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I have someone who is begging for a conversation with me about infinity. He thinks that Cantor got it wrong, and suggested to me that Gauss did not really believe in infinity, and would not have tolerated any hierarchy of infinities.

I can see that a constructivist approach could insist on avoiding infinity, and deal only with numbers we can name using finite strings, and proofs likewise. But does anyone have any knowledge of what Gauss said or thought about infinity, and particularly whether there might be any justification for my interlocutor's allegation?

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    Gauss was involved with convergence tests for infinite series.2012-05-04
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    If there ever was a question fitting for the [infinity] tag, this is it.2012-05-04
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    To be fair to Gauss you should consider also what his contempories thought about completed (vs. potential) infinity. To properly evaluate Gauss' remark requires extensive knowledge of the mathematics of that era (and an ability to effectively "forget" what you know of today's math when need be). Neither of these are commonplace.2012-05-04
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    I strongly agree with @Bill. It is a common mistake for people to evaluate historical events and quotes as if they were occurring in present time. To fully understand something that had happened three centuries ago, one has to understand the spirit of the era before attempting to understand the event itself.2012-05-04
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    Well, it looks as though the whole thing is more interesting than I really imagined. There are "tamed infinities" involved in mathematical objects like the Projective Plane and the Riemann Sphere. GH Hardy writes in Pure Mathematics "$\infty$ by itself means nothing, although phrases containing it sometimes mean something" [sect 55 page 117 tenth edition] and proceeds to use it liberally e.g. as a limit of integration.2012-05-05
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    @UnreasonableSin Do you have any context or details of how Gauss described or understood what he was doing?2012-05-05
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    I've added a preemptive protection, since history taught us that this topic can be a crank magnet. :-)2015-01-17
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    @AsafKaragila Thank you!2015-01-17

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Here is a blog post from R J Lipton which throws some light on this question. Quoting from a letter by Gauss:

... so protestiere ich zuvörderst gegen den Gebrauch einer unendlichen Größe als einer vollendeten, welcher in der Mathematik niemals erlaubt ist. Das Unendliche ist nur eine façon de parler, indem man eigentlich von Grenzen spricht, denen gewisse Verhältnisse so nahe kommen als man will, während anderen ohne Einschränkung zu wachsen gestattet ist.

The blog gives this translation:

... first of all I must protest against the use of an infinite magnitude as a completed quantity, which is never allowed in mathematics. The Infinite is just a mannner of speaking, in which one is really talking in terms of limits, which certain ratios may approach as close as one wishes, while others may be allowed to increase without restriction.
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    This is not particularly new, the deleted account which posted on this page might have been owned by an insufferable troll, but in this case it was just quoting Gauss. (And posting spam in a now-deleted answer...)2014-08-05
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Your interlocutor seems to oppose infinity (and attribute similar views to Gauss) on finitist or constructivist grounds. If this is the case, he would probably similarly oppose infinitesimals. This is because specifying an infinitesimal typically involves an infinite amount of data, at least in modern theories.

Here he would be wrong to assume similar beliefs on Gauss's part because Gauss specifically and routinely used infinitesimals in his development of differential geometry. A detailed discussion of this may be found in the book by Michael Spivak on Differential Geometry, Third edition, volume 2, chapter 4. The discussion starts on page 62 as follows: "Gauss now nonchalantly introduces infinitely small quantities..."

Your interlocutor also mentioned hierarchies of infinities. On page 75 in Spivak's translation of Gauss, one finds products of infinitesimals, and an expression for curvature in terms of these. These are second order infinitesimals. Thus Gauss dealt with a hierarchy of infinitesimals.

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You are right. Gauss did not believe in finished infinity. He would have condemned Cantor's ideas.

(Was nun Ihren Beweis anbelangt), „so protestiere ich zuvörderst gegen den Gebrauch einer unendlichen Größe als einer vollendeten, welcher in der Mathematik niemals erlaubt ist. Das Unendliche ist nur eine facon de parler, indem man eigentlich von Grenzen spricht, denen gewisse Verhältnisse so nahe kommen als man will, während anderen ohne Einschränkung zu wachsen gestattet ist. [C. F. Gauß, Briefwechsel mit Schumacher, Bd. II, p. 268 (1831)]

Translation from comments

"I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction"

