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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
Apr
14
comment Math Database For Problem Descriptions In An App.
Thanks for catching the typo. I fixed it.
Apr
14
comment What philosophical consequence of Goedel's incompleteness theorems?
Another link is here. This is more speculative, which is why I didn't put it in the answer below, but it is a great attempt to show how incompleteness arises naturally in decision theory, and in particular w.r.t. Newcomb's paradox. This is a highly philosophical question, and it appears that incompleteness relates to it non-trivially. Overall, I'm advocating that more credence ought to be given to the proposition that the incompleteness theorems have legitimate philosophical side effects. That's all.
Apr
14
comment What philosophical consequence of Goedel's incompleteness theorems?
Look, no one is disputing that people can draw erroneous conclusions from Godel's theorem. But the questions you ask about alternate logics and so on are exactly why it is valid to ask philosophical questions about Godel's theorem. Do you believe that human mathematicians can "see" the consistency of whatever axiomatic theory they are equivalent to? A lot of people think the answer is yes, even some mathematicians. I think a person can argue the answer is no, especially if you believe in computationalism or reductionism, but it's by no means a settled philosophical question at all.
Apr
14
comment What philosophical consequence of Goedel's incompleteness theorems?
That's totally disingenuous reasoning, and I didn't think it was unreasonable to extrapolate the "obvious" that I claimed was part of your argument above.
Apr
14
comment What philosophical consequence of Goedel's incompleteness theorems?
Or, maybe to be more mathematical, what about, say information theory? Here is a draft of a paper that I wrote on applications of information theory and the machine learning idea of boosting to philosophy. Should we also say that Leslie Valiant's work on evolvability has no applications to philosophy? I think you should read the Aaronson paper that I linked below. Just because one person invokes something and it gets debunked through hard work doesn't mean there are no applications.