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Dedekind’s 1858 introduction of Dedekind Cuts as a method of defining real numbers is a bona fide use of sets as mathematical objects. He is using sets to define mathematical objects (real numbers) as a pair of sets.

Prior to Dedekind, Boole worked with sets in his works Mathematical Analysis of Logic of 1847 and Investigations of the Laws of Thought of 1854. Here, although Boole is using sets and defining operations on sets, it is with the aim of defining logical operations. Since logical operators are not mathematical objects it is not clear to me that Boole’s works qualifies as a bona fide use of sets as mathematical objects.

Prior to Boole, the waters become murkier.

Q: Who was first to introduce sets as bona fide mathematical objects?

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The reason it gets murky is that the kind of use of which Dedekind cuts are prototypical became characteristic only after the arithmetization of analysis at the end of 19th century. Dedekind did not publish his 1858 work until 1872, by which time Cantor already started publishing his papers on point sets with similar ideas, including his own construction of real numbers as classes of rational sequences. This was a key difference between them and their closest precursor, Bolzano, who could not prove existence of say least upper bounds "rigorosly" because he lacked such constructions. We can bootstrap some prior uses into this but then what qualifies as "bona fide" becomes a judgement call. This is not uncommon, modern concepts have their precise meanings only in the modern context, when we try to apply them before that we are stretching it. But of course the modern context does not emerge out of nothing. Paradoxically, in history there are often no "firsts".

The tradition of geometric loci predates Euclid. A locus is a figure singled out by a property ("symptom", as Apollonius called it), e.g. the locus from which exactly two normals to a parabola can be drawn, considered by Apollonius. Aristheus (c.350 BC) wrote a book Plane Loci, not extant but mentioned by Pappus, which served as a stepping stone to the work of Euclid and Apollonius on conic sections. We now identify loci with "sets of points", but this is not exactly how Greeks saw it, lines and planes were not dissociated into points then. Descartes brought these ideas closer to modern use in La Geometrie (1637), when he coordinatized the plane and identified loci with "zero sets" of polynomials, but even he does not treat them as "sets of coordinate pairs" as we do today.

First Euler (1750, unpublished until 1849) and then more explicitly Gauss in Disquisitiones Arithmeticae (1798, published 1801) introduced congruence classes (Euler called them residues and preferred to think of them as remainders), and algebraic manipulations with them, the modular arithmetic, see Is Gauss the first who introduced congruences? These are perhaps closer in spirit to Dedekind cuts, indeed the whole idea of arithmetizing analysis was in part motivated by importing "Gauss's rigor" from arithmetic.

The tradition Boole was following can be traced at least as far back as Leibniz, and with some charity to Raymundus Lullius or even Aristotle. This is complicated by the conflation of intensions with extensions which persisted until the early 20th century, so in the early sources what is represented are more concepts than classes or sets they define. In any case, they were treated as "objects" with extensional connotation due to representation by circles or other figures in diagrams, what we now call "Venn diagrams". The use of diagrams was popularized by Euler, but as Baron writes in A Note on the Historical Development of Logic Diagrams:"Despite Euler's stature as a mathematician he apparently made no attempt to develop these relations algebraically. Nonetheless, through him, knowledge of the diagrams became widespread and they had some considerable influence in the nineteenth century". Boole was one of the first to start thinking of what the circles represent as "classes" (extensions) rather than concepts, Venn, who gave the most comprehensive account of the diagrams and their manipulation, followed him. But the conflation is still present even in Russell and Zermelo, the purely extensional use is clearly articulated only in Hausdorff's Grundzuge der Mengenlehre (1914), see How did mathematicians notate the empty set before $\varnothing$?

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    $\begingroup$ I think you have hit the nail on the head here. From a modern point of view we can interpret historical work in a different light. It is so true. And with mathematics being a "uniquely" accretive discipline, it is often impossible to know where do we draw the line. $\endgroup$ – Nick Oct 24 '16 at 19:37
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Leibniz (1679) in an appendix to a letter sent to Huygens used sets of points to define geometric objects. For example: A(X) 8 AV is a constraint on the list of all points a,b,c, ... called {X} such that Aa, Ab, ... is congruent to line segment AV. They share an element A. So {X} a sphere. And so on. He uses () not {} but ...

This was republished about three times in opera omnia type collections prior Dedekind and Cantor (and part of a question posed by Moebius to which Grassmann wrote his second monograph introducing inner products. Leibniz did nothing with this because Huygens was not interested.

(It was also the first abstract geometry. Without defining a metric as Riemann would later, any triangle is congruent to any other in this notation. Indeed, this confused Grassmann, and he suggested his own system winning the essay prize. Grassmann (1847) summarized the system at the start of his own book. This might have spread the notation.)

Leibniz, Gottfried, 1679. In 1969, Philosophical Papers and Letters, Leroy Loemker, trans., Dordrecht: Reidel, pp. 251--253.

Grassmann, Hermann, 1847, Geometrische Analyse geknüpft an die von Leibniz erfundene geometrische Charakteristik, Leipzig: Weidmann, pp. 4--5.

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  • $\begingroup$ I'm guessing this is probably the letter of 8 Sept 1679 where Leibniz attaches to his letter a copy of his essay on his "investigations into quadrature". $\endgroup$ – Nick Oct 25 '16 at 15:03
  • $\begingroup$ Yes. That is the one. $\endgroup$ – Guido Jorg Oct 25 '16 at 15:08
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The first to introduce the terminus set (in German: Menge) in its mathematical meaning was Bernard Bolzano in "Paradoxien des Unendlichen", Reclam, Leipzig, 1851, p. 2.

Euclid's "Therefore, prime numbers are more than any assigned multitude of prime numbers." certainly belongs to the first uses of the idea of set.

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  • $\begingroup$ I should have mentioned Bolzano in my question. Thanks for reminding me. He was a very interesting figure. $\endgroup$ – Nick Oct 28 '16 at 21:46
  • $\begingroup$ All the more so as “Menge” and “plēthos” (El. IX 20) both have exactly the same meaning: “multitutude”. $\endgroup$ – fdb Oct 29 '16 at 18:03

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