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
respectively with dimension 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 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).
Usual geometries are second-order theories made of
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).
- 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 :
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).
- 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.
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 (for physics) some more powerful geometries than those
linked to RCF are used,
involving tools of analysis from second-order arithmetic, 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
- 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
(*) 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.
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Up: 5. Geometry
Set Theory and Foundations of Mathematics