6. Foundations of Geometry
6.1. Introduction to the foundations of geometry
What is geometry
For pure mathematics, 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),
ignoring their possible connections to physics; and this is
even needed as a theoretical background to better understand the cases
relevant to physics.
The first two classically considered geometries were those of “the plane”
and “the space” as they seem. They are now called Euclidean
geometries(*),
respectively with dimension 2 (for the plane geometry), and 3 (space
geometry).
The 3-dimensional Euclidean geometry is the first and most obvious
theory of physics,
describing our physical space. To form a clear 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. Any two points are regarded as
either equal or different, and between any two distinct points there
are other points and so on, which logically implies that any segment
contains infinitely many points. This infinity just comes as a good
approximation, as points very close to each other are physically harder
and harder to distinguish, with no clear distinction of any scale where
this distinction may become impossible or meaningless (it may
be argued to be the Planck length, far too small to be measured as
it is much smaller than nuclei of atoms).
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).
Usual geometries are second-order theories made of - Types and
structures forming a weak
second-order theory :
- One base type called the "space" whose elements are the "points";
- Some types of structures over it: traditional ones are kinds of figures which are sets of points
(straight lines, circles or spheres...) but other types will be needed.
- Axioms include a second-order one, of "continuity". Like the induction axiom of
arithmetic, first-order logic can interpret it by an axiom
schema restricting its range of application to sets of points definable with parameters.
This forms the logically complete system of Tarski's first-order axioms of
Euclidean plane geometry.
The use of real numbers
The use of second-order axioms for geometries by which they escape first-order logic, turns out
to be reducible to a specific case: the theory of the system ℝ of real numbers, that is
an algebra whose language is made of constants 0,1
and binary operations +,⋅, from which subtraction and order can be defined : x
≤ y ⇔ (∃z, y = x + z⋅z).
While geometries admit formalizations without it, the type ℝ can anyway be restored
from them by a construction.
Geometries can be conveniently formalized in two parts :
- a self-sufficient description of ℝ
- a description of all the rest by first-order logic using it, with finite lists of types, symbols
and axioms which suffice insofar as ℝ is assumed to keep its standard interpretation.
This "use of ℝ with its standard interpretation" sums up all the needed framework
for geometry beyond first-order logic. Some geometries may be
interestingly generalized, keeping many similar properties, by replacing ℝ by
another system of numbers (especially the set ℂ of complex numbers).
In second-order logic, ℝ has a semantically complete
description, whose second-order axiom (of continuity) describes its order as
topologically complete. The reduction (restriction) of the axiom
of topological completeness as an axiom schema of first-order logic,
gives the logically complete first-order theory of real closed fields (RCF).
ℝ can also be constructed from ℘(ℕ),
using the decomposition of positive real numbers in integral part and fractional part,
and representing the latter by its binary expansion as a subset of ℕ. Even the
RCA0 subsystem of second-order
arithmetic suffices to ensure the needed properties for this first-order
development of ℝ out of ℘(ℕ), whose standardness is equivalent to
the standardness of the resulting ℝ.
But the structures (0,1, +,⋅) of ℝ are weaker than those from that developed second-order
arithmetic, as they do not suffice to distinguish integers (define either ℕ or ℤ) as a
subset of ℝ: each standard
number can be individually defined there but ℕ cannot be defined any better than the set of its
standard elements ("x∈ℕ", defined as
"x=0 ∨ x=1 ∨ x=2 ∨...and so on to infinity", is not a formula).
This weakness explains why RCF can be logically complete without
contradicting the incompleteness theorem
of arithmetic: RCF cannot express the undecidable formulas of arithmetic.
The diversity of non-standard real closed fields (non-isomorphic to ℝ) is illustrated
though not exhausted by those constructed as above from non-standard models of second-order
arithmetic. Namely, some contain non-standard numbers (larger than any standard
natural number, so their inverses are infinitesimals), while others, having an
(elementary) embedding into the standard ℝ, miss some undefinable numbers from it.
As RCF can only define numbers as zeros
of its polynomials (solutions of algebraic equations), whose existence is deduced from topological
completeness, the poorest real closed field is the set of real algebraic numbers (zeros
of polynomials with standard rational coefficients, while the zeros of polynomials with real
algebraic coefficients are among those of more complicated polynomials with rational coefficients).
Ironically for a foundation of geometry, it does not contain the number π (which is
not an algebraic number). In practice, physical applications involve some more
powerful framework, with tools of analysis
from RCA0 beyond RCF, such as trigonometric functions and
exponentials, including π.
