How do mathematicians even begin writing a proof?

Question

Something I've been curious about for a while. All the proofs I've seen are so complicated and convoluted that I don't see how anyone ever manages to write one. And devoting their whole lives to something so probably impossible seems quite risky, since it could all easily be a waste. So they must have some tangible hopes for success as motivation to continue.

Answer 1

You might be interested in reading this interview with Andrew Wiles (Fermat's Last Theorem proof):

http://www.pbs.org/wgbh/nova/proof/wiles.html

For instance, in one section:

"NOVA: And during those seven years, you could never be sure of achieving a complete proof.

AW: I really believed that I was on the right track, but that did not mean that I would necessarily reach my goal. It could be that the methods needed to take the next step may simply be beyond present day mathematics. Perhaps the methods I needed to complete the proof would not be invented for a hundred years. So even if I was on the right track, I could be living in the wrong century."

I hope this helps some.

Answer 2

Well despite the fact that I have to do what I consider to be a fair amount of math, being in a University Engineering program, I still don't *really* know how those crazy mathies do what they do.
Usually these proofs are an attempt to solve a well known, and long-standing problem (and as you mentioned, usually extremely convoluted).
Having a lot of math experience will give you an insight into how you should approach a problem... Sadly I do not have that insight yet - the reason I am not so great on the math exams...
And with regards to the possibility of failure... most mathies enojy what they do, and do not worry about failure.. it is all about the process, all about the "beautiful" math that they are working on.
Believe me I don't understand it, but I do appreciate it.
If you are up for it, it can be a rewarding career (from what I have heard)

Answer 3

The proofs usually aren't nearly as bad as they look, once you understand what's going on. In university math classes, you regularly write tons of proofs, and it's really not that big of a deal.

The key is to understand what's going on. Once you understand it, you'll generally have a pretty good idea of how to approach writing a proof - and the process does get a lot easier as you get some experience with it.

If you know that the thing you're proving is true, and you understand why it's true, then writing the proof usually isn't that hard. Not always, though: sometimes the most obvious things are the toughest to prove. However, usually it's not that bad.

Answer 4

OK, I guess the educational system is in worse shape than anyone previously thought. When I was in high school we wrote proofs all the time. Go talk to your schools geometry teacher and if they can't explain this to you scream to every one who will listen that your community needs a new school board, superintendent, principals and all new teachers. This is very basic stuff.

Answer 5

they think how they can make it the most miserable for math students, and what will cause the most pain and torture, haha

Answer 6

I can answer this query from personal experience!

But first of all, some of your responders have just NOT seen the point of your question. I take it you're REALLY interested in the PROCESS of ORIGINALLY coming up with a SIGNIFICANT problem whose solution and/or proof you'd like to obtain, and not simply writing out a (possibly more polished) pre-existing solution or proof that someone else has already discovered and that can be found in a book somewhere.

Here's how it happened with me. I was a Cambridge wrangler, which means I'd obtained a First Class degree in mathematics at Cambridge. Like many people specialising ultimately on the applied maths side, I'd then become an astrophysicist. I'd been interested for many years in how Newton discovered the law of gravity and gone on to explore its consequences, and had mastered most of his quite challenging geometrical proofs and propositions in the Principia. This was a very personal interest, even before I was assigned to teach a university course on Gravity, which I did for some 25 years.

Getting back to Newton, unlike most of his proofs and derivations, his own published and geometrically challenging determination of the law of gravity from Kepler's Laws is NOT QUITE fully self-contained. At one point, rather than prove everything he needed in his own way (as was his custom), he uncharacteristically made an appeal to a known result from classical Greek geometry. That was something "... demonstrated by the writers on conic sections," namely that "All parallelograms circumscribed about any conjugate diameters of a given ellipse or hyperbola are equal among themselves," meaning that, for a given ellipse or hyperbola, all such parallelograms have equal area.

The geometry used in this and other derivations by Newton is quite beyond the capabilities of most of today's students, so over the years I had set myself the task of coming up with the simplest and shortest NON-CALCULUS derivations that WOULD be accessible to them. I was successful quite early in finding such non-calculus derivations. However, a compact, understandable derivation of the law of gravity itself from elliptical motion had always thwarted me. As I continued thinking that there had to be some angle or approach that I was missing, my admiration for Newton's geometrical proof grew.

I knew that uniform motion in a circle involves appreciating the role that the curvature of the circle plays in both the kinematics and the dynamics. So, I began to vaguely appreciate that if I could perhaps find something, indeed ANYTHING to relate the curvature of the ellipse at a point to some other simple property or properties at that point (I was becoming desperate by now), maybe THAT would provide the KEY to unlocking this problem. Over the years I made one or two desultory starts along those lines, but seemed to get nowhere.

Finally, however, inspiration struck, out of nowhere. I was sitting at my kitchen table, waiting for Bill Clinton to give his first inaugural speech, and thought I'd while away some empty air time by looking at this problem again. I thought back to my inspiring old English Grammar School (= U.S. High School) Maths teacher, and thought about what I'd learned from him in coordinate geometry classes.

"Dammit," I thought, "I'd better just grit my teeth and work out the curvature of the ellipse at a general point in the most pedestrian British Schoolboy way (remember, my motivation was: NO CALCULUS)! So what was a step-by-step method that I could use to tackle this? : Move the axes to a general point P on the ellipse, rotate them to become the tangent and normal at P, examine the small variable local approximation in that latest tangent/normal frame, and then psyche out from the form of that approximation what the radius of curvature is."

When I did that somewhat tedious but straightforward set of steps, you could have knocked me down with a feather. For, the formula for the radius of curvature that I found in that way intriguingly contained the CUBE of a factor that I realized I'd obtained in a formula for SOMETHING ELSE at a general point P, long before, a result that at that time had seemed to lead nowhere further. The next step was now OBVIOUS! So, cubing that earlier result and eliminating the now common cube term from what I'd just found that morning, a beautiful and extremely simply stated new theorem emerged, a relationship absolutely GEARED for the purpose I'd had in mind. With this new theorem, the derivation of the law of gravity from Kepler's first two Laws of Planetary Motion was reduced to only two or three lines of schoolboy algebra!

I naturally contacted several of my mathematical friends about the new theorem I'd discovered. All of them were completely unaware of itsexistence. Several suggested other mathematicians I should contact. One of them however said "Why not go right to the top and call up X, the world's leading expert on conics and quadric surfaces."(I had in fact studied X's books in Grammar School, 40 years earlier.) I followed this advice, X was very intrigued, and the upshot was that I was invited to publish both my derivation of the theorem and its application to discovering the law of gravity in a specialist mathematical journal.

What was particularly gratifying about all this was that I myself had DISCOVERED this theorem used NOTHING but the methods I'd learned in Grammar School!

How long was it from when I'd first begun thinking about this problem until I actually found the solution? About 12 years. How long did it take me to find my derivation once I gritted my teeth and really buckled down to it? About 1 - 2 hours at most. I'd had many other preoccupations in the intervening years, of course, and it's possible that the problem was bubbling away somewhere in my brain all that time. But sometimes, there just come circumstances and a moment when everything is propitious, the obscuring mental clouds clear, and the solution to a problem comes to the fore as though it were naturally ordained.

And there you have it --- a personal account of mathematical discovery.

[LATER: Considering the question that you posed, why on Earth has someone given this account a "thumbs down"?!
Did he or she just get out of the wrong side of the bed this morning?]

Live long and prosper.