The logics commonly known and Euclidean geometry have their basic structure in common. There are some axioms and some rules how to deduce new true statements from other ones. Logics describe reasoning processes, geometry describes objects and their positions. Both are purely formal systems that are used to assert the triuth of certain statement given the truth of certain axioms.

So neither logics nor geometry really deduce properties of objects. Only if the formalization is correct, i.e. the axioms and rules describe the real word, then we can deduce something abou the real world. But there cannot be any proof that the formalization is correct...

The Euclidean geometry is a logical construction, because any geometrical object in the Euclidean geometry can be constructed of blocks. One can use segments of a straight line as such blocks. Property of such a block can be described as a set of rules (axioms). Besides, there are rules describing displacement of these blocks. The rules, describing properties of blocks and the rules of their displacements form axioms of the Euclidean geometry. Using these axioms, one can formulate the rules of any geometric object construction. These rules form statements of the Euclidean geometry. These rules have a form of logical construction, because the ancient Greeks considered logic as the most respectable science. If a geometry has a restricted divisibility, not all geometrical objects of such a geometry can be constructed of blocks. Such a geometry cannot be a logical construction, generally speaking.

Peter, in my statement "there cannot be any proof that the formalization is correct" refers to the model plus logic. I do not see how to prove that the model models the real word correctly. Here only "empirical induction" is used: for the given model we have found all the predictions we have deduced and checked against reality to be true.

For example, we have not found any parallel lines that intersect. So we say Euclidean geometry models the relations of objects in the real world. Now suppose the universe is finite and at its end all lines that are long enough converge and thus intersect (purely invented example). Then we might find out  one day that our model was not good, not correct in the sense I meant. For this kind of correctness or maybe better adequacy I do not believe there can be a formal proof. Because we cannot use "the world" in proofs, only models thereof. 

Unfurtuanately, you do not answer the question about connection between the logic and Eucledean geometry. The qquestion arose, because there are space-time geomeries, which are not a logical construction. Why can the Euclidean geometry be a logical consruction, whereas another real geometries are not a logical construction? What properties of the Euclidean geometry admit one to consider it as a logical construction? 

"..space-time geometries which are not a logical construction." I think you are not saying that what Gauss, Bolyai, Lobachevsky and Riemann did more than 150 years ago was not logical. So what are you saying?

As I understand it, the foundation on which Euclidean geometry as Euclid originally formulated it stands doesn’t arise from logic. His five “postulates” come not from logic but from intuitions arising from the way we experience objects in the real world. Euclid takes for granted that we already know, from experience of the real world, what a straight line is, what the “length” of a line means, what a “right angle” is, what it means to “extend a line segment indefinitely in a straight line”, and so on. Logic lies not at the foundations, but is employed only in the construction of “proofs”: the postulates imply “theorems”, derived by logical deductions of the form: if the propositions are true, then such and such theorems follow.

The modern axiomatic approach to geometry proceeds from undefined entities and relations between those entities and between sets of entities. There is no a prori assumption that the entities and the relations correspond to anything in the real world. Euclid’s geometry can be formulated in this purely abstract way. Only then can it be said to be a strictly logical construction. But Euclidean geometry is no more and no less “logical” in this sense than any other geometry (eg, Lobachevski’s hyperbolic geometry, elliptic geometry, affine geometry, conformal geometry, projective geometry, Riemannian geometry, Minkowskian geometry, etc), so I have no idea what you mean when you say “other real geometries are not logical constructions.”

Geometry does not deduce properties of “real” objects. There can be correspondences between the properties of abstract objects belonging to a geometry and objects that we encounter in the real world. This is found to be true of Euclidean geometry for objects we normally encounter but, as Peter Leopold pointed out, there can be no proof of that. And according to general relativity, it is not valid at large scales…

Traditional Euclidean geometry is known to contain hidden assumptions based on human intuition. These intuitions are involved in reasoning on pictures. A true formalization (and inventory) of what is really used was obtained by Alfred Tarski and collaborators, and constitutes a first-order theory with 10 axioms and one axiom scheme. See

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.27.9012

It uses the "primitives" of point and congruence (of line segments). Formally, it does not allow to talk about "lines" unless the axiom scheme is converted into a second order statement. Tarski's system cannot be finitely axiomatized.

