# What did Gauss think about infinity?

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?

-
If there ever was a question fitting for the [infinity] tag, this is it. –  Asaf Karagila May 4 '12 at 21:16
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. –  Bill Dubuque May 4 '12 at 22:02
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. –  Asaf Karagila May 4 '12 at 22:23
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. –  Mark Bennet May 5 '12 at 5:16
I've added a preemptive protection, since history taught us that this topic can be a crank magnet. :-) –  Asaf Karagila Jan 17 at 22:14

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.
-
While this link may answer the question, it is better to include the essential parts of the answer here and provide the link for reference. Link-only answers can become invalid if the linked page changes. –  Yes Aug 5 '14 at 21:44
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...) –  Asaf Karagila Aug 5 '14 at 22:55

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)]

"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"

-
@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" –  user17762 May 4 '12 at 21:29
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. –  André Nicolas May 4 '12 at 22:18
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. –  Bill Dubuque May 4 '12 at 22:59
Why am I not surprised to see Contra refer to Mückenheim? –  Jyrki Lahtonen May 5 '12 at 13:36
@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. –  Bill Dubuque May 5 '12 at 20:43

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.

-

## protected by Asaf KaragilaJan 17 at 22:13

Thank you for your interest in this question. Because it has attracted low-quality answers, posting an answer now requires 10 reputation on this site.