Talk:Synthetic geometry

message: 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. — Preceding unsigned comment added by 12.221.98.167 (talk • contribs) 04:19, 9 December 2004 (UTC)
thread_title: Untitled

message: 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)
thread_title: Definition is content-free

message: Is this the same thing as synthetic geometry? Many sources, such as the geometry article, define synthetic geometry as the use of certain drawing instruments to do geometry, with the present discussion referred to as axiomatic geometry. Is a title change needed, or do Hilbert and others indeed refer to their discipline as "synthetic"? -- Cheers, Steelpillow (Talk) 17:04, 12 April 2010 (UTC)
replies: text: As far as I'm aware, axiomatic geometry means the same as synthetic geometry.  The geometry article no longer says otherwise :-)  What other sources have you seen? Jowa fan (talk) 10:03, 9 May 2010 (UTC)
replies: text: Same here. I am familiar with the philosophical idea of logical synthesis via axioms - of which I assume synthetic geometry is (or should be) a sub-discipline. A while ago I read Hilbert & Cohn-Vossen's "Geometry and the imagination", Greenberg's book (title forgotten) and Coxeter's "Projective geometry", and their axiomatic approach was consistent with the logical one, but I cannot remember to what extent they might have used the term "synthetic geometry" rather than "axiomatic geometry". -- Cheers, Steelpillow (Talk) 18:45, 9 May 2010 (UTC)
thread_title: Axiomatic geometry

message: Yeah, thats what I thought when I read that opening paragraph 
If I can calculate something, and then measure it in "real" life that’s one thing
If I use a set of rules A is B B is not C etc and then predict something that seems unreleated out of the morass that seems reality that is called either logic or philosophy and seems a bit more magical. But why does that seem more magical?
I would suggest it is the same "THING" where you either use a standard library function or roll your own, viz a sense of discovery i.e. you have learned WHAT it means and not just "this is how it is done"
I think this is called meta-thinking 
The question remains; I have given you the facts but do you know what the facts ARE do you know what the facts MEAN can you EXPAND the facts?
It all suggest a form of compression as in LZW no PSI but it might feel the same!
(Another thought; any compression relies on an agreed encoding scheme; so it might appear as a consequence of our evolution as a group predator, e.g. we are all singing from the same hymn book)
Stalinvlad May 2010
80.7.79.102 (talk) 23:48, 16 May 2010 (UTC)
thread_title:  Epistemology 

message: The following part of History was challenged.
:Karl von Staudt showed that synthetic geometry can be used to express the axioms of the field of rational numbers on a line. His profound study was continued by Mario Pieri and David Hilbert, among others. The results have set a new foundation to mathematics.
Consideration of the links given, particularly the last one to Foundations, will show that the statement is reasonable. It may be expanded by giving the results of David Hilbert in his book Foundations of Geometry which addresses algebraic properties deduced from geometric.Rgdboer (talk) 20:55, 3 September 2012 (UTC)
replies: text: The link "foundation to mathematics", which redirects to foundations of mathematics#Projective geometry, simply explains that projective geometry can be used to describe the axioms for a field.  It does not say anything about this being considered as a foundation of mathematics.  More information is needed here! Jowa fan (talk) 00:43, 4 September 2012 (UTC)
thread_title:  Staudt 

message: There is a subtle difference between synthetic geometry and axiomatic geometry in my opinion. First of all synthetic is the older and predominant term (2,160 vs. 808 hits on Google scholar). The use of axiomatic seems to be endemic amongst the math education and physics communities. One usually refers to the synthetic treatment of a subject, essentially meaning a coordinate-free approach to the topic. When a geometry is examined in this way, it may be called a synthetic geometry. To make the arguments of a synthetic geometry rigorous you essentially have no choice but to use the axiomatic method. However, if rigor is not a concern or the foundations of the topic are not important, then the axiomatics can be pushed into the background and you would still have a synthetic treatment. For instance, Pascal's theorem on conics can be easily proved by introducing coordinates or can be proved synthetically by using perspectivities. Axiomatics has almost no role in deciding how one wants to prove this result, unless you want to develop the result from first principles. 
Given this viewpoint, I am having a hard time trying to decide how much axiomatics to include in the article. Those who view synthetic and axiomatic geometry as synonyms (the current viewpoint of the article) would want more axiomatics than I would. I would prefer to devote more (some?) space to an example of using synthetic methods to prove something and contrasting that to a coordinate treatment. Regardless of this issue, the current section titled Logical synthesis is a poorly written fragment of what it should be if you really wanted to talk about axiomatics and I am not inclined to do the rewrite to bring it up to standard. Any thoughts on this? Bill Cherowitzo  (talk) 22:19, 12 September 2013 (UTC)
replies: text: I asked this question a while ago - see above. What we personally believe must take second place to the more reliable references. I have yet to revisit even those I mentioned above. Until someone does, we may note that Axiomatic geometry redirects here, so unless and until someone can divide the material into distinct standalone topics, we are left with this one article to explain both terms. Or, perhaps we can best understand axiomatics as the abstract mode of synthesis and merely include it here as a subtopic. Either way, axiomatic logic (aka logical synthesis?) is a distinct discipline and there should be no need to explain it in depth here, merely to summarise the relevant aspects and explain their application to geometry. — Cheers, Steelpillow (Talk) 09:13, 13 September 2013 (UTC)

text: That makes sense, feel free to remove or condense my recent additions once you have changed that redirect. — Cheers, Steelpillow (Talk) 19:48, 13 September 2013 (UTC)
thread_title:  Logical synthesis 

message: I added a unreferenced section section tag to the section computational 
synthetic geometry. What is meant by "a computational synthetic geometry 
has been founded"? Was it lying dormant in the closet? To the best of my 
knowledge there are different ways to make synthetic geometry amenable to 
computer treatment. This here shows me already 3 different ways.
:a synthetic approach
::Hilbert's axioms: points, lines, planes + geometric axioms
::Tarski's axioms: points + geometric axioms
::Euclid's axioms
http://geocoq.github.io/GeoCoq/
Are they related to matroids? Maybe, maybe not. This book below defines,
which is rather a study in a link between synthetic and analytic:
"Computational synthetic geometry deals with methods for realizing abstract 
geometric objects in concrete vector spaces. This research monograph considers 
a large class of problems from convexity and discrete geometry including 
constructing convex polytopes from simplicial complexes, vector geometries 
from incidence structures and hyperplane arrangements from oriented matroids. 
It turns out that algorithms for these constructions exist if and only if 
arbitrary polynomial equations are decidable with respect to the underlying field."
:Computational Synthetic Geometry, © 1989
:Autoren: Bokowski, Jürgen, Sturmfels, Bernd   
:https://www.springer.com/de/book/9783540504788
Jan Burse (talk) 02:12, 4 February 2019 (UTC)
thread_title:  Unreferenced Section Computational synthetic geometry