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    Do you have any reference/context for this?2012-05-04
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    I have edited my answer.2012-05-04
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    I'll what a little to see what else people say. In the meantime, I'm glad a learned German at school ... thanks2012-05-04
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    Translation, please!2012-05-04
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    @PeterTamaroff "I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction"2012-05-04
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    @Marvis Thank you very much.2012-05-04
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    @RahulNarain The last part was really off.2012-05-04
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    @RahulNarain I didn't see your translation. Did you delete it?2012-05-04
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    @Marvis It was google-translated.2012-05-04
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    I wonder how Gauss would have reacted to 20th Century mathematics with Hilbert and Cantor and all that following dealing with infinities and infinite-dimensional spaces...2012-05-04
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    Probably would have been horrified for $15$ minutes. Might have taken him a day or two to really make good use of the new tools.2012-05-04
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    It makes no sense to write something like [Gauss] "would have condemned Cantor's ideas". Almost 80 years passed between Gauss' Disq. Arith. and Cantor's work on set theory. If Gauss had been a contemporary of Cantor and, so, had knowledge of mathematics of that later era, then he may well have praised Cantor's work. But no one can know. It is disrespectful to Gauss to write such highly speculative remarks. We don't need more romanticized math history in the style of E.T. Bell. Nowadays much is on line, and you can read the *true* history.2012-05-04
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    @BillDubuque If you have any links to what you're talking about (last sentence) I'd like to visit them.2012-05-05
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    @Peter What I meant is that, in the pre-computer era of E.T. Bell, it was not easy for readers to verify historical information without access to a good university library. But nowadays there are many journals freely available online, so it is much easier for readers (and authors!) to check the veracity of historical assertions.2012-05-05
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    I suppose the question becomes "what in Cantor might have caused Gauss a problem?" - and I expect my knowledge of the original work of Cantor is less than that of Gauss. And would the notion of bijection between sets been comprehensible? It all seems obvious now, but the use of "unrestricted comprehension" in early set theory was the source of paradox, and membership of infinite sets is specified by formulae. Cantor's ideas were rejected by some notable contemporaries.2012-05-05
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    @ Bill Dubuque: How can you be sure that Gauss would have "praised Cantor's work"? Poincaré, Brouwer, Wittgenstein knew it and did _not_ praise it but condemn it. I say: If Gauss could see the present state of mathematics, he would react like Christ would have done if he had returned in the 15th century and had seen the Vatican. But neither of us has proofs for his assertions.2012-05-05
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    @ Mark: It is not correct that "all seems obvious now", on the contrary. But differing opinions are simply suppressed in some media. Look here for example: http://www.hs-augsburg.de/~mueckenh/Infinity/Open%20Letter.pdf2012-05-05
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    Why am I not surprised to see Contra refer to Mückenheim?2012-05-05
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    @Contra - I don't understand the link or, on reflection, your phrase "finished infinity". I cited GH Hardy in my comments on the question, and my feeling is that his comments in Pure Mathematics, of which I quoted only one, share the spirit of Gauss's statement, which you have so helpfully quoted. I wonder what Gauss would have made of the Riemann Sphere, which looks to me like what I might once have called a finished infinity. Perhaps you could elaborate or clarify.2012-05-05
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    @Contra Please read more carefully. I never said that Gauss *would* have praised Cantor's work if he were a contemporary. Rather, I said that he *might* have. My point is that no one can possibly know what Gauss would have thought. Your assertion that Gauss "would have condemned Cantor's ideas" is as unfounded as an assertion that Euclid would have condemned work on noneuclidean geometry.2012-05-05
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    @Mark: Here is a quote from Cantor, collected works, Zermelo (ed.), Springer (1932) p. 175: "Zu dem Gedanken, das Unendlichgroße nicht bloß in der Form des unbegrenzt Wachsenden und in der hiermit eng zusammenhängenden Form der im siebzehnten Jahrhundert zuerst eingeführten konvergenten unendlichen Reihen zu betrachten, sondern es auch in der bestimmten Form des Vollendet-unendlichen mathematisch durch Zahlen zu fixieren..." Here and on many other occasions Cantor speaks of the "completed infinity". Gauß only used the potential infinity, i.e. the limits of convergent series, endless plane etc.2012-05-05
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    @ Bill Dubuque: Sorry, I read too superficially. Your statetment is impeccable.2012-05-05
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    @WolfgangMueckenheim - that's interesting. I wonder whether the objection is that "infinity" is meaningless because it does not correspond to anything in the "real world" - which was part of the mistaken objection to non-euclidean geometry, and is a philosophical objection (or theological if "only God is infinite"). Or whether the core would be that infinity as an abstract object is insufficiently well-defined to be mathematically useful - in which case it could be argued that Cantor and his successors repaired the deficit, and recovered a mathematically useful concept of abstract infinity.2012-05-06
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    Why is it acceptable to post a quote in German and expecting people to understand?2014-08-10