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 Apr 21 comment Cluster Analysis Terminology question In the applied machine learning literature, 'hypergraph' is the appropriate term for most applications, and it is meant in the full mathematical sense. You see that in everything from stats papers to computer vision to social networks. See my answer below. Apr 21 answered Cluster Analysis Terminology question Apr 21 answered Solving $Ax = B$ when $A$ has a large condition number. Apr 21 answered Weird conclusion about variance/covariance from differentiating Apr 21 comment Weird conclusion about variance/covariance from differentiating I think something is being lost in translation in the OP. The question should be asking: suppose $a$ is chosen to optimize portfolio risk between $X$ and $Y$, and is chosen such that $\text{COV}(X,Y) = a\text{Var}(X)$. Then show that $a$ must be 1. Apr 18 answered Books for Understanding Bayesian probability from the Beginning Apr 18 comment Can't see how $e^{\operatorname{Log}(z)} = z$ in these notes What is $Re[z]$ and $Im[z]$ in terms of $|z|$ and the angle $z$ makes with the positive real axis? It's just polar coordinates, basically. Apr 16 comment How can Radon-Nikodym and Borel-Cantelli be used to calculate Probability distribution? The paragraph on the Radon-Nikodym connection makes sense to me. After all, that's the precise way in which the Cantor function (which is a valid CDF) is shown to have no PDF. Apr 15 comment How to expand undifferential function as power series? Check out the basics linked here. Your question was something that spurred decades of math research and resulted in radically changing the notion of a function. Fourier series is one method to get some kinds of convergence properties for series expansions of non-differentiable functions, when the points causing non-differentiability are sufficiently well-behaved. Apr 15 comment How to expand undifferential function as power series? Re-derive Fourier series? Apr 15 comment Why bother with Mathematics, if Gödel's Incompleteness Theorem is true? @CarlMummert I totally agree. If you follow my link to the answer to the other question, you'll see a long passage that discusses a rebuttal to Penrose's ideas. Actually, Penrose took that idea from Lucas, who proposed it much earlier, I think in the 60s, without all of the quantum gravity hoopla. Apr 15 comment Why bother with Mathematics, if Gödel's Incompleteness Theorem is true? @Didier, I agree with you 100%. That component of this question should be isolated from question that the title of the post asks. Godel himself asked the question that is in the title and the only answered that seemed to satisfy him was "well, let's hope $P\neq{NP}$". But the stuff about approximating absolute truth, etc., is not appropriate for Godel's theorems. That's a whole different philosophical beast and the analogy with Newtonian-to-Quantum physics isn't a good one given the state of the art in the philosophy of truth. Apr 14 comment What philosophical consequence of Goedel's incompleteness theorems? Be closed-minded about it if you want to, but the bulk of the uses of Godel's theorems in philosophy come from mathematicians, especially decision theorists. It's a non-trivial part of the philosophy of mathematics. You are certainly free to disagree with it if you want, but it's nothing but disingenuous to claim that there are virtually no philosophical ramifications of Godel's theorem. Apr 14 comment What philosophical consequence of Goedel's incompleteness theorems? Look, you argued above that because GIT is a theorem, what matters here is actually verifying that its hypotheses are met by aspects of reality before we can say it has philosophical ramifications. Yet, with respect to things like Aumann's Agreement Theorem (the hypotheses of which are surely not met by non-Bayesian human brains), experts in decision theory and the epistemology of probability still argue that it has serious philosophical ramifications. I'm sorry if you didn't understand why my critique of your theorem comment was valid, but it was. Apr 14 comment What philosophical consequence of Goedel's incompleteness theorems? @IasafroMaesman This is very similar to the kinds of things that decision theorists study. I love both the logical approach to it, mentioned with your naive example, and also the practical approach, via things like the Kahneman and Tversky work on cognitive biases (scope insensitivity seems relevant to your specific question). I'm glad that you clarified your thoughts. It is a useful contribution to this thread. Apr 14 comment What philosophical consequence of Goedel's incompleteness theorems? And I willfully admit that I haven't read Franzen's book. I should do so given what you are saying, but all my peers who study decision theory laugh about that book. They essentially consider it a useless straw-man type argument against really dumb uses of the incompleteness theorems. I'm not advocating dumb uses; I'm advocating legit uses (legit in the sense that they have been deemed legit by the same peer review process that deems any other math research legit). Apr 14 comment What philosophical consequence of Goedel's incompleteness theorems? I'm not referring to creationists. I was more referring to the arguments that a pure Bayesian reasoner could be viewed (just philosophically, not biologically, due to computability arguments) as a fixed point of Darwinian evolution. I do not consider creationist claims valid enough to even merit a response. And you still haven't responded on points about Kritchman and Raz except to say that you don't think it's philosophical. Well, that's great, but doesn't jive with the math community at large. Apr 14 comment What philosophical consequence of Goedel's incompleteness theorems? Also, Arrow's Impossibility Theorem and Aumann's Agreement Theorem are also theorems, but it is widely accepted that these have many philosophical ramifications. The agreement theorem in particular has many interesting consequences in Bayesian decision theory, which is a branch of formal philosophy. Status as a theorem doesn't preclude something from being relevant to philosophy. See this for example. Apr 14 comment What philosophical consequence of Goedel's incompleteness theorems? Maybe don't use quotes if you're not inventive enough to look for things that Google can't hand you directly. Here's a useful link to get you started. You may want to read the sequence on reductionism first to get anything out of the sequence on metaethics. Apr 14 revised Math Database For Problem Descriptions In An App. edited body