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Mathematical Infinity

 
 
Reply Fri 9 Apr, 2010 05:01 pm
I want to fire up a discussion of infinity in the mathematical sense. Is it a number? A concept? Is it thinkable? Is it usefully vague? Is infinity a sort of anti-number? After all, isn't number as number essentially specific? Is infinity some strange informal function?

Is thought essentially finite? Is the concept of infinity a finite concept that refers away from its own nature? I think we can easily include the infinitesimal in this discuss, as it seems to be the reciprocal of infinity, or 1/infinity.

Thoughts?
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Night Ripper
 
  1  
Reply Fri 9 Apr, 2010 05:23 pm
@Reconstructo,
Reconstructo;150068 wrote:
After all, isn't number as number essentially specific?


One place infinity pops up is in irrational numbers, numbers such as pi and the square root of two which are given by an infinite decimal representation like 3.141596535... and so on forever without repeating.
Reconstructo
 
  1  
Reply Fri 9 Apr, 2010 05:29 pm
@Night Ripper,
Night Ripper;150079 wrote:
One place infinity pops up is in irrational numbers, numbers such as pi and the square root of two which are given by an infinite decimal representation like 3.141596535... and so on forever without repeating.


Indeed. And isn't that strange? Recently I learned the difference between algebraic irrationals and transcendental irrationals. The transcendentals seem to be the strangest of the strange. If I had known how poetic mathematics is, I would have dived in long before.

Euler ("oiler" and I love the sound of that.) reminds me of Picasso of Schoenberg. He did some crazy things with numbers, beautiful prankish things. i to the power of i. or x to the power of x to the power of x....to infinity. or e to the power of pi times i. But I digress.
0 Replies
 
platorepublic
 
  1  
Reply Fri 9 Apr, 2010 05:31 pm
@Reconstructo,
I think infinity is actually a discrete number to be honest.
Reconstructo
 
  1  
Reply Sat 10 Apr, 2010 01:14 pm
@platorepublic,
platorepublic;150082 wrote:
I think infinity is actually a discrete number to be honest.


Well, that's an interesting statement. I think you should present your reasons, for on the surface it sounds paradoxical. After all, finite and discrete are synonyms in my book. Welcome, in any case, to the discussion. And please elaborate. Smile
platorepublic
 
  1  
Reply Sat 10 Apr, 2010 04:49 pm
@Reconstructo,
Reconstructo;150313 wrote:
Well, that's an interesting statement. I think you should present your reasons, for on the surface it sounds paradoxical. After all, finite and discrete are synonyms in my book. Welcome, in any case, to the discussion. And please elaborate. Smile

In this world, for example, infinity is a number - it is albeit a very huge number. We are limited by quanta, huh.

Even our minds could only think of a number that is very small compared to this number, but it is discrete - we just couldn't think of it due to our small brains.
rhinogrey
 
  1  
Reply Sat 10 Apr, 2010 05:33 pm
@Reconstructo,
Go, then, go in search of the Fountain of Youth; proclaim at the top of your lungs when at last you find it. You have heard none before you cry out in joy of its capture; and you will hear none in the future. You will hear only the echoing in your own mind. The Fountain of Youth is precisely that which causes your bones to whither and return to dust.
0 Replies
 
Reconstructo
 
  1  
Reply Sat 10 Apr, 2010 07:03 pm
@platorepublic,
platorepublic;150372 wrote:
In this world, for example, infinity is a number - it is albeit a very huge number. We are limited by quanta, huh.

Even our minds could only think of a number that is very small compared to this number, but it is discrete - we just couldn't think of it due to our small brains.

What number is it?
trismegisto
 
  1  
Reply Sat 10 Apr, 2010 08:03 pm
@Reconstructo,
Reconstructo;150068 wrote:
I want to fire up a discussion of infinity in the mathematical sense. Is it a number? A concept? Is it thinkable? Is it usefully vague? Is infinity a sort of anti-number? After all, isn't number as number essentially specific? Is infinity some strange informal function?


How does a number have no beginning nor an end? Even PI begins with a three. A number cannot go on for infinity as numbers always start somewhere. They might go on for a long time, as with PI, but a long time is not infinity.

