If similarity geometry means similarity-invariant geometry, what are its objects? Google has a lot to say about similarity spaces, none of it relevant to similarity invariance. Sticking to finite dimensions, a Euclidean space is standardly defined as an inner product space over the reals. As such it has predicates recognizing those line segments of unit length, and those lines passing through the origin, neither of which I remember from Euclid, who seems rather to be writing about similarity geometry. Euclid does however talk about right triangles and line segments of equal length (and equal angles but that follows from the other two). So clearly Euclid is doing more than just affine geometry. If Euclid's plane is not the Euclidean plane, what is it? If Euclid was doing similarity geometry it should be called a similarity space (certainly not a vector space or a Euclidean space or an affine space or a projective space or a metric space or a topological space). If it's called something else what is it? If Euclid was doing some other kind of geometry than similarity geometry what kind was it? Vaughan
Some helpful discussion with Dana Scott and Fred Linton got me to the point where I felt I could say reasonably succinctly exactly what a Euclidean space is (or ought to be). The following summarizes my present understanding, which is a lot clearer than 24 hours ago. I've seen only two essentially different approaches so far, bottom-up (adding structure) and top-down (forgetting structure). Bottom-up: A Euclidean space is an affine space transformable only by similarities. Since affine spaces are already understood to be transformable only by affinities, the further restriction to similarities is equivalent to the preservation of circles, or of right isosceles triangles. Affine spaces are in turn definable as an affine-closed subspace of a projective space transformable only by affinities. Constraining the morphisms in this way is morally equivalent to adding structure (of some unspecified kind), whence "bottom-up". Top-down: A Euclidean space is a torsor for a Euclidean inner product space E. A torsor, or principal homogeneous space, is (in this instance) a generalized metric space with vector distances in place of real distances so as to make the triangle inequality an equality, with d(x,y) = -d(y,x) and d(x,y) = 0 iff x = y. Like metric spaces torsors have no origin. E supplies the distances. I put "Euclidean" as a modifier for "inner product space" to connote the liberalization of morphisms thereof to preserving inner product only up to a constant factor (as opposed to the presumed default of on the nose). This liberalization accommodates scaling, and can be considered as forgetting the scale. That and torsors for forgetting the origin makes this approach "top-down." I've used both approaches in software for things like surveying and computer graphics, using respectively top-down and bottom-up. Top-down is simple at a low level, is primarily vector oriented, involves little or no multiplication, and works fine with single-precision floating point or even fixed point arithmetic. Bottom-up is simple at a high level (if you don't try to follow the individual arithmetic steps when debugging), uses an extra dimension to express projective geometry (needed to represent the part at infinity dropped when passing to affine space), is primarily matrix oriented, involves lots of multiplication, and benefits from double precision. All high-end video cards today use bottom-up exclusively, understandable for rendering which needs projective rather than Euclidean geometry, but its benefits for rigid solid modeling over top-down are less clear to me, though for articulated solid modeling bottom-up seems preferable. What other approaches to defining Euclidean space have been proposed? Vaughan
Funny you should ask this just as I was trying to find a satisfying answer to this question. Now we are so accustomed that an Euclidean space has some orthonormal system plonked down somewhere in it, or that it has at least a fixed origin and the scalar product that defines lengths and angles. But, like Eve and Adam were created without navels, Euclid's space was created without an origin; also it was created completely flat (with curvature 0 in modern lingo), so that there is no 'canonical' unit of length; and it was created withot a priory orientation. But you can measure one line segment with another in it, you can also mesure angles, and you can _choose_ one of the two possible orientations and call it positive. To capture all this I cooked up some structure; you decide if it does the job. First, an Euclid's space (let's call it that, to avoid confusing it with an Euclidean space) is a real affine space (V,P), where V is a real vector space of _vectors_ in the space, and P is the set of its _points_. There are also two operations +: P >< V -> P and -: P >< P -> V; the first is an action of the additive group of V on P, so it satisfies a + 0 = a and (a + u) + v = a + (u + v) for all points a and all vectors u, v. Moreover, the two operations are 'local inverses' of each other in the sense that (a + u) - a = u and a + (b - a) = b for all points a, b and all vectors u. This definition eliminates the need for an a priory origin. To make life