Wikipedia:Reference desk/Archives/Mathematics/2013 January 12

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January 12

Computing the CDF of the Student's T-distribution

Is there a formula that approximates the CDF of the Student's t-distribution to at least five decimal places that doesn't use integrals, infinite sums, or expressions as long as your arm? --67.160.38.148 (talk) 02:37, 12 January 2013 (UTC)

Probably not, except that Student's t-distribution#Special cases gives simple formulas when the number of degrees of freedom is one or two. Duoduoduo (talk) 14:03, 12 January 2013 (UTC)

π = √10?

Is pi's value equal to that of the square root of 10? --Yashowardhani (talk) 07:59, 12 January 2013 (UTC)

No. See the article Pi, which makes this clear (this specific case is mentioned in the Properties section). As you could confirm using a calculator, they differ already in the second digit after the decimal point. — Quondum 08:41, 12 January 2013 (UTC)

Agreed. If you want an approximation, 22/7 is much closer. StuRat (talk) 09:58, 12 January 2013 (UTC)

It was common to cancel π with √10 (as if they were equal) if you were aiming for a rough approximation in the days before calculators. Dbfirs 08:27, 14 January 2013 (UTC)

Sum of products

Hello again. Can you prove the following inequality: $\sum _{\vec {\alpha }}\prod _{i=1}^{K}\alpha _{i}^{\alpha _{i}}\leq A^{A+{\frac {K-1}{2}}}$ where the sum $\sum _{\vec {\alpha }}$ is taken over all $K$-dimensional vectors whose entries are nonnegative integers which sum to $A$?--AnalysisAlgebra (talk) 12:30, 12 January 2013 (UTC)

You may like to use an Iverson bracket ([true]=1 and [false]=0) to write your LHS expression like this

$\sum _{\alpha \in \mathbb {N} ^{K}}\left[A=\sum \alpha \right]\prod _{i=1}^{K}\alpha _{i}^{\alpha _{i}}$

Note that $\alpha _{i}\leq A$ for 1 ≤ i ≤ K such that

$\left[A=\sum \alpha \right]\alpha _{i}^{\alpha _{i}}\leq \left[A=\sum \alpha \right]A^{\alpha _{i}}$

and

$\left[A=\sum \alpha \right]\prod _{i=1}^{K}\alpha _{i}^{\alpha _{i}}\leq \left[A=\sum \alpha \right]\prod _{i=1}^{K}A^{\alpha _{i}}=\left[A=\sum \alpha \right]A^{\sum \alpha }\leq A^{A}$

such that

$\sum _{\alpha \in \mathbb {N} ^{K}}\left[A=\sum \alpha \right]\prod _{i=1}^{K}\alpha _{i}^{\alpha _{i}}\leq A^{A}\sum _{\alpha \in \mathbb {N} ^{K}}\left[A=\sum \alpha \right]=A^{A}{\binom {A+K-1}{A}}$

Bo Jacoby (talk) 01:35, 13 January 2013 (UTC).

would it be a proof by construction of the possible existence of God?

Suppose a strong atheist says, "it is impossible for God to exist". We define terms extremely clearly for what this would mean. By these terms, I write a computer program (it can be quite complex, model a universe with physical laws, etc) whatever it takes to create something substantial.

If, by the atheist's definition, I am the "God" over this universe, then have I proven, by construction, the possible existence of God? (In that I have produced such a relationship and case.) — Preceding unsigned comment added by 178.48.114.143 (talk) 23:22, 12 January 2013 (UTC)

All you are saying there in mathematical terms is if there is a definition of X and one can show X exists then does X exist. That is rather trivial mathematics. Perhaps you want something else, in which case maybe the Science or Humanities reference may be able to help you better. Dmcq (talk) 01:09, 13 January 2013 (UTC)

Not exactly what you asked, but you may be interested in Gödel's ontological proof of the existence of God. --Trovatore (talk) 01:22, 13 January 2013 (UTC)

I don't think any "strong atheist" says that it's impossible for God to exist. Strong atheists say that the evidence is overwhelming that God does not exist. Duoduoduo (talk) 01:51, 13 January 2013 (UTC)

No, I don't think that's true — some of them claim that the concept of God is logically contradictory, and therefore it's not just impossible, but actually logically impossible, for God to exist. My personal opinion is that both they and, on the other side, the ones who claim to prove by logic alone that God does exist, are missing the point, but nevertheless it is a position that a nontrivial number of people take. --Trovatore (talk) 01:55, 13 January 2013 (UTC)

Also, anyone who claims to have evidence for the non-existence of God or anything else is talking out their backside. Absence of evidence is not evidence of absence. Non-believers are utterly unable to prove the non-existence of God by recourse to evidence, just as believers are utterly unable to prove the existence of God by recourse to evidence. It's all entirely about belief or lack thereof. There is nothing else. -- Jack of Oz ^[Talk] 07:56, 13 January 2013 (UTC)

@Trovatore: Could you give a reference I could read that says some people believe the existence of God is logically impossible (as opposed to logically incompatible with some set of evidence)?

