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Did Gauss think that the size of infinite was incomparable?

Or did he leave some opinions about infinity or infinitesimals?

Did he have any letters or unpublished research materials that can tell us about his views on infinite?

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    $\begingroup$ "I protest first of all against the use of an infinite quantity as a completed one, which is never permissible in mathematics. The infinite is only a facon de parler, where one is really speaking of limits to which certain ratios come as close as one likes while others are allowed to grow without restriction", Gauss's 1831 letter to Schumacher. This was a common position going back to Aristotle that very few deviated from before Cantor. For context, see Waterhouse, Gauss on infinity. $\endgroup$
    – Conifold
    Commented May 26 at 3:16
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    $\begingroup$ Incomparable to what? $\endgroup$ Commented May 26 at 19:04

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Gauss was "against the use of an infinite quantity as a completed one" as pointed out in the comments. Such an attitude goes against the grain of modern set-theoretic mentality where one sometimes postulates the completed infinity of $\mathbb N$ even before doing $2+2$. But Gauss' attitude was common among mathematicians in the 17-19 centuries. His contemporary Cauchy similarly spoke against completed infinity in his Turin lectures in 1833. A century and a half earlier, Leibniz rejected completed infinity ("infinite wholes") in mathematics as early as the 1670s. It is easy to criticize such attitudes from the modern viewpoint, but note that even from Leibniz's viewpoint, this comment by Gauss contains a conflation of separate notions of infinity: an infinite quantity or magnitude on the one hand, and an infinite multitude (as formalized by Cantorian infinities), on the other.

Leibniz distinguished between the notion of a bounded infinity (infinitum terminatum) and an unbounded infinity (infinitum interminatum). As already mentioned, the latter is formalized today in terms of Cantorian infinities. Meanwhile, the former is formalized by the notion of infinite (more precisely, unlimited) number in modern number systems incorporating such ideas, such as nonstandard analysis. Leibniz used such "bounded infinities" (and their reciprocals, infinitesimals) in his analysis of geometric problems and other problems of the calculus. So one would have to include Leibniz among the "very few" mentioned by Conifold in the comments. For details on the infinita terminata in Leibniz, see for example this publication:

Katz, M.; Kuhlemann, K.; Sherry, D.; Ugaglia, M. "Leibniz on bodies and infinities: rerum natura and mathematical fictions." Review of Symbolic Logic 17 (2024), no. 1, 36-66. https://doi.org/10.1017/S1755020321000575, https://arxiv.org/abs/2112.08155

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