Author Topic: Philosophy of Mathematics Question  (Read 1068 times)

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Offline StGeorge

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Philosophy of Mathematics Question
« on: March 05, 2010, 01:16:03 AM »
Are there any mathematical theories in which the increasing distance and absolute opposition of positive numbers and negative numbers is challenged? 

For example, we are taught to believe that the higher the positive number, the farther it is away from the negative numbers.  Positive infinity is the absolute polar opposite of negative infinity, and the gap between them is infinite. 

However, might it be that a positive number reaches a point in which it actually becomes a negative number?  Where, say, positive infinity = negative infinity?  And where there is no gap but an overlapping?   

In philosophy, there is the question of whether time is linear or cyclical.  Can the same question be applied to mathematics? 

Thanks! 
« Last Edit: March 05, 2010, 01:35:31 AM by StGeorge »

Offline Entscheidungsproblem

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Re: Philosophy of Mathematics Question
« Reply #1 on: March 05, 2010, 01:39:58 AM »
I know that if you were to look at the projectively extended real number line, negatives and positives will converge at a single infinity.  But, since infinity is undefined, I don't see how one could "roll over", and have a number change its sign.  Infinity would have to be defined.  

The projectively extended real number line is more of a short cut than anything else, and has some extra limitations placed upon it.

I'm not a maths major though, so don't quote me.
« Last Edit: March 05, 2010, 01:44:22 AM by Nebelpfade »
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Offline chrevbel

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Re: Philosophy of Mathematics Question
« Reply #2 on: March 05, 2010, 06:06:02 PM »
For example, we are taught to believe that the higher the positive number, the farther it is away from the negative numbers. 
True enough, but remember that infinity is not a number.  Infinity is no farther away from negative infinity than it is from one-thousand.

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Positive infinity is the absolute polar opposite of negative infinity, and the gap between them is infinite.
The gap between infinity and any other quantity is infinite. 

Quote
However, might it be that a positive number reaches a point in which it actually becomes a negative number?  Where, say, positive infinity = negative infinity?
Not really, I don't think; but there are situations where it's kind of illustrative to imagine that this is the case.  Look at the graph of the tangent function, for example.  It has "tails" running both to positive infinity and to negative infinity.  If you imagine the graph wrapped around an infinitely large cylinder, they would "meet".