Every thing and every thought begins as a concept, however, infinity does not begin so can infinity itself be a concept? I am not sure, to be honest. But thoughts of infinity are definitely concepts.

I do not believe it would be possible to have this conversation if concepts of infinity were not thinkable. Every thing real and imagined originates as a thought. Infinity is no different.

Is infinity useful? Only when discussing the Infinite Supreme it neither applies nor exists anywhere else.

As far as an anti-number, infinity is to numbers as darkness is to light.

Reconstructo;150068 wrote:
Is thought essentially finite? Is the concept of infinity a finite concept that refers away from its own nature? I think we can easily include the infinitesimal in this discuss, as it seems to be the reciprocal of infinity, or 1/infinity.

Thoughts?


All thoughts are finite, they begin and they end. Think of thoughts as the wise blind men fondling the elephant. Thoughts on specific ideas don't always overlap nor can they be applied universally to the entire idea. We can only have thoughts about aspects of ideas, we never have grasp the whole.
Reconstructo
 
  1  
Reply Sat 10 Apr, 2010 10:03 pm
@trismegisto,
trismegisto;150410 wrote:
How does a number have no beginning nor an end? Even PI begins with a three. A number cannot go on for infinity as numbers always start somewhere. They might go on for a long time, as with PI, but a long time is not infinity.

We should acknowledge, I think, that infinity is indeed useful in math. For instance, 1/x. As x tends toward infinity, the expression tends toward zero. What I love about math, among other things, is that it reveals the structure of human thought. No magnitude is the final magnitude. And this is fascinating. Therefore the invention of mathematical infinity. Are you familiar with Cantor? Very fascinating stuff. I just read a good book on him. He used the aleph, which is the first letter of the Hebrew alphabet, to represent his transfinite numbers. This aleph is connect to the Kabbalah, and the Ein Soph, and also to Elohim. And yet this mysticism inspired mathematics is real mathematics, and from what I've read, Cantor's vision is set theory is the standard now. Cantor showed that there are as many numbers between 0 and 1 as there are between 1 to infinity, as strange as that may sound. And some other amazing things as well. His technique was to use a diagonal method and put numbers into one-to-one correspondence.

---------- Post added 04-10-2010 at 11:04 PM ----------

trismegisto;150410 wrote:


I do not believe it would be possible to have this conversation if concepts of infinity were not thinkable.


I agree. But my point would be that our concept of infinity is nothing more than the negation of our concept of the finite. -finite, you might say. Of course this negative of finity must be understood in terms of the number spectrum. It's a piece in the lingual-numerical-spatial game of mathematics.

---------- Post added 04-10-2010 at 11:08 PM ----------

trismegisto;150410 wrote:

Is infinity useful? Only when discussing the Infinite Supreme it neither applies nor exists anywhere else.

As far as an anti-number, infinity is to numbers as darkness is to light.


Well, it is quite useful here and there. It's an abstraction that functions usefully as shorthand. Let's compare one to google (which is one followed by one hundred zeros) to one to infinity. Infinity is that strange concept that names the pseudo-end of an endless spectrum. THe concept is a bit paradoxical. And yet calculus, to my knowledge, was invented by using its reciprocal, the infinitesimal, and calculus seems to be perhaps the great leap in the history of mathematics, as the real world is in motion. I think it all goes back to Zeno's paradoxes.
ughaibu
 
  1  
Reply Sat 10 Apr, 2010 10:11 pm
@trismegisto,
trismegisto;150410 wrote:
How does a number have no beginning nor an end? Even PI begins with a three. A number cannot go on for infinity as numbers always start somewhere. They might go on for a long time, as with PI, but a long time is not infinity.
Take a triangle with two 72 degree angles, bisect one of these angles with a line meeting the opposite side of the triangle. You'll have drawn a similar triangle. This process can be continued infinitely, thus demonstrating the irrationality of the golden ratio.
0 Replies
 
Reconstructo
 
  1  
Reply Sat 10 Apr, 2010 10:11 pm
@trismegisto,
trismegisto;150410 wrote:


Is infinity useful? Only when discussing the Infinite Supreme it neither applies nor exists anywhere else.

