Scottish Nous

Cantor Refuted

At arXiv.org.  Laurent Germain claims to have shown that the cardinality of the continuum is ℵ0.  That is, he claims to show that there is a one-one onto mapping between the set of natural numbers and the set of real numbers, despite the fact that Cantor's diagonalization argument establishes that this is impossible with perfect certitude. [via Good Math, Bad Math]

You might wonder what natural number Germain maps π onto.  Peasy!

For example, the node (3,1,4,1,5,9,2,6) can be defined by the decimal number (3,1415926) which is the number π.

One would have thought that, given the construction, the node (3,1,4,1,5,9,2,6) would define the number 3.1415926, which is demonstrably less than the number π.  And since no number is less than itself, this is not the number π.  But if he really does map π onto (3,1,4,1,5,9,2,6), what will he map 3.1415926 onto?  Oh shit!  Worse still, his construction doesn't even show how to define a function that takes each rational number to a unique natural number.

Open Courseware and Illegal Books

There's a nice collection of links to Open Courseware here.  I might as well also here mention Gigapedia.  It's an incredible linkfarm to high quality PDFs of mostly copyrighted material, including a HUMONGOUS number of philosophy books.  (Who scans these?)  You can login anonymously using BugMeNot, if you prefer that to registering.  (Logging in is required in order to search the site.)  And remember, it's not illegal - surely! - to download a copy of a book you already own!  Of course, I can't be blamed if you succumb to temptation, but honestly, the free dissemination of information not everyone can afford access to isn't that bad,  it?  And let's be serious.  What professor of philosophy would object to you downloading their book?  The objection couldn't possibly be on the basis of the royalties they would otherwise receive.