/sci/ - Science & Math

[Anonymous No.6394336]

This is something I genuinely don't understand:
How can mathematical, 0-dimensional points form dimensional shapes, such as lines, circles, etc? Doesn't there always have to be an infinite amount of points next to each other for them to form any shape?
If there's infinite points, then how can there be shapes with differing lengths? Is there a differently infinite amount of points in two lines with the lengths 5 and 8, for example?

[Anonymous No.6394355]

>>6394336
>Is there a differently infinite amount of points in two lines with the lengths 5 and 8, for example?

yes and no. It's kind of depending, what kind of approach you are using.

in set theory we can make statements about the cardinality (the size) of sets.
we say, one a set X has a higher cardinality than a set Y if there exists a surjective ( http://en.wikipedia.org/wiki/Surjective_function ) mapping from X to Y.
In that sense, the intervall [0,1] (this represents the line between 0 and 1) has the same cardinality as the set [0,2], since the function f(x) = 2x is surjective.

for actually measuring lengths, we have to use a different approach, but I can't be assed to write that down, too because it's a little more complicated.
But if you are interested, you can read http://en.wikipedia.org/wiki/Measure_theory

[Anonymous No.6394359]

>>6394336
Natural numbers and even numbers are both infinite sets, they're even both countable sets, and yet the latter is strictly contained in the former.

But this really hasn't much to do with "lenght". Read up something about measure.

[Anonymous No.6394365]

>>6394336
The answer is that lines have more structure than just the underlying sets. One way to write down this structure is by introducing measure theory, yes, but for a more elementary approach, look into topology and in particular metric spaces.

[Anonymous No.6394367]

a line is a collection of points such that wherever you look on the line, there's a point right there, precisely where you decided to look, no matter where the point you decided to look is