easier, assume V is finite dimensional. Morphism (V,P) -> (U,Q) of affine spaces is a pair of maps h: V -> U and f: P -> Q, where h is a linear map and f(a + v) = f(a) + h(v) for all points a in P and all vectors v in V. Now, the metric of the Euclid's space. There are positive-definite (symmetric bilinear) forms on V. Call two such forms similar if they differ by a constant positive factor; a similarity class is therefore a ray (open half-line with the endpoint the zero form) in the space of all symmetric bilinear forms on V. Now consider the structure E = (V, P, m), where (V,P) is a real affine space and m is a similarity class of positive-definite forms on V: this is my proposed structure of Euclid's space. Let S be the group of all automorphisms (h,f) of the affine space (V,P) that preserve m: for every g in m, g(h(u),h(v)) = c g(u,v) for some positive constant c (depending on h, not depending on u and v) and all vectors u, v. These are the similarity trasformations of E, defining the similarity geometry of E. If A is any structure built from vectors and points (and lines etc) in E, then the orbit of A under S is the object studied in this geometry. Let R be the group of all automorphisms (h,f) of the affine space (V,P) that preserve some form in m, and therefore preserve _every_ form in m: g(h(u),h(v)) = g(u,v) for all g in m and all vectors u, v. This is the group of rigid motions of E. Orientation: if dim V = n, then there is a one-dimensional space of alternating n-linear forms on V; choosing one of the two rays (half-lines with the endpoint at the zero form) in this space chooses the orientation. -- France
If similarity geometry means similarity-invariant geometry, what are its objects? Google has a lot to say about similarity spaces, none of it relevant to similarity invariance.
Sticking to finite dimensions, a Euclidean space is standardly defined as an inner product space over the reals. As such it has predicates recognizing those line segments of unit length, and those lines passing through the origin, neither of which I remember from Euclid, who seems rather to be writing about similarity geometry.
Euclid does however talk about right triangles and line segments of equal length (and equal angles but that follows from the other two). So clearly Euclid is doing more than just affine geometry.
If Euclid's plane is not the Euclidean plane, what is it? If Euclid was doing similarity geometry it should be called a similarity space (certainly not a vector space or a Euclidean space or an affine space or a projective space or a metric space or a topological space). If it's called something else what is it? If Euclid was doing some other kind of geometry than similarity geometry what kind was it?
Vaughan
-- Dr. France Dacar Email: france.dacar@ijs.si Intelligent Systems Department Phone: +386 1 477-3813 Jozef Stefan Institute Fax: +386 1 425-3131 Jamova 39, 1000 Ljubljana, Slovenia
Euclides was doing Euclidean Geometry !
If similarity geometry means similarity-invariant geometry, what are its objects? Google has a lot to say about similarity spaces, none of it relevant to similarity invariance.
Sticking to finite dimensions, a Euclidean space is standardly defined as an inner product space over the reals. As such it has predicates recognizing those line segments of unit length, and those lines passing through the origin, neither of which I remember from Euclid, who seems rather to be writing about similarity geometry.
Euclid does however talk about right triangles and line segments of equal length (and equal angles but that follows from the other two). So clearly Euclid is doing more than just affine geometry.
If Euclid's plane is not the Euclidean plane, what is it? If Euclid was doing similarity geometry it should be called a similarity space (certainly not a vector space or a Euclidean space or an affine space or a projective space or a metric space or a topological space). If it's called something else what is it? If Euclid was doing some other kind of geometry than similarity geometry what kind was it?
Vaughan
Vaughan Pratt writes:
I've seen only two essentially different approaches so far, bottom-up (adding structure) and top-down (forgetting structure).
Where does Euclid's approach fit in this scheme? Starting from say Hilbert's formulation of Euclidean geometry it is certainly possible to get to an inner product space and all the rest, but it's certainly not direct and not easy. -- Bob -- Robert L. Knighten RLK@knighten.org
Vaughan Pratt wrote:
Top-down:
A Euclidean space is a torsor for a Euclidean inner product space E.
A torsor, or principal homogeneous space, is (in this instance) a generalized metric space with vector distances in place of real distances so as to make the triangle inequality an equality, with d(x,y) = -d(y,x) and d(x,y) = 0 iff x = y. Like metric spaces torsors have no origin. E supplies the distances. I put "Euclidean" as a modifier for "inner product space" to connote the liberalization of morphisms thereof to preserving inner product only up to a constant factor (as opposed to the presumed default of on the nose). This liberalization accommodates scaling, and can be considered as forgetting the scale. That and torsors for forgetting the origin makes this approach "top-down."
