math experts needed on this problem

[DrPizza]

Quick background:
Godel's proof, etc.
All mathematical systems are based on assumptions which therefore cannot be proven within that system.

I don't know if one of my calculus students really knows/understands that, but I allowed her to ramble on today.

Odd examples: in non-Euclidean space, you can draw 2 parallel lines (using the definition that parallel lines remain the exact same distance apart) Yet, the 2 parallel lines, which are side by side, are also the same line. (you're thinking, "huh??" - 2 parallel lines on a mobius strip)

Another example: non-Euclidean geometry (Einstein, anyone?) based on a different set of assumptions than Euclidean geometry.

Okay, here's what a student suggested: +infinity and -infinity are just directions, like East and West. "okay," I said, in complete agreement at this point. Well, she continued, suppose that the number system is curved, like space, and that eventually you get back to where you started. Or, it's like the Earth - if you continue going west, you wind up east. And, if you continue going east, you wind up west....

As nonsensical as this is, my mind started wandering. Of course, she's wrong, we can prove her wrong using our number system. But, what if our assumptions (as intuitive as they feel, like parallel lines never touching - again, then there's non-Euclidean geometry, disagreeing with this assumption) that we base our entire counting system are wrong.

Or, perhaps more interesting, could a new branch (or does one exist) of mathematics, based on this bizarre assumption of hers, or one based on assumptions that lead to her bizarre conclusion, exist? I didn't get enough sleep last night... I thought about this for only a minute or so. My mind just sort of went fuzzy and I now have a headache. Anyone want to clarify this?

 

[kleinwl]

I think I understand what she is implying, however I would counter that numbers are discrete like Euclidean geometry. No matter how far one goes to the positive or the negative, the numbers can not "loop". I understand how much of a problem this is on Earth, where one can travel 360 degrees and end up at his starting position... and would say that a different mathimatical model for this would be helpful. And in fact would be very interesting, to be able to count from 0-360 and then start over (similar to a base 360 numbering system). However, this does not mean that the way mathematics are designed is flawed. For example, no matter how large the universe grows, it will not loop around to touch itself (unless my understanding of the expansion of the universe if flawed). Perhaps a better example is no matter how many millions (and more millions) of dollars you earn, you will not suddenly become poor, once you have earned too much (unless your playing a computer game and exceed the number buffer or have the IRS jump on you for tax evasion or something).

Sorry I ramble a bit... but, the design of mathematics and +/- infitinity are decrete functions that are correct, in their applications. The fault comes when they are stretch to describe events that are not decrete, such as the geometry of the globe where 360=720=1080. When they are so poorly used, the faults become obvious. In your example, your student is incorrect, however she is (correctly) pointing out the limits of the system and the need for more advanced mathematics to decribe looping systems (such as the globe).

 

[Gibsons]

Originally posted by: DrPizza
Quick background:
Godel's proof, etc.
All mathematical systems are based on assumptions which therefore cannot be proven within that system.

I don't know if one of my calculus students really knows/understands that, but I allowed her to ramble on today.

Odd examples: in non-Euclidean space, you can draw 2 parallel lines (using the definition that parallel lines remain the exact same distance apart) Yet, the 2 parallel lines, which are side by side, are also the same line. (you're thinking, "huh??" - 2 parallel lines on a mobius strip)

Another example: non-Euclidean geometry (Einstein, anyone?) based on a different set of assumptions than Euclidean geometry.

Okay, here's what a student suggested: +infinity and -infinity are just directions, like East and West. "okay," I said, in complete agreement at this point. Well, she continued, suppose that the number system is curved, like space, and that eventually you get back to where you started. Or, it's like the Earth - if you continue going west, you wind up east. And, if you continue going east, you wind up west....

As nonsensical as this is, my mind started wandering. Of course, she's wrong, we can prove her wrong using our number system. But, what if our assumptions (as intuitive as they feel, like parallel lines never touching - again, then there's non-Euclidean geometry, disagreeing with this assumption) that we base our entire counting system are wrong.

Or, perhaps more interesting, could a new branch (or does one exist) of mathematics, based on this bizarre assumption of hers, or one based on assumptions that lead to her bizarre conclusion, exist? I didn't get enough sleep last night... I thought about this for only a minute or so. My mind just sort of went fuzzy and I now have a headache. Anyone want to clarify this?

I recommend aspirin and maybe some caffeine for the headache.

I can't say I really 'grok' the idea of infinity in the first place, so this ends my contribution to this thread.

