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So many of these proofs are of the form: 1. Assume the desired conclusion is false. 2. Make an error. 3. Derive a contradiction. And by the way, most mathemat

So many of these proofs are of the form: 1. Assume the desired conclusion is false. 2. Make an error. 3. Derive a contradiction. And by the way, most mathematicians have made plenty of bad proofs. I told my thesis adviser I had a "proof" months before I actually did. Years before that, I found a short proof of Fermat's Last Theorem (which was however too long for the margin of my book ...). Generally, the way "big" proofs go is that there's a whole lot of new mathematics, and some handwaving at various points along the way. Then the work starts of checking whether the handwaves are actually accurate. The better the mathematician, the more likely it is that they actually are ...

raverbashing 14y ago
Yes Because proofs (or disproofs) by contradiction are "the easy way out" Proving something is right is much harder if not impossible (hi Mr. Goedel)
- CurtMonash 14y ago
  That wasn't my point. Rather, I was suggesting that proofs by contradiction can easily be in error (because if you make an error, you have a good chance of eventually getting to your contradiction). Anyhow, when I did mathematics, I didn't run into people caring about proof by contradiction vs. direct proof. However, I did occasionally run into a preference for constructive vs. non-constructive proofs. E.g., my thesis used a prior result that included a funky existence proof, and my adviser pushed me to find an alternate proof that was directly constructive.
  - raverbashing 14y ago
    Oh yes, I had understood that "proofs by contradiction can easily be in error" but I suppose that's because people try the contradiction proof first, because it "should be easier" And of course, constructive proofs are usually more complicated, in terms of proof structure as well as 'where to go and how to get there'
    - CurtMonash 14y ago
      My comment was based on a more precise point, namely that #3 in the process below is very natural: 1. Assume the theorem is false. 2. Make some calculations based on that assumption. 3. Eventually, make an error. 4. Pursue that error to a contradiction. 5. Claim victory.

raverbashing 14y ago

Yes Because proofs (or disproofs) by contradiction are "the easy way out" Proving something is right is much harder if not impossible (hi Mr. Goedel)

CurtMonash 14y ago
That wasn't my point. Rather, I was suggesting that proofs by contradiction can easily be in error (because if you make an error, you have a good chance of eventually getting to your contradiction). Anyhow, when I did mathematics, I didn't run into people caring about proof by contradiction vs. direct proof. However, I did occasionally run into a preference for constructive vs. non-constructive proofs. E.g., my thesis used a prior result that included a funky existence proof, and my adviser pushed me to find an alternate proof that was directly constructive.
- raverbashing 14y ago
  Oh yes, I had understood that "proofs by contradiction can easily be in error" but I suppose that's because people try the contradiction proof first, because it "should be easier" And of course, constructive proofs are usually more complicated, in terms of proof structure as well as 'where to go and how to get there'
  - CurtMonash 14y ago
    My comment was based on a more precise point, namely that #3 in the process below is very natural: 1. Assume the theorem is false. 2. Make some calculations based on that assumption. 3. Eventually, make an error. 4. Pursue that error to a contradiction. 5. Claim victory.