現代数学の系譜 工学物理雑談 古典ガロア理論も読む65
■ このスレッドは過去ログ倉庫に格納されています
この伝統あるガロアすれは、皆さまのご尽力で、
過去、数学板での勢いランキングで、常に上位です。
このスレは、現代数学のもとになった物理・工学の雑談スレとします。たまに、“古典ガロア理論も読む”とします。
それで宜しければ、どうぞ。
後でも触れますが、基本は私スレ主のコピペ・・、まあ、言い換えれば、スクラップ帳ですな〜（＾＾
最近、AIと数学の関係が気になって、その関係の記事を集めています〜（＾＾
いま、大学数学科卒でコンピュータサイエンスもできる人が、求められていると思うんですよね。
スレ主の趣味で上記以外にも脱線しています。ネタにスレ主も理解できていないページのURLも貼ります。関連のアーカイブの役も期待して。
話題は、散らしながらです。時枝記事は、気が向いたら、たまに触れますが、それは私スレ主の気ままです。
スレ46から始まった、病的関数のリプシッツ連続の話は、なかなか面白かったです。
興味のある方は、過去ログを（＾＾
なお、
小学レベルとバカプロ固定
サイコパスのピエロ（不遇な「一石」https://textream.yahoo.co.jp/personal/history/comment?user=_SrJKWB8rTGHnA91umexH77XaNbpRq00WqwI62dl 表示名:ムダグチ博士 Yahoo! ID/ニックネーム:hyperboloid_of_two_sheets （Yahoo!でのあだ名が、「一石」）
（参考）http://blog.goo.ne.jp/grzt9u2b/e/c1f41fcec7cbc02fea03e12cf3f6a00e サイコパスの特徴、嘘を平気でつき、人をだまし、邪悪な支配ゲームに引きずり込む 2007年04月06日
（なお、サイコの発言集「実際に人を真っ二つに斬れたら 爽快極まりないだろう」、「狂犬」、「イヌコロ」、「君子豹変」については後述（＾＾； ）
High level people
低脳幼稚園児のAAお絵かき
上記は、お断り！
小学生がいますので、18金（禁）よろしくね！（＾＾
（旧スレが1000オーバー（又は間近）で、新スレを立てた） >>735-738 >>744
>スレ主よ、conglomerabilityとは何か説明してみろｗ
>ちなみに俺は理解したぞｗ
ピエロちゃんども
これはこれは、珍しく本格的だなｗ（＾＾
いいね〜（＾＾
長いけど、Alexander Pruss氏の本から抜粋するよ
ベースがないと、議論が上滑りだからね
https://books.google.co.jp/books?id=RXBoDwAAQBAJ&;pg=PA77&lpg=PA77&dq=%22conglomerability%22+assumption+math&source=bl&ots=8Ol1uFrjJQ&sig=ACfU3U1bAurNGJm5872wDblskzsSgsU0iA&hl=ja&sa=X&ved=2ahUKEwioiPyV_IPiAhXHxrwKHUeaArUQ6AEwCXoECEoQAQ#v=onepage&q=%22conglomerability%22%20assumption%20math&f=false
Infinity, Causation, and Paradox 著者: Alexander R. Pruss Oxford University Press, 2018
P75
(抜粋)
2.5.3 COUNTABLE ADDITITVITY AND CONGLOMERABILITY
In the setting of classical probability theory, Good's Theorem (Good 1967) guarantees that it never pays for a perfectly rational agent who ha, 110 reason to fear loss of rationality to refuse free information in order to make better decisions.
Our paradoxes, however, do not contradict Good’s Theorem, since classical probability theory assumes countable additivity of probabilities, which is violated by countably infinite fair lotteries.
Indeed, the paradoxes we just discussed are fundamentally due to the lack of countable additivity in the lottery probabilities.
A probability function P is countably additive provided that whenever E1, E2,・・・are disjoint events,
then P(E1 ∨ E2 ∨・・・ ) =P(E1 ) + P(E2)+・・・ Classical mathematical probability theory assumes all probability functions to be countably additive.
But in the countably infinite fair lottery, we do not have countable additivity.
つづく つづき
The reason we do not have countable additivity differs depending on whether the probability of. particular ticket winning is zero or infinitesimal.
If the probability is exactly zero, then we lack countable additivity because
1 = P(E1 ∨ E2 ∨・・・) if En is the probability of ticket n being picked (it’s certain that some ticket or other is picked)
whereas P(E1) + P(E2) +・・・ = 0 + 0 + ・・・ = 0.
If on the other hand, P( En ) = α for some (positive) infinitesimalsα, then things are more complicated.
The standard systems for construction of infinitesimal do not in general define a countable in finite sum of infinitesimals, at least in our case where the summands are the same. Thus, the required equation P(E1 ∨ E2 ∨・・・ ) =P(E1 ) + P(E2)+・・・ does not hold,
since although the left .hand side is defined, the right-hand side is not.
In our infinite fair lottery case, we can intuitively see why we shouldn't be able to have a meaningful sum.