Cantor was a believer in some kind of God, and talked more and more like his unprovable equation was given to him by God. He ended up going mad, it seems. I wonder if the movie Pi was inspired by Cantor. Surely its author was aware of the story. Check this out.
Continuum hypothesis - Wikipedia, the free encyclopedia
Georg Cantor - Wikipedia, the free encyclopedia

---------- Post added 04-10-2010 at 11:12 PM ----------

ughaibu;150441 wrote:
Take a triangle with two 72 degree angles, bisect one of these angles with a line meeting the opposite side of the triangle. You'll have drawn a similar triangle. This process can be continued infinitely, thus demonstrating the irrationality of the golden ratio.



Nice. I didn't know of that. I'm going to have to google that for some images.
0 Replies
 
trismegisto
 
  1  
Reply Sun 11 Apr, 2010 12:23 pm
@Reconstructo,
Reconstructo;150434 wrote:
We should acknowledge, I think, that infinity is indeed useful in math. For instance, 1/x. As x tends toward infinity, the expression tends toward zero. What I love about math, among other things, is that it reveals the structure of human thought. No magnitude is the final magnitude. And this is fascinating. Therefore the invention of mathematical infinity. Are you familiar with Cantor? Very fascinating stuff. I just read a good book on him. He used the aleph, which is the first letter of the Hebrew alphabet, to represent his transfinite numbers. This aleph is connect to the Kabbalah, and the Ein Soph, and also to Elohim. And yet this mysticism inspired mathematics is real mathematics, and from what I've read, Cantor's vision is set theory is the standard now. Cantor showed that there are as many numbers between 0 and 1 as there are between 1 to infinity, as strange as that may sound. And some other amazing things as well. His technique was to use a diagonal method and put numbers into one-to-one correspondence.

---------- Post added 04-10-2010 at 11:04 PM ----------



I agree. But my point would be that our concept of infinity is nothing more than the negation of our concept of the finite. -finite, you might say. Of course this negative of finity must be understood in terms of the number spectrum. It's a piece in the lingual-numerical-spatial game of mathematics.

---------- Post added 04-10-2010 at 11:08 PM ----------



Well, it is quite useful here and there. It's an abstraction that functions usefully as shorthand. Let's compare one to google (which is one followed by one hundred zeros) to one to infinity. Infinity is that strange concept that names the pseudo-end of an endless spectrum. THe concept is a bit paradoxical. And yet calculus, to my knowledge, was invented by using its reciprocal, the infinitesimal, and calculus seems to be perhaps the great leap in the history of mathematics, as the real world is in motion. I think it all goes back to Zeno's paradoxes.



I will definitely look into Cantor, sounds interesting.

As far as 1 to infinity goes. Maybe I am using a different definition.
As far as I am concerned infinity has neither beginning nor end: Boundless.
So, for me, you cannot start at 1 and end at infinity. It is only infinity to infinity, in fact , it is only infinity.

When you say 1 to infinity, I hear 1 to eternity. An eternity can continue on forever just as infinity does the difference being an eternity has a beginning whereas infinity does not.

---------- Post added 04-11-2010 at 11:27 AM ----------

ughaibu;150441 wrote:
Take a triangle with two 72 degree angles, bisect one of these angles with a line meeting the opposite side of the triangle. You'll have drawn a similar triangle. This process can be continued infinitely, thus demonstrating the irrationality of the golden ratio.


Again I would agree with everything you say if only you substituted "infinitely" for eternally.

Infinity has no beginning nor end. Eternity is epochal.
It may seem like nitpicking but eternity and infinity are two vastly different concepts.
jack phil
 
  1  
Reply Sun 11 Apr, 2010 01:14 pm
@Reconstructo,
Nullity and infinity and negative infinity and the imaginary infinite all pickled my interest for some time. I noted that the negative and the imaginary were both adjectives unto infinity and so figured them likewise in some fashion. So it seems Mr. Buckminster Fuller was on the right path in regards to mathematics.

