Stefan Walter
Reputation
2,264
Next privilege 2,500 Rep.
Create tag synonyms
 Mar 9 awarded Popular Question Nov 25 comment What is the purpose of showing some numbers exist? @Vandermonde: "the assertion of the existence within $\mathbb{R}$ of $\sqrt{2}$ feels like what it's really claiming is the existence within $\mathbb{R}$ of an element with 'the $\sqrt{2}$ property'" I agree, but I don't think the OP understood that. If it's clear what you mean, you can certainly talk of the existence of $\sqrt{2}$. I still don't like it, but that's a matter of taste. Oct 26 comment How much do we really care about Riemann integration compared to Lebesgue integration? @Ennar: I agree with the quote, but Dieudonné certainly didn't speak for all "experts". The overwhelming majority of university professors seems to still believe in the didactic value of the Riemann integral. Oct 22 asked Conditions on a permutation of $\mathbf{N}$ such that rearrangements of series don't change convergence Oct 19 comment What is the purpose of showing some numbers exist? Understood, thanks. I agree that all these "elementary" notions should be carefully defined with rigorous proofs. However, most students have never thought about if, why and in what manner numbers exist. That's just not an issue at school. So, instead of "we'll now prove something you've always known to be true", I would rather stress that the "existence" of $\sqrt{2}$ isn't some deep fact, but simply a way of talking about, let's say, approximate rational solutions of $x^2=2$. Oct 19 comment What is the purpose of showing some numbers exist? IMO @Max's comment is the correct answer to this question. It is a misconception to think that existence is a property that numbers may or may not have. Before you even write $\sqrt{2}$ you must define what you mean by that. The proof shown by the professor has to be done before (or be a part of) this definition. Oct 19 comment What is the purpose of showing some numbers exist? @Max: You can't just define $\sqrt{2}:=p$, you have to establish uniqueness first. In this case there are two numbers with that property, one negative and one positive, and we define $\sqrt{2}$ to be the positive one. Oct 19 comment What is the purpose of showing some numbers exist? "You believe $\sqrt{2}$ is a number... so the question is whether or not it's a real number." I don't understand that sentence. What kind of numbers are you referring to? Surely the OP doesn't know about complex numbers yet. Sep 13 awarded Yearling Sep 9 comment Why does $e$ have multiple definitions? This is different from the question in that taking any of these as the definition of $1$ would indeed be circular. Jun 21 comment Can any symmetric bounded bilinear mapping occur as the second derivative? @I disagree: Thanks a lot. The assumption that $E$ is finite-dimensional does not seem to be necessary, as one can simply define $f(x)=\frac{1}{2}\nu(x,x)$ in the general case. Would you like to write that as an answer or should I? Jun 19 comment Can any symmetric bounded bilinear mapping occur as the second derivative? Feel free to change the tags, as I'm not sure. Jun 19 asked Can any symmetric bounded bilinear mapping occur as the second derivative? Jan 20 awarded Nice Question Dec 9 awarded Caucus Dec 3 awarded Notable Question Sep 22 awarded Favorite Question Sep 13 awarded Yearling Sep 10 reviewed Approve Prove by induction: $1(1!)+\cdots + n\cdot n!$ = (n+1)! - 1 Aug 29 reviewed Approve Infimum and supremum of the empty set