How many kinds of infinity are there ?

mart

selfobsessednewguy said:
...

https://en.wikipedia.org/wiki/Euler's_identity

(euler's identity) here we have an irrational number e which related to exponential growth and another irrational number pi, and also we have a complex identity called i which is defined as the 'imaginary number' used in complex analysis, defined mathematically as sq(-1). if we raised e to the power i*pi, we get fucking -1, an integer. AN INTEGER.
...

If you like this you'll probably like the result of raising e to the power of pi times the square root of 163. I've still not managed to really get my head round why that is so nearly an integer.

Whats more fun with this is to then ask what people think "raising e to the power of ..." actually means. Everyone thinks they know what powers mean, until you ask what it means to take something to an irrational power. What does 2 to the root 2 mean for example? Your calculator will happily give a numerical answer, but what does it mean? It's a fun story to tell.

There's a whole beautiful set of ideas linked together - powers, integration, logs, and you need them all to make sense of what's going on.

A5D5E5

ICBM said:

57Deluxe said:

isn't thinking that the expansion of the universe is slowing and at some point will contract back. In which case infinity does not exist at all.

No, current thinking is that the mass of the universe is far too small - even including the dark matter - to do that and the universe will continue to expand forever into emptiness and darkness. Which is a cheery thought.

More than that, the rate of expansion is increasing!

A5D5E5

selfobsessednewguy said:

ChrisMusic said:

selfobsessednewguy said:

so how fascinating is it that the series

1 + 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 + ... = PI SQUARED OVER 6 ?????????

this is the inverse squared series. the Basel problem back in the day was 'what is the limit of the infinite inverse squares series?' because it was used in physics but german physicists got tired of approximating it using that sum and they wanted to know what the limit is in some kind of closed form. and Euler came along with a beautiful and very unconventional but relatively simple proof that the limit is exactly pi ^ 2 over 6.... so this series can not exceed that precise value. what the fuck is pi doing here? well i dont know. maths.

more to the point defining one unresolvable number in terms of another unresolvable number !

I think all this points a finger in the direction of a fundamental misconception at the heart of our understanding of maths.

Crack the intrigue and paradoxes and maybe you have the next layer of understanding of the reality which the language maths describes ?
(maybe we don't have the full alphabet / lexicon yet?)

It would be interesting to come back and view the progress in this, in say 500 years time, once computing and algorithmic power has come of age and is roped in to enhance our conceptual understanding, rather than build reality based on weird and almost prehistoric approximations as we seem want to do now.  Really, we are better than that...

well actually, we have been working with pi for much longer than 500 years. approximations of the ratio of a perfect circle to its diameter have roots going back more than 1000 years. we are very advanced in our development of mathematics, and in fact we know that pi is not an 'unresolvable' or somewhat mysterious number as many view it, it is what we call an irrational number, which is a number which cannot be expressed as a fraction because it is non-repeating. 

many numbers are irrational. that doesn't make them unresolvable or mysterious. square root of 2 is irrational, which means if we have a right angle triangle with sides of length 1, then the longest side has length sq(2), which is an irrational number. in fact, if we measure anything in the real world to a very high accuracy, you will find that any measurement will be non-repeating as the real world is not as perfect as mathematicians describe... so irrational numbers are everywhere.

the mystery here with the infinite sum is how the inverse squared series, if we take finitely many terms of ths series (making it resolvable) then it will be always under pi^2/6. however, i am saying mystery but it's not really a mystery to any mathematician who has read and understood the proof... but it still is.

however this is a very small thing when we compare it to some of the most beautiful results in mathematics. another one of my favourites is the fact that

https://en.wikipedia.org/wiki/Euler's_identity

(euler's identity) here we have an irrational number e which related to exponential growth and another irrational number pi, and also we have a complex identity called i which is defined as the 'imaginary number' used in complex analysis, defined mathematically as sq(-1). if we raised e to the power i*pi, we get fucking -1, an integer. AN INTEGER.





A FUCKIN INTEGER

I learned this as:

Little Jack Horner sat in the corner trying to work out pi.  He said, "It's minus the log of minus one all raised to the power of i".

ChrisMusic

Speaking of algorithms, I just watched this.  Interesting, non ?

The Secret Rules of Modern Living: Algorithms

http://www.bbc.co.uk/iplayer/episode/p030s6b3/the-secret-rules-of-modern-living-algorithms

on iPlayer for 19 days, until the inevitable repeats bring it back.
(I wonder what algorithm controls Auntie Beebs decisions?)  

ChrisMusic

Thanks for posting Euler's product formula @selfobsessednewguy , an elegant conundrum indeed.  I'm not sure whether I envy you your daily mindfuck ration or not.  Actually maybe I do just a bit, if life had followed a different path.  "What just happened?" sounds like a magnificent challenge.  Good luck with your master's  

57Deluxe

but - can you make your delay auto-oscillate into infinity?

selfobsessednewguy

ChrisMusic said:

The "say 500 years" had no significance other than to moot the potential exponential growth in the trilogy of the power of computing, the increasing sophistication of algorithms, and the advent of 'proper' artificial intelligence.  Once those systems are left to develop themselves, then I think there is the potential for a quantum shift in our understanding of the inner workings of the universe.  IMHO, we have nowhere near the sophisticated understandings we like to credit ourselves with as a species, but we are developing the tools to furnish the never-ending curiosities the human race is blessed with.  