@JackofOz: Try to keep it nice, Jack. I do assert that, and I'm not talking out of my backside. And remember that the refdesk is not a forum for your views. The way it works is this: (Existence of God) ⇒ ~A; we empirically observe A; therefore ~(Existence of God). One often asserted example of A is evil. And no, I don't want to debate it with you; my post just observed that people assert something, not that this is the place to debate whether they're right. Duoduoduo (talk) 14:39, 13 January 2013 (UTC)

I don't know that I have a secondary source, but it's implicit in any number of primary ones. Not feeling like looking them up right now, but I'm sure you can find them if you want to. --Trovatore (talk) 20:15, 13 January 2013 (UTC)

Duoduoduo, I never claimed you were the one talking in that, er, special way. But if, by the use of "we", you are putting yourself in the camp of those who assert they have evidence of the non-existence of God, then what I said would also apply to you. -- Jack of Oz ^[Talk] 10:25, 14 January 2013 (UTC)

Proof aside, evidence is often important though, to believers and nonbelievers alike, for instance see argument from inconsistent revelations. -Modocc (talk) 17:20, 14 January 2013 (UTC)

I interpret the OP's question as enquiring about logical possibility, not logical truth. To paraphrase it: It seems that we can construct, in principle, a logically possible world in which God exists. Can we conclude that such a world is logically possible?. This appears to be tautological, but in answer to the OP, the "seems" in my paraphrase is such a Pandora's box that we cannot start to answer even this simple question in the affirmative. Implied in the question are unvalidated assumptions that "God" and "world" can be defined in a meaningful and logically consistent way. We have difficulty with definition of the concepts in Russell's paradox; these are worse. From an intuitive perspective, Gödel's incompleteness theorem might shed some light: as soon as we require these definitions to have any meaningful attributes, the attempt to conclude logical possibility is likely to fail. — Quondum 07:29, 13 January 2013 (UTC)

OK, the misuse of the incompleteness theorems is one of my peeves, so don't take anything personally here. People hear vague statements of the theorems and read things into them that they just don't say. I strongly suspect that's what you're doing. If not, then what is the relevant formal theory you're describing? Can you show that Robinson arithmetic is relatively interpretable in that theory? Can you show that the theory is computably enumerable? From the conclusion of the first theorem (the theory, if consistent, fails to prove some true statement of arithmetic) or of the second theorem (the theory, if consistent, does not prove a formal statement of its own consistency), how exactly do you conclude that you can't demonstrate the logical possibility of the existence of God? --Trovatore (talk) 07:41, 13 January 2013 (UTC)

(By the way, you're in good company — one person who has committed this error in public, in an absolutely blatant and completely erroneous way, is Stephen Hawking.) --Trovatore (talk) 07:46, 13 January 2013 (UTC)

Heh – I didn't make myself clear enough: I intended the incompleteness theorems as an analogy of how intuitive preconceptions about the possibility of completeness/consistency (of logical systems in the case of the theorems, and of definitions in this case) can be flawed. I am aware that the incompleteness theorems in no way apply directly here (notice my preface "From an intuitive perspective..."), but I'm happy enough to concede the charge of abuse, at least on the grounds of possibly misleading presentation thereof and marginal relevance. I also did not intend to imply that no demonstration of the logical possibility of the existence of God exists, only that at present any such attempted demonstration is likely to be highly implausible because the issues I mentioned (firstly suitable definitions, e.g. one for a god that is adequately distinguishable from one for a wiggly-woo, and secondly logical consistency of the assumptions) need serious effort to address. In the OP's argument particular, the first is only addressed as a god in some sense akin to a creator of a toy universe, and the second is not addressed at all. — Quondum 09:19, 13 January 2013 (UTC)

Well, here we get back to what I was saying about missing the point. The question is not "can I come up with some abstract definition of God and show that there is/is not a being matching the definition, or that his existence is/is not logically/otherwise possible". The question is "that which I perceive as God (or perhaps understand as God), is he real?". If you don't perceive God, or if you do but don't realize that what you perceive should be called God, then that question won't make sense to you at all; this subjective component is the most fundamental reason that it misses the point to argue the question from the standpoint of abstract logic. --Trovatore (talk) 20:15, 13 January 2013 (UTC)

Huh? I cannot find this subjective aspect in the question, despite careful re-reading. — Quondum 08:55, 14 January 2013 (UTC)

I was referring to my first response to Duoduoduo, not so much to the original question. However, I do feel that the original question misses the point in this way. --Trovatore (talk) 09:06, 14 January 2013 (UTC)

Oh, maybe here's the confusion: When I say "the question is not...", I don't mean "the original question is not...". I mean "the {correct|useful|interesting} question is not...". --Trovatore (talk) 09:08, 14 January 2013 (UTC)

Okay, but that angle perhaps belongs elsewhere. — Quondum 09:18, 14 January 2013 (UTC)

(By the way, I should maybe clarify that the statement I was berating Hawking for was not about God. It was about a theory of everything.) --Trovatore (talk) 20:53, 13 January 2013 (UTC)