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Can you point me to some books on the history of the foundations of mathematics? At the moment I'm searching for something light because of my lack of mathematical maturity and also the fact that I'm self-learning this, my intention is to make a two-step program: First a light book about it then one a little more serious on the topic.

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Logicomix? –  Asaf Karagila Jan 19 '13 at 9:00
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Logicomix -- fun but not very reliable on the actual history (and indeed, doesn't pretend to be). –  Peter Smith Jan 19 '13 at 10:00
    
Oh, this seems cute. –  Vÿska Jan 19 '13 at 14:17

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It depends whether you want pure history or (so to speak) a rational reconstruction of some of the leading ideas.

An obvious recommendation for the history is Ivor Grattan-Guinness, The Search for Mathematical Roots, 1870 - 1940, (Princeton) which is subtitled "Logics, Set Theories and the Foundations of Mathematics from Cantor through Russell to Gödel". This is reliable and clearly written though perhaps not the most exciting read ever (it weighs in over 500 pages).

Also very good is José Ferreirós, Labyrinth of Thought: A History of Set Theory and its Role in Modern Mathematics (Birkhauser).

Neither, however, could be called "light" though they don't require much mathematical maturity to read.

If you want something that tells you less about the detailed history, but which highlights and assesses some of the Big Ideas that were in play, Marcus Giaquinto's The Search for Certainty (OUP) is much shorter, and quite beautifully readable. This book could make a very good way in to starting to finding out a little about some of the foundationalist projects of the late 19th and 20th centuries.

As a footnote, I'd add that there is still both illumination and fun to be had reading Bertrand Russell's Introduction to Mathematical Philosophy (surely the most readable of the works from the Founding Fathers!).

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I liked From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, edited by Jean van Heijenoort. It contains many of the original crucial papers, such as the Frege axioms that turned out to be inconsistent, the letter that Russell sent him pointing this out, and the sad epilogue he wrote to his book afterward; Gödel's original incompleteness paper, and a lot of crucial developments in between.

I am a big fan of reading original materials when possible. They are often groundbreaking not just because of their results but because of the presentation. It's often interesting to compare the actual original ideas with what people now say the original ideas were; these often don't match at all. Original materials give a better sense of the historical development of ideas, which often brings its own insight into why things are done the way they are. Famous mathematicians become famous in part because of the brilliance of their thought and clarity of their expression of it, and so the original exposition is often brilliant and clear where an exposition by some later, lesser person is not. On the other hand, sometimes the original exposition is confusing, because the ideas we not completely clear yet. But if you are trying to understand the history, there is no substitute for the original documents.

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It can be illuminating to go back to originals. But that is no substitute for reading a good history (unless you immerse yourself in a lot more than is in van Heijenoort!). To take just one example, Gödel's incompleteness paper is a masterpiece of compression and even great logicians like Zermelo initially didn't understand it. And you need to know the history to understand that Gödel's concern was much more with the Hibertians than the logicists. Etc. –  Peter Smith Jan 19 '13 at 20:07

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