Can Geometry Exist Beyond Our Physical Reality?

[roger]

I wanted to know if euclidean geometry is to do with the real world ?

generalisations of vector space to anything that satisfies the axioms for a vector space can be made, but how can geometry be studied without reference to the real world ?

roger

 

[HallsofIvy]

By doing what mathematics always does: Setting up axioms, "undefined" terms, and definitions and deriving whatever theorems can be proved from those.

In fact, Euclidean geometry, like any mathematical system, does NOT describe the "real world" perfectly. In order to apply any mathematical system to a "real world" problem, you have to select "real world" things to associate with the undefined terms. Then you have to show that the axioms are still true when referred to those "real world" things. But that's NEVER true. "Real world" things are subject to imprecise measurements. The best we can ever hope for is that the mathematical structure will approximately match the "real world".

By the way, the reason I keep putting "real world" in quotes is because I am not certain what you mean by it. In my "real world", I might well have to take a test on geometry next week!

 

[roger]

What do mathematicians mean by the phrase; '' has nothing to do with the real world'' ?

I don't know what is meant by real world either, but I expected most mathematicians to, since I've read about them debating such issues.

so for example why would it be easier for me to do geometry in 2d or 3d than 10d ?

 

[Nimz]

It would be easiest (and most trivial) to do geometry in 0d. I was about to say most pointless, but since 0d is a point, that wouldn't quite be accurate It's probably easier to do geometry in 2d than 3d because 2d is a lot easier to draw, and it's easier to gain an intuition for either 2d or 3d than 10d, since our eyes can't see anything beyond 3d.