Wikipedia:Reference desk/Archives/Mathematics/2013 August 4
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August 4
editmath
editwhat is the greatest common factor of32 and 28 plase ans — Preceding unsigned comment added by 39.53.158.255 (talk) 16:12, 4 August 2013 (UTC)
- Why don't you read Greatest common divisor and see if you can't work it out for yourself? Rojomoke (talk) 16:58, 4 August 2013 (UTC)
- The greatest common factor of 32 and 28 is . Widener (talk) 17:38, 4 August 2013 (UTC)
- GCD(32,28) = GCD(28,32 Mod 28) = GCD(28,4) = GCD(4,28 Mod 4) = GCD(4,0) = 4. Count Iblis (talk) 19:31, 4 August 2013 (UTC)
probability
editIn Littlewood's Miscellany, he talks about the difficulty of setting out a formal axiomatic theory of probability. He talks about a possible "axiom of probability" and calls it "A". He says:
We have not merely failed to justify a workable A ; we have failed even to state one which would work if its truth were granted. It is generally agreed that the frequency theory won't work. But whatever the theory it is clear that the vicious circle is very deep-seated: certainty being impossible, whatever A is made to state can only be in terms of ' probability '. One is tempted to the extreme rashness of saying that the problem is insoluble . . ."
What work has been done in the intervening time on this issue? What reputable sources are there that discuss this? thanks, Robinh (talk) 20:38, 4 August 2013 (UTC)
- Honestly, this doesn't sound like an issue about axiomatics at all — you can certainly write down "axioms" of probability that are basically a rephrasing of the definition of probability space. What Littlewood seems to be talking about is interpretation. That's a much more difficult question. There are two broad schools, frequentism and Bayesianism. --Trovatore (talk) 20:49, 4 August 2013 (UTC)
- (OP) thanks Trovatore. I guess interpretation is what I'm asking about. Littlewood was wondering about the number of trials needed before the Law of large numbers kicked in. I think he was concerned that there was nothing to prevent one requiring graham's number of trials to ensure enough regularity for useful results. Robinh (talk) 21:00, 4 August 2013 (UTC)
- Well, sure. Actually it's worse than that. The law of large numbers tells you only what happens with probability 1, but probability 1 is not the same as certainty (see almost surely), so if you're trying to base your interpretation on the law of large numbers, then there's an inherent circularity that you have to either get around somehow or ignore. For me that's the killing argument against frequentism, and makes Bayesianism attractive by comparison. On the other hand Bayesianism has its own issues, as you then seem to owe an account of what "prior probability" means. --Trovatore (talk) 21:20, 4 August 2013 (UTC)
- Thanks for this Trovatore. This is what I wanted to know. Presumably something has been written on this inherent circularity since Littlewood's comments? What literature is out there on this topic? Robinh (talk) 21:31, 4 August 2013 (UTC)
- I don't think I can really help you there. Anyone else have any leads? --Trovatore (talk) 21:46, 4 August 2013 (UTC)
- Thanks for this Trovatore. This is what I wanted to know. Presumably something has been written on this inherent circularity since Littlewood's comments? What literature is out there on this topic? Robinh (talk) 21:31, 4 August 2013 (UTC)
- Well, sure. Actually it's worse than that. The law of large numbers tells you only what happens with probability 1, but probability 1 is not the same as certainty (see almost surely), so if you're trying to base your interpretation on the law of large numbers, then there's an inherent circularity that you have to either get around somehow or ignore. For me that's the killing argument against frequentism, and makes Bayesianism attractive by comparison. On the other hand Bayesianism has its own issues, as you then seem to owe an account of what "prior probability" means. --Trovatore (talk) 21:20, 4 August 2013 (UTC)
- (OP) thanks Trovatore. I guess interpretation is what I'm asking about. Littlewood was wondering about the number of trials needed before the Law of large numbers kicked in. I think he was concerned that there was nothing to prevent one requiring graham's number of trials to ensure enough regularity for useful results. Robinh (talk) 21:00, 4 August 2013 (UTC)
- Honestly, this doesn't sound like an issue about axiomatics at all — you can certainly write down "axioms" of probability that are basically a rephrasing of the definition of probability space. What Littlewood seems to be talking about is interpretation. That's a much more difficult question. There are two broad schools, frequentism and Bayesianism. --Trovatore (talk) 20:49, 4 August 2013 (UTC)
There are two interpretations: the probability for a future event to happen, or that some possibility is true. Using the word credibility for the second concept may reduce confusion. So you may talk about the credibility that there is intelligent life on the moon. Bo Jacoby (talk) 13:37, 5 August 2013 (UTC).
- After reading the above, I see this as basically a question about the philosophy of mathematics. I usually go the the very good Stanford_Encyclopedia_of_Philosophy for this sort of thing. Here's their page on the topic, which has several refs younger than Littlewood [1]. You may also check out the similar articles from SEP's competitors, listed at the bottom of our article. SemanticMantis (talk) 15:18, 5 August 2013 (UTC)