fom
Posts:
1,969
Registered:
12/4/12


Re: Matheology § 230
Posted:
Mar 24, 2013 10:45 AM


On 3/24/2013 7:51 AM, WM wrote: > > > > Matheology § 230 > > Cantor's theory of infinite sets, developed in the late 1800's, was a > decisive advance for mathematics, but it provoked raging controversies > and abounded in paradox. One of the first books by the distinguished > French mathematician Emile Borel (18711956) was his Lecons sur la > Théorie des Fonctions [Borel, 1950], originally published in 1898, and > subtitled Principes de la théorie des ensembles en vue des > applications à la théorie des fonctions. > This was one of the first books promoting Cantor's theory of sets > (ensembles), but Borel had serious reservations about certain aspects > of Cantor's theory, which Borel kept adding to later editions of his > book as new appendices. The final version of Borel's book, which was > published by GauthierVillars in 1950, has been kept in print by > Gabay. That's the one that I have, and this book is a treasure trove > of interesting mathematical, philosophical and historical material. > One of Cantor's crucial ideas is the distinction between the > denumerable or countable infinite sets, such as the positive integers > or the rational numbers, and the much larger nondenumerable or > uncountable infinite sets, such as the real numbers or the points in > the plane or in space. Borel had constructivist leanings, and as we > shall see he felt comfortable with denumerable sets, but very > uncomfortable with nondenumerable ones. > > The idea of being able to list or enumerate all possible texts in a > language is an extremely powerful one, and it was exploited by Borel > in 1927 [Tasic, 2001, Borel, 1950] in order to define a real number > that can answer every possible yes/no question! > You simply write this real in binary, and use the nth bit of its > binary expansion to answer the nth question in French. > Borel speaks about this real number ironically. He insinuates that > it's illegitimate, unnatural, artificial, and that it's an "unreal" > real number, one that there is no reason to believe in. > Richard's paradox {{s. KB090826}} and Borel's number are discussed > in [Borel, 1950] on the pages given in the list of references, but the > next paradox was considered so important by Borel that he devoted an > entire book to it. In fact, this was Borel's last book [Borel, 1952] > and it was published, as I said, when Borel was 81 years old. I think > that when Borel wrote this work he must have been thinking about his > legacy, since this was to be his final booklength mathematical > statement. The Chinese, I believe, place special value on an artist's > final work, considering that in some sense it contains or captures > that artist's soul. If so, [Borel, 1952] is Borel's "soul work." [...] > Here it is: Borel's "inaccessible numbers:" Most reals are > unnameable, with probability one. Borel's oftenexpressed credo is > that a real number is really real only if it can be expressed, only if > it can be uniquely defined, using a finite number of words. It's only > real if it can be named or specifed as an individual mathematical > object. [...] So, in Borel's view, most reals, with probability one, > are mathematical fantasies, because there is no way to specify them > uniquely.
http://en.wikipedia.org/wiki/Philosophy_of_science#Scientific_realism_and_instrumentalism
http://en.wikipedia.org/wiki/Idealism#Idealism_in_the_philosophy_of_science
Everyone is entitled to an opinion.
Any given opinion can be portrayed as someone else's fantasy by better men than me.
But, there is nothing that is not a fantasy when the topic of discussion is WM's theory of monotonic inclusive crayon marks.

