Thoughts on Infinity

[tcepsa]

If I'm remembering correctly, one of the generally accepted concepts in mathematics is that two lines in a plane will never intersect if and only if they are parallel. They go on forever without crossing. However, if they are out of parallel by the tiniest amount then at some point they must cross.

What point is that?

Comments

[girlonaroad.livejournal.com]

Affine plane is just a plane (points and lines connecting points) which satisfies certain properties - axioms really. Geometry is completely based around axioms - think Euclid's "The Elements." So, if you have an axiomatic system, it always turns out some things are independent of axioms (like the "axiom" of choice is independent of axioms underlying algebra, logic, etc). So, this parallel line axiom is one of those independent ones. that means we can decide, "nevermind - parallel lines DO intersect - they intersect HERE, at the point at infinity/ line at infinity", and now we'll have a new type of geometry, in this case projective geometry.

RP2 mentioned in the article really wraps some of these concepts up nicely. Take a the unit circle in 2 dimensions. So, you have the origin and all points at a distance < 1 from the origin. How many independent vectors do you have? Well, anything parallel is really the same vector, right, so we can reduce what we are considering to just lines that pass THROUGH the origin. Okay, now if we want all parallel lines to intersect, the only way for us to do this is to decide that when a line heads out from the origin and hits the boundary circle, it'll connect to where that line intersects the boundary int he other direction. So basically opposite points on the boundary circle are defined to be the same point - "identified with one another." You can do geometry on this space - you can study its topological properties, etc.

[tcepsa.livejournal.com]

Awesome, thanks for clarifying :) While it still kind of bends my brain to try to conceive of any two points directly across from each other on a circle as being the same point (very tricky to visualize, which is how I prefer to do things ;) I think that I have a better understanding of what that means.

To see whether I understand the idea behind the "line at infinity," it sounds like it is basically something that somebody came up with (a new axiom, because it wasn't derivable from the existing set of axioms?) to allow a different geometry (projective geometry) to exist which then makes it possible to address certain types of problems that weren't solveable (or were extremely messy) in the geometry defined by the previous set of axioms. Am I close? :)