Intuition and Proofs: A Scientist's Perspective

[3.141592654]

Homework Statement

This isn't a homework question so I apologize if I'm in the wrong section, but I'm wondering if proofs are 'easy' or 'intuitive' to you. I recently took a linear algebra course in which I was sometimes able to get through the proofs without any trouble but was completely confounded by several proofs that I encountered. After thinking for a while about the proofs I would often ask for help and upon seeing the answer the logic seemed intuitive and obvious. I often felt ridiculous for not getting it on my own the first time around. I noticed that as the semester wore on and I became more familiar with the material I tended to get better at this but there were still some problems which seemed utterly unsolvable at first that actually ended up being fairly straight forward. It was typically some key piece of information that I hadn't thought to consider that once incorporated made the problems manageable. Is this a shared experience or do some people just "get it"?

 

[gb7nash]

Maybe a few people are so smart that they immediately understand everything that's presented to them. However, for most people, being able to write and understand different kinds of proofs is an acquired skill. There's a reason why there exists a class devoted to writing proofs.

 

[Damidami]

I agree with 3.14...
I think most mathematicians have a geometric interpretation of both definitions, theorems and proofs.
That's what allows them to write proofs in the blackboard, they don't know every hipothesys "by memory", they just can "see" them, by thinking of them in a more or less geometrical way.

For example, when explaining system of linear equations, they know the interpretation (in $\mathbb{R}^2$) of two lines that intersect, or are equal, or don't intersect. I think that's the kind of way they visualize every other concept (no matter how "advanced" it is).

So, while proofs require to be rigurous, a formed mathematician can "see" the truth of it in some intuitive way.

 

[Deveno]

in algebra (my favorite subject), i think in terms of "qualitative behavior".

so when i think of a matrix A, for example, i think of it as shrinking its nullspace to 0, and taking everything else (the linearly independent columns) to an image pretty much 1-1 (maybe moved around a little).

this is sort of a hybrid geometrical/symbolic "picture" i get. it's not purely geometrical, because i do this for things that have more dimensions than i can visualize. sometimes i think of A with an internal hammer that "squashes" its nullspace, and everything it misses "gets away" unharmed.

so algebraic entities are like "blobs" and "blips" that die, or survive. 0 is the "great graveyard" (or "grand collector of indistinguishable things" (which is why a name like "unity" or "identity" makes sense for an identity element, to me, at any rate)).

on a slightly higher level, the rules of a structure have a certain "shape" or from. the idea is to be able to label a form in a useful way, to help us sort things out. these things commute, those things don't..ok, now we have two "bags" to put stuff in. some of my "sorting bins" for matrices:

the identity
scalar multiples of the identity
diagonal
similar to diagonal (diagonalizable)
upper-triangular
nilpotent
normal
orthogonal/unitary
symmetric/hermetian
anti-symmetric/skew-hermetian

each of these has a certain "shape" in my mind, it fits in a certain kind of "parking space", with certain properties that allow me to take "short-cuts" on the path of the "ugly general case".

 

[rachellcb]

@ 3.141... I can definitely relate to the feeling you described. Often I find it helps to collect all the definitions, theorems, lemmas etc on a particular object given in the problem in one place, and then draw on them to solve the problem. After a while they start to form part of your intuition about the subject!