distributed.

Of course any random string of sufficient length will eventually contain substrings that don't themselves match a uniform distribution, but that's where scratch's thinking seems to make use of the expected amount of time before such strings occur. If the notion of "truly random" isn't primarily concerned with arbitrary substrings, but only with truncations, then it makes a difference how long you wait before you encounter a recurring string of digits.

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Re: A Thread for scratch123's Random Math Questions

Postby gmalivuk » Wed Feb 13, 2013 1:17 am UTC

But the thing about scratch's notion of randomness is that he seems to have rendered it impossible, because every infinite sequence is going to have smaller bits that are (or at least seem) pattern-like.
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Re: Why do e, pi, and sqrt(2) look less random when multipli

Postby scratch123 » Wed Feb 13, 2013 3:23 pm UTC

mike-l wrote:Ok, here's 2 numbers. One is from one of your 'less random' versions of pi and e, and one is a random number I just made in excel. Which is which?

7797204405
7737121582


There is nothing wrong with this question and it provided an interesting puzzle. What I decided to do was take the absolute value of the difference of consecutive integers in each sequence and then doing the same for the new sequence. This technique is actually used in some compression algorithms except the numbers are in binary instead. Here are the results:

1. 7797204405
022524045
20332441
2301203

2. 7737121582
044611436
40250313
4235322

I could have done this until there was only one number left but the pattern was becoming too obvious. The first sequence only uses the numbers 0-3 when compressed 3 times which is simpler than the 2-5 the second sequence uses. In addition to this the last 3 numbers in sequence 1 are just a slight permutation of the first 3. Sequence 1 is clearly less random. This also matched my intuitive judgement of which sequence was less random before I started analyzing them.

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Re: Why do e, pi, and sqrt(2) look less random when multipli

Postby eSOANEM » Wed Feb 13, 2013 4:19 pm UTC

scratch123 wrote:No having different definitions in various fields means people can't decide what randomness really means. Like the wikipedia article said sometimes it means lack of pattern and sometimes it means lack of predictably. I am starting to get an idea from the people in this topic that people lean toward the "lack of predictability" definition while I prefer the "lack of pattern" definition. I am also not introducing my own definition of randomness either. It happens to be a definition that I have read about and happened to agree with. It is well known that humans are good at pattern recognition which is why they can beat computers at many tasks. It has to do with the fact that humans process stuff in parallel while computers have to do tasks one at a time.


No.

It simply means that different definitions are appropriate in different fields. The reason for this is that if I start a new field and notice a property in it which is similar but not quite the same as one in a different field, I can either come up with a new name or name it by analogy. Usually people prefer the latter because it provides some intuition of what the property means and because it doesn't lead to an enormous explosion of jargon. Unfortunately this does mean you get multiple definitions. Usually though, only one applies at a time (and where more than one applies, usually context makes it clear or else people add adjectives to clarify which definition they're using:

e.g.

a homogeneous differential equation could be one of the form dy/dx=f(y/x) or it could be of the D(y)=0 where D is some differential operator but it's pretty clear which one I mean from the equation I'm describing

sum can mean a binary operation on scalars, vectors, matrices, tensors etc. or can be an operation on a whole sequence. Combine this with the fact that, in physics, magnitudes of vector quantities are often referred to by the same name as the vector and you could have confusion: by the sum of two forces, do I mean the vector sum or their scalar sum? Well, I just solved the ambiguity right there for you.
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Re: Why do e, pi, and sqrt(2) look less random when multipli

Postby jestingrabbit » Wed Feb 13, 2013 9:43 pm UTC

scratch123 wrote:
mike-l wrote:Ok, here's 2 numbers. One is from one of your 'less random' versions of pi and e, and one is a random number I just made in excel. Which is which?

7797204405
7737121582


There is nothing wrong with this question and it provided an interesting puzzle. What I decided to do was take the absolute value of the difference of consecutive integers in each sequence and then doing the same for the new sequence. This technique is actually used in some compression algorithms except the numbers are in binary instead. Here are the results:

1. 7797204405
022524045
20332441
2301203

2. 7737121582
044611436
40250313
4235322

I could have done this until there was only one number left but the pattern was becoming too obvious. The first sequence only uses the numbers 0-3 when compressed 3 times which is simpler than the 2-5 the second sequence uses. In addition to this the last 3 numbers in sequence 1 are just a slight permutation of the first 3. Sequence 1 is clearly less random. This also matched my intuitive judgement of which sequence was less random before I started analyzing them.


So, no matter what the numbers, this is how we work out which is more random? Why? Because there's a compression algorithm that uses it? What about all the other compression algorithms that do something completely different. Why are compression algorithms involved in this at all?

