# Talk:Reinhold Baer

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## Projective line

Reinhold Baer is a polemical figure in geometry. His Linear Algebra and Projective Geometry (1952) sets out to "establish the essential structural identity of projective geometry and linear algebra." For evidence, he states:

It has, of course, long been realized that these two disciplines are identical.
Projective geometry < — > linear manifolds
Projectivities < — > semi-linear transformations
Collineations < — > linear transformations
Dualities < — > semi-bilinear forms

On page 71 he writes:

A line [= linear manifold of rank 2] has no geometrical structure, if considered as an isolated or absolute phenomenon, since then it is nothing but a set of points with the number of points on the line as the only invariant.

The audacity of challenging projective harmonic conjugate as the basis for projective geometry is clear in his caveat "if considered as an isolated or absolute phenomenon". Since the phrase "a line has no geometrical structure" has been used by editors, the complete quotation has been given.

The classical ideas of affine and projective geometry are durable stages in advance of understanding, and the proponents of Baer’s truncation of this knowledge are identifiable.Rgdboer (talk) 23:07, 23 June 2014 (UTC)

Finite geometry has elucidated many fundamental ideas, and dismissing the harmonic conjugate approach of classical authors is another outcome. Buekenhout geometry with its diagrams has provided wide perspective in the manner advocated by Reinhold Baer, so that projective geometry need not follow the classical line. The 1998 text Projective Geometry by Albrecht Beutelspacher and Ute Rosenbaum demonstrates the capacity of the modern approach.Rgdboer (talk) 22:57, 18 October 2016 (UTC)

See projective harmonic conjugate#Galois tetrads to confirm that "a line has no geometric structure" is false even in finite geometry. Rgdboer (talk) 22:02, 23 November 2016 (UTC)