[FOM] 713: Foundations of Geometry/1

[Harvey Friedman]

There has been a great deal of work on so called "foundations of
geometry" from a variety of perspectives, from the deeply
philosophical to the deeply mathematical, and lots of
interdisciplinary levels in between.

I want to address a range of issues in geometry from a distinctly
foundational perspective, where I try to make the foundational
motivations as clear as I can. I will be reasonably flexible as to
what I mean by "geometry". Generally, I will be operating at a much
too fundamental level philosophically/foundationally to be identifying
"geometry" with its development in modern mathematics as identified by
the mathematical areas such as (see
https://en.wikipedia.org/wiki/List_of_geometry_topics)

Absolute geometry
Affine geometry
Algebraic geometry
Analytic geometry
Archimedes' use of infinitesimals
Birational geometry
Complex geometry
Combinatorial geometry
Computational geometry
Conformal geometry
Constructive solid geometry
Contact geometry
Convex geometry
Descriptive geometry
Differential geometry
Digital geometry
Discrete geometry
Distance geometry
Elliptic geometry
Enumerative geometry
Epipolar geometry
Finite geometry
Fractal geometry
Geometry of numbers
Hyperbolic geometry
Incidence geometry
Information geometry
Integral geometry
Inversive geometry
Inversive ring geometry
Klein geometry
Lie sphere geometry
Non-Euclidean geometry
Numerical geometry
Ordered geometry
Parabolic geometry
Plane geometry
Projective geometry
Quantum geometry
Reticular geometry
Riemannian geometry
Ruppeiner geometry
Spherical geometry
Symplectic geometry
Synthetic geometry
Systolic geometry
Taxicab geometry
Toric geometry
Transformation geometry
Tropical geometry
Topology. (my addition)

However, as the issues I raise - and of course many others - get
substantially addressed, there will be considerable overlap with
various of these kinds of "geometry" above in this list. The
difference here is that I will try to always focus on the
philosophical/foundaitonal aims, rather than any usual kind of
mathematical goals. Thus naturally, what is considered particularly
important from the philosophical/foundational perspective, cannot be
expected to match what is considered particularly important from the
mathematical perspective.

In particular, I expect that at east for a while, a lot of what I ask
and do is "essentially known" if reformulated in purely mathematical
terms.

INTUITIVE PLANE GEOMETRY

There are many very appropriate starting points for plane geometry for
various purposes, including essentially starting from nothing, not
even any kind of numbers. I would like to engage with just about any
of this.

But in this first posting, I want to start with the usual completely
ordered 2-divisiable abelian group of real numbers.

his itself is subject to interesting analysis, and fits into the
foundations of geometry as Intuitive Line Geometry. However, that will
be the subject of later postings. An important aspect of Intuitive
Line Geometry is Aristotle's conception of the continuum, which uses
intuitive intervals as opposed to points ("intervals" do not have
endpoints). But not here.

So in this development, we take the points in the Plane to be the
ordered pairs of real numbers.

We now want to determine what the lines are in R^2 (as subsets of
R^2), and also the betweenness relation, and distance function. We
develop a small number of highly intuitive principles that determine
these three notions.

In this approach, we first treat R^2 with lines only. Here are the
three conditions on the lines in R^2, which are certain subsets of
R^2.

1. Every line contains at least two points, and has at most one point
in common with every other line.
2. Every line is disjoint from a line.
3. Every line containing (a,b), (c,d) also contains ((a+c)/2,(b+d)/2).
4. If the points in a line come arbitrarily numerically close to point
x in both coordinates simultaneously, then x is on the line.

THEOREM 1. There is exactly one set of subsets of R^2 for "lines"
satisfying conditions 1-4. These are the usual lines.

We now define Betweenness explicitly.

5. (c,d) lies between (a,b) and (e,f) if and only if (a,b), (c,d),
(e,f) are on a line, and (a < c < e or e < c < a or b < d < f or f < d
< b).

The simplest characterization of distance that we know of is as follows.

6. The distance between (0,a),(0,b) is |a-b|.
7. Every line preserving permutation of R^2 whose set of fixed points
form a line is distance preserving.