For this thread not to "hang in the air", it should refer to this or other (usually not first-order) axiomatizations in discussing the  logic foundations of Euclidean geometry. Most (all?) mathematical theories are ultimately founded in logic. Is this thread about the hardships in reducing Euclidean geometry to first-order logic?

Yuri ~

“…The Euclidean geometry is a logical construction, because any geometrical object in the Euclidean geometry can be constructed of blocks…. there are rules describing displacement of these blocks…”

This looks to me like nothing more than a different formulation the axioms of Euclidean geometry. The construction of the blocks and their rules of displacement depend on the usual axioms of Euclidean geometry, so it comes as no surprise that we can get back the usual Euclidean concepts by starting out from the properties of “blocks”. It’s a circular argument. I cannot see how this makes Euclidean geometry more logical than any other geometry. Simpler in some sense, perhaps, but not more logical.

I think that misunderstandings in this discussion are arising because you have not made clear to us exactly what “logical” means to you in this context.

Dear Eric, It is evident that logic and construction of blocks are connected things. But what is prime? As far as a geimetry is a science on geimetrical objects, then construction of geometrical objects from blocks is prime. If there are geometries, where not all geometrical objects can be constructed of blocks, then there are geometries, which are not a logical construction. This conclusion is very important, because it is used to think, that any geometry is to be a logical construction.

Dear Marcel,  The method of axiomatization and a choice of axioms are not the points, which are discussed. Are there geometries, which cannot be axiomatized at all? It is the real question.  If, yes, then how can one construct such geometries?

Yuri ~

I presume your concept of building up Euclidean geometry from “blocks” relies on the idea of “congruence”; there is a group of motions (rotations, reflections and translations) that leave the geometrical properties of an object unchanged. But Euclidean geometry is not unique in that respect – other “geometries” have their own characteristic groups of motions. Some analogue of “congruence” therefore exists in these other geometries (with, arguably, the possible exception of differential geometries). The idea of approaching geometrical concepts by building up the space from assembling “blocks” can be applied to geometries other than Euclidean geometry.

Dear Yuri,

1. "The method of axiomatization and a choice of axioms are not the point": There is much tradeoff in choosing axioms. Tarski proposed many variant axioms for Euclidean geometry. Some axioms combine better than others. Some are more intuitive than others. The exact choice of axioms is often a technical matter and even a matter of taste.

2. "Are there geometries, which cannot be axiomatized at all?" Any mathematical theory, from set theory to topology, from ordered sets to (semi)lattice theory, from groups to fields, from real calculus to Banach spaces, all can be formalized to axioms and logic. In elementary geometry, there is a particularly rich supply of proposals, due to Pasch, Peano, Veblen, Hilbert, Whitney, Prenowitz, Bryant-Webster and others. In some cases, the process of axiomatization has been a hard one. This is the case not only for Euclidean geometry, but also for relation algebra (the algebra of binary relations on a given set). In the latter case, it was Tarski (again) who succeeded in producing a convincing axiom system  (around 1940, pretty late anyhow).

I cannot imagine how a mathematical discipline can fare well without first principles. Also, Euclid's early proposal of axioms doesn't meet modern standards.

Dear Marcel, It is quite natural, that you do not believe in existence of nonaxiomatizable geometries. As far as I know, all mathematicians believe that only axiomatizable geometries do exist. However, if in some geometry the equivalence relation is intransitive, such a geometry cannot be a logical construction, because in any logical construction the equivalence relation is transitive. Let consider a monistic conception of Euclidean geometry, when all geometrical quantities and statements are expressed in terms of one quantity (world function). Such a monistic conception is possible. Let us now replace the Euclidean world function by another world function in all geometrical statements of Euclidean geometry. We obtain another monistic conception of another geometry. This new geometry will be not a logical construction, generally speaking, because deformation destroys logical connections between statements of Euclidean geometry. Such space-time geometry exists in reality. Unfortunately, mathematicians do not accept such geometries. It is a great mistake.

