# The origin of sets?

The history of set theory from Cantor to modern times is well documented. However, the origin of the idea of sets is not so clear. A few years ago, I taught a set theory course and I did some digging to find the earliest definition of sets. My notes are a little scattered but it appears that the one of the earliest definition that I found was due to Bolzano in Paradoxien des Unendlichen:

There are wholes which, although they contain the same parts $A$, $B$, $C$, $D$,..., nevertheless present themselves as different when seen from our point of view or conception (this kind of difference we call 'essential'), e.g. a complete and a broken glass viewed as a drinking vessel. [...] A whole whose basic conception renders the arrangement of its parts a matter of indifference (and whose rearrangement therefore changes nothing essential from our point of view, if only that changes), I call a set.

(The original German text is here, §4; I don't remember where I got the translation.)

According to my notes, Bolzano wrote this in 1847. Since Boole's An Investigation of the Laws of Thought was published a just few years later in 1854, it seems that the idea of sets was already well known at that time.

What was the earliest definition of 'set' in the mathematical literature?

Historical queries of this type are hopelessly vague, so let me give some more specific criteria for what I am looking for. The object doesn't have to be called "set" but it must be an independent container object where the arrangement of the parts doesn't matter.

• It should also be fairly general in what the set can contain. A general set of points in the plane is probably not enough in terms of generality but if the same concept is also used for collections of lines then we're talking.
• It shouldn't have implicit or explicit structure. Line segments, intervals, planes and such are too structured even if the arrangement of the parts technically doesn't matter.
• It should be an independent object intended to be used and manipulated for its own sake. For example, the first time a collection of points in general position was considered in the literature doesn't make the cut since there was no intent to manipulate the collection for its own sake.
• It should be a definition. Formal definitions as we see them today are a relatively new phenomenon but it should be fairly clear that this is the intent, such as when Bolzano says "I call a set" at the end of the quote above.
• It should be mathematical concept. The strict divisions we have today are very recent but it should be clear that the sets in question are intended for mathematical purposes. Paradoxien des Unendlichen is perhaps more of a philosophical treatise than a mathematical one, but it is clear that Bolzano is considering sets in a mathematical way.

That said, any input that doesn't quite meet all of these criteria is welcome since the ultimate goal is to understand how the modern idea of set came to be.

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This is a borderline case for CW. I don't expect a definite answer, but it is not intended as a collection of resources to be sorted and the idea of a useful answer makes sense. What do people think? – François G. Dorais Dec 22 '12 at 21:35
I don't think it should be CW. If only because a good answer would definitely be worthy of every vote and vote given to it. It's a very interesting question too! – Asaf Karagila Dec 22 '12 at 21:49
I've read that Dedekind also made important early contributions. I don't know the chronology of the contributions themselves, but Dedekind was 14 years older than Cantor and 50 years younger than Bolzano. Probably you know much more about this than I do. – Tom Leinster Dec 22 '12 at 21:52
Find some examples by looking for "set" at MathWord jeff560.tripod.com/mathword.html – Gerald Edgar Dec 22 '12 at 22:16
Thanks Gerald! There is only one example there that significantly predates Bolzano: "In 1796, William Frend used the phrase “set of numbers” in The Principles of Algebra. This use of the word was found by Stanley Burris, who wites, 'This was certainly not an influential book since Frend did not accept negative numbers, but it suggests the use of the word set in math texts may have been common.'" – François G. Dorais Dec 22 '12 at 22:26

Euler in Lettres à une princesse d'Allemagne sur divers sujets de physique et de philosophie, 17-24 feb 1761, writes about objects he calls spaces (my emphasis):

As a general notion encompasses an infinity of individual objects, one regards it as a space within which all these individuals are enclosed: thus, for the notion of man, one makes a space (fig. 39) in which one conceives that all men are comprised. For the notion of mortal, one also makes a space (fig. 40), where one conceives that everything mortal is comprised. Then, when I say that all men are mortal, that comes down to the former figure being contained in the latter.

(...)

These round figures or rather these spaces (for it doesn't matter what shape we give them) are very well-suited to facilitating our reflections (...)

etc., and illustrates this with what we would call ensemblist diagrams (fig. 39 to 89), famously reproduced on Swiss banknotes. The applications he gives here are to everyday logic, so perhaps less mathematical than intended by the question. (I don't know if he ever wrote again on the subject.)

Edit: Something I didn't know is that these diagams were themselves anticipated by Leibniz, as can be seen in his manuscript De formæ logicæ comprobatione per linearum ductus ($\simeq$ "Testing logical forms through line drawings") published in "Opuscules et fragments inédits de Leibniz: extraits des manuscrits de la Bibliothèque Royale de Hannover", Olms Verlag (1986), pp. 292-321.