Dear Vaughan, I too thought about torsors, but couldn't see round the problem that they fix a unit length. (Suppose E is an inner product space and X a torsor for it, then for any x, y in X there is a unique v in E taking x to y, and so the length of v gives the distance from x to y.) Does "liberalizing the morphisms" in the way you suggest really do the trick? That seems to require a new notion of torsor, and I can't see how it would work technically. Regards, Steve.
Dear Colleagues, The artificial choice of unit of length should indeed be avoided in fundamental considerations since, among other things, it trivializes the relation between length & area, etc. Over a given rig R, a Euclidean space E seems to be 1) a torsor over an R-module V where V is equipped with 2) an isomorphism V->Hom(V, L) 3) where L is an invertible R-module. The fact that L itself has no given rig structure can be compared with the general idea of metric (TAC Reprints 1) as valued in a monoidal category (which has one covariant binary operation, NOT two.) The R-modules R, L, L@L, etc, may be non-isomorphic, and there may even be other invertibles corresponding to time, force, etc. However, this Picard group will have all its elements of order two (i.e.,each invertible module will have its own R-valued pairing) if R is real in the sense that a sum of several squares is invertible if one of them is (as shown a few years ago by Steve Schanuel). That result is for the category of abstract sets, destroying my hope that the free abelian group on three generators might occur for a suitable spatial topos of affine algebraic geometry. Of course, a unit of M, where M is an invertible module, should be an isomorphism R->M, but in general such isomorphisms exist only locally. To see nontrivial invertible modules, look at classical arithmetic or complex analysis, or in the present geometric vein, look at rigs and R-modules not in abstract sets, but in more cohesive or variable toposes. Garrett Birkhoff recommended the topos of G-sets for a certain group G of homogeneities, obtaining (in a rather tautological way) the result that the group of dimensional analysis occurs as a Picard group. Looking forward to your thoughts Bill Quoting Steve Vickers <s.j.vickers@cs.bham.ac.uk>:
Top-down:
A Euclidean space is a torsor for a Euclidean inner product space E.
A torsor, or principal homogeneous space, is (in this instance) a generalized metric space with vector distances in place of real distances so as to make the triangle inequality an equality, with d(x,y) = -d(y,x) and d(x,y) = 0 iff x = y. Like metric spaces torsors have no origin. E supplies the distances. I put "Euclidean" as a modifier for "inner product space" to connote the liberalization of morphisms
Vaughan Pratt wrote: thereof
to preserving inner product only up to a constant factor (as opposed to the presumed default of on the nose). This liberalization accommodates scaling, and can be considered as forgetting the scale. That and torsors for forgetting the origin makes this approach "top-down."
Dear Vaughan,
I too thought about torsors, but couldn't see round the problem that they fix a unit length. (Suppose E is an inner product space and X a torsor for it, then for any x, y in X there is a unique v in E taking x to y, and so the length of v gives the distance from x to y.) Does "liberalizing the morphisms" in the way you suggest really do the trick? That seems to require a new notion of torsor, and I can't see how it would work technically.
Regards,
Steve.
the sense (a joke) of my posting was lost because some how the title of it was lost in the way my posting had the following title: "an stupid answer"
Euclides was doing Euclidean Geometry !
If similarity geometry means similarity-invariant geometry, what are its objects? Google has a lot to say about similarity spaces, none of it relevant to similarity invariance.
Sticking to finite dimensions, a Euclidean space is standardly defined as an inner product space over the reals. As such it has predicates recognizing those line segments of unit length, and those lines passing through the origin, neither of which I remember from Euclid, who seems rather to be writing about similarity geometry.
Euclid does however talk about right triangles and line segments of equal length (and equal angles but that follows from the other two). So clearly Euclid is doing more than just affine geometry.
If Euclid's plane is not the Euclidean plane, what is it? If Euclid was doing similarity geometry it should be called a similarity space (certainly not a vector space or a Euclidean space or an affine space or a projective space or a metric space or a topological space). If it's called something else what is it? If Euclid was doing some other kind of geometry than similarity geometry what kind was it?
Vaughan
From: Steve Vickers
That seems to require a new notion of torsor, and I can't see how it would work technically.