 

[harrkev]

Originally posted by: kleinwl
For example, no matter how large the universe grows, it will not loop around to touch itself (unless my understanding of the expansion of the universe if flawed).

Actually, many believe that the universe DOES curve back upon itself. This is because: A) The universe is expanding, B) The expansion is occuring evenly in all directions, and B) The universe looks the same in all directions.

There are three possible explanations for this:
1) We are at the exact center of the expanding universe.
2) There is NO center, and the universe looks the same everywhere, which implies that there is no edge. This means two posibilities:
. . . 2a) The universe is infinite, but expanding uniformly.
. . . 2b) The universe is finite, but with not edges. Kind of like a 3-D sphere. This implies that you can travel in one direction and eventually get back to your starting point.
3) Something else strange is going on which we can't even guess at right now.

If you do not believe in a Creator, then #1 is quite unlikely, and #3 is not even worth worrying about. So #2 it is...

If course, if you DO believe in God, #1 and #2 are still on the table.

I do not know if there is any reason to assume a finite universe over an infinite one, though.

 

[cquark]

Originally posted by: DrPizza
Quick background:
Godel's proof, etc.
All mathematical systems are based on assumptions which therefore cannot be proven within that system.

I don't know if one of my calculus students really knows/understands that, but I allowed her to ramble on today.

Odd examples: in non-Euclidean space, you can draw 2 parallel lines (using the definition that parallel lines remain the exact same distance apart) Yet, the 2 parallel lines, which are side by side, are also the same line. (you're thinking, "huh??" - 2 parallel lines on a mobius strip)

Another example: non-Euclidean geometry (Einstein, anyone?) based on a different set of assumptions than Euclidean geometry.

Okay, here's what a student suggested: +infinity and -infinity are just directions, like East and West. "okay," I said, in complete agreement at this point. Well, she continued, suppose that the number system is curved, like space, and that eventually you get back to where you started. Or, it's like the Earth - if you continue going west, you wind up east. And, if you continue going east, you wind up west....

As nonsensical as this is, my mind started wandering. Of course, she's wrong, we can prove her wrong using our number system. But, what if our assumptions (as intuitive as they feel, like parallel lines never touching - again, then there's non-Euclidean geometry, disagreeing with this assumption) that we base our entire counting system are wrong.

Or, perhaps more interesting, could a new branch (or does one exist) of mathematics, based on this bizarre assumption of hers, or one based on assumptions that lead to her bizarre conclusion, exist? I didn't get enough sleep last night... I thought about this for only a minute or so. My mind just sort of went fuzzy and I now have a headache. Anyone want to clarify this?

Algebra on elliptic curves works somewhat like that, using the point at infinity (positive and negative infinity are considered to be equivalent) as the identity element of the elliptic curve group.

I suspect that you would find a match to your student's structure in Lie Groups, which are a type of group tied to differentiable manifolds. The manifold is a non-Euclidean structure that can allow curves like the one suggested above, while the Lie Group will provide an algebraic structure to work on that curve.

 

[eigen]

Originally posted by: cquark

Originally posted by: DrPizza
Quick background:
Godel's proof, etc.
All mathematical systems are based on assumptions which therefore cannot be proven within that system.

I don't know if one of my calculus students really knows/understands that, but I allowed her to ramble on today.

Odd examples: in non-Euclidean space, you can draw 2 parallel lines (using the definition that parallel lines remain the exact same distance apart) Yet, the 2 parallel lines, which are side by side, are also the same line. (you're thinking, "huh??" - 2 parallel lines on a mobius strip)



Another example: non-Euclidean geometry (Einstein, anyone?) based on a different set of assumptions than Euclidean geometry.

Okay, here's what a student suggested: +infinity and -infinity are just directions, like East and West. "okay," I said, in complete agreement at this point. Well, she continued, suppose that the number system is curved, like space, and that eventually you get back to where you started. Or, it's like the Earth - if you continue going west, you wind up east. And, if you continue going east, you wind up west....

As nonsensical as this is, my mind started wandering. Of course, she's wrong, we can prove her wrong using our number system. But, what if our assumptions (as intuitive as they feel, like parallel lines never touching - again, then there's non-Euclidean geometry, disagreeing with this assumption) that we base our entire counting system are wrong.

Or, perhaps more interesting, could a new branch (or does one exist) of mathematics, based on this bizarre assumption of hers, or one based on assumptions that lead to her bizarre conclusion, exist? I didn't get enough sleep last night... I thought about this for only a minute or so. My mind just sort of went fuzzy and I now have a headache. Anyone want to clarify this?

