
How I do Proofs - duck
http://bentilly.blogspot.com/2011/01/how-i-do-proofs.html
======
lkozma
Nice writeup, certainly helps in organizing your thoughts.

It reminded me of the Feynman method a bit though :)

1\. Write down the problem.

2\. Think real hard.

3\. Write down the solution.

More seriously, the classic book by Gy. Polya "How to solve it" is also very
useful:

[http://www.amazon.com/How-Solve-Aspect-Mathematical-
Method/d...](http://www.amazon.com/How-Solve-Aspect-Mathematical-
Method/dp/0691023565) (non-affiliate)

[http://www.amazon.com/How-Solve-Aspect-Mathematical-
Method/d...](http://www.amazon.com/How-Solve-Aspect-Mathematical-
Method/dp/0691023565?tag=laszkozm-20) (affiliate)

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euccastro
As a neophyte, the only thing I find missing is proof by cases, exhaustive
proof (sometimes there are just a few cases that can be trivially proven
separately, yet a comprehensive proof is elusive), and (while it's a bit
tangential) a couple tips on how to read others' proofs (e.g., what do things
like "clearly" and "without loss of generality" precisely mean in this
context, and what to do when you encounter them).

~~~
btilly
I didn't include proof by exhaustion because it wasn't going to come up in the
course. But in principle it is the same as any other technique, but the trick
is figuring out which cases to break it into. There is an art to that which
varies by topic, and I don't feel up to trying to explain it in general. :-)

On how to read proofs, my hope was that I wouldn't be presenting any proofs
that are hard to read. And given their level of skill, I didn't want them
trying such shortcuts either. That said, the trick is to recognize weasel
words where the author's laziness makes the reader do work, and then do that
work.

When someone says "clearly" what they mean is "the proof is routine, and I
don't want to clutter my line of reasoning digressing into a proof of it." If
you don't find it clear, then you have to produce that proof (or at least an
outline of it). Filling in these details is often one of the hardest parts of
reading someone else's proofs.

"Without loss of generality" means, "I'm about to set up a bunch of stuff that
looks like it could be a special case, but it really isn't." As a reader it is
then your job to figure out why it isn't a special case. Once you are
convinced that it really isn't a special case, then you can accept the
specific setup that just got made.

And "similarly" means "the proof for this piece is pretty much the same as
what you just saw, and I don't feel inclined to write it all out again." In
that case you need to convince yourself that this is true.

------
btilly
It should be noted that this is in some way a follow-up to
<http://news.ycombinator.com/item?id=850485>, which is a follow-up to
<http://news.ycombinator.com/item?id=818367>.

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mturmon
I usually start with, "Do I believe the proposition is true? If so, why?"

This is sort of like the "Do I understand it" part of the flowchart in the OP.

But in my experience, if you think in detail and can justify why you believe
the proposition is true, that helps break down the proof into steps.

------
duck
The actual process is shown here:
[https://docs.google.com/document/pub?id=1_uwl3WDZk_BxNOUL7W0...](https://docs.google.com/document/pub?id=1_uwl3WDZk_BxNOUL7W0FiPMMxdmi7w4OoP4prUcIs2s)

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justlearning
(As a thought experiment) I find it very interesting that the same
flow/dissection can be applied to outside math, as set of tools to solve any
problem - even to make those indecisive moments interesting. I just went
through the flow to make a 'decision'.

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bialecki
Tangential, but I don't understand this reaction after he tells them in class
they're going to go through a proof together:

'You should have seen the shock on their faces! Some started complaining. So I
said, "No, seriously. You will all prove this. Just wait and see. It will
work."'

Are people really that timid when all they're being asked to do is try?

~~~
btilly
These students in all likelihood had never, ever, been asked to produce a
mathematical proof. They were being told that they were about to write one as
a group, after having a general description of how to do it, without seeing
any worked out examples first. It is quite possibly the first time that they
had ever been expected to do anything for a first time in a math class without
seeing a worked example first.

Furthermore they were staring at a theorem that was complicated and abstract
enough that none of them had any idea why it was true. They had just been
given a method of attack, but had no particular reason to have any confidence
in it.

I expected and wanted this surprise. I wanted them to take this handout
seriously. I thought that the experience of seeing how easy an apparently
impossible problem really was would make them pay attention.

