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New entry synthetic projective geometry and also sort of disambiguation and history page synthetic geometry making a distinction with synthetic differential geometry.
Related algebraic entry quasifield.
Thanks.
I have added hyperlinks to the technical terms. Also I have added links to your new entries from relevant entries, so as to increase the chance that people can find the entries when browsing.
I have created a stub projective geoemtry just so that the link doesn’t appear grey-ed.
Concerning content:
after your discussion of the difference between the attitude of synthetic geoemtry and that of Lawvere’s SDG I have added the following remark (here):
(But this is different for his later development of cohesive geometry which includes the original synthetic differential geometry via differential cohesion. Here no analysis enter the axioms.)
I have a question on your next sentence after that, where you say
Lawvere also notes the failure (for physical applications) of the correspondence between evolution paths and corresponding maps in differential geometry,
What is this referring to?
What is this referring to?
Lawvere was complaining that the physical reasoning about evolution can be done in two ways: either consider evolution of a particle as a map from interval to a space or as a point in the mapping space. In practical analysis, these are not equivalent as the equivalence works only under heavy analytic assumptions; it works however in SDG.
But this is different for his later development of cohesive geometry which includes the original synthetic differential geometry via differential cohesion. Here no analysis enter the axioms.
I agree that cohesive geometry per se is far more synthetic in its essence; though it is hard to compare to the universe of notions of synthetic geometers of 19th century. Also most of the classical geometries of synthetic geometry would not qualify to raise a topos (for example, the subject of combinatorial projective planes, as a typical representative),
On the other hand, sorry, but the phrase “original synthetic differential geometry” makes no sense. SDG is invented by Lawvere and his school, period. Synthetic in 19th century had nothing to do with differential calculus, it was denoting the part of axiomatic geometry dealing only with properties expressible in the language of points, lines, and so on (no metrics, angles, functions, coordinates, neighborhoods, particularly no infinitesimally small quantities and distances). Word synthetic means non-analytic (both in the sense of Lagrange’s coordinate methos and Newton-Lebniz-Cauchy infinitesimal calculi). The terminology SDG conflated the 3 classical needs: for pre-Cauchy intuitive analysis, for analysis allowing analysis on mapping spaces, and for axiomatic relations without coordinate model. What you allude is just the third aspect only. Adding the terminology and consideration of infinitesimals is not digestable as synthetic geometry in the sense of 19th century, it is a neoterminological shift, conflation.
the phrase “original synthetic differential geometry” makes no sense. SDG is invented by Lawvere and his school, period.
in “Axiomatic cohesion” he begins to talk about infinitesimals in the “original” SDG without invoking the “original” axioms of SDG, but instead using cohesion.
about infinitesimals in the “original” SDG
Then he likely talked about his early attempts which are not published. SDG was not a word in 19th century.
Edit: see my next post.
OK, I understood. The original is referring to non-cohesive description of Lawvere/Kock/Reyes/Dubuc SDG.
Cohesive geometry is more in line with spirit of SG of 19th century. But it describes a different subject, a geometry which has more than just the synthetic aspect (in the sense of non-Lawvere SG).
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