Logic and Infinity

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Democritus
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Re: Logic and Infinity

Post by Democritus » Mon Jun 07, 2004 12:09 am

Ibn Taymiyyah wrote: 5) Therefore they will never become infinite in size because they will never become equal (point (1)).

Deep inside I know that there is something wrong with this argument but I can’t put my finger on it.
Well, it didn't disprove infinity, but it did demonstrate that you have to be careful when thinking about in infinity. Infinity is not a number. You cannot ever "reach" infinity.

For any number x you can imagine x+1. So you cannot imagine a number x such that x=infinity. There is no such number. But you can imagine incrementing x forever, so that x increases towards infinity.

Ibn Taymiyyah wrote: 1) We cannot imagine two objects that are infinite in size to be anything but equal in size.
If an object is "infinite in size," it means that we can't evaluate its size. Whatever number x we propose as being the size, there is always another number x+1, larger than the one we propose. Ordinarily, we cannot say that two objects of "infinite size" are equal in size, because we have no sizes to compare.

But check out the link annis posted about Georg Cantor and transfinite numbers.

Here is one of Canor's observations: Imagine the set N of all natural numbers (1,2,3,4,5,6.....). Now eliminate all the even numbers, so we are left with set O=(1,3,5,7,9...). Common sense tells you that set O is half as big as set N, right? But both O and N are still "infinite in size," even though we removed every other item from set N to get set O. In fact, for every item in N, we can find another item in O to pair it with, as if they were equal in size. Weird, huh?

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Post by threewood14 » Mon Jun 07, 2004 10:27 pm

Here is one of Canor's observations: Imagine the set N of all natural numbers (1,2,3,4,5,6.....). Now eliminate all the even numbers, so we are left with set O=(1,3,5,7,9...). Common sense tells you that set O is half as big as set N, right? But both O and N are still "infinite in size," even though we removed every other item from set N to get set O. In fact, for every item in N, we can find another item in O to pair it with, as if they were equal in size. Weird, huh?
I think infinity is not a number. Its really hard to use it anywhere without running into a lot of trouble. I totally agree with you. That is weird :?
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Post by Emma_85 » Tue Jun 08, 2004 4:56 pm

Infinity can't be a number, but it's a useful thing in mathematics. But we should always remember that we made the rules of mathematics. Infinity is not something we can imagine, it's something we came up with too. I'd like to know who came up with the idea of infinity? Was it a philosopher, mathematician, or is this a concept we don't know the origins of because they are so far back?
Mathematics has come up with some 'strange' things, zero, negative numbers and such like, but those concepts have always existed, mathematicians just formulated them. What about infinity? Anyone know? :?
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Post by threewood14 » Wed Jun 09, 2004 1:02 am

I had a thought that kind of has to do with infinity.

Everything that has a begining has an end. Its from the Matrix actually, but I think its a good concept.

If you think about it, everything that has had a begining has had an end. Look at the Romans, the Greeks, the Nazis, and of course many other powers. It doesn't exclude modern day countries either. Look at people. People are born, and they die. They are not infinite.

I guess the only stand one could take would to use a geometric ray. But if you think about it, where does a ray exist other that in our minds? It really is a mathematical concept in our heads. I don't think that there are any objects that are rays in the universe. You can still specify a ray, but its really just a bunch of points in space time put together with a starting point. It's geometry.

Other than that, what would you say?
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Post by mingshey » Wed Jun 09, 2004 11:14 am

Infinity is a helplessness for a fish. :P

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mercutio
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shooting Zeno

Post by mercutio » Thu Jun 10, 2004 2:02 pm

threewood14 wrote:you gotta remember this about infinity. look at a ruler. how much space is between the one unit mark and the 2 unit mark. if you are using metric, probably about 1cm. now look at it this way. how many point in space time are between those two lines. well first of all, a point in space time is infinitly small. therefore, there are an infinite amount of points in space time between those two objects. we always look at our surroundings at human intervals. now take time. how many point in time are in one second. infinite right? one second is just a measurement of time at the rate we humans experience it.

