Algebraic intuition vs geometric intuition

Mathguy15

This has been a curiosity of mine lately. I am wondering about what makes an algebra person an algebra person. I know geometers(at least it seems like it) seem to have a keen ability of spatial visualization. What characterizes the abilities of an algebra person? To clarify, I'm not just talking about say elementary algebra (I'm only fifteen). I'm thinking about linear algebra and commutative algebra also. I am wondering if any of you could shed some light on this curiosity of mine. Any thoughts?

sincerely,

Mathguy

Anonymous217

That's quite a bit of an exaggeration. I'd recommend reading Thurston's "On Proof and Progress in Mathematics" if you want more insight in perspective and intuitions within knowing mathematics (or fields thereof). Several of these meta-mathematics papers by famous mathematicians are practically must-reads. They really shed light into the motivation of mathematics itself.

Mathguy15

Yes, I've read a part of Thurston's essay before. He had some interesting things to say about the nature of mathematics research. In particular, I remember how he said that a mathematician's job is to make humans understand mathematics better. He also said something about how proofs are not necessarily all mathematicians do.

symbolipoint

Students will find at the foundations level of Mathematics, that some truths about Geometric items can help explain corresponding truths in Algebra of Real Numbers. Two examples are The Triangle Inequality Theorem, and Completing The Square for finding roots for quadratic functions. Yet, some people are predonimantly either algebra people or geometry people.

lavinia

Mathematics is based on insight. Some people are gifted with geometric insight just as some people have perfect pitch or photographic memories. But I think that all people are capable of the deep concentration that leads to insight whether it be geometrical, algebraic, or analytic.