@BOOK{gray:book04,

author = “Jeremy Gray”,

title = “Janos Bolyai: Non-Euclidean Geometry and the Nature of Space”,

publisher = “Burndy Library”,

year = “2004”,

address = “Cambridge, MA, USA”}

*“He calls such a move a “Euclidean rescue” by analogy with the case of Euclidean geometry – when Einstein and others gave evidence showing that physical space did not obey Euclidean geometry, geometry was reinterpreted as being about abstract points and lines, ratehr than physical ones as had always been presupposed.”*

I’ve not read Resnik’s book, so I’m not sure if my comment is a criticism of his argument or of your summary. But the historical record is other than what this statement says. Hilbert published his axiomatic treatment of geometry (in which geometry was about abstract entities) in 1899, before Einstein’s re-conception of physical space as non-Euclidean from 1905.

The historical impetus to the re-interpretation of geometry as being about abstract entities was not the failure of physical space to conform to the axioms of Euclidean geometry, but was the presence of multiple competing axiom systems for geometry (Euclidean and non-Euclidean). These were discovered in the early 19th century and discussed widely among mathematicians from the mid-19th century. In other words, the revision of the beliefs of mathematicians about the nature of geometry was entirely due to evidence from mathematics, and not due to any evidence from the real-world.

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