But there is a certain idolatry going on with the presentation of infinity as actually a number when our base ten system will never reach anything other than what it already contains-- and never reach any number we might call infinity.

Are we wishing to speak of different systems when we bring infinity to the table? It doesn't seem that infinity is the answer to any question.

If infinity is contained in the number, then no number could capture infinity.

I guess an analogy could be made: If God is contained in the word, then no word captures God.
Reconstructo
 
  1  
Reply Sun 11 Apr, 2010 03:04 pm
@jack phil,
jack;150570 wrote:
Nullity and infinity and negative infinity and the imaginary infinite all pickled my interest for some time. I noted that the negative and the imaginary were both adjectives unto infinity and so figured them likewise in some fashion. So it seems Mr. Buckminster Fuller was on the right path in regards to mathematics.

But there is a certain idolatry going on with the presentation of infinity as actually a number when our base ten system will never reach anything other than what it already contains-- and never reach any number we might call infinity.

Are we wishing to speak of different systems when we bring infinity to the table? It doesn't seem that infinity is the answer to any question.

If infinity is contained in the number, then no number could capture infinity.

I guess an analogy could be made: If God is contained in the word, then no word captures God.


Good post. Yes, our system can never write it out, this infinity. In a math sense it functions as something abstract. As x tends to infinity...that would be a common use. And x can tend to infinity for eternity. So infinity is something like the bottomless pit in revelation. If infinity were not directional (positive or negative), it would be utterly absurd. But it does manage to be useful as a strange sort of abstraction. Asymptotes are strongly related, and in some cases the lemniscate is the ideal symbol to use. Just as pi is used for radians (which is brilliant) and yet pi is also impossible to present in terms of positional notation. Same with e. To me, this is the most beautiful aspect of math. It jumps off the positional system, and requires an ideogram. Cantor brought in the aleph, which is different but related. Also quite beautiful. He strikes me as a mystic who yet could succeed as a rigorous math-man. A rational mystic, you might say. Maybe Witt could be described as a rational mystic. (TO see the world as a limited whole, it is that which is the mystic...) And rational numbers are rational exactly because they can be written as the fraction of whole numbers. Interesting....
0 Replies
 
north
 
  1  
Reply Sun 11 Apr, 2010 10:36 pm
@Reconstructo,
Reconstructo;150400 wrote:
What number is it?


three dimensional
0 Replies
 
ughaibu
 
  1  
Reply Sun 11 Apr, 2010 10:41 pm
@trismegisto,
trismegisto;150557 wrote:
Infinity has no beginning nor end.
Bigger similar triangles can also be constructed, infinitely.
north
 
  1  
Reply Sun 11 Apr, 2010 11:01 pm
@jack phil,
jack;150570 wrote:
Nullity and infinity and negative infinity and the imaginary infinite all pickled my interest for some time. I noted that the negative and the imaginary were both adjectives unto infinity and so figured them likewise in some fashion. So it seems Mr. Buckminster Fuller was on the right path in regards to mathematics.

But there is a certain idolatry going on with the presentation of infinity as actually a number when our base ten system will never reach anything other than what it already contains-- and never reach any number we might call infinity.

Are we wishing to speak of different systems when we bring infinity to the table? It doesn't seem that infinity is the answer to any question.

If infinity is contained in the number, then no number could capture infinity.

I guess an analogy could be made: If God is contained in the word, then no word captures God.


this is my delimma

mathematical infinity is really rather , irrelevent

it is the infinity of energy and matter , that matters most:D
0 Replies
 
trismegisto
 
  1  
Reply Mon 12 Apr, 2010 01:52 pm
@ughaibu,
ughaibu;150709 wrote:
Bigger similar triangles can also be constructed, infinitely.


You mean eternally.
Reconstructo
 
  1  
Reply Mon 12 Apr, 2010 03:09 pm
@Reconstructo,
The word eternal can mean the negation of time or endless infinite time. I like the bottomless pit in the book of Revelation. Same concept, only spatially, and downward. It's strange that calculus is based on the limit concept. It's a weird part of human thinking. The binding of unbound in language/symbol.
0 Replies
 
 

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