Philosophy Now Forum

One of the problems with number patterns is that the short ones can have more than one possible answer when it comes to determining the next number in sequence (for me there must be at least six listed numbers in sequence before I proceed forward with a candidate sequence).

I can't think of any sequences off the top of my head for this thread, but I will do an internet search for an example. In the meantime, do you have anything you have to say that's relevant?

PhilX

Yes, why do you think this philosophy of mathematics rather than just mathematics?

Arising_uk wrote:Yes, why do you think this philosophy of mathematics rather than just mathematics?

Basically I'm asking at what point does a collection of numbers cease being a mere collection of numbers and turn into a definite sequence whereby you can make a prediction of what the next number will be at either end of the proposed sequence (e.g. the numbers 1, 2 is too short to say that the next number is three because one can argue that the next number can be 4 because double 1 is 2 and double 2 is 4 unless you further specify what the rule is regarding which numbers you can attach to either end of the string of numbers so how many numbers do you need to have a definite rule that will tell you what the next number(s) will be?

That's my question in a nutshell.

PhilX

I would think that would depend upon the persons pattern recognition with respect to numbers and whether the sequence has a rule.

Seems to depend. E.g. I tend to recall four numbers which I think are 1, 2, 4 and 8. Now one would normally expect the next number up to be 16. Yet a valid argument said the next number can be 15. A true sequence shouldn't have more than one valid rule. So I think my question becomes how can one tell that you have a true sequence of numbers with only one rule attached? (I've just put up my serrated triangle which I think helps one to find series of prime numbers, three [true?] sequences of which I listed composed of prime numbers, those series I've extended past the borders of the triangle by recognizing a pattern common to all three series)

The question I'm posing in this thread is of great importance to the philosophy of math since sequences are commonly taught (e.g. calculus - btw the numbers in the last [completed] series I think are identical with the results of Euler's equation for 40 prime numbers).

PhilX

Any sequence of digits follows this rule:-

'The next digit is entirely random'.

So you can never know for sure.




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Any arbitrarily long sequence of digits will eventually contain the sequence 'Hitler'.

Metazoan wrote:Any sequence of digits follows this rule:-

'The next digit is entirely random'.

So you can never know for sure.




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Any arbitrarily long sequence of digits will eventually contain the sequence 'Hitler'.

Only one question. How do you know the next digit is "entirely" random? (e.g. there's the transcendental number .101001000100001...)

PhilX

Hi Phil,

Philosophy Explorer wrote:Only one question.

Philosophy Explorer wrote:How do you know the next digit is "entirely" random?

I don't, that was not my point. My point is that you can't know that it isn't.

I took your question to be: Can I deduce the rule behind a partial sequence to be certain of the next digit in that sequence?

You can deduce a partial sequence from a rule, but you cannot deduce a rule from a partial sequence.

There will always be the possible rule: 'The next digit is entirely random' and that, by definition doesn't define the next digit and also includes all possible digits.

You can define a sequence as a representation of pi and you can always deduce the next digit.

Given a sequence that looks like pi but you have no rule defining it as such, then unless you check every digit (which you can't do) then you cannot know it is pi.

Correlation does not prove causation.


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I seldom accept kisses from people named Phil.

Hi Metazoan,

The example I gave is defined so if, e.g., you count up to the third one, then you know by definition that it has three zeros immediately following that one, etc. A counterexample to what you were talking about before.

PhilX

Hi Phil,

Philosophy Explorer wrote:The example I gave is defined so if, e.g., you count up to the third one, then you know by definition that it has three zeros immediately following that one, etc. A counterexample to what you were talking about before.PhilX

I see that as an example of what I said, not a counterexample.

That makes me suspect that what you think I meant by what I said is not what I think I meant by what I said.

If you wish to persue this, can you write your interpretation of what you think I meant so that we can clear it up.

Just to be mischievous:-

Your sequence:the transcendental number .101001000100001...
You said:The example I gave is defined so if, e.g., you count up to the third one, then you know by definition that it has three zeros immediately following that one, etc.

I fully agree that if I count up to the third one then I know by definition that it has three zeros immediately following it. However that is simply because they are already revealed. (See above.)

If I counted up to the fifth (last revealed) one then, I suggest, things get a little tricker.

We know this is a number, and we know it is transcendental.

(I know the next digit is not random, but I never said it was. But you can only know that from the definition, not the number. There are an infinite number of random numbers that start with those digits.)

To get the next unknown digit there are two things I appear not to know, the number base which the string is using, (is it binary? Decimal? Sexydecimal?) and that there is exactly one transcendental number that starts with those digits.

Your definition must define all the digits in the sequence unambiguiously and doing that is trickier than you might think.

Just try arguing that 1+1=2 and see how far you get.


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This is a string of entirely random characters and has no meaning.

Metazoan asked:

"To get the next unknown digit there are two things I appear not to know, the number base which the string is using, (is it binary? Decimal? Sexydecimal?) and that there is exactly one transcendental number that starts with those digits."

The number base is decimal and there is exactly only one transcendental number that starts with those digits (and implicitly has an infinite number of 0's and 1's, the number of 0's depending on how many 1's are ahead of the 1 under consideration which then determines the number of 0's).

PhilX

Hi Phil,

A clear and direct answer. That took me by suprise, thanks. I have been lurking here too long.

Out in interest and ignorance, being a mathematical numpty, how would I prove that there were no transcendental numbers between 0.1010010001000010' and 0.1010010001000010000010'? (Which is a sizeable chunk of numbers just below your putative sequence but still in capture range of it. Or that is what I intended it to mean.)

ETA: Or 0.101001000100001((1...9)...) is not transcendental?

There are an infinite number of them and none of them is transcendental?

My understanding of transcendental roughly means non-algebraic.

What you are saying then is that all numbers between the two above are algebraic.

To quote wikipedia:-

"In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a non-zero polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e. Though only a few classes of transcendental numbers are known (in part because it can be extremely difficult to show that a given number is transcendental), transcendental numbers are not rare. Indeed, almost all real and complex numbers are transcendental, since the algebraic numbers are countable while the sets of real and complex numbers are both uncountable. All real transcendental numbers are irrational, since all rational numbers are algebraic. The converse is not true: not all irrational numbers are transcendental; e.g., the square root of 2 is irrational but not a transcendental number, since it is a solution of the polynomial equation x^2 − 2 = 0."

I read that as saying given any two, non identical, numbers there are always an infinite number of transcendental numbers between them.

Phew.

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After all that you want a trite remark? Gimme a break.

Hi Metazoan,

I don't know how one would go about proving a number transcendental and I suspect such proofs to be technical, i.e. long and filled with terminology.

Just wanted to add that

"That makes me suspect that what you think I meant by what I said is not what I think I meant by what I said."

is quite a mouthful.

PhilX

I wanted to add that the numbers from my third string (the one starting with 197) I've confirmed coincides with Euler's quadratic equation.

Here's an example of something (concerning factorials) that looks like a pattern, but isn't:

1! = 1
3! = 2•3
5! = 4•5•6
7! = 7•8•9•10

But 9! doesn't equal 11•12•13•14•15.

PhilX

Question: how do I express a power? E.g. x[sup]2[/sup] doesn't work.

PhilX

Edit: Just learned that the above can be expressed as X^2. Can we do better at this forum, like superscripts? (if this particular forum were to be used for math, then I'd like my readers not to stress their eyes to understand equations).