# 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?

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Gauss was involved with convergence tests for infinite series. –  Unreasonable Sin May 4 '12 at 21:04
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
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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. –  André Nicolas May 4 '12 at 22:18