I don't know if it's a new notion, but the following construction of what I'll call Aff(C), the affinitization (affination?) of C, should work for any category C concrete over Ab via a faithful functor U: C --> Ab. (Ab(Z,-): Ab --> Set automatically makes Ab concrete over Set.) The subcategories of Vct_k we've been talking about, in particular Euc with origin but liberal morphisms, are all instances of such a C. If this is different from torsors I'd appreciate some insight into the difference. Definition: Take the objects of Aff(C) to be those of C. A morphism in Aff(C) from c to d is a pair (h, m) where m: c --> d is a morphism of C and h: U(c) --> U(d) is any group homomorphism such that h(x) - h(y) = U(m)(x - y) for all x, y in U(c). End definition. The effect of this construction is to make the objects of Aff(c) generalized metric spaces. Reading between the lines of the above, an implicit metric D(x, y) is given by the group structure in the form x + D(x, y) = y, or equivalently D(x, y) = y - x; there is no need to make D explicit with its own symbol. All four Frechet axioms follow after modifying them to make the triangle inequality an equality and D(x, y) = -D(y, x) (symmetry becomes antisymmetry). The morphism (h, m) maps the points of the metric space via h subject to coherently (over the whole space) maintaining the distance with m. That should do it for Euc. France Dacar did things in the other order: torsor first, liberate from scale second. I guess they commute ... but: To be sure we've been talking about the same category Euc of what France called Euclid's spaces, for the Euc I have in mind, Euc(E_m,E_n) has (m+1)n - (m+1 choose 2) + 1 degrees of freedom when 1 <= m <= n. Outside that range Euc(E_m,E_n) has n degrees of freedom, with rigidity obliging E_m for m > n to collapse to E_0. When m = n > 0 this reduces to 1 + n + (n choose 2), corresponding to the one dilatation, n translations, and rotation group of order (n choose 2) = n(n-1)/2 in E_n. Here's a table. . 0 1 2 3 4 5 6 7 8 9 ----------------------------------------- 0 | 0 1 2 3 4 5 6 7 8 9 1 | 0 2 4 6 8 10 12 14 16 18 2 | 0 1 4 7 10 13 16 19 22 25 3 | 0 1 2 7 11 15 19 23 27 31 4 | 0 1 2 3 11 16 21 26 31 36 5 | 0 1 2 3 4 16 22 28 34 40 6 | 0 1 2 3 4 5 22 29 36 43 7 | 0 1 2 3 4 5 6 29 37 45 8 | 0 1 2 3 4 5 6 7 37 46 9 | 0 1 2 3 4 5 6 7 8 46 In contrast Aff_R(A_m,A_n) (affine spaces) has (m+1)n degrees of freedom, while Vct_R(V_m,V_n) (vector spaces) has mn. If Euler invented affine geometry as being somehow cleaner than Euclidean geometry he was going in the right direction, with vector spaces making things simpler yet via familiar mxn matrices. Regarding Bob Knighten's question on approaches to synthetic Euclidean geometry such as Hilbert's axiomatization, has anyone axiomatized more than E_3 (Euclid Books 11-13)? A satisfactory definition of Euclidean space needs to account for all finite dimensions. Vaughan
Bourbaki in his Algebra, and MacLane & Birkhoff in their Algebra, have similar definitions. I do not know the one you cite. However, the definition I have given describes an affine space as an honest equationally presented algebraic structure; usually they just assert that given any two points a and b, there exists a unique vector v such that a + v = b. With the difference of points an explicitly given operation of the structure, the axiom a + (b - a) = b gives the existence, and the axiom (a + v) - a = v the uniqueness. Using an n-dimensional real space instead of beloved R^n gets rid of the `canonical' basis. Observing vectors as translations of the space of points does away with an a priory origin. My contribution was that to do without an a priory unit of length, let the metric of the Euclid's space be given by a class of mutually proportinal positive definite forms, instead of a single form.
"An affine space is, roughly speaking, a vector space, but without a particular vector being chosen as zero." Or shall we say that an affine space is a perfect geometric communism where all points are equal, while in a vector space one vector is more equal than others.
-- France Ellis D. Cooper wrote:
FYI, Hassler Whitney, "Geometric Information Theory", 1957, Appendix I, Section 10, presents the notion of "affine space" essentially as you describe: "An affine space is, roughly speaking, a vector space, but without a particular vector being chosen as zero."
Funny you should ask this just as I was trying to find a satisfying answer to this question. Now we are so accustomed that an Euclidean space has some orthonormal system plonked down somewhere in it, or that it has at least a fixed origin and the scalar product that defines lengths and angles. But, like Eve and Adam were created without navels, Euclid's space was created without an origin; also it was created completely flat (with curvature 0 in modern lingo), so that there is no 'canonical' unit of length; and it was created without a priory orientation. But you can measure one line segment with another in it, you can also mesure angles, and you can _choose_ one of the two possible orientations and call it positive. To capture all this I cooked up some structure; you decide if it does the job.