For consider our infinite sum:
α +α+α+α+ ・・・
= (α +α) +(α +α) +・・・
=2α+2α+ ・・・
=2(α+α+・・・ ).
If the value of this sum is x, then x =2x, But if x is not zero, then we can divide both side, by x to yield 1 = 2, and so x must be zero. However, x cannot be zero since it must be at least as big as α, and hence a contradiction follows, from the assumption that the sum has a value.
The lack of countable additivity in the case of an infinite lottery is responsible for a phenomenon known as non-conglomerability.
A probability function P is conglomerable with respect to a partition E1,E2,・・・(a partition is a collection of pairwise disjoint event such that their disjunction is the whole space of possibilities ) provided there is no event A and real number such that for all I we have P(A｜Ei ) <= a and yet P(A) > a.
Conglomerability is a very plausible properly.
つづく >>751
つづき
Suppose you are certain that some event in the partition will occur.
If you also know for sure that whatever event in that partition you learn occurs, your probability for A will be at most a, then how could your rational probability for A be more than a ?
Conglomerability is closely related to van Fraassen’s very plausible Reflection Principle which says that if one is rationally certain that one will have a certain rational credence, one should already have that credence now (van Fraassen 1984).
But typically, where there is no countable additivity, there is lack of conglomerability (Shervish, Seidenfeld, and Kadane 1984).
In the case of the countably infinite fair lottery, we can see the lack of conglomerability directly.
Let E be the event that the ticket picked will be even and O the event that it will be odd.
By finite additivity,
P(E) + P(O) = 1,
so at least one of the two events must have probability at least 1/2,
(Intuitively, they both have probability exactly 1/2, but I don't need that for the argument.)
Suppose that P(E) >= 1/2 (the argument in the case where P(O)>= 1/2 will be very similar).
つづく >>752
つづき
Then consider the partition provided by the following sets:
E1= {2, 1, 3)
E2= {4, 5, 7 }
E3= {6, 9, 11 }
E4. = {8, 13, 15 }
Observe now that each En contains exactly one even number and two odd ones.
Thus, by the fairness of the lottery, P( E| En) = 1/3.
Thus, P(E|En) < 1/2 for all n, but by assumption P(E) >=1/2, and conglomerability is violated.
Where conglomerability is absent, one gets strange results such as reasoning to a foregone conclusion and paying not to receive information (Kadane, Schervish,and Seidenfeld 1996), just as we saw in Section 2.5.
And the symmetry puzzle in Section 2.4 is also a non-conglomerability puzzle.
Taking the original two-ticket version, the probability that my ticket number is bigger than yours is initially within an infinitesimal of 1/2.
But the conditional probability that my ticket number is bigger than yours given what my ticket number is - whatever that may be - is at most an infinitesimal, and so conglomerability is violated.
One possible response to my preceding paradoxes is that non-conglomerability needs to be accepted when dealing with countably infinite fair lotteries, and non-conglomerability just happens to have a number of paradoxical consequences.
But the cost of accepting non-conglomerability is high, namely many paradoxical consequences.
It is better to take non-conglomerability in these lotteries to be both a paradox in its own right and the mathematical root of a number of other paradoxes.
(引用終り) >>750
＜conglomerabilityとは？＞
１）>>750 P(E1 ∨ E2 ∨・・・ ) =P(E1 ) + P(E2)+・・・ Classical mathematical probability theory assumes all probability functions to be countably additive.
But in the countably infinite fair lottery, we do not have countable additivity.
２）>>751 The lack of countable additivity in the case of an infinite lottery is responsible for a phenomenon known as non-conglomerability.
A probability function P is conglomerable with respect to a partition E1,E2,・・・(a partition is a collection of pairwise disjoint event such that their disjunction is the whole space of possibilities ) provided there is no event A and real number such that for all I we have P(A｜Ei ) <= a and yet P(A) > a.
３）Conglomerability is closely related to van Fraassen’s very plausible Reflection Principle which says that if one is rationally certain that one will have a certain rational credence, one should already have that credence now (van Fraassen 1984).
But typically, where there is no countable additivity, there is lack of conglomerability (Shervish, Seidenfeld, and Kadane 1984).
In the case of the countably infinite fair lottery, we can see the lack of conglomerability directly.
４）Thus, P(E|En) < 1/2 for all n, but by assumption P(E) >=1/2, and conglomerability is violated.
Where conglomerability is absent, one gets strange results such as reasoning to a foregone conclusion and paying not to receive information (Kadane, Schervish,and Seidenfeld 1996), just as we saw in Section 2.5.
And the symmetry puzzle in Section 2.4 is also a non-conglomerability puzzle.
５）ということで、以上の要点抜粋をまとめると、無限個の宝くじのように、可算無限の微小な和を加えると
conglomerabilityが保証されないので、paradoxになると
６）こういう説明を、数学Dr Alexander Pruss氏は、例のmathoverflowでもしているんだな
タイプアップに時間が勝ったが
取り敢ずこんなところだな（＾＾
以上 ■ このスレッドは過去ログ倉庫に格納されています
