Truth, Reason and Love

Post by **Griffon64** » Fri Mar 16, 2007 5:56 am

I really enjoy it when that guy starts talking about numbers

Post by **Primula Baggins** » Fri Mar 16, 2007 6:55 am

Are pi and e examples of such numbers, Faramond?

I've read the suggestion that in other universes those constants may have different values. But it is so interesting to me that they are both (here, anyway) irrational numbers. And yet they describe the real and the measurable.

Really interesting post. I must come back and read it again when I'm not on Nyquil.

(I have always loved the idea of the different sizes of infinities—the set of all whole numbers, which is ten times the size of the set of all whole numbers divisible by 10, though both are infinite; and both infinities are much smaller than the infinite set of all real numbers . . . I wish I'd had the gumption to study these things seriously.)

Post by **Faramond** » Sat Mar 17, 2007 12:02 am

Primula Baggins wrote:Are pi and e examples of such numbers, Faramond?

Oh no! These numbers are perfectly describable. For example:

e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + ...
( n! = n*(n-1)* ... *3*2*1, for example, 5! = 5*4*3*2*1, and 0! is defined as equal to 1 )

pi/6 = 1 + 1/(2^2) + 1/(3^2) + 1/(4^2) + 1/(5^2) + ...
(Lidless talked about this series a bit over on B77)

I alluded to a "pattern" in the decimal representation last post, but this was a bit misleading. The decimal representations of e and pi do appear random, but I can finitely describe what each number is, even if I'm talking about infinite sums, as here. The infinite sums have patterns!

I can't give you an example of a number that can't be pointed to, because if I could, then that number could be communicated and described finitely. But we can prove there are numbers that can never be described finitely, that can't ever be pointed to in any way, and yet they exist under the commonly held axioms of mathematics.

It would take forever to write out all of pi or e, of course. But it only takes a finite amount of time for me to tell someone these numbers in such a way that they could turn around and describe these numbers to any precision of decimal points in a finite amount of time.

I've read the suggestion that in other universes those constants may have different values. But it is so interesting to me that they are both (here, anyway) irrational numbers. And yet they describe the real and the measurable.

Well, I don't know if the constants would have different values in other universes. Now, if you define pi as the ratio of a circle's circumference to diameter, then you will get different values of pi depending on the curvature of space where you make the measurement. The value of pi we use is the one you get from circles in Euclidean space, with curvature of zero, where parallel lines exist and never get any further from each other.

I believe that space in the solar system has a very small curvature, though for most practical purposes it's Euclidean space. I think pi has to be defined as the ratio of circumference to diameter in zero curvature space, though. The value of pi turns up in all kinds of physics equations that don't seem to have anything to do with circles!

I can't imagine how e would be different in another universe, but ... I guess anything's possible in another universe.

(I have always loved the idea of the different sizes of infinities—the set of all whole numbers, which is ten times the size of the set of all whole numbers divisible by 10, though both are infinite;

These are the same size infinity, though.

Mathematicians count two sets to be the same size if they can be put into one-to-one correspondence with each other.

1 --- 10
2 --- 20
3 --- 30
.
.
.
10 --- 100
11 --- 110
.
.
.

This is a one-to-one correspondence between the sets, so these infinities are the same.

and both infinities are much smaller than the infinite set of all real numbers . . . I wish I'd had the gumption to study these things seriously.)

Just google "Cantor diagonalization" if you want to know why the real numbers make a larger infinity!

Post by **vison** » Sat Mar 17, 2007 4:51 am

Post by **Faramond** » Sat Mar 17, 2007 5:18 am

Maybe I shouldn't go off on tangents!

Split it off if it needs to be, of course.

Post by **vison** » Sat Mar 17, 2007 5:20 am

Faramond wrote:Maybe I shouldn't go off on tangents!

Split it off if it needs to be, of course.

Oh. It's a tangent.

Hm. Not a cosine?

*has betrayed ancient schooling in very, very, extremely very elementary stuff*

*wishes was math genius*

*gazes fondly at old logarithim tables on bookshelf*

Sat Mar 17, 2007 5:22 am

Faramond wrote:Maybe I shouldn't go off on tangents!

Split it off if it needs to be, of course.

My inclination is to not split it off, unless enough people feel strongly that it should be. Yes its a tangent in a way, but I think it does relate to the topic of the thread, such as it is.

Post by **Faramond** » Sat Mar 17, 2007 6:37 am

I like the pun, vison.

V, did you know that pi and e are transcendental numbers, at least in the mathematical sense?

Sat Mar 17, 2007 6:54 am

Vaguely, in the deep recesses of my mind (there was a time while I was an undergraduate when I was briefly a double major in Math and English. Then I dropped both of them and became a Philosophy major.)

