Introduction to the foundations of geometry
This text is under construction.
What is geometry
Geometry is the family name of a fuzzy range of mathematical
theories, some of whom have useful relevance to describe the
space of our physical universe (to describe possible figures there,
and relate their diverse measures), and other aspects of the laws of physics.
Diverse geometries can be explored for the sake of pure mathematics
(as studies of abstract realities),
disregarding their possible connections to physics, and this is
even needed as a theoretical background to better understand
those relevant to physics as particular cases.
The first two geometries classically considered, were those of “the plane”
and “the space” as they naturally appear. They are now called Euclidean
by their dimension, respectively 2 (for the plane geometry), and 3 (space
The 3-dimensional Euclidean geometry is the first and most obvious
theory of physics,
describing our physical space. To be a clear (simple) and complete
mathematical theory, this description must be idealized, accepting the simple properties
that seem to fit with ordinary physical measures, as if they were exactly
true, regardless the physical limitations of any attempt
at accuracy verifications. For example, it must regard any two points as
either equal, or different, and that between any two different points there
are other points and so on, which logically implies that any segment
contains infinitely many points. This infinity so comes as a mere good
approximation, as intermediates between points very close to each
other are physically harder and harder to distinguish, with no clear way
to figure out at which scale may this distinction become impossible or
meaningless (this may be argued to be the Planck length, which is far too
small to be measured anyway, much smaller than nuclei of atoms).
Usually, geometries are weak
second-order theories with one base type called the "space" whose
elements are the "points", seen as pure elements. Traditionally considered other types are kinds
of figures which are sets of points (straight lines, circles or spheres...) but
types beyond sets of points will be needed for diverse reasons.
Actually, this geometry is known to be inaccurate as a description of
physical space, since General Relativity theory is established as a more accurate
description of the physical space-time, which comes down to Euclidean
Geometry as a usually quite good
approximation (of its pure space aspect, part of Special
Relativity as a description of space-time).
While not exactly a physical theory, Euclidean plane geometry is useful for
educational purposes (being more directly visible and simpler than Euclidean
space geometry but with some complexity, thus a good intermediate step to
understand it), its relevance to aspects of physical space (describing flat
surfaces: screens, papers, pieces of land), and as an understanding of
complex numbers, which play key roles both for pure mathematics, and for
physics (especially the formalisms of waves and quantum physics).
Multiple automorphisms and models
Most geometries can be characterized as having a big
group of automorphisms in each model.
Such theories are not directly obtained by developing theories
without automorphisms (ZF with only sets and no pure elements, or
second-order arithmetic), but obtaining them from there requires
to also forget some structures, and notice how some
permutations then become automorphisms.
By its general definition, the automorphism group for any theory with a finite language
is a second-order construction. In classical geometries (but not topology), it is
even first-order constructible from the space. For example, rotations are automorphisms in Euclidean geometries,
and can be specified by just a few parameters (depending on the dimension: center or axis, angle... or the image of one tuple).
Anyway geometries generally admit their own automorphism group either as a type, or
as a subset of a type (especially the type of endomorphisms, if different).
Like in a set theoretical framework, for geometries with nontrivial automorphisms,
multiple models (spaces) may be interestingly considered in each isomorphic class
(instead of taking one to represent them all), for the following reason.
Any isomorphism f between 2 systems E and F induces a bijection
g ↦ f०g from Aut(E) to the set of isomorphisms
between E and F. Thus, a plurality of automorphisms
of E means a plurality of isomorphisms between E
and F. This plurality makes the difference between E and F
meaningful, as an object in E may correspond to several possible ones in
F, depending on the choice of isomorphism, so choosing an object in E does
not mean choosing an object in F in any invariant way. By contrast, a unique
isomorphism between models makes them unambiguously play the role of each other (copies of
each other), rendering their plurality superfluous.