And furthermore, if you follow through a little bit on your calculations, ie do one more line, you get

1. 2301203 -> 132223 -> 21001 -> 1101 -> 011 -> 10 -> 1
2. 4235322 -> 212210 -> 11011 -> 0110 -> 101 -> 11 -> 0

which seems to reverse what you were claiming would happen. On the next iteration, seqence 1 has 3s, and sequence 2 doesn't. But frankly, I think you're completely ad hoc and without clear or good reasons for doing this. "An algorithm does something like this" isn't a reason.
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Re: A Thread for scratch123's Random Math Questions

Postby mike-l » Wed Feb 13, 2013 11:48 pm UTC

To be fair, he did pick the one from e correctly, though of course he had a 50% chance of doing so. Also those numbers weren't picked with any bias, I just said I'm going to press more digits n times and scroll to the bottom, and the random number was the first try, pure coincidence that they both started with 77.
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Re: Why do e, pi, and sqrt(2) look less random when multipli

Postby scratch123 » Thu Feb 14, 2013 5:13 pm UTC

jestingrabbit wrote:
scratch123 wrote:
mike-l wrote:Ok, here's 2 numbers. One is from one of your 'less random' versions of pi and e, and one is a random number I just made in excel. Which is which?

7797204405
7737121582


There is nothing wrong with this question and it provided an interesting puzzle. What I decided to do was take the absolute value of the difference of consecutive integers in each sequence and then doing the same for the new sequence. This technique is actually used in some compression algorithms except the numbers are in binary instead. Here are the results:

1. 7797204405
022524045
20332441
2301203

2. 7737121582
044611436
40250313
4235322

I could have done this until there was only one number left but the pattern was becoming too obvious. The first sequence only uses the numbers 0-3 when compressed 3 times which is simpler than the 2-5 the second sequence uses. In addition to this the last 3 numbers in sequence 1 are just a slight permutation of the first 3. Sequence 1 is clearly less random. This also matched my intuitive judgement of which sequence was less random before I started analyzing them.


So, no matter what the numbers, this is how we work out which is more random? Why? Because there's a compression algorithm that uses it? What about all the other compression algorithms that do something completely different. Why are compression algorithms involved in this at all?

And furthermore, if you follow through a little bit on your calculations, ie do one more line, you get

1. 2301203 -> 132223 -> 21001 -> 1101 -> 011 -> 10 -> 1
2. 4235322 -> 212210 -> 11011 -> 0110 -> 101 -> 11 -> 0

which seems to reverse what you were claiming would happen. On the next iteration, seqence 1 has 3s, and sequence 2 doesn't. But frankly, I think you're completely ad hoc and without clear or good reasons for doing this. "An algorithm does something like this" isn't a reason.


But you can compare compression algorithms based on how simple they are. Iterating all the way to one number makes things more complicated. Also you made a mistake on #1. The next number should be 131123.

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Re: A Thread for scratch123's Random Math Questions

Postby Talith » Thu Feb 14, 2013 5:41 pm UTC

You can't just start throwing words around like 'simple' without defining them. If you're trying to have a serious mathematical discussion, don't use terms with ambiguous meaning unless you're going to clarrify the specific definition you're using. It does no good to confuse someone because you're using the definition that suits you and it does even less good if you're not consistant in the deifnition you use.

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Re: A Thread for scratch123's Random Math Questions

Postby mike-l » Thu Feb 14, 2013 8:16 pm UTC

This also isn't even remotely a compression, as it's not reversible. It's a hash function.

2301203 could also have been

1388663561
250203215
35223114
2301203

or many many many others.
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Re: A Thread for scratch123's Random Math Questions

Postby gmalivuk » Thu Feb 14, 2013 8:55 pm UTC

mike-l wrote:This also isn't even remotely a compression, as it's not reversible.
Which means that, even if scratch is defining "pattern" (to the extent that he's defining anything at all in this discussion) in terms of compressibility rather than predictibility, it still fails in this case.

And of course even when you can compress some small bit of a longer sequence in a proper (i.e. reversible) way, if you can' do so the same way with longer bits of the sequence, then whatever "compression" you're doing doesn't actually demonstrate that there's a pattern to the whole thing. Just that there are small, pattern-like substrings in *any* sufficiently long sequence, whether it's random or not.
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Re: A Thread for scratch123's Random Math Questions

Postby Stickman » Thu Feb 14, 2013 9:47 pm UTC

In particular, incompressible strings of length n can contain substrings compressible by an O(log n) amount. This is especially true of infinite strings. Randomness of an infinite string depends on the near incompressibility (up to a constant factor) of all but a finite number of prefixes - it's certainly possible for a Kolmogorov random infinite string to start with 10,000 '9's, but even a randomly selected Kolmogorov random string will have some prefixes compressible by some amount, and will surely contain such substrings. In fact, a string randomly generated by selecting consecutive digits from some multinomial distribution on {0,..,9} (positive probability for each digit) will almost surely contain any given finite subsequence!