First, an Euclid's space (let's call it that, to avoid confusing it with an Euclidean space) is a real affine space (V,P), where V is a real vector space of _vectors_ in the space, and P is the set of its _points_. There are also two operations +: P >< V -> P and -: P >< P -> V; the first is an action of the additive group of V on P, so it satisfies a + 0 = a and (a + u) + v = a + (u + v) for all points a and all vectors u, v. Moreover, the two operations are 'local inverses' of each other in the sense that (a + u) - a = u and a + (b - a) = b for all points a, b and all vectors u. This definition eliminates the need for an a priory origin. To make life easier, assume V is finite dimensional.
Morphism (V,P) -> (U,Q) of affine spaces is a pair of maps h: V -> U and f: P -> Q, where h is a linear map and f(a + v) = f(a) + h(v) for all points a in P and all vectors v in V.
Now, the metric of the Euclid's space. There are positive-definite (symmetric bilinear) forms on V. Call two such forms similar if they differ by a constant positive factor; a similarity class is therefore a ray (open half-line with the endpoint the zero form) in the space of all symmetric bilinear forms on V. Now consider the structure E = (V, P, m), where (V,P) is a real affine space and m is a similarity class of positive-definite forms on V: this is my proposed structure of Euclid's space.
Let S be the group of all automorphisms (h,f) of the affine space (V,P) that preserve m: for every g in m, g(h(u),h(v)) = c g(u,v) for some positive constant c (depending on h, not depending on u and v and the choice of g in m) and all vectors u, v. These are the similarity trasformations of E, defining the similarity geometry of E. If A is any structure built from vectors and points (and lines etc) in E, then the orbit of A under S is the object studied in this geometry.
Let R be the group of all automorphisms (h,f) of the affine space (V,P) that preserve some form in m, and therefore preserve _every_ form in m: g(h(u),h(v)) = g(u,v) for all g in m and all vectors u, v. This is the group of rigid motions of E.
Orientation: if dim V = n, then there is a one-dimensional space of alternating n-linear forms on V; choosing one of the two rays (half-lines with the endpoint at the zero form) in this space chooses the orientation.
-- France
-- Dr. France Dacar Email: france.dacar@ijs.si Intelligent Systems Department Phone: +386 1 477-3813 Jozef Stefan Institute Fax: +386 1 425-3131 Jamova 39, 1000 Ljubljana, Slovenia
Vaughan wrote in part:
[A "Euclidean space"] has predicates recognizing those line segments of unit length, and those lines passing through the origin, neither of which I remember from Euclid, [...].
Euclid does however talk about right triangles and line segments of equal length (and equal angles but that follows from the other two).
You seem to have answered your main question, but here are two side points: I have seen a definition of "Euclidean space" better than the above (as a real inner product space): a real affine space with a compatible metric and the parallelogram identity. This is equivalent to a torsor of a real inner product space, or to a real affine space modulo length-preserving transformations. I agree that your definition is correct and this is wrong, since Euclid had only relative lengths, not absolute length. But it's worth knowing that the term "Euclidean space" has varied meanings. I can try to track down the reference if you wish. I also think that it's enlightening to look at Tarski's axioms for geometry. These specify a set of points equipped with a ternary and a quaternary relation (betweenness: A lies on the line segment BC; and AB is congruent to CD). Most of the axioms are independent of dimension, and the betweenness- and congruence-preserving bijections are precisely the affine similarities, as you want. (Indeed, preserving betweenness makes it affine; preserving congruence makes it a similarity.) But one final axiom sets the dimension; this states (in effect) that there exist n + 1 points, affinely independent and affinely spanning the space. If you label the first n points (1,0,0,...) through (...,0,0,1) and the last point the origin, then every point has a unique label, and the space becomes \R^n. Thus the difference between the bad "Euclidean space" as \R^n (down to having a specific basis, with no nontrivial automorphisms) and the good Euclid's space as you want to define it is precisely whether the isomorphisms also fix these n + 1 points. (Allowing the isomorphisms to permute the n non-origin points gives us the common "Euclidean space" as a real inner product space; allowing them to permute all of the n + 1 points gives us "Euclidean space" as a torsor of a real inner product space, as in the first section of this post.) --Toby
I wrote in part:
I also think that it's enlightening to look at Tarski's axioms for geometry. [...]