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Reminds me of that drunken clown Zeno & his paradoxes, the arrow one this time. To go from a to b it must past through c, to go from a to c it must pass through d, to go from a to d it must go through e, blah blah blah ad infinitum et absurdum. Therefore the arrow can never arrive at b because there are an infinite amount of points it must pass through before it gets there. Are you right, Mr Zeno, are you right? No, you are wrong. Why? Because if you were alive I would shoot an arrow at you & it would not pass through an infinite amount of points & never hit you, it would hit you & you would die. Again. RIP.

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Post by Michaelyus » Fri Jul 02, 2004 10:11 pm

Remember: the arrow has a distance (a length) too.
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Re: shooting Zeno

Post by messalina » Thu Jul 08, 2004 7:04 am

mercutio wrote:
threewood14 wrote:you gotta remember this about infinity. look at a ruler. how much space is between the one unit mark and the 2 unit mark. if you are using metric, probably about 1cm. now look at it this way. how many point in space time are between those two lines. well first of all, a point in space time is infinitly small. therefore, there are an infinite amount of points in space time between those two objects. we always look at our surroundings at human intervals. now take time. how many point in time are in one second. infinite right? one second is just a measurement of time at the rate we humans experience it.

__________
threewood14
Reminds me of that drunken clown Zeno & his paradoxes, the arrow one this time. To go from a to b it must past through c, to go from a to c it must pass through d, to go from a to d it must go through e, blah blah blah ad infinitum et absurdum. Therefore the arrow can never arrive at b because there are an infinite amount of points it must pass through before it gets there. Are you right, Mr Zeno, are you right? No, you are wrong. Why? Because if you were alive I would shoot an arrow at you & it would not pass through an infinite amount of points & never hit you, it would hit you & you would die. Again. RIP.

It's the mathematicians vs. the engineers here -
so the joke goes, a room full of mathematicians and engineers are asked to line up against a wall, each across from a large stack of money. they are told that if they are able to cross the room to the money they can keep it, but the catch is that each step they take must be half the distance of the previous step. All the mathematicians step down, knowing that this is impossible and that they will never get to the money. the engineers, on the other hand, all stride across the room confidently, pick up the money, and shrug their shoulders saying, "eh, close enough!"

note: i don't mean to insult the practical knowledge of mathematicians, or the theoretical knowledge of engineers here, i just think it's a funny joke. :D

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Post by messalina » Thu Jul 08, 2004 7:26 am

threewood14 wrote:Here is something that is mind boggling. If something moves an infinite small distance, it would take an infinite amount of time for it to get from point A to point B whereever they may be. Let us say that we give it 100 years to move one inch. Since the distance it is moving is infinitly small, it would obviously not get there by then. But, it would have not moved at all. Because an infinitely small distance plus an infinitely small distance equals and infinitly small distance. If it did move a distance, then it would take a finite amount of time to get from a to b. But since it doesn't, an infinitly small distance can be thought of as 0!!
It can practically be thought of as 0, but it is still not quite 0, as long as it has distance, and thus a size. if we think of the distance as the length 1/n (for n=1,2,3,...etc.) and then let n approach infinity, then we know that this length will never actually be 0 (since 1/anything is nonzero), but 0 can be considered as a limit for the distance we travel, since the length of the distance will never be less than 0 either.

rereading this, i'm not sure if this clarifies or confuses anything further, but i'll throw it out and see if anyone has anything to say. It's been a long time since i've read anything Zeno wrote, so maybe he dealt with this after all and i have forgotten/not read it.. :)

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Post by Michaelyus » Thu Jul 08, 2004 5:14 pm

Zeno, Zeno, Zeno. Motion is relative; did you know that your ink bottle there is travelling at over 100km/h? Spacetime is continuous and it cannot be grouped into class intervals. It may be possible to split the distance up, but there is a continuous flow throughout the points. Things don't happen at various points; they happen during/through it.
Democritus wrote:You cannot ever "reach" infinity.
So true. Infinity is not specified, although it is the quotient of anything and zero. This is why dividing by zero is illegal in mathematics; it is a concept including everything, anything and nothing (but not something); it cannot be split (a true [face=SPIonic]atom[/face] then).
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