

Logic and Mathematics - an overview. - RiderOfGiraffes
http://www.solipsys.co.uk/new/LogicAndMathematicsOverview.html?HN

======
ionfish
Be warned that the term 'overview' is apt. In particular the section on
philosophy of mathematics is extremely brief, and misses out many important
positions and issues in contemporary philosophy of mathematics. A better
survey appears in the Stanford Encyclopedia of Philosophy.

<http://plato.stanford.edu/entries/philosophy-mathematics/>

There are also some strange passages, for example where Simpson suggests that
Hilbert could have profited from examining Aristotle's distinction between
potential and actual infinities. It is strange because Simpson does not say
_how_ Hilbert's finitism would have benefited from this distinction, although
it is certainly a historically important one. Hilbert's immediate influence in
this regard was Cantor, who essentially rebelled against the dogma of his day
which held strongly to the Aristotelian line (albeit with exceptions; for
examples see [1]).

Cantor's set theory treated infinite collections as 'completed' infinities
which could be studied and manipulated mathematically just as finite objects
can be. However, in a sense he merely tweaked Aristotle's doctrine, pushing
allowable cardinalities far into the transfinite, but stating that 'absolute'
infinity (such as the mathematical universe as a whole) was unattainable.
Michael Hallett's book _Cantorian Set Theory and Limitation of Size_ contains
a good exposition of Cantor's position. A lecture on Cantor's philosophy was
given last year at Bristol University by Leon Horsten, which you can download
as an mp3. [2]

[1] <http://plato.stanford.edu/entries/settheory-early/>

[2]
[http://www.bris.ac.uk/philosophy/podcasts_html/Cantor_by_Leo...](http://www.bris.ac.uk/philosophy/podcasts_html/Cantor_by_Leon_Horsten.mp3)

------
RiderOfGiraffes
I've put this link in a web page and pointed you at that rather than
submitting the PDF URL diectly, because I refuse to submit a PDF as a link,
knowing that it would get scribd'd.

(edited for clarity)

~~~
flatline
Interesting: I just realized that with Chrome I no longer notice PDF links as
any sort of problem, due to the built-in rendering engine.

------
davidmathers
The best overview I've seen is Mathematics: Form and Function by Saunders Mac
Lane: <http://www.amazon.com/dp/0387962174/>

It has the benefit of being written by one of the top mathematicians of the
20th century.

There are downloadable copies in the series of tubes.

~~~
2arrs2ells
Gödel, Escher, Bach is a bit more idiosyncratic, but covers some of the same
topics (in an incredibly interesting way).