Dear Eric, You are quite right, that there are another geometries having other motion groups. But it does nothing. The problem is lies in the question, whether such geometries exist, which forms a logical construction. Do such geometries exist or not?

Yuri ~

"The problem lies in the question, whether such geometries exist."

If you mean whether such geometries correspond to properties of the "real world", then the answer depends on which particular aspects of the real world you are investigating or describing. Four dimensional Minkowskian geometry (pseudo-Euclidean) geometry corresponds to the real physical world within the limits of applicability of special relativity. A much simpler example is the application of projective geometry by architects and artists when three-dimensional (Euclidean) objects are represented "in perspective" on a plane.

Exotic geometries are studied by mathemeticians because they are intrinsically interesting as imaginative logical structures; they "exist" as logically consistent abstract structures. The "real world" is irrelevant to these developments. Many important discoveries in "pure" mathematics (geometry not excepted) have no immediately obvious practical applications. That doesn't make them "illogical". Some of the "exotic" geometries provide very effective ways of understanding and dealing with algebraic concepts and methods that are applicable by theoretical physicists to investigations of the real world.

If we consider the universe at large and accept Einstein's gravitational theory, then we are forced to recognize that Euclidean geometry fails to describe "reality" at large scales. There are no "straight lines" and no "parallel lines" in the universe at large. Therefore, according to your useage of the words  "existence" and "logical" we would have to conclude that Euclidean geometry is illogical and doesn't exist...

Dear Peter, I shall not discuss different versions of axiomatization, because the construction method, using axiomatization, is not effective. If for any new geometry, one needs to prove numerous theorems to obtain statements of the new geometry, it is very difficult, although mathematicians like to play such deductive games. Having constructed the first geometry by the axiomatization method, one should construct any new geometry by means of a deformation of the first (Euclidean) geometry. The Euclidean geometry is presented in the monistic form. It means that all definitions and all geometrical objects of Euclidean geometry are presented in terms of the world function. To obtain the same definitions and the same geometric objects are obtained from the Euclidean ones, one needs to replace  the Euclidean world function by the world function of the new geometry in all Euclidean definitions. 

This method of construction is very simple. It does not test compatibility of new axioms, because there is only one fundamental quantity: world function. All other geometrical quantities are expressed via the world function. Of course, obtaining of these expression needs a work, but it is a problem of the Euclidean geometry. Besides, these expressions are obtained only one time.

Constructing axioms of a new geometry, one modifies axioms of Euclidean geometry, and one cannot distinguish between general geometric properties and special Euclidean properties. For instance, dimension of geometry, betweenness, congruence are special Euclidean properties which can be absent in a new generalized geometry. The general geometric properties are to be expressed via world function and only via world function. I know only three general geometric definitions:

(1) (Scalar product (AB.CD) of two vectors AB and CD

(AB.CD) = \sigma (A,D) + \sigma (B,C) - \sigma (A,C) - \sigma (B,D)

(2) Parallelism (AB||CD) of two vectors AB and CD (AB.CD) = |AB| |CD|

(3) Linear dependence of vectors AB1,AB2, …ABn They are linear dependent, if the Gram’s determinant det||(ABi.ABk)||, i,k=1,2,..n vanishes.

Note, that all these definition are formulated in the coordinateless form. They contains only the world function \sigma . In the proper Euclidean geometry these definitions coincide with conventional definitions in terms coordinates. They are supposed to be valid in any physical geometry. The physical geometry is a monistic conception of a geometry, when all geometric properties and objects are described in terms of the world function.

Already the case of the geometry of Minkowski (pseudoEuclidean geometry with signature (1,-1,-1,-1)) gives an interesting example. There are two definition of equality of vectors AB and CD.

(1) conventional definition vectors are equal, if their coordinates are equal.

(2) coordinateless definition, vectors AB and CD are equal, if (AB.CD) = |AB| |CD| and |AB| = |CD|.

In the proper Euclidean geometry both definitions are equivalent. In the Minkowskian geometry both definitions are equivalent for timelike vectors, although the conventional definition contains four equations, whereas the coordinateless definition contains only two equations.. For spacelike vectors they are not equivalent. The convention definition is single-variant and transitive, whereas the coordinateless definition is multivariant and intransitive. Let us consider three spacelike vectors a = {0,0,0,z}, b = {r,r,0,z}, c = {r,0,r,z}. According to the coordinateless definition a = b and a = c, but b is not equal to c. According to conventional definition all three vectors a, b, c are different.

