Quote: Original post by nullsquared

Quote: Original post by cowsarenotevil
To be honest, though, a person would have to be kind of dumb to see a proof that starts with the statement that is supposed to be proven and not at least consider the possibility that it is, in fact, a correct proof written "backwards."

The question is: does that make it a valid proof? I can write this sentence backwards, but does that make it a valid sentence? After all, it has all the same words and punctuation.

In both cases the answer is "yes, as long as you correctly notate the way it's meant to be interpreted."

Quote: Original post by Sneftel

Quote: Original post by aryx
Actually, we do disagree to a certain extent. By starting with sin^2 + cos^2 = 1 and leading up to a trivially true statement (sin^2 + cos^2 = sin^2 + cos^2), you haven't really proved anything.

Sure you have. Given E=E, you've shown that since E=D, D=C, C=B, and B=A, E=A. I guess the notation's a bit confusing if you haven't seen it and are used to two-column proofs, but it's a convenient form for constructing identities.

The problem comes in when you're not starting from E=E - that's where the whole issue stems from in this thread.

Quote:
In both cases the answer is "yes, as long as you correctly notate the way it's meant to be interpreted."

Okay now I understand you.

Quote: Original post by aryx

Quote: Original post by Sneftel
I don't think we disagree, except on notation. I've often seen the "only change one side" form, but it's limited to proving equations, so it's pretty uncommon among proofs as a whole. In either case, the sufficiency of the proof is clearly verifiable from the statements given.

Actually, we do disagree to a certain extent. By starting with sin^2 + cos^2 = 1 and leading up to a trivially true statement (sin^2 + cos^2 = sin^2 + cos^2), you haven't really proved anything. In particular, what you've shown is that by assuming sin^2 + cos^2 = 1 is true, you can show that sin^2 + cos^2 = sin^2 + cos^2.

One could simply in this instance interpret the implications as going both ways.

This is a common problem solving strategy. One starts with the claim to be proved and performs only reversible steps to obtain something evidently true.

But I agree that it is best to explicitly indicate stuff like this in proofs.

[Edited by - nilkn on March 12, 2010 11:44:26 PM]

Quote: Original post by nilkn
One could simply in this instance interpret the implications as going both ways.

This is a common problem solving strategy. One starts with the claim to be proved and performs only reversible steps to obtain something evidently true.

Sorry to nitpick, but I like to make sure things are clear :)

Indeed, I did state earlier that if these were specified as equivalencies (i.e., implications going both ways) then things would be fine. Nevertheless, if not explicitly shown in your deduction, implications are often assumed, and if they are then beginning with your conclusion is a logical fallacy. Even if you do show them as equivalencies, it's probably best to present your proofs such that you end up at your conclusion; "Better safe than sorry" they say.

I think cowsarenotevil said it best with "yes, as long as you correctly notate the way it's meant to be interpreted." I think being explicit is especially important in proofs because you avoid any possible misinterpretation. Maybe I'm just another one of those nitpicky students with a pure math degree, but it's kind of hard not to be nitpicky when you have to prove things like a + c = b + c ==> a = b.

[EDIT]
And you added "But I agree that it is best to explicitly indicate stuff like this in proofs." I think this is the most important thing to take from this thread.

To reply to the original post of nullsquared: It seems like you're on the right track as far as logic is concerned. Keep thinking for yourself. Your approach was correct for the case where cos x \neq 0, and the final statement is meaningless / ill-defined for the case where cos x = 0. So while you could have made this more precise, your solution is exactly as valid as the teacher's.

You are also right that, in order to prove something, you have to start at a trivially true statement and work your way up from there.

As others have already pointed out, this can all be made much clearer by using implication arrows. A good proof should look like this:

A (trivially true)=> B=> C=> D (what you want to prove)

(Of course, the implication arrows are not always written down explicitly, but you should think of them as always being there implicitly.)
Whereas the proofs you've shown look like this:

D<= C<= B<= A

Now in many cases when dealing with equations, your transformations are actually equivalences (or "if and only if), i.e.:

D<=> C<=> B<=> A

which justifies this mode of working at least sometimes.

However, while working backwards is often useful for finding a proof, I would strongly advise against writing proofs down in this "wrong" order. The reason is that it is very easy to make mistakes when working in the wrong order by thinking that some statements are equivalent when they're really not. As an assistant, I see even masters level students of mathematics making really stupid and unnecessary mistakes when trying to write down proofs in the "wrong" order (heck, even professors of mathematics aren't immune). It's best if you just try to keep the proof structure straightforward, flowing only from true to true statements using implications in the forward direction.

(Like all rules, this rule may be broken, but never ignored. And as long as you're not sufficiently advanced, you should probably never break it. That your teacher writes down proofs in the wrong order is a potential sign of a bad teacher. He should have at least explained these proof-order issues.)