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Logic and Mathematics - an overview.

RiderOfGiraffes 16y ago
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 16y ago
  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.
ionfish 16y ago
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/ 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/ http://plato.stanford.edu/entries/settheory-early/ [2] http://www.bris.ac.uk/philosophy/podcasts_html/Cantor_by_Leon_Horsten.mp3 http://www.bris.ac.uk/philosophy/podcasts_html/Cantor_by_Leo...
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davidmathers 16y ago
The best overview I've seen is Mathematics: Form and Function by Saunders Mac Lane: http://www.amazon.com/dp/0387962174/ 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 16y ago
  Gödel, Escher, Bach is a bit more idiosyncratic, but covers some of the same topics (in an incredibly interesting way).

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 16y ago
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.

ionfish 16y ago

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/ 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/ http://plato.stanford.edu/entries/settheory-early/ [2] http://www.bris.ac.uk/philosophy/podcasts_html/Cantor_by_Leon_Horsten.mp3 http://www.bris.ac.uk/philosophy/podcasts_html/Cantor_by_Leo...

deleted 16y ago

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davidmathers 16y ago

The best overview I've seen is Mathematics: Form and Function by Saunders Mac Lane: http://www.amazon.com/dp/0387962174/ 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 16y ago
Gödel, Escher, Bach is a bit more idiosyncratic, but covers some of the same topics (in an incredibly interesting way).