I can't imagine mathematics without sets, but the question "what was mathematics like before there were sets" is not answerable, I think. Instead, a good answer to the title question should cover a certain aspect of the more general question.
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$\begingroup$ I think a notion to found the basis of mathematics was also at play. Whether Cantor new about this I am not sure. $\endgroup$– Ali CaglayanOct 30, 2014 at 13:23
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2$\begingroup$ I don't know whether this link is already provided in this thread but I think I should share it here. ias.ac.in/resonance/Volumes/19/11/0977-0999.pdf $\endgroup$– ankitMar 27, 2015 at 5:30
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$\begingroup$ Thank you @ankit, it's a very nice and absolutely relevant article. $\endgroup$– BenMar 27, 2015 at 21:22
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2$\begingroup$ Of course you can't imagine mathematics without sets - mathematics before formal set theory isn't the same as "mathematics before there were sets". Like algorithms existed forever although their formalization is no 150 years old, people were always using intersections of collections (sets) and so on. $\endgroup$– Nikolaj-KDec 16, 2015 at 20:03
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3 Answers
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An immediate motivation of Cantor to work on what became set theory was his earlier work on trigonometric series. To solve a problem in that domain he considered the set (a closed set) of zeros of such a function, then the derived set of this set, the derived set of this set and so on. This is all still classical, but then had to go a step beyond that to consider first the intersection of all these sets, and then the derived set of that set and so on.
So he came to consider transfinite ordinals.
This is discussed in various places, including "Set Theory and Uniqueness of Trogonometric Series" by Kechris or "Uniqueness of trigonometric series and descriptive set theory, 1870–1985" by Roger Cooke (Archive for History of Exact Sciences, 1993)
The original paper is (I think) "Ueber die Ausdehnung eines Satzes ais der Theorie der trigonometrischen Reihen (Math. Annalen, 1872)"
Another motivation was his earlier work on number theory. Using what is now called a diagonalisation argument he was able to prove results on the existence of transcendental numbers. This is in his 1874 paper "Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" ("On a Property of the Collection of All Real Algebraic Numbers")
In brief, the original motivation was to have better tools for making progress on existing problems.
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$\begingroup$ Do you have an references for the first point? $\endgroup$ Oct 30, 2014 at 12:03
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2$\begingroup$ Besides the references you suggest, the standard place to read about this is the preface by Jourdain to his translation of Cantor's Math. Annalen memoirs, Contributions to the founding of the theory of transfinite numbers. $\endgroup$ Nov 8, 2014 at 21:56
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3$\begingroup$ The most detailed discussion I know of in English for Cantor's trigonometric series papers is Dauben's The trigonometric background to Georg Cantor's theory of sets. Regarding Cantor extending the countability argument from the rationals to the algebraic numbers, this originated from Dedekind in letters to Cantor. English translations of the relevant letters are on pp. 844-850 of Ewald's book (reference [7] here). See also pp. 177-186 of Ferreirós' 1999 book and his 1993 Historia Math. paper. $\endgroup$ Nov 14, 2014 at 19:47
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Actually Cantor was working on a specific problem from the theory of trigonometric series, the so-called uniqueness problem (I cannot be more specific until MathJax is introduced to this site). This problem led him to consideration of arbitrary sets on the real line. I mean more complicated sets than finite sets or finite union of intervals. At that time there was no tools and no terminology to study arbitrary sets, so all this had to be created.
In the process of this study he created not only the set theory but also what is called now General topology. (In is interesting to notice that the original problem about trigonometric series has no complete solution to this day:-)
The original method of proof, the so-called "diagonal procedure" goes back to Cantor's predecessor, Paul du Bois Reymond, who was also studying trigonometric series.
answered Nov 8, 2014 at 13:11
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$\begingroup$ Sorry for the nit-pick but it is the second time I notice it: MathJax not MathJack. $\endgroup$– quidNov 8, 2014 at 15:12
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1$\begingroup$ You are absolutely right. The diagonal procedure was used for the "orders of infinity" type questions. But du Bois-Reymond also studied trig series, just an interesting coincidence:-) $\endgroup$ Nov 8, 2014 at 23:59
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$\begingroup$ @quid: Thanks! You can actually edit the text when you spot misprints. $\endgroup$ Nov 9, 2014 at 0:01
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2$\begingroup$ Unfortunately I do not yet have enough points here to edit, and for suggested edits there is a character limit. $\endgroup$– quidNov 9, 2014 at 0:11
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According to Cantor himself it was his desire to replace the mechanical explanation of nature by a more complete theory. See several aspects in What of Cantor's claims has become true??