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

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)

Axiomatic geometry

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)

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)

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)

Epistemology

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)

Staudt

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)

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)

Yes, checking shows this aspect of Hilbert's work is not yet told in his article. Staudt's algebra of points had a significant impact; more references will be added. As for the reference to special relativity, that has been mentioned as affine geometry where there is a History section detailing references previously in this article. The development there seems more appropriate than under Synthetic geometry.Rgdboer (talk) 02:59, 4 September 2012 (UTC)

Logical synthesis

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)

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)

Agreed. I am currently working on Foundations of geometry as the axiomatic approach to geometry. When I am finished, I'll redirect Axiomatic geometry to that page. You might want to note how little overlap there is between what is currently on that page and this one (but to be honest, the major overlap areas have yet to be written on that page). The properties of axiom systems—consistency, independence, completeness, categoricalness—while important in comparing various axiom systems, does not have a role once a system has been chosen and the geometry is developed synthetically. That material belongs, and will eventually make it to, the foundations article. This article should be concerned with what one does after an axiom system has been selected. Bill Cherowitzo (talk) 17:01, 13 September 2013 (UTC)

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)

Tables, chairs and beer mugs

Although this famous "quote" of Hilbert can be found all over the web and has several citations in reliable sources, it nearly rises to the level of urban myth. The statement seems to come from Blumenthal, Hilbert's former student and biographer, but there are some discrepancies in his account which have led some to think that Blumenthal may have made this up himself, because it does reflect Hilbert's thinking so well. For a discussion of this, with references see this Math Stackexhange question. Due to the ambiguity, I had inserted the phrase "It is claimed that ..." before the Hilbert statement. This was reverted by Steelpillow with the reason that the statement was cited and not claimed. Curiously, in a previous edit he had changed "one" -> "logicians", giving a rendering that I have not seen anywhere else, including the cited work of Greenberg. This is a pretty minor point and not worth belaboring, but I did want to set the record straight. Bill Cherowitzo (talk) 00:17, 14 September 2015 (UTC)

On the minor edit from "one" to "logicians", another editor had tagged the section with a reminder that Wkipedia articles are not written in a personal way using pronouns such as "you", "one", "we", etc. Since this was not a direct quotation from source, it could - and should - be edited to conform to Wikipedia's house rules. It might be argued that I should have changed it to "geometers" rather than "logicians" but that is a different issue.

I think that the IP was more concerned with the tone of the passage (as seen from the various attempts to get an appropriate tag) than the specific use of pronouns (the tag finally picked didn't make any sense, there was only one use of "you" in the passage). In any event "one" is not considered a personal pronoun and can be used freely in keeping with an encylcopedic tone. I think that replacing it by anything more specific would be considered WP:SYNTH. (Maybe I am just being a little sensitive on the issue. A while back I made a similar innocent edit that recently got picked up and published. I got reamed by some colleagues for that incident.)Bill Cherowitzo (talk) 17:06, 14 September 2015 (UTC)

If "one" is acceptable than I have no strong opinions either way. — Cheers, Steelpillow (Talk) 11:54, 16 September 2015 (UTC)

Ok. I'll change that back. Bill Cherowitzo (talk) 19:07, 16 September 2015 (UTC)

However the anecdote itself is from a reliable source. It is so widely repeated that not to include it would be remiss. Wikipedia's policy of verification is driven by what reliable sources say and not by the truth of what they say. If there is adequate sourcing for the doubt about whether Hilbert actually said it or what he might really have meant by it, then that might be worth explaining somewhere. However mere suspicions expressed by one or two writers are not enough on their own.

I hope this addresses the issues raised. — Cheers, Steelpillow (Talk) 09:00, 14 September 2015 (UTC)

I did not advocate not reporting the anecdote, I was just trying to make the statement a more accurate reflection of the literature. The rub here is that the anecdote is repeated so often in tertiary sources (and that includes Greenberg), but only rarely in secondary sources. (A secondary source would be one that cites Blumenthal, while Blumenthal's article itself would be considered a primary source in this instance.) One of these is the article by Grattan-Guiness, a well known and highly respected math historian. As secondary sources, and not tertiary sources, are supposed to be the basis for reliability (WP:PSTS), more weight (and not less as you suggest) should be given to them, and that is the reason for my edit. I was aware of this issue before I found the Stackexchange question on-line, so I would say that this concern is a bit more widespread than one or two writers. Bill Cherowitzo (talk) 14:59, 15 September 2015 (UTC)

Thank you for explaining about the sources. My less detailed knowledge was the reason I wrote "If there is adequate sourcing..." without trying to prejudge the issue. I am still unclear about that. The suggestion that Hilbert has been misunderstood begs the question as to what he really meant and I think we would need a sensible source for that. Otherwise, we can only state a weaker caveat along the lines of, "It has been claimed that Hilbert is misunderstood", and I am not convinced that is a significant enough claim to be worth including. — Cheers, Steelpillow (Talk) 11:54, 16 September 2015 (UTC)

Agreed. I wouldn't go down that path. I have no idea of what Grattan-Guiness meant by that misunderstood remark. The bigger picture is that there is only one source (Blumenthal) for the famous remark and no written confirmation. This has been picked up, embellished and repeated so often that it is treated like gospel, but in reality its reliability is very weak. To say that "It is said that Hilbert said ..." rather than "Hilbert said ..." is a more accurate statement of what is in the secondary sources. I don't think that this article is the place to go into the details, but I haven't found the "quote" mentioned in the articles about Hilbert or his axiom system. Bill Cherowitzo (talk) 19:07, 16 September 2015 (UTC)

As I recall, the interpretation as relating to axiomatic treatment is the one which the quotation is most often attached to. Having to peddle an urban myth without adequate evidence as to the truth of the matter is always galling. — Cheers, Steelpillow (Talk) 09:34, 18 September 2015 (UTC)

Synthetic vs axiomatic geometry

I am wondering again is Synthetic and Axiomatic geometry really the same thing? or should we split them up into two seperate articles.