What of the two definitions of equality are true. From general viewpoint the coordinateless definition is more realiable. In the dynamics a use of coordinateless definition leads a possibility of tachyons and explanation of the dark matter problem on the basis of the tachyon gas. See for details "Dynamic equations for tachyon gas", Int. J. Theor. Phys. 52, 133( 10), 3683- 3695, (2013), the paper in the electronic form http://gasdyn-ipm.ipmnet.ru/~rylov/detg.htm. In the general relativity a use of the physical space-time geometry leads to extended general relativity (EGR), where, the black holes  are  absent. In the microcosm the physical discrete space-time geometry leads to skeleton conception of elementary particles.

I notice that Peter and Eric have the same problem as I have in understanding Yuri's question and the concepts he suggests.

"non-axiomatizable geometries": Euclidean geometry undoubtedly originated from solutions of real-world problems related with architecture and agriculture. In this early stage, it could be seen as non-axiomatized geometry. Euclid turned it into a mathematical discipline by organizing the knowledge and methods into axioms and logic manipulations.

"Logic" originated as the art of drawing undeniable conclusions from accepted premises. So, if you link "construction of blocks" with logic, you enter a game of premises and conclusions. If you do not have axioms, then your premises must come from the real world.  At best, that is applied mathematics.

Dear Eric, You are quite right, that the logical consistency and problem of the geometry existence are different things. But at construction of geometry by a modification of the Euclidean geometry axioms it is very important to distinguish between the special properties of the Euclidean geometry and general geometric properties. One cannot use the special properties of the Euclidean geometry. For instance, the straight line segment T[AB] between the points A and B is described in terms of the world function \sigma as follows

T[AB] ={R | d(A,R)+d(R,B) = d(AB)}     where d(A,B) = \sqrt(2 \sigma(A,B)).

This definition is valid in any physical geometry (i.e. geometry, described completely by world function). In general, at arbitrary \sigma it is a (n-1)-dimensional surface in the n-dimensional space. However, in the proper Euclidean geometry this geometrical object T[AB] is one-dimensional line in the space of any dimension. It is a very special property of the Euclidean geometry. But it is used to think that one-dimensionality of T[AB] takes place in any generalized geometry. It is used to introduce this property into axiomatics of any generalized geometry. Of course, it cannot lead to a consistent generalized geometry. For instance, the Riemannian geometry is not consistent. A use of the physical geometry removes problems connected with logical consistency of geometrical statements, because logical consistency is an attribute of the geometry construction, but not an attribute of the geometry itself.

Dear Peter, I see, that you cannot accept the viewpoint, when space-time geometry cannot be nonaxiomatizable. You are not alone with such a viewpoint. I know only one mathematician, who accepted my viewpoint. His name Poznyak. He was a leader of the seminar on “geometry as a whole”. Unfortunately, he is not with us now. As to other mathematicians of Lomonosov university, their relation is as follows. When I submitted a report on geometry, their reaction is as follows: “ There are no persons who dealing with such things, the report will not interesting for participants of our seminar."  Your last post is unclear for me. I think, you are understanding this.

It is interesting to hear your impressions from the consideration of intransitivity of spacelike vectors equivalence in the geometry of Minkowski. This example was very useful in explanation of the problem of dark matter, which is considered now as a very important problem of astrophysics.

I should note, that I was teached, that the nonaxiomatizable geometry is impossible. I needed thirty years to overcome this delusion. By the way about delusions, if you have time, look at my last paper about delusions and their overcoming “ Nature of some conceptual problems in geometry and in the particle dynamics”                   http://gasdyn-ipm.ipmnet.ru/~rylov/nscpgpd6e.pdf

I should like to add, that there is a mistake in the address of the last reference. The true address is  http://gasdyn-ipm.ipmnet.ru/~rylov/nscpgpd1e.pdf

As I am not familiar with the concept of a "world function" (a concept from physics), I consulted some papers with mathematical explanations. The overall picture is differential geometry: lines become geodesics which, in general, are only locally available and unique between two points. Affine transformations survive only as local parallel transport; rotation is problematic in general. The area is far away from elementary geometry, the basic concepts of which are close to prime concepts in mathematics and where axiomatic reasoning is fruitful.