The Galois connection (Aut, sInv) and the double meaning of "invariance"
For any set E, the relation of strong
RelE and the symmetric group ⤹E
induces a Galois connection (sInv, Aut) between their powersets (similar to
(Inv, End) previously seen): for any set
G⊂ ⤹E of permutations, the
set of strong invariants of G is
sInv (G) = Inv (G
∪ -G) = Inv (G) ∩ Inv(-G)
If G is a group then G = -G, thus sInv (G) = Inv (G). As we know,
the set Inv (G) of invariants of G is made of orbits of tuples,
and any unions of these.
∀L⊂RelE, ∀G⊂ ⤹E, L⊂sInv
(G) ⇔ G⊂AutL(E).
We initially conceived invariant structures as those
defined by expressions without parameters. This implies invariance by
automorphisms. Like proofs explore truths (theorems), that is the closure of the set of axioms,
whose addition to axioms preserves the class of models,
from a language L⊂R explore its closure
Inv(AutL(E)) ⊃ L, whose addition to
L leaves unchanged the set of automorphisms of each model
(among other isomorphisms between models).
Constructions leave unchanged the groups of automorphisms as abstract
groups, just providing more types on which these groups act.
While the completeness
theorem gave a converse for the similar
concept of provability, such a converse no more strictly holds here.
Let us shortly explain this subtle difference between both concepts of invariance,
before undertaking to confuse them in a way which will fit
for most needs of studying geometry at a basic level.
Our last scheme
of constructions (set of all structures
defined by an expression with all values of parameters), was that of an
invariant second-order structure that is a set of non-invariant first-order
structures. Well-defined by that formula, its invariance by
automorphisms means that automorphisms preserve that set while acting as
transformations on it.
Two tuples can be distinguished by a structure
r ∈ Inv(AutL(E)), when they cannot be moved one
to the other by an element of AutL(E). Intuitively, it
suggests that these tuples are not similar with respect to L, unless
there is a trouble to choose
an automorphism witnessing their similarity.
One might then hope for this dissimilarity, and thus r
itself, to be expressible from structures in L. But this may
require infinitely complex definitions, inexpressible with finite
formulas. For example, there is no nontrivial automorphism in the system
of real numbers (ℝ, 0,1,+,⋅), or in a model of second-order arithmetic,
yet objects of an uncountable
type (real numbers or sets of integers) cannot be all defined without parameter
(but, as objects, they are definable with a parameter: themselves).
There are still 2 viewpoints which may make both concepts of
invariance somehow coincide :
Now starting from a permutation group G intended as the set of
automorphism of a space E, can we rebuild its geometry, looking
for a good list L of a few structures from Inv G,
which would suffice to define other useful invariant structures (not
the whole set of them, which is uncountable), at least so that
This was a core idea of Felix Klein's Erlangen
Program for the foundations of geometry.
- In first-order logic, the theory admits non-standard models
with automorphisms moving some
undefinable real numbers to some infinitely
close neighbors (by moving non-standard positions of decimals...); but
these automorphisms only exist outside the model.
- In second-order logic, we may admit "second-order"
ways of defining structures where any
subset of ℕ is "defined" by the infinite data of all its
Completeness of first-order geometry
The difficulties in choosing a logical framework for geometries, turn out
to be reducible to a specific case: the theory of the set ℝ of real numbers.
Indeed, while formalizations of geometries without numbers are
possible, the type ℝ can anyway be restored from them by a
Then they can be conveniently formalized using ℝ as a base type aside
the space, beyond which all the rest can be formalized in first-order logic, with finite lists of types, symbols
and axioms which suffice insofar as ℝ is assumed to keep its standard interpretation.
So, this way of "using ℝ with its standard interpretation" may be regarded as all we
need in guise of framework for geometry beyond first-order logic.
The system of real numbers is essentially unique in the sense that
This "framework" (use of real numbers) suffices to prove that diverse
geometries have existing models of (by constructing a type of "points", as tuples of real numbers
or more complicated things... and forgetting some
structures) and are semantically complete: that between any two models
of any given geometry with the same dimension there is an isomorphism
(definable with parameters, typically by
choosing coordinate systems and identifying points with the
same tuple of values of coordinates). For first-order logic, the isomorphism classes of
models of each complete geometry follow the isomorphism classes of real closed fields.
