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The most pictorial geometric thinking I find in Dedekind is the intuitive-geometric idea of a continuum which he criticizes as too vague before he gives his account of the continuum based on cuts.

In sharp contrast to Riemann, who Dedekind whole-heartedly praises, Dedekind and Weber prove the Riemann-Roch theorem without so much as using the word "Curve (Kurve);" and using the word "surface (Fläche)" only in describing Riemann's own work.

Dedekind was a great geometrizer in the sense of formalizing ideas of dimension and genus and differentials. But so far as I have found he really does not see these pictorially.

It is not just that he does not trust pictorial thinking in proofs. Lots of people who do not trust pictures in proofs still give pictures with their proofs.

Is there some pictorial thinking in Dedekind that I have missed?

ADDED: K.B.'s answer points to a peculiarly non-visual visualization, where Dedekind says he cannot see the elements of a set. The best source I can find is a editorial note by Emmy Noether in Dedekind's Gesammelte mathematische Werke vol. 3 p. 449.

Dedekind expressed himself on the concept of a set saying: one represents a set as a closed sack, containing completely determinate things, but one cannot see them, and one knows only that they are determinate and are there.

The German is a bit more subtle:

Dedekind äußerte hinsichtlich des Begriffs der Menge: er stelle sich eine Menge vor wie einen geschlossenen Sack, der ganz bestimmte Dinge enthalte, die man aber nicht sehe, und von denen man nichts wisse, außer daß sie bestimmt und vorhanden seien.

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    $\begingroup$ I think to redefine the real axis you really had to have a problem with geometry in your childhood ;) $\endgroup$ – VicAche Apr 29 '15 at 9:20
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According to Bernstein, Dedekind imagined a set like a closed sack containing certain defined things. [Becker: Grundlagen, S. 316] Isn't that an example of pictorial thinking?

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  • $\begingroup$ This would seem to be pictorial but not pictorial-geometric thinking, and the latter is what the question is about. $\endgroup$ – Rory Daulton Jul 3 '17 at 0:03

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