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Apr
15
comment What philosophical consequence of Goedel's incompleteness theorems?
Do you have any good secondary sources? In general I agree with this answer. The only problem I have is the way you ask the question 'How can we be sure something is true just because we see a mathematical proof of it?'. When I read about mathematics of the late 19th/early 20th century I don't see that mathematicians as Hilbert, Frege, Russell, even Goedel are principally concerned about truth. They wanted to formalize mathematics into consistent and complete formal systems and eliminate contradictions, but were they really concerned about truth?
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Apr
13
comment What philosophical consequence of Goedel's incompleteness theorems?
@Arturo Magidin: of course you cannot generalize it to each topic in mathematics. In my opinion (and so it was in my course) goedel's incompleteness theorems are amongst the top 10 theorems in mathematics you can write a philosophical essay about.
Apr
13
comment What philosophical consequence of Goedel's incompleteness theorems?
Yes I know. But I decided to write an essay (for a course philosophy of science) about the incompleteness theorems and their philosophical consequences. so i need something where there is no counterargument that can be stated in two sentences
Apr
13
asked What philosophical consequence of Goedel's incompleteness theorems?
Apr
13
asked Differences between between concepts related to Gödel's Incompleteness theorems: self-referencing, diagonalization and fixed point theorem?
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10
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asked Books like Concrete Mathematics for Mathematicians
Apr
4
comment Limit of alternating sum with binomial coefficient
You are right, I changed it to $\log(n)$. Thanks for your help.
Apr
4
comment Limit of alternating sum with binomial coefficient
I don't know how to make a comment to the previous answer. But I think you forgot the $\frac{1}{x}$ in $\int_{r}^{r+1} e^{-x \log n} (1+O(1/n)) \Gamma(x+1) dx$, or you should use $\int_{r}^{r+1} e^{-x \log n} (1+O(1/n)) \Gamma(x) dx$ instead. So the answer should be (starting the sum from $0$): $\sum_{k=0}^n (-1)^k \binom{n}{k} \log(a+bk) =\frac{\Gamma(r)}{n^r \log n} (1+O(1/\log n)$? Is it ok to write $\sum_{k=0}^n (-1)^k \binom{n}{k} \log(a+bk) \approx \frac{\Gamma(r)}{n^r \log n}$, and so actually $\lim_{n \to \infty} \sum_{k=0}^n (-1)^k \binom{n}{k} \log(a+bk) = 0$ for $r$ fixed?
Mar
28
comment Limit of alternating sum with binomial coefficient
Thanks a lot. I will try to understand it. It seems somehow easier than the proof in the previous question (just $\log(k)$), right? Yes, the $-\log(a)$ in your last line cancels out, or same I could from the beginning have started the sum at 0.
Mar
27
comment Limit of alternating sum with binomial coefficient
Thanks, I will look through it. But in fact I know this limit (sorry should have mentioned this). Though I haven't yet tried to understand the proof because it seemed hard and I am not sure if it helps me with my equation. The serie in my question arises from a generalization where the easy case leads to the sum $\sum\limits_{k=1}^n \binom{n}{k}(-1)^k \log k$.
Mar
27
asked Limit of alternating sum with binomial coefficient