Algebra on elliptic curves works somewhat like that, using the point at infinity (positive and negative infinity are considered to be equivalent) as the identity element of the elliptic curve group.

I suspect that you would find a match to your student's structure in Lie Groups, which are a type of group tied to differentiable manifolds. The manifold is a non-Euclidean structure that can allow curves like the one suggested above, while the Lie Group will provide an algebraic structure to work on that curve.

Good answer.

You may also want to think about groups modulo.In this sense the numbers loop around and you can even say "How far" you have to go around before you are back at the beginning. i.e the order of the group.

 

[CycloWizard]

Originally posted by: kleinwl
I understand how much of a problem this is on Earth, where one can travel 360 degrees and end up at his starting position... and would say that a different mathimatical model for this would be helpful. And in fact would be very interesting, to be able to count from 0-360 and then start over (similar to a base 360 numbering system).

This is exactly what I was thinking of as well. There actually is a system that uses this methodology, at least for statistical purposes. When measuring angles, you obviously can't have numbers outside a certain range. Thus, there is a distribution analagous to the Gaussian distribution (which goes from -inf to +inf in linear space): the von Mises distribution. It goes from zero to pi (or -pi to +pi) and is continuous at the limit. This compensates for the impossibility of an infinite tail in a circular distribution. I'm not familiar enough with how these things were derived to know how far the theory extends, but it's definitely interesting that a high schooler is actually that thoughtful.

 

[eLiu]

You could define a mathematical system with that property...but I'm not sure how much you could do with it, being as how it's going to lack a whole lot of properties that we tend to enjoy working with...

But I'm going to guess that some advanced knowledge of set/group theory would be necessary to successfully define such a result...I have no such knowledge, and I'm going to shutup before I make an ass of myself.

Also, +inf & -inf are only directions if you consider a line. Jump to say, the complex plane and these notions of infinity suddenly don't make sense anymore--i.e. what is the Pi/4-direction infinity or the 3pi/13 direction infinity or the 4pi direction infinity? Is the 4pi direction different from 2pi different from 0, etc?

 

[halfpower]

If you are on a surface, you can go west and loop back to east.

If you are deep in financial debt, you can always go deeper.

 

[RossGr]

I do not think that any of this has anything to do with Gödel, a better statement of Gödel?s incompleteness is: In any sufficiently complex logical system there exist statements which can not be proven.

This does not say that no statement can be proven, but that there are statements which cannot be proven, big difference.

Now, consider the motion of the image formed by a simple convex lens. The image position is given as

1/i = 1/f - 1/o

where i is the image distance, o is the object distance, and f is the focal length of the lens.

Consider an object at an infinite distance from the lens, it will form an image at the focal length of the lens (ever burn a bug with a magnifying glass?) , now as you move the object in from infinity, the image will move away from the focal point until finally with object at the focal length of the lens its image will be at positive infinity. The interesting thing, and the whole point of this post, is what happens to the image, as you move the object INSIDE the focal point, suddenly the sign on the image distance changes to negative so image is now at NEGITIVE infinity. With smooth continuous motion of the object the image has jumped from positive to negative infinity. This is observable simply by looking at a distant object through a magnifing glass and moveing the lens towards your eye, when your eye is outside the focal length, you will observe and inverted real image, now move the lens towards your eye, as you go through the focal length, the image goes crazy,then when your eye is inside the focal length the image is now upright and virtual.

This is a very real concrete example of something moveing from postitive to negative infinty, without going through all of the points between.

 

[ZeroNine8]

This sounds like something analogous to a parametric equation. If you consider your 'movements' along a number line to be the variable and your resulting equation values to be your absolute position on the number line, you could arrive at a similar situation. In this way of thinking, we can simulate a situation in which incrementing a value does not result in an equal increment of absolute change in value.

Unit Circle: (spinning off of the original example)

x = cos(t), y = sin(t)

where x,y are your 'absolute' value relative to some reference, and t can progress to infinity while x,y remain finite.

The question is, then, do we assess a value on changes in t or as changes in absolute x,y position (with the conventional system we think we do both, but in actuality, who knows). Mathematically, this is nothing new, but I don't really know how one would detect whether or not the 'real' number line is straight, curved, or otherwise. The parametric reference frame and the cartesian reference frame are interchangeable within certain constraints (i.e. 0<=t<=2Pi), so the same may be true of number systems. As long as we never have an absolute vantage point, we won't know the difference .