First of all the link axiomatic geometry in the lead Synthetic geometry (sometimes referred to as axiomatic or even pure geometry) links not to synthetic geometry but to the article Foundations of geometry.

I modified the lead a bit to include a bit from what is in the section Logical synthesis and that was promptly removed.

Maybe it is better to split it in two articles:

• Axiomatic geometry that is working with axioms (and thus in principle with primitive notions, logical deductions formula, and that ilk)
• Synthetic geometry that is working with straightedge, compass and maybe even marked straightedges.

Euclid can easely be seen as an Synthetic geometer, for him circles do intersect eachother something that can bee seen in the construction but could not be locically proven.WillemienH (talk) 21:07, 2 January 2016 (UTC)

I reverted your change because it missed the mark. The essence of synthetic geometry is that it is coordinate free and not that it is axiomatic. If you don't use coordinates you must base your reasoning on some ground and that is, most frequently, the axiomatic method. However, this does not mean that they are the same thing (to be precise, I am referring to being coordinate-free vs. using the axiomatic method). Peano arithmetic is also axiomatically presented, but no one refers to it as "synthetic arithmetic". Axiomatic geometry is a modern phrase used to describe synthetic geometry, probably cooked up by some educator who didn't think that the term "synthetic" was a good choice, and wanted a more meaningful expression. The link in the lead is actually correct in the following sense - the link is on axiomatic and should point to an article that talks about the axiomatic method as applied in geometry ... this article is the Foundations of geometry article. It is unfortunate that "axiomatic geometry" can be interpreted either generically, as "geometry studied axiomatically", or as a proper name, an alternative to synthetic geometry. Perhaps we should make axiomatic geometry a disambiguation page! In any event, something should be done about fixing this up. Bill Cherowitzo (talk) 05:10, 3 January 2016 (UTC)

I think your remark about "synthetic arithmetic" misses the point there is no established study of "coordinate based arithmetic" comparable with the difference between coordinate geometry and synthetic geometry, but I think we could discuss that for a long time. In Descartes time I guess there was no such thing as axiomatic geometry, there was only Euclidean geometry with compass and straightedge constructions, Decartes added axial coordinates to it. I do think there is a difference between compass and straightedge constructions and axiomatic / logical formal proofs and would like to have seperate articles for them. (but then are there no other "coordinate free geometries "?) I guess we could discuss that for a long time. I am wondering about what is written at Coordinate-free , first of all i think it should be renamed to Coordinate-free geometry and are vectors really coordinate free? (given that it is based on unit vectors and so ) I changed axiomatic geometry to a disambiguation page. WillemienH (talk) 11:20, 3 January 2016 (UTC)

Projective geometry is a topic which is sometimes treated axiomatically (e.g. Hilbert), sometimes via constructional methods (e.g. Edwards), sometimes via its own (coordinate-free) algebra, and sometimes by imposing coordinates. I think it might be a useful test case to see whether our definitions; a) make sense and b) can be found in the literature. — Cheers, Steelpillow (Talk) 12:10, 3 January 2016 (UTC)

Projective geometry is a fine example of a subject that can be studied synthetically (in a coordinate free manner) or analytically (using coordinates) and several texts use precisely those terms in describing aspects of their treatments. (For instance, Beutelspacher & Rosenbaum, Projective Geometry - Chapter 1 is called Synthetic Geometry and Chapter 2 is called Analytic Geometry and in Bumcrot, Modern Projective Geometry we find the statement "In the previous section we generalized the synthetic version of P₂(R). Now we shall generalize the analytic (coordinatized) version." (pg. 30)). The important thing to note here is that we are not talking about different types of geometry but rather about different ways of studying a geometry. The term "coordinate-free geometry" does not make any sense to me. You can study any geometry in a coordinate-free manner ... this says nothing about the type of geometry you are examining. "Synthetic geometry" is the historical term used to describe the study of geometry in a coordinate -free manner (see Klein's definition quoted in the article). Initially this only applied to Euclidean geometry to contrast it from the treatment following Descartes (i.e., using coordinates), but when other types of geometry began to emerge the term was broadened to include any type of geometry studied without the use of coordinates.

The point I was trying to make earlier was that "synthetic" and "axiomatic" (in its generic sense) are not synonyms. If they were, then talking about "synthetic arithmetic" would make sense, but, as has been pointed out, such terminology is not used because there is no coordinatized version of arithmetic. This just underscores what I was saying, "synthetic" corresponds to not using coordinates while "axiomatic" does not have that connotation. Unfortunately, some authors have confounded the two terms and use "axiomatic geometry" (as a proper name) as a synonym of "synthetic geometry". We are not in the position of correcting the mistakes of others on Wikipedia pages, so we need to report what is in the literature and that is why the term "axiomatic" is given in the lead (as it should be). Bill Cherowitzo (talk) 20:18, 3 January 2016 (UTC)

Unreferenced Section Computational synthetic geometry

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)