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I am curious about von Neumann's "Continuous geometry", but found no recent text or survey on it. Does anyone know the book and would be so nice to share their impression, and if/how the concept of such geometries fits into contemporary tries to generalize geometry?

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4 Answers 4

At the conference on Non-commutative geometry and operator algebras, at Vanderbilt in 2009 (focused on $\mathbb{F}_1$-geometry), Connes gave a series of lectures. In the 4th or 5th lecture he sketched the idea of von Neumann's "continuous geometry" and said how it could fit into the picture of $\mathbb{F}_1$-geometry.

This was long ago and I can't recall what was the exact connection. Possibly the point was that one can see $\mathbb{F}_1$-geometry as limit case of the geometry of Tits buildings at the different primes and von Neumann's geometry was meant to provide a "continuum of geometries" to fill in the space between the primes...

Connes' lectures were filmed by Norihiko Minami, but I know of no place where they appear online. You may ask Minami or the organizers of the conference, who got a copy, I think.

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I think there were only ever 3 books published on Continuous Geometry, all over 50 years ago. There's Skornyakov's book mentioned in the answer above, and before that von Neumann's notes, as linked to in the question. But the very first book published on Continuous Geometry was written in Japanese by Fumitomo Maeda. There is an official German translation Kontinuierliche Geometrien but no official English translation as far as I know. But if you want an idea of what Maeda's book contains you can see the short excerpt I posted together with my own English translation at the bottom of my web page.

I have not read any of these books cover to cover, but my impression is that the books by Maeda and Skornyakov are more up to date. Von Neumann's notes are based on his lectures in the late 30's, when the theory was still being developed, particularly in the reducible case, and could not be updated when they were finally published posthumously 20 years later by von Neumann's former student Halperin.

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In plane projective geometry, there are "subspaces" of dimension 0 (points) dimension 1 (lines) and dimension 2 (the whole space). We may also add dimension -1 (the empty set). Axioms may be given concerning incidence among these various subspaces. There is a "duality" that switches points and lines; the whole space and the empty set; reverses inclusion.

In $n$-dimensional projective geometry, there are "subspaces" of dimension $-1,0,1,\dots,n$. Axioms may be given concerning incidence again. And duality.

The idea of continuous geometry is to allow subspaces of dimensions in more general totally ordered sets. The main example is $[0,1]$. Still there are axioms concerning incidence. And duality.

As noted in the other answers, it did not turn out to be an important idea.

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Skornyakov L.A., Complemented modular lattices and regular rings, 1964,

Ulrich BREHM, Marcus GREFERATH, Stefan E. SCHMIDT, Projective Geometry on Modular Lattices (In: F. Buekenhout, Handbook of incidence geometry. Buildings and foundations, 1995)

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