Talk:Synthetic geometry
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Article makes the statement that discovery of non-Euclidean geometry can be considered a success or failure. Would someone more knowledgable on this subject than me clarify this? If it's an opinion, it should be removed or arguments for both sides developed. Thank you.
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[edit] Definition is content-free
The first sentence claims a distinction between systems using "theorems and synthetic observations to draw conclusions" and those using "algebra to perform geometric computations and solve problems." The terms "synthetic" and "geometric" occur in both the definiens and the definiendum with no evident stopping condition for the implied recursion and therefore contribute nothing. What remains is one positive characteristic, "using theorems", and three negative: algebraic, computational, and problem oriented. Euclid and Descartes both prove theorems, so that's not a helpful distinction. Euclid is surely "problem oriented" (his was the only geometry available for engineering problems before Descartes), so that doesn't help either. "Algebraic" doesn't seem to help---Boolean algebra proves its theorems algebraically, does that make it synthetic or analytic? "Computational" doesn't help either, theorem-proving is heavily computational. The definition conveys nothing.
What does seem to be true is that synthetic geometry customarily refers to the geometry of Euclid and its refinements by Hilbert and others, based on the axiomatic definition of such geometric primitives as points, lines, polygons, and circles in terms of their relationships. Hilbert's student Otto Blumenthal quoted his advisor as having said in conversation "Man muß jederzeit an Stelle von 'Punkte, Geraden, Ebenen' 'Tische, Stühle, Bierseidel' sagen können"---one must always be able to say instead of 'point, line, plane' 'table, chair, beer mug'. Taken literally this might seem vacuous in light of the successful translation of Euclid into other languages, but one takes Hilbert's point to be that the content of an axiomatization of geometry must not depend in any way on our preconceived notions of those terms.
The traditional antithesis to synthetic geometry is Descartes's analytic geometry, which expresses all geometric entities relative to a coordinate frame consisting of d orthogonal axes intersecting in a common origin so as to define a d-dimensional space of points each a d-tuple of real numbers. The primitive geometric entities of analytic geometry are the points themselves together with the solution spaces of equations between polynomials associating one variable with each axis, some but not all of which have their counterparts in the constructions of Euclid. A key difference from Hilbert's beer mugs is that Cartesian geometry does not start from a semantic vacuum but takes several notions as given a priori including the real numbers, the operations +, −, ×, and ÷, the association of variables with axes, and the solution space concept.
If there are no objections I will revise the article accordingly. --Vaughan Pratt 02:27, 18 September 2007 (UTC)
I came looking for a definition/understanding of what synthetic geometry is and didn't find it. As far as I'm concerned, feel free to rewrite.--Eujin16 (talk) 04:43, 19 November 2007 (UTC)
