Proofs

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To me, a lot of mathematical proofs just look like a bunch of handwaving. I have almost no clue how to "prove" most things, because there doesn't seem to be any rhyme or reason to it. Can anyone make sense of it for me?...

They want it written down sa you progress to the answer. I've been faced with a few questions I knew the answer to it, and I made up how I got there. The answer would usually be right, but the "story" isn't

Yeah, that's kind of how it is for me. I can tell whether the statement is true or not just by looking at it, but apparently some of the methods I use aren't "legitimate".

It's been a few years, but as I remember the essence is:

1. You start with a mathematical statement. It may or may not be true.

2. The aim is to draw logical mathematical statements from the original statement and reduce those to a very trivial statement such as 1 = 1 therefore the original statement was valid and is now proved, or if you end up with an absurdity such as 1 = 0 then the original statement was invalid.

3. In going from (1) to (2) you aim to logically change the statement in small logical steps. Each logical step must be indisputable logic; or the whole basis of the proof falls like a house of cards and the original statement may or may not be true.

Maybe it's because we are visual thinkers. We have the answers structured in our minds, we have a very detailed visual image of the whole system, yet we don't understand the abstract way most mathematicians write. And unluckily it seems that we'll have to learn this language if we want people to even believe what we have done.

I have faced that problem too, i got the answer but, "How on earth did you do it? I still can't figure it out".

PS: I have been struggling lately with learning that formal and symbolic language.

There is rhyme. There is reason. You just don't have the chops to grasp it (by your own admission). Mathematical proof is the highest exercise of the human intellect. It is what elevates us close to godhood. That is the good news. The bad news is that we exercise pure reason maybe ten percent of the time.

ruveyn

Of course, each step in practice isn't. Ever try proving basic arithmetical propositions using Peano Axioms and formal rules of inference? There are isolated reports, scarcely more than rumors, that several people have actually died of boredom while attempting such a task.