But one final axiom sets the dimension; this states (in effect) that there exist n + 1 points, affinely independent and affinely spanning the space. If you label the first n points (1,0,0,...) through (...,0,0,1) and the last point the origin, then every point has a unique label, and the space becomes \R^n.
I just realised (reading my post again on the mailing list) that Tarski's axiom doesn't give the relative distances between these points, so these labels (and the later remarks about fixing them) are invalid. Of course, given Tarski's n + 1 points, you can find my n + 1 points, and even do this algorithmically (using the Gram-Schmidt process), so my other remarks hold only if you actually do this (so that preserving the points means preserving the results only after the Gram-Schmidt process has been applied). Thus, the situation is a little less elegant than I implied (but maybe that's Tarski's fault for phrasing his axiom so liberally ^_^). --Toby
Vaughan Pratt writes:
Regarding Bob Knighten's question on approaches to synthetic Euclidean geometry such as Hilbert's axiomatization, has anyone axiomatized more than E_3 (Euclid Books 11-13)? A satisfactory definition of Euclidean space needs to account for all finite dimensions.
Postings crossing in the ether make this a bit confusing, but Toby Bartels effectively answered this with his reference to Tarski's axioms so my only additional contribution is a reference where the details for all finite dimensions are discussed: Alfred Tarski and Givant, Steven, 1999, "Tarski's system of geometry," Bulletin of Symbolic Logic 5: 175-214. -- Bob -- Robert L. Knighten RLK@knighten.org
A lot of people long before Tarski, or Hilbert, knew how to extend this kind of axiomatization to any finite dimension. But is it "satisfactory"? Specifically, what about (n-dimensional) volumes. Euclid 11--13 does not well axiomatize the method of exhaustion he uses for the volume of solids. And Dehn's theorem that already for polyhedra the theory of volume will not reduce to equidecomposition. I should probably already know this, but what is known about the theory of n-dimensional volume in axiomatic n-dimensional Euclidean geometry? Colin ----- Original Message ----- From: Robert L Knighten <RLK@knighten.org> Date: Sunday, September 16, 2007 11:26 am Subject: categories: Re: Stupid question: what space was Euclid working in? To: categories list <categories@mta.ca>
Vaughan Pratt writes:
Regarding Bob Knighten's question on approaches to synthetic
Euclidean > geometry such as Hilbert's axiomatization, has anyone axiomatized more
than E_3 (Euclid Books 11-13)? A satisfactory definition of Euclidean > space needs to account for all finite dimensions.
Postings crossing in the ether make this a bit confusing, but Toby Bartelseffectively answered this with his reference to Tarski's axioms so my only additional contribution is a reference where the details for all finitedimensions are discussed:
Alfred Tarski and Givant, Steven, 1999, "Tarski's system of geometry," Bulletin of Symbolic Logic 5: 175-214.
-- Bob
-- Robert L. Knighten RLK@knighten.org
Colin McLarty writes:
A lot of people long before Tarski, or Hilbert, knew how to extend this kind of axiomatization to any finite dimension. But is it "satisfactory"?
Specifically, what about (n-dimensional) volumes. Euclid 11--13 does not well axiomatize the method of exhaustion he uses for the volume of solids. And Dehn's theorem that already for polyhedra the theory of volume will not reduce to equidecomposition.
I should probably already know this, but what is known about the theory of n-dimensional volume in axiomatic n-dimensional Euclidean geometry?
. . .
Alfred Tarski and Givant, Steven, 1999, "Tarski's system of geometry," Bulletin of Symbolic Logic 5: 175-214.
Of course Tarski's axiomatization had precursors going all the way back to Euclid, and as is discussed in the Tarski and Givant paper none of his axioms are new with Tarski. There is some discussion of the actual development of Euclidean geometry from the axioms in that article, but I believe (though I do not actually have the book) the detailed development was presented in Schwabhauser, W. and Szmielew, W. and Tarski, A., Metamathematische Methoden in der Geometrie, Springer-Verlag, 1983 But the development is in showing how to go from the axioms to analytic geometry not in developing a direct extension of Euclid's methods. -- Bob -- Robert L. Knighten RLK@knighten.org
participants (8)
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Colin McLarty -
Eduardo Dubuc -
France Dacar -
Robert L Knighten -
Steve Vickers -
Toby Bartels -
Vaughan Pratt -
wlawvere＠buffalo.edu