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?- love(math) is unrequited. true.


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.
Apr
14
comment What philosophical consequence of Goedel's incompleteness theorems?
@Arturo That's true; you did not say the word 'obvious', but here you are extrapolating from the fact that Penrose can be disagreed with to claim that it means Godel's theorem has no philosophical implications. For that matter, Darwin's theory of evolution can be argued with (like any scientific theory), so should we believe it has no philosophical applications? Any given alleged philosophical application of evolution can be disagreed with, often successfully, so we are to conclude that there exist no applications to philosophy? That seems to be the reasoning you're giving.
Apr
14
comment What philosophical consequence of Goedel's incompleteness theorems?
@Quinn, you've never heard of moral computationalism?
Apr
14
comment What philosophical consequence of Goedel's incompleteness theorems?
That sounds like confirmation bias to me. Because I think that very complicated arguments can be made to counter against one the most talented physicists of the last century, you think it supports the idea that Godel's theorem obviously doesn't have Penroses' alleged consequences. Hrmm.. Though I do believe Penrose can be argued with, I think one can only do so if willing to really think about the philosophical extent of Godel's theorem. It's not simple to say humans can't reason about the formal system they are. And, to boot, none of this says anything about the Kritchman and Raz paper.
Apr
14
comment What philosophical consequence of Goedel's incompleteness theorems?
Godel's theorem has most definitely added to philosophy, in varied and interesting ways, no less.
Apr
14
comment What philosophical consequence of Goedel's incompleteness theorems?
These comments are amazingly short-sighted. There is actually a wealth of legitimate philosophical consequences of Godel's theorems. See my answer from below.
Apr
14
comment What philosophical consequence of Goedel's incompleteness theorems?
This is a totally legitimate answer. It's upsetting that it was downvoted. As I mentioned in my answer above there was a popular math paper published just last year on Godel's theorem and its application to the surprise examination paradox.
Apr
14
comment What philosophical consequence of Goedel's incompleteness theorems?
@Iasafro: your answer is totally a valid one. Don't let the snarky naysayers nor downvoters discourage you. The entire field of machine ethics, a la Nick Bostrom and Steve Omohundro, uses very serious maths and logic to address exactly this kind of philosophical problem in strong A.I. And, as I indicated in my answer above, there are a ton of ways that Godel's theorem is relevant to philosophy.
Apr
14
answered Why bother with Mathematics, if Gödel's Incompleteness Theorem is true?
Apr
14
comment Why bother with Mathematics, if Gödel's Incompleteness Theorem is true?
These arguments are funny. Godel himself wrote a letter to Von Neumann in which he basically said "If P = NP, then nothing will be left for human mathematicians apart from the postulation of axioms." (He used different language because P v NP wasn't really known at the time). See this paper and my answer to another Godel question for more.