Such fields can be non-standard (non-isomorphic to ℝ) either because they have
nonstandard numbers (larger than any standard
natural number) or because some "finite" but undefinable numbers are missing.
Its description in second order logic (which, beyond first-order axioms, uses the second-order axiom
that its order is complete)
is sematically complete (all its standard models are isomorphic);
- Its first-order version (with weak
translation as a schema of first-order axioms) is the logically complete
first-order theory of real
- Having no nontrivial automorphisms in its standard interpretation, the uniqueness
of the isomorphisms between its standard models, makes these models essentially the same
Geometries may be interestingly generalized by replacing ℝ by
something else such as the set ℂ of complex numbers, still
keeping many things work the same.
ℝ can be built from P(ℕ), using binary expansions.
Its non-standard models either contain non-standard numbers, or fail to contain
some undefinable numbers. As the first-order theory of real closed fields only
has algebraic equations as a means to define
numbers, the poorest model is just made of algebraic numbers.
We get an expression of any geometry as a complete
first-order axiomatic theory, by means of its relation with ℝ
together with such an axiomatization of ℝ.
This quality of completeness comes in contrast with the incompleteness theorem
of arithmetic. It avoids contradiction with it, by the fact that
formulas of arithmetic (and especially the undecidable ones) cannot
be expressed as formulas in the first-order theory of real numbers.
Because this theory cannot express the predicate over real numbers
"to be an integer", as the formula "x=0 or x=1 or
or...(and so on to infinity)", well, is not a formula.
The theory of real closed field obliges its models (the real
closed fields) to contain all algebraic numbers (solutions of
algebraic equations with (standard) rational or integral
coefficients). However, it cannot express the existence of
any number that is not algebraic (because any finite list of
first-order properties satisfied by a non-algebraic number, is
also satisfied by some algebraic number).
In particular, such a theory cannot express the number π (which is
not an algebraic number). What a pity for a geometry !
At first, we shall assume geometrical spaces and real
numbers as known from experience (from secondary-level
mathematics). Fully rigorous (set theoretical and axiomatic)
foundations will be presented later.
The dimension n of a space can be intuitively understood as
the arity of tuples of real numbers (called coordinates), needed to give the
position of a point. In other words, the space has a bijective and continuous
correspondance to ℝn
(this correspondence is called a coordinates system).
For each natural number n (say, nonzero to make something
non-trivial) there is one n-dimensional Euclidean geometry, but also
diverse other n-dimensional geometries, with different concepts and
properties. Even infinite-dimensional geometries may be considered.
As some geometries (such as the spherical geometry) have no
good global coordinates system, at least we put this as a local requirement :
the existence of correspondences of its small enough regions with those of
By the action of the automorphism group, the space of points usually has only one orbit (but
vector spaces have 2 orbits, one being a singleton), so that when regarding the group of
automorphisms as a space itself, it has even higher dimension.
This happens for many finite-dimensional spaces of interest, but
yet not all:
- Some spaces, such as arbitrary Riemannian manifolds, may have
no automorphism except the identity.
- Other ones, such as topological spaces, have a "space" of
automorphisms that can best be qualified as
(*) in honor to the Greek geometer Euclid, who around 300
BC, published the Elements, mathematical treaty gathering
and methodologically ordering, with logic, axioms and proofs, the
main mathematical knowledge of that time, with a special focus on
plane and space geometries. It remained the most famous mathematical
work until the 19th century, when its axioms and proofs were found
incomplete. Only in 1899 a really complete axiomatic expression of
Euclidean plane geometry was published by Hilbert.
(**)This bridge to Inv(AutL(E))
by means of extended kinds
of "definitions" (with infinite amounts of data) has been
investigated by Borner, Martin Goldstern, and Saharon Shelah : short
presentation in .ps - full article.
Set Theory and Foundations of Mathematics