In the setting that Yuri aims at, one has to live with with integrals and differential equations. As Peter observed, it ultimately uses an axiomatic description of the real line and, in fact, a considerable portion of ZFC. Yuri may be right in that it makes little sense to think in terms of axioms in such a mathematically advanced and highly "computational" setting.

The subject is also too far away from my competence fields to enter into a more detailed discussion.

Logic is not particularly strong or particularly weak for geometric models. It just affords model theory as with any other object. One just takes a geometric model (for example R3  , the sets of all lines and all planes as different sorts, the relation of incidence for points, lines, planes, the euclidian distance, etc...) and considers the set of all first order quantified sentences true in the model. This set of sentences is the Euclidian Geometry. Tarski and Szmielew have shown that this first order theory is completely axiomatizable with a recursive axiomatization, so is a decidable theory. This fact has almost nothing to do with logic, but with the fact that the theory of real closed fields is itself recursively axiomatizable and decidable. Logic is just a system to write down mathematical statements and to formalize proofs starting from axioms. The whole rest comes from the object which is studied, here the euclidean geometry. Spheric geometry, hyperbolic geometry, etc. are also complete and decidable theories. 

"Euclidean geometry [...] is completely axiomatizable with a recursive axiomatization, so is a decidable theory. This fact has almost nothing to do with logic, but with the fact that the theory of real closed fields is itself recursively axiomatizable and decidable."

The fragment "has almost nothing to do with logic" is probably a slip of the pen. When using terms like "complete" and "decidable", the conversation is entirely about logic. The picture is, that with many elementary geometries (e.g., affine, Euclidean, or even projective), a dominant number systems appears to be associated: ordered fields (affine), real closed fields (Euclidean), or simply fields (projective). The corresponding "plane" (or space: 3-dimensional, n-dimensional,..) is then the dominant type of model .

The core of Yuri's idea seems to be that geometry of geodesics is far from elementary in the above sense.

If one insists on axiometization, one needs at minimum set theory. For continuity purposes, specifically ZFC+AC: Zormelo-Fraenkel set theory plus the axiom of choice. However in modern logic there is a whole school (the pluralists) that maintain, in varying degrees, that mathematics doesn't need the justification presumably afforded by axioms because its results are reproducible (or verifiable or consistent). In the above several different logics have been cited. In the attempt to apply a logic to the real world, one requires empirical input. For example if your model requires positive, zero or negative curvature then you need hyperbolic, Euclidean or spherical geometry. Also (I can't pose aas an expert here), I undestand that moderm mathematics has, for many practical purposes, reduced geometry to algebra and algebra to geometry, though much work here is on-going.

James ~

“I undestand that moderm mathematics has, for many practical purposes, reduced geometry to algebra and algebra to geometry…”

This raises the question: what do we mean by “geometry”? How is it defined; how is it distinguished from other kinds of mathematics? The word comes, of course, from the Greek: literally, “Earth measuring”, but that’s not much help!

The characteristic of geometry that distinguishes it from other kinds of mathematics such as algebra or number theory (et cetera), and that in a sense serves as a definition of “geometry”, is that it appeals to the imagination through intuitions arising from the way we perceive and experience the “real world”. This remains so even when geometrical ideas wander into realms of higher dimensions and structures that are hard – or impossible – to “visualize”. So long as the link to spatial intuitions remains, however tenuous, we are still dealing with “geometry”. Once that link is gone, it is no longer “geometry”, it is algebra - where the patterns and structures are abstract and have a different kind of appeal. But there is no sharp boundary.

Well, for example, differential geometry, very useful in physics, is a geometrical way of thinking but it's development is analysis.

James ~

The development of differential geometry began with Gauss's study of the geometry of curved surfaces in E3. Algebra and analysis and are the needed tools. Anyway, categorising branches of mathematics has limited usefulness - there are no sharp boundaries.