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I'm writing/starting with my bachelor thesis, the subject is about "infinity": what it is, why do we accept it, etc., but most of all my goal is to give an overview of the history of the interpretation from mathematicians (or maybe philosophic intellectuals).

So my concrete question is: I am looking for great 'history of mathematics' books, where the books focus on 'infinity' (ancient Greeks, etc.), so that I can show how the interpretation has changed through the past millenia, and why it changed. And what about the concept of infinity now? There are today still math professors who do not want to accept infinity.

Does anybody have good resources for the 'historical view of infinity'? Later, if I have time, I will extend it with some philosophical side tracks, but first I need the basis to fully understand, mathematically, what the concept is, how it developed etc.

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    $\begingroup$ You can add to the list below also : AW Moore, The Infinite (1990) $\endgroup$
    – Mauro ALLEGRANZA
    Commented Dec 15, 2014 at 10:44
  • $\begingroup$ I recently posted several references in this discussion group that would be useful for your question. See my answer to Did Galileo's writings on infinity influence Cantor?. Several of the older references (all freely available on the internet) are especially useful (e.g. Couturat's book, Vivanti's papers, Young's book) because of their references to literature that you otherwise wouldn't easily find out about. $\endgroup$
    – Dave L Renfro
    Commented Dec 15, 2014 at 20:22
  • $\begingroup$ You can probably find a lot of things similar to what you're trying to do by googling the words history infinity student thesis Cantor. For example, on the 2nd page of hits I found a 2009 Masters thesis A Finite History of Infinity by Amy Whinston, and there are probably many other undergraduate and Masters level theses freely available online. (I hope your faculty advisor is aware of how much is out there ...) $\endgroup$
    – Dave L Renfro
    Commented Dec 15, 2014 at 20:40
  • $\begingroup$ I wouldnt think you could use this as a direct source, but check out the book White Light by Rudy Rucker. It's a very quick read and gave a weirdly insightful view into the concept of infinity. It's fun, and there's quite a lot of references to Cantor and Hilbert, too. $\endgroup$
    – galois
    Commented Dec 15, 2014 at 20:59

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The way I would go about this would be to take references from decent online papers and journals on the subject, and then research which of those might seem worth your money. I would pay particular attention to books about Georg Cantor.

Here is an example of such a book chase. A paper titled A History of Infinity was written at Texas A&M by a professor in the math department (Dr. Allen). In the references to that paper, I see at least two stand out book sources:

Dauben, Josheph Warren, Georg Cantor, His Mathematics and Philosophy of the Infinite, Princeton University Press, Princeton, New Jersey, 1979

and

Lavine, Shaughan, Understanding the Infinite, Harvard University Press, Cambridge, MA, 1994.

If possible see if you can't survey 20 such references in this manner, secure from your library, possibly through inter library loan, and buy the ones you like the best after making your mind up as to whether the book is something you should own.

You will know when it is time to stop surveying and start picking acquisitions. Now you are ahead by at least two sources, but you might want to look at Dr. Allen's other references in that paper above as that looks like a pretty decent paper. You can certainly cite that online paper as well, it being from a reputable mathematician with the stamp of a university on it.

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For the distinction of potential and actual infinity see Transfinity, in particular chapter I, but also a lot of sources in chapter V, and for some statements without references but easily to be found in the internet and pointing to further sources: the lecture slides https://www.hs-augsburg.de/homes/mueckenh/HI/HI11.PPT and https://www.hs-augsburg.de/homes/mueckenh/HI/HI12.PPT

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