Dimensions and semantic completeness
The dimension n of a space can be intuitively understood as
the arity of tuples of real numbers needed to give the position of a
point (called its coordinates), according to some map which is a
continuous correspondence of the space to ℝn,
called a coordinates system (it will be bijective in our first considered
geometries but may be non-bijective in others).
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.
The "framework" of the use of ℝ to formalize geometries, suffices to prove for
"completely specified" geometries (specifying the dimension, which suffices
for Euclidean geometry but not always otherwise):
- Semantic consistency: the existence of models (built using
ℝn)
- Semantic completeness : between any two models,
isomorphisms can be defined with parameters
(namely, choices of coordinate systems). First-order
logic still sees them as incomplete, with isomorphism classes following those of RCF
Multiple automorphisms and models
Most geometries, except RCF, have a big group
of automorphisms in each model.
To produce such spaces from a framework without automorphisms
(ZF with only sets and no pure elements, or second-order arithmetic),
requires not only developments
but also to forget some structures, and notice how some permutations then become
automorphisms.
Spaces with non-trivial automorphisms will be interestingly seen as
non-unique in each isomorphic class (instead of picking one to represent them all),
for the following reason. Any isomorphism
f between 2 systems E and F induces a bijection
g ↦ f০g 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, the uniqueness of
isomorphism between standard models of ℕ or ℝ makes them unambiguously
play the role of each other (copies of each other), making their plurality superfluous.
The double meaning of "invariance"
Unlike provability which by the completeness
theorem ranges over all truths of models, definability by language L without parameters
may not fill Inv(AutL(E)), for the following reason.
Structures r ∈ Inv(AutL(E)), distinguishing any
tuple t∈r from those outside r, are those for which no element of
AutL(E) can move t out. This suggests
that t is not similar to anyone outside r with respect to L, unless
the trouble is to choose
an automorphism witnessing the similarity.
But this dissimilarity, and thus r itself, may be formally inexpressible from L
as it may require infinitely complex descriptions.
For example, there is no nontrivial automorphism in ℝ, or in a model of second-order arithmetic,
yet not all objects of an uncountable
type (real numbers or sets of integers) can be defined without parameter
(but, as objects, they are definable with a parameter: themselves).
Both concepts of invariance can be still reconciled in different ways:
- For first-order logic, non-trivial automorphisms may externally exist over
non-standard models of second-order arithmetic, thus also non-standard real closed fields
(moving undefinable real numbers to infinitely close neighbors);
- 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
elements(**).
A set of axioms of a theory can be seen as a mere tool to approach an intended
range of models, on the other side of the
(Mod,Tru) connection.
Similarly, a language may be a mere tool to approach an intended concept
of invariance better defined by a group G on the other side of the (Aut, sInv)
connection. Specifying the geometry of a space E by a permutation group
G of E intended as Aut(E), was a core idea of Felix Klein's
Erlangen
Program.
Thus, "invariance" in geometry will be
meant by G, and equivalently qualify structures defined with
parameters in ℝ or among structures over ℝ only.
Now starting with a mere concrete
category C of intended spaces whose class of morphisms is given by intuition,
let us review methods to produce invariant structures. We already gave the construction of all invariant relations,
elements of Inv(End E) or Inv(Aut E), from the trajectories or orbits of tuples.
A language L ⊂ Inv G of first-order invariants may not suffice to
approach the concept of invariance defined by a given group :
- A well-describable L independent of E may not fill Inv G
(which is usually uncountable, but an uncountable L
may be accepted either directly or as an additional type), in which case invariants outside
L may be undefinable from L.
- We need G = AutL E, which will fail even for
L = Inv G in the case of topology.
For this and more reasons which will appear later, the formalization of geometries needs
second-order invariants.
For example, Euclidean space geometry admits the type "plane", each plane is a set of points,
and automorphisms can move any plane to any other plane. On Earth,
"the sea level" names a plane, distinguishing its points from those above or below it,
but Euclidean geometry does not see it as invariant, i.e. does not accept the name "sea level"
in its language. A language accepting it, would form a different geometry.
In spaces of classical geometries with given finite
dimensions, the endomorphism monoids and automorphism groups (which are second-order invariants)
are first-order constructible, by a number of parameters which depends on the dimension, such as
the image of one tuple with enough elements.
For example, rotations (automorphisms in Euclidean geometry) can be specified by their
center or axis, angle... but such a first-order constructibility no more holds for infinite dimensional spaces.
(*) 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.
6. Geometry
Next : 6.2. Affine spaces
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