0
votes
2answers
44 views

assuming the conclusion

A natural deduction proof goes from premmisses to conclusion, and under normal circumstances you will not assume the conclusion. Sometimes you may assume the negation of the conclusion and do some ...
0
votes
2answers
86 views

What are some good introductory books on mathematical proofs?

There was a time when I avoided math proofs, but now I am starting to enjoy them. I am taking Intro to Linear Algebra and am falling in love with proofs. Are there any introduction to mathematical ...
1
vote
1answer
77 views

Have I understood the whole Incompleteness business correctly?

I am reading Gödel-Escher-Bach and a good dialogue by Eliezer Yudkowsky and I think I might have understood the nature of the Completeness and Incompleteness theorems (at least regarding Peano ...
3
votes
1answer
82 views

Success of Hilbert's Axioms

We know Euclid's axioms were found to be having many loopholes as in there were still many assumptions which weren't being stated in his system of axioms . Are Hilbert's axioms today completely ...
2
votes
1answer
141 views

A pedantic question about defining new structures in a path-independent way.

Sometimes there are multiple equivalent ways of defining the same structure; for example, topological spaces are determined by their open sets, but also by their closed sets. I'm looking for a way of ...
13
votes
7answers
1k views

Is a counterexample considered a rigorous proof that a property is not true?

This is my follow-up question to my own query earlier: How can I algebraically prove that $2^n - 1$ is not always prime? Almost half of the answers said that I provided my own proof by giving ...
3
votes
3answers
298 views

What do we mean by an “Elegant Proof”? [closed]

What do we mean when we say that a mathematical proof is elegant? Of course one can say that the proof is beautiful, but what do we precisely mean when we say that a proof is beautiful ? Is there a ...
12
votes
6answers
1k views

Why do statements which appear elementary have complicated proofs?

The motivation for this question is : http://math.stackexchange.com/questions/4066/rationals-of-the-form-fracpq-where-p-q-are-primes-in-a-b and some other problems in Mathematics which looks as if ...