Oh, and thanks for your posts @selfobsessednewguy , that's the most stimulation I've enjoyed with my clothes on for a very long time  

No doubt, I agree that we have a ways to go. We can't even figure out how to factorise large numbers efficiently, which is the method by which our bank accounts stay secure

also you're welcome, i love maths (actual maths, what we have been talking about, not pre-A level rubbish) and there are topics which i could talk for a very long time about.

one of them is the most beautiful formula in existence in my opinion, the Euler product formula

selfobsessednewguy

ChrisMusic said:

The "say 500 years" had no significance other than to moot the potential exponential growth in the trilogy of the power of computing, the increasing sophistication of algorithms, and the advent of 'proper' artificial intelligence.  Once those systems are left to develop themselves, then I think there is the potential for a quantum shift in our understanding of the inner workings of the universe.  IMHO, we have nowhere near the sophisticated understandings we like to credit ourselves with as a species, but we are developing the tools to furnish the never-ending curiosities the human race is blessed with.  

Oh, and thanks for your posts @selfobsessednewguy , that's the most stimulation I've enjoyed with my clothes on for a very long time  

yeah we still have a long ways to go in the world of maths. we can't factorise large numbers in an efficient way yet which is why we use it for encryption when we deal with money online, and the clay mathematics institute have 7 problems which are unsolved which, if you provide a proof, gain you $1 million. i'm not kidding

also you're welcome, i like maths (actual mathematics, not the rubbish you're taught in high school) a lot so it's something i can talk a lot about. 

the most beautiful formula in existence in my opinion is 



which looks confusing right? well there are two huge symbols, one on each side, which mean two different things. the one on the left means 'SUM from 1 to infinity with n' and the one on the right means 'take a PRODUCT over all prime numbers p' of the things next to the symbols. so what this formula looks like without the notation is

1/(1^s) + 1/(2^s) + 1/(3^s) + 1/(4^s) + ... = 1/(1-2^(-s)) * 1/(1-3^(-s)) * 1/(1-5^(-s)) * ...

where * is multiply, and ^ means 'to the power of'. 

why is the euler product formula the most beautiful formula in existence? it takes a sum over all numbers on the left hand side, (as in, n runs over 1, 2, 3, 4, 5, ...) and a product over all PRIME NUMBERS on the right hand side (p runs over 2, 3, 5, 7, 11, 13, ...)... how incredible is that? prime numbers are seen as being mysterious most of the time and we don't fully understand why they are distributed the way that they are (the Riemann hypothesis is a problem relating to these things, and you get $1 million if you solve it) but here we have a formula which EQUATES some kind of relationship between a sum over all natural numbers and a product over all prime numbers. how cool is that?



awesome. maths yeeeee

selfobsessednewguy

ChrisMusic said:

The "say 500 years" had no significance other than to moot the potential exponential growth in the trilogy of the power of computing, the increasing sophistication of algorithms, and the advent of 'proper' artificial intelligence.  Once those systems are left to develop themselves, then I think there is the potential for a quantum shift in our understanding of the inner workings of the universe.  IMHO, we have nowhere near the sophisticated understandings we like to credit ourselves with as a species, but we are developing the tools to furnish the never-ending curiosities the human race is blessed with.  

Oh, and thanks for your posts @selfobsessednewguy , that's the most stimulation I've enjoyed with my clothes on for a very long time  

yeah we still have a long ways to go in the world of maths. we can't factorise large numbers in an efficient way yet which is why we use it for encryption when we deal with money online, and the clay mathematics institute have 7 problems which are unsolved which, if you provide a proof, gain you $1 million. i'm not kidding

also you're welcome, i like maths (actual mathematics, not the rubbish you're taught in high school) a lot so it's something i can talk a lot about. 

the most beautiful formula in existence in my opinion is 



which looks confusing right? well there are two huge symbols, one on each side, which mean two different things. the one on the left means 'SUM from 1 to infinity with n' and the one on the right means 'take a PRODUCT over all prime numbers p' of the things next to the symbols. so what this formula looks like without the notation is

1/(1^s) + 1/(2^s) + 1/(3^s) + 1/(4^s) + ... = 1/(1-2^(-s)) * 1/(1-3^(-s)) * 1/(1-5^(-s)) * ...

where * is multiply, and ^ means 'to the power of'. 

why is the euler product formula the most beautiful formula in existence? it takes a sum over all numbers on the left hand side, (as in, n runs over 1, 2, 3, 4, 5, ...) and a product over all PRIME NUMBERS on the right hand side (p runs over 2, 3, 5, 7, 11, 13, ...)... how incredible is that? prime numbers are seen as being mysterious most of the time and we don't fully understand why they are distributed the way that they are (the Riemann hypothesis is a problem relating to these things, and you get $1 million if you solve it) but here we have a formula which EQUATES some kind of relationship between a sum over all natural numbers and a product over all prime numbers. how cool is that?



awesome. maths yeeeee