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gjm11
This is (1) probably not useful to anyone with even rudimentary problem-
solving skills and (2) more "How to do anything" than "How to prove theorems".
But it's _for_ people with scarcely any problem-solving skills. It's kinda sad
that a university mathematics course is full of people with scarcely any
problem-solving skills, but that's hardly btilly's fault.

I wouldn't expect there to be a lot of HN participants who need it, though.

~~~
btilly
I agree that it is just divide and conquer, presented in detail, with some
specifics thrown in on techniques that are useful in mathematical proofs.

However I think that this is useful, even for students with decent problem
solving skills.

The biggest issue that I was trying to solve is clearing up misconceptions
about what a proof actually is. A beginning student knows that proofs have to
be utterly convincing. They know that even many basic and obvious things
require proof. So they are uncertain what techniques are OK to use. Worse yet,
all of their liberal arts education pushes them towards believing that the way
to make something convincing is to add verbiage, examples, supporting
references, and so on. All of which is the exact opposite of what you actually
want to do in a mathematical proof.

A second issue that I was trying to address is the art of reading problems and
figuring out what it actually says. This is why I included specific advice on
how some common phrases actually cover two separate claims.

A third issue is that students arrive at a first course where they have to do
proofs having seen (and usually ignored) lots of examples of worked proofs,
but never having done one. Even if you use divide and conquer elsewhere, there
can be a barrier to realizing that this is the technique you need to use here.

And that assumes that the students are competent at divide and conquer. But
there are many paths through academia where you can arrive at the first course
where you have to do mathematical proofs without having explicitly learning to
divide and conquer. You can complain about that situation, or do something
about it. I chose to do something about it.

------
mariusK
... and that's how I do em: <http://www.eekaproof.com>

------
sz
So the only way to solve a problem is by applying known techniques,
apparently.

~~~
mhb
Yes. And one of the techniques is _Try to Find a New Technique_

~~~
sz
Are you sure? Look at the flowchart again.

~~~
gjm11
Huh? It's right there: second column, fourth box down. You could quibble that
it isn't "one of the techniques", but the effect of that is to invalidate your
original complaint that all the chart tells you to do is to apply known
techniques.

~~~
sz
Look at the arrows.

~~~
gjm11
OK, I looked at the arrows. (1) What's your point? (2) Is there any actual
reason why you haven't said already what it is, rather than all this "Look at
the flowchart" / "Look at the arrows" nonsense?

There is an arrow from "Do known techniques apply?", labelled "No". It leads
to "Try to find a new technique". There's an arrow back from there to "Do
known techniques apply?". Obviously, if you succeeded in finding a new
technique applicable to the problem, the set of "known techniques" has
expanded and you then follow the "Yes" arrow from "Do known techniques
apply?".

Is it your opinion that there is something wrong with this? If so, what?

(If you respond with another passive-aggressive "Look at X" reply, I shall
ignore it unless looking at X immediately convinces me that you've been right
all along and I've been missing something.)

~~~
sz
(1) When I see the phrase "known techniques" in the context of math it
generally refers to techniques known in the field. Novel techniques are not
"known" in this sense. The different, literal, trivial interpretation of
"known" makes the word superfluous. That was the motivation for my first
comment. In both cases (your point of view being the second) my comment is
true; by this flowchart a technique must be known to be applied. The only way
I could see someone disagreeing is if they're confused about what "known"
means.

(2) It seemed obvious, and moreover too trivial a point to merit more than
just pointing it out. I apologize if I've caused any offense.

* To clarify, I don't consider "find a new technique" to be a technique, which is why I said to look at the arrows.

~~~
btilly
Please don't dismiss the "literal, trivial interpretation". If you read the
explanations that came with the flowchart, they try to make it very clear that
"known" means "known to you".

Also consider the context. This was a handout to students in a first linear
algebra class, that was meant to help them learn to do basic proofs on their
homework problems. Nobody expected them to be engaged in original research.
Any useful technique they needed to "discover" was very likely to be well-
known to lots of people, including me.

Finally it is not clear to me why you think that the word "known" is
superfluous. There is a world of difference between the stage where you are
running through the techniques you know, trying to find one that fits, and the
stage where you're engaged in expanding your list of available techniques. I
was trying to get at that difference.

~~~
sz
I didn't read the descriptions before, they do make it clear.

I was just saying that you can't apply a technique you don't know.