Post by **Primula Baggins** » Sat Mar 17, 2007 7:13 am

Thanks for the explanations, Faramond! That's interesting. Not sure I'll be Googling that for a few months, though I will try to do it eventually.

The one-to-one correspondence thing makes sense, I guess, but seems unfair somehow. Why doesn't the same thing go for the set of all real . . . oh, wait. Is it because there are an infinite number of real numbers between any two integers? So there can't be a one-to-one correspondence?

Probably not.

But I have fun thinking about it.

I do remember when I was 12 or 13 making a chart where I tried to work out the logical progression for how many vertices and faces and right angles a square, cube, hypercube, etc., had. I think I got to six dimensions before I short-circuited, but it was fun (and I learned later that my answers were right; there is a logic to the progression).

My problem is loving math but having a qualitative mind.

Rather like loving Impressionist painting while totally colorblind. Or actually probably more like loving the idea of dance, while having two left feet with weak ankles in lead-lined boots.

Or perhaps it's just an absurd fondness for analogies.

Yes, Officer, I'll go quietly.

Post by **Jnyusa** » Sat Mar 17, 2007 8:10 am

Don't split the thread. What could be closer to God than math?

Post by **anthriel** » Sat Mar 17, 2007 4:10 pm

I just have to share that I believe "Cantor Diagonals" would be a great name for a rock band.

And the only thing closer to God than math would HAVE to be music.

Post by **Griffon64** » Sat Mar 17, 2007 5:03 pm

Music and math is weaved together anyway, aren't they?

Sat Mar 17, 2007 5:46 pm

Post by **Misha** » Sat Mar 17, 2007 6:39 pm

Aren't all math teachers God incarnate? I certainly thought so in middle school. They seemed all knowing and all powerful to me. And then there was that red pen.....

Gosh..... I had to read back a few pages..... so this is what happens if one isn't paying close attention around here!

Mathematics was created to explain the world around us in measurable ways. Theories were/are developed and "believed" to be true, but much of scientific experimentation meets with failure. The true scientist must have a measure of faith to sustain the pursuit of his proof.

Are truth, reason and love applicable in the scientific mind and pursuit? I believe they most certainly are. Faith in the unknown? To a degree. Faith in the certainty that the unknown can be discovered, with the right set of metrics. Faith in the process of seeking, waiting, exploring. Not blind acceptance though.

Einstein did not believe in god as a personal diety but as a force that we might call Nature. Thanks for the NPR link VT

He was enthralled with the "mathematical elegance" of the universe and believed that there was "something deeply hidden behind everything." In his work Einstein sought to understand what god was thinking.

For some, science is religion; a certain "religious" awe makes for good science. The thrill of the unknown drives discovery. Mystery is inherent to both science and religion.

"Science without religion is lame and religion without science is blind." -Albert Einstein

Post by **Frelga** » Sat Mar 17, 2007 7:01 pm

Anthriel wrote:I just have to share that I believe "Cantor Diagonals" would be a great name for a rock band.

And the only thing closer to God than math would HAVE to be music.

Maybe.

But in the beginning there was Word.

Post by **Primula Baggins** » Sat Mar 17, 2007 7:21 pm

Not in Tolkien, though. . . .

Post by **Primula Baggins** » Sun Mar 18, 2007 7:25 am

Astronomy Picture of the Day for March 18. . . .

http://antwrp.gsfc.nasa.gov/apod/

(If it isn't March 18th any more, go find it on the site—it's easy.)

Post by **Misha** » Sun Mar 18, 2007 11:23 am

http://antwrp.gsfc.nasa.gov/apod/archivepix.html

The archives are indeed awe inspiring!

Sun Mar 18, 2007 5:59 pm

Misha wrote:Aren't all math teachers God incarnate? I certainly thought so in middle school. They seemed all knowing and all powerful to me. And then there was that red pen.....

Gosh..... I had to read back a few pages..... so this is what happens if one isn't paying close attention around here!

Mathematics was created to explain the world around us in measurable ways. Theories were/are developed and "believed" to be true, but much of scientific experimentation meets with failure. The true scientist must have a measure of faith to sustain the pursuit of his proof.

Are truth, reason and love applicable in the scientific mind and pursuit? I believe they most certainly are. Faith in the unknown? To a degree. Faith in the certainty that the unknown can be discovered, with the right set of metrics. Faith in the process of seeking, waiting, exploring. Not blind acceptance though.

Einstein did not believe in god as a personal diety but as a force that we might call Nature. Thanks for the NPR link VT

He was enthralled with the "mathematical elegance" of the universe and believed that there was "something deeply hidden behind everything." In his work Einstein sought to understand what god was thinking.

For some, science is religion; a certain "religious" awe makes for good science. The thrill of the unknown drives discovery. Mystery is inherent to both science and religion.

"Science without religion is lame and religion without science is blind." -Albert Einstein

What a great post, Misha! Thank you very much.