The idea here is that now that we know exactly what the lines are, we
can mathematically determine all of the line preserving permutation
whose set of fixed points form a line, and hence the scope of 6.
However we can avoid having to "do the math' and use the following
alternative form of 6:

7'. Every line is the set of fixed points of some line and distance
preserving permutation of R^2.

Thus 6' can be readily visualized by picturing the reflection about
the given line, and "see" that it is line and distance preserving.

THEOREM 2. There is exactly one "distance function" satisfying 6 and
7, There is exactly one "distance function" satisfying 6,7'. This is
the usual Euclidean distance function in R^2.

If we have lines and distance, we can define Betweenness explicitly by

5'. y is between x and z if and only if x,y,z are distinct and lie on
a line, and the distance from x to y is less than the distance from x
to z.

INTUITIVE R^k GEOMETRY
k >= 3

We work with R^k and again first determined the lines only, as certain
subsets of R^k.

k1. Every line contains at least two points, and has at most one
point in common with every other line.
k2. Every union of k-1 lines is disjoint from a line.
k3. Every line containing x,y also contains (x+y)/2.
k4. If the points in a line come arbitrarily numerically close to
point x in all k coordinates simultaneously, then x is on the line.

THEOREM 3. There is exactly one set of subsets of R^k for "lines"
satisfying conditions k1-k4. These are the usual lines in R^k.

We now define Betweenness explicitly.

k5. y lies between x and z if and only if x,y,z are on a line, and for
some 1 <= i <= k, x_i < y_i < z_i or z_i < y_i < x_i.

The simplest characterization of distance that we know of is as follows.

k6. The distance between (0,...,0,a) and (0,...,0,b) is |a-b|.
k7. Every line preserving permutation of R^k whose set of fixed points
form a line is distance preserving.

We can also use the more directly visual

k7'. Every line is the set of fixed points of some line and distance
preserving permutation of R^k.

Thus 6' can be readily visualized by picturing the reflection about
the given line, and "see" that it is line and distance preserving.

Of course, visual clarity does in a sense require k <= 3.

THEOREM 4. There is exactly one "distance function" satisfying k6 and
k7, There is exactly one "distance function" satisfying k6,k7'. This is
the usual Euclidean distance function in R^k.

f we have lines and distance, we can define Betweenness explicitly by

k5'. y is between x and z if and only if x,y,z are distinct and lie on
a line, and the distance from x to y is less than the distance from x
to z.

FIRST AND SECOND ORDER AXIOMATIZATIONS

We regard the above as a second order characterization of Euclidean
geometry on R^k, k >= 2. It is not really an official second order
axiomatization since it presupposes the usual completely ordered
2-divisiable abelian group of real numbers.

However, the characterization can be modified to give new or newish
second order axiomatizations of Euclidean geometry. This will be taken
up later. Also, I would like to use these ideas to give a first order
axiomatization of Euclidean geometry which might be incomparably
simpler than the usual.

***********************************************
My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
https://www.youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ
This is the 713th in a series of self contained numbered
postings to FOM covering a wide range of topics in f.o.m. The list of
previous numbered postings #1-699 can be found at
http://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/

700: Large Cardinals and Continuations/14  8/1/16  11:01AM
701: Extending Functions/1  8/10/16  10:02AM
702: Large Cardinals and Continuations/15  8/22/16  9:22PM
703: Large Cardinals and Continuations/16  8/26/16  12:03AM
704: Large Cardinals and Continuations/17  8/31/16  12:55AM
705: Large Cardinals and Continuations/18  8/31/16  11:47PM
706: Second Incompleteness/1  7/5/16  2:03AM
707: Second Incompleteness/2  9/8/16  3:37PM
708: Second Incompleteness/3  9/11/16  10:33PM
709: Large Cardinals and Continuations/19  9/13/16 4:17AM
710: Large Cardinals and Continuations/20  9/14/16  1:27AM
711: Large Cardinals and Continuations/21  9/18/16 10:42AM
712: PA Incompleteness/1  9/2316  1:20AM

Harvey Friedman