Wednesday, October 11, 2006

Geometry and the Foundations of Mathematics

Modern mathematics lays its foundations on logic and set theory. Mathematicians have expended a lot of effort to make their study rigorous. Nothing seems more rigorous than founding the subject matter on logic.

Despite this, one might still claim that geometry lies at the heart of mathematics. Geometry is basic to our everyday experience. Everyone is an excellent geometer even though we seldom recognize or think about it. We walk about the world each day and make prodigious and unconscious geometric calculations. If we were not excellent geometers, we would most assuredly die in a hurry. Think about walking the length of the Magnificent Mile in Chicago at rush hour. You had better know a lot about space and time, or you will be road kill before you know it.

Geometry is even useful when thinking about logic and set theory. The ability to draw a picture can be very useful when doing things like cardinal arithmetic or understanding the set theoretic hierarchy.

Much of the power of mathematics comes from formal symbol manipulation, yet drawing a picture comes in handy when trying understand what you are doing. Pictures, however, can lead us into logical errors, especially if we draw them sloppily. That is not the fault of geometry.

Anyway, I say that geometry is the foundation of mathematics in the fuzzy sense I have sketched. I will leave it there for right now.

2 Comments:

At 5:06 PM, Blogger janus said...

Geometry is even useful when thinking about logic and set theory. The ability to draw a picture can be very useful when doing things like cardinal arithmetic or understanding the set theoretic hierarchy.

good post; i agree with you.

speaking loosely, i liken geometry to 'what we can visualise' and algebra or logic as 'following well-defined rules.'

they aren't independent of one another, of course. drawing a picture gives you intuition, but nothing certain; in that way, logic keeps us honest. on the other hand, the rules are not given to us, and geometric intuition provides us motivation for formulating useful rules.

i've wondered before if one can remove all geometry from mathematics, and strangely enough, i believe it is possible. the best method of testing this might be whether computers can be programmed to prove new theorems, and not resigned merely to verify. at the moment, i don't think we have many computers that can 'visualise.'

maybe it means that mathematics is ultimately human, because it is geometric ..

 
At 12:42 PM, Blogger Lynn said...

janus -

I agree that geometry might be the thing that ultimately makes mathematics human and embeddied in our physical makeup. It is one of those ideas I cannot seem to give up on.

 

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