That said, Kolmogorov complexity / incompressibility are not a good way of approaching the "randomness" of pi, e, and sqrt(2), since it seems like all of those infinite strings have very low Komogorov complexity. Each can be represented by simple finite algorithms (i.e. series representations).

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Re: A Thread for scratch123's Random Math Questions

Postby 4=5 » Sun Feb 17, 2013 5:05 am UTC

Actually could I borrow scratch123 for a moment? I'd like to automate his pattern recognition abilities and limitations for my own purposes.

scratch123: could you let me know of all the non random patterns in each of these five sequences? The things that catch your attention and seem unusual.

Code: Select all

111100001010000001000001011010


101110011001100111100011110110


000010011101110100011001101001


111110110100111110111011011101


000011110110111101001100001100

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Re: A Thread for scratch123's Random Math Questions

Postby scratch123 » Wed Feb 27, 2013 12:25 am UTC

4=5 wrote:Actually could I borrow scratch123 for a moment? I'd like to automate his pattern recognition abilities and limitations for my own purposes.

scratch123: could you let me know of all the non random patterns in each of these five sequences? The things that catch your attention and seem unusual.

Code: Select all

111100001010000001000001011010


101110011001100111100011110110


000010011101110100011001101001


111110110100111110111011011101


000011110110111101001100001100


I think the 4th one is the least random since it has 2 sequences of 5 straight 1's and it has many more 1's than 0's. No other sequence has a 1 and 0 ratio as great as that one.

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Re: A Thread for scratch123's Random Math Questions

Postby flownt » Wed Feb 27, 2013 4:45 am UTC

Hooray for using a consistent test statistic!!!

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Re: A Thread for scratch123's Random Math Questions

Postby gmalivuk » Wed Feb 27, 2013 5:10 am UTC

scratch123 wrote:I think the 4th one is the least random since it has 2 sequences of 5 straight 1's and it has many more 1's than 0's. No other sequence has a 1 and 0 ratio as great as that one.
Why are sequences of 1's important, but sequences of 0's aren't? Why is a high 1 to 0 ratio important, but a high 0 to 1 ratio isn't?
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Re: A Thread for scratch123's Random Math Questions

Postby scratch123 » Thu Feb 28, 2013 2:01 am UTC

gmalivuk wrote:
scratch123 wrote:I think the 4th one is the least random since it has 2 sequences of 5 straight 1's and it has many more 1's than 0's. No other sequence has a 1 and 0 ratio as great as that one.
Why are sequences of 1's important, but sequences of 0's aren't? Why is a high 1 to 0 ratio important, but a high 0 to 1 ratio isn't?


Well there is nothing like a sequence of 5 straight 0's 2 times in any of the other numbers either. Even if you look at the 0 to 1 ratio in the other numbers it isn't very high.

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Re: A Thread for scratch123's Random Math Questions

Postby Dason » Thu Feb 28, 2013 3:09 am UTC

scratch123 wrote:
gmalivuk wrote:
scratch123 wrote:I think the 4th one is the least random since it has 2 sequences of 5 straight 1's and it has many more 1's than 0's. No other sequence has a 1 and 0 ratio as great as that one.
Why are sequences of 1's important, but sequences of 0's aren't? Why is a high 1 to 0 ratio important, but a high 0 to 1 ratio isn't?


Well there is nothing like a sequence of 5 straight 0's 2 times in any of the other numbers either. Even if you look at the 0 to 1 ratio in the other numbers it isn't very high.


Can you explain why you're using that stuff as part of your criteria though? Can you tell me what you would expect the longest string of either 0s or 1s to be if the sequence was completely random?
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Re: A Thread for scratch123's Random Math Questions

Postby gmalivuk » Thu Feb 28, 2013 3:24 am UTC

scratch123 wrote:
gmalivuk wrote:
scratch123 wrote:I think the 4th one is the least random since it has 2 sequences of 5 straight 1's and it has many more 1's than 0's. No other sequence has a 1 and 0 ratio as great as that one.
Why are sequences of 1's important, but sequences of 0's aren't? Why is a high 1 to 0 ratio important, but a high 0 to 1 ratio isn't?


Well there is nothing like a sequence of 5 straight 0's 2 times in any of the other numbers either.
The first number has 6 straight zeros followed by 5 straight zeros.
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