Wikipedia:Reference desk/Archives/Mathematics/2024 February 26

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February 26

"Credence"

If I understand correctly, "credence" is the same as "subjective probability". Is this term really widely used/preferred? If not, it would make more sense to rename that article (currently "subjective probability" redirects to a section in "Bayesian probability") and link it to d:Q25382706. — Mikhail Ryazanov (talk) 00:46, 26 February 2024 (UTC)

One relevant question – to which I don't have the answer – is whether an encyclopedic treatment of the notion of "subjective probability" requires the Bayesian interpretation of probability.* The lead of our article on interpretations of probability states that there are two broad categories of probability interpretations, which can be called "physical" and "evidential" probabilities, and then goes on to claim that evidential probability is "also called Bayesian probability". Now I for one am very hesitant to attach the label "evidential probability" to subjective probability, given my subjectively high probability estimate for people happily attaching subjective probabilities to events without any evidential basis. I must confess, though, I have no evidence for this estimate. The actual treatment of the Bayesian interpretation then follows in the section Probability interpretations § Subjectivism, which begins with Main article: Bayesian probability and does not discuss other interpretations. I quote the first two sentences, because they are immediately relevant to the question posed here:

Subjectivists, also known as Bayesians or followers of epistemic probability, give the notion of probability a subjective status by regarding it as a measure of the 'degree of belief' of the individual assessing the uncertainty of a particular situation. Epistemic or subjective probability is sometimes called credence, as opposed to the term chance for a propensity probability.

This is all unsourced. To complicate the matter, the lead of our article on Bayesian probability states that Bayesian probability belongs to the category of evidential probabilities. This is a strange formulation if Bayesian probability is the only member of that category. Evidential probability is a red link, while Epistemic probability redirects to Uncertainty quantification, which is interpretation-agnostic.

There is clearly some need of more terminological precision and clarity here, which should be based on sourced material, while care should be taken not to assume that if one author equates two terms, it represents a generally accepted view.  --Lambiam 09:04, 26 February 2024 (UTC)

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*One datapoint: the article Субъективная вероятность (Subjective probability) on the Russian Wikipedia does not refer to Bayesian probability in the main text, but only indirectly by referring to I. J. Good's monograph The Estimation of Probabilities: An Essay on Modern Bayesian Methods as a significant contribution to the development of the theory of subjective probability. It states that the first formulation of subjective probability was given by Frank Ramsey in 1926, presumably in his paper Truth and Probability (which does not mention Bayes, but may perhaps nevertheless be Bayesian in its approach).

Can these numbers have these names?

Can these numbers have these names?

1. (sequence A153501 in the OEIS): Abundant-perfect numbers (compare with (sequence A271816 in the OEIS): Deficient-perfect numbers)
2. (sequence A005382 in the OEIS): Sophie-Germain primes of the second kind
3. (sequence A005383 in the OEIS): Safe primes of the second kind
4. (sequence A112715 in the OEIS): Proth primes of the second kind
5. (sequence A005105 in the OEIS): Pierpont primes of the second kind

Like "Cunningham chain of the second kind", also like that Woodall primes may be called "Cullen primes of the second kind" 36.233.209.202 (talk) 10:20, 26 February 2024 (UTC)

Is there a simple proof that all groups with order less than 60 are solvable?

Is there a simple proof that all groups with order less than 60 are solvable? 114.35.106.188 (talk) 10:45, 26 February 2024 (UTC)

Burnside's theorem means you need at least three prime factors. Then one can use Sylow's theorem - see the section small groups are not simple. And no it isn't simple ;-) NadVolum (talk) 13:21, 26 February 2024 (UTC)

Burnside is a big help, but its proof uses representation theory so I don't know if using it really qualifies as "simple". As I recall you can use a variety of tricks from elementary group theory, for example that every p-group has a non-trivial center, the Sylow theorems, etc., to reach 60 without Burnside. Either way, if you're just using basic tools then there's a lot to do and it's more like a project than an exercise. --RDBury (talk) 15:12, 26 February 2024 (UTC)

OEIS: A001034 gives the first few orders for which there is a simple group that is insolvable. NadVolum (talk) 18:25, 26 February 2024 (UTC)

OK, I saw this sequence, is there a simple proof that all groups with order less than 1092 and divisible by neither 60 nor 168 are solvable? Also I have a question, prove or disprove that all groups with order not divisible by 12 are solvable. 218.187.66.141 (talk) 17:13, 28 February 2024 (UTC)

In the previous OEIS sequence, the number 29120 is not divisible by 12. However, according to Isaac Saffold's comment, all numbers in the sequence are divisible by 4, so it would appear that all groups with order not divisible by 4 are solvable. GalacticShoe (talk) 17:30, 28 February 2024 (UTC)

That there are no non-solvable groups of odd order is the Feit–Thompson theorem. That there are no simple groups of order 2n with n odd is provable using elementary methods. Together these prove that all groups with order not divisible by 4 are solvable. There are stronger, somewhat advanced theorems, e.g. a non-Abelian simple group of even order must have order divisible by 12 or 16, but IIRC they require advanced representation theory. I'm not sure what "simple proof" is supposed to mean here. Accessible to a high school student with no exposure to abstract algebra? Accessible to an undergraduate with a semester of elementary group theory? Accessible to a graduate student with several semesters of advanced group theory? Less then a page long? And what are you allowed to assume in this proof? One "simple" proof is to look at the list in the Classification of finite simple groups and verify all the non-cyclic ones have order less that 60; would that count? As it stands I think the question is too vague to be answered. --RDBury (talk) 00:53, 29 February 2024 (UTC)

Correction, the stronger theorem I mentioned should be that a non-Abelian simple group of even order must have order divisible by 12, 16 or 56. It was actually proved by Burnside in the late 19th century, so the proof is not as advanced as I was thinking. --RDBury (talk) 01:28, 29 February 2024 (UTC)