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Merci beaucoup! Euler doesn't quite capture the lack of structure as well as Bolzano. Also, as you observe, it's not clearly a mathematical object. Nevertheless, this is an important example! – François G. Dorais Dec 23 '12 at 2:35
And those figures rondes are Venn diagrams a century before Venn :-) – Mariano Suárez-Alvarez Dec 23 '12 at 2:47

This isn't meant entirely seriously as an answer to your question, but: on page 344 of Practical Foundations of Mathematics, Paul Taylor writes:

Adam of Balsham (1132) observed that the difference between finite and infinite sets is that the latter admit proper self-inclusions, such as $n \mapsto 2n$.

Obviously this is staggeringly early and it would be astonishing if this dude Adam had anything like our present-day conception of set. Paul doesn't appear to give a reference, but perhaps he (Paul, not Adam) will see this and tell us more.

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This is often referred to as Galileo's paradox. Galilei discusses it in some detail, but it goes back further. The following web site gives references going back as far as Plutarch: earlham.edu/~peters/writing/infinity.htm – Michael Renardy Dec 23 '12 at 1:14
Adam of Balsham is mentioned in Styazhkin's "History of Mathematical Logic from Leibniz to Peano". I don't have access to that book now, or any other notes on this topic. Historical references like this in my book were meant simply to point out that many important ideas are much older than the followers of Cantor would have you believe. I would not be surprised to be told that some Arabic mathematician or philosopher had made this comment much earlier. Or even Archimedes. – Paul Taylor Dec 23 '12 at 8:28
Paul, using the sentence "much older than the followers of Cantor would you have believe." makes it sound like you claim there is some conspiracy to make Cantor a great mathematician, even if he wasn't. And that there is some shadow society which works hard to keep in the shadow all those who discussed sets before Cantorian times... I just want to say that if there is such secret society, and they are reading this, I would like to be a member! :-) – Asaf Karagila Dec 23 '12 at 11:14
Adam of Balsham was known as Parvipontanus because he taught near the Petit Pont in Paris. He wrote a book called Ars Disserendi. Regarding Cantor, I am not sure whether he is officially regarded as a Great Mathematician, say alongside Dedekind, but there is most certainly a conspiracy regarding his ideas: just try getting a job in a Pure Mathematics department if you are an atheist regarding Set Theory. I am not sure if Asaf is declaring himself an atheist too, but we are not alone. – Paul Taylor Dec 23 '12 at 21:21
Paul I don't know what you mean about "atheist regarding set theory". It is provable that if ZFC is consistent then we can develop most mathematics in any model of ZFC ("most" because we know some of the recent mathematics requires slightly more - often in the form of large cardinals). But if you mean to say that you don't believe that ZFC is consistent or "true" then I don't see how that is relevant to mathematics. It's a philosophical bent. I don't believe that we can answer this question very well, so I prefer to waste my time doing something less not-useful than mathematical theism. – Asaf Karagila Dec 24 '12 at 13:12

"A new and compendious system of practical arithmetick", by William Pardon in 1738, contains the passage:

Here if the first Series or Set of Numbers increases by 1, and the second decreases by 1; the third increases by 2, ...

The emphasis is in the original, so that it is not set that is being described. So in 1738, it's meaning was already taken for granted.

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Published in 1738 according to WorldCat - worldcat.org/title/… – François G. Dorais Dec 22 '12 at 22:54
I don't know if emphasis was used in that way in 1738. – François G. Dorais Dec 27 '12 at 4:19

From the Wikipedia article on Euler diagrams:

"The first use of "Eulerian circles" is commonly attributed to Swiss mathematician Leonhard Euler (1707–1783)."

"Venn diagrams are a more restrictive form of Euler diagrams. A Venn diagram must contain all $2^n$ logically possible zones of overlap between its $n$ curves, representing all combinations of inclusion/exclusion of its constituent sets, but in an Euler diagram some zones might be missing if they are empty sets."

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Possibly an instance of Baez's theorem, where 'Venn diagram' I've seen applied to what are apparently called Eulerian diagrams. – David Roberts Dec 23 '12 at 9:33
I'm afraid this meets none of the criteria in the question. – François G. Dorais Dec 23 '12 at 20:22
Was this intended as a comment to Mariano Suarez-Alvarez's comment to François Ziegler's answer? – François G. Dorais Dec 23 '12 at 23:21
Sort of. I wanted it to be a standalone answer so it would include the image, but the image hasn't appeared. I was not aware until recently that there was a more specific definition of Venn diagrams. I had never heard of Euler circles until I read Wikipedia. – Stxmqs Dec 24 '12 at 7:34
Looking back at my old school textbook I see that all the examples given are indeed proper Venn diagrams. However the book doesn't give any actual definition so one could easily come away thinking that Euler diagrams are also Venn diagrams. – Stxmqs Dec 24 '12 at 7:48

## protected by Andrés E. CaicedoOct 5 '13 at 23:00

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