FOM: More Axiomatization of Geometry

[Harvey Friedman]

This is a better version of the previous posting 1:16PM  1/31/99,
Axiomatization of geometry. It supercedes it.

This abstract was motivated by three talks (by three different people) I
have heard over a period of years concerning geometrical reasoning by
diagrams. In these talks, the speakers wish to avoid following the modern
method of reducing plane geometry to algebra, and instead dwell on the
original Greek approach through diagrams. This abstract addresses the
relevant axiomatic issues in efficient mathematical terms - down to its
mathematical essence. Philosophers interested in this topic will probably
want to recast this development somewhat to interface better with various
views of diagrams and diagrammatic reasoning.

I am not an expert in the literature on axiomatic geometry, and
consequently I would appreciate comments on this approach. To what extent
has this topic been addressed in the literature?

AXIOMATIZATION OF EUCLIDEAN PLANE GEOMETRY BASED ON EQUIDISTANCE
by
Harvey M. Friedman
February 1, 1999

It will be convenient to consider Euclidean plane geometry normalized with
three distinguished points 0,1,i. We can think of 0 as the origin (0,0) and
1 as (0,1), and i as (1,0).

In the algebraic approach to geometry, one defines the plane and its
Euclidean metric in the usual first order way in the field of real numbers.
In this way, the first order Euclidean plane geometry (with distinguished
0,1,i) reduces to the first order algebra of real numbers.

Under the geometric approach, we consider structures such as (R^2,0,1,i,E),
where E(x,y,z,w) holds if and only if d(x,y) = d(z,w); i.e., the Euclidean
distance between x and y is the same as the Euclidean distance between z
and w.

We can define the field of real numbers in (R^2,0,1,i,E). In fact, much
more is true. This is almost surely known - but I don't know a reference.

THEOREM 1. The sets definable in (R^2,0,1,i,E) are exactly the
semialgebraic subsets of R^2. By Tarski, these are also the subsets of R^2
definable in (R,0,1,+,x). The same is true for 0-definable.

(Here definable means definable with any number of parameters. And
0-definable means definable with no parameters).

We also consider interpretability.

THEOREM 2. (R,0,1,+,x) is interpretable in (R^2,0,1,i,E) and vice versa.

According to Tarski, the first order theory of (R,0,1,+,x) has a beautiful
axiomatization via the real closed field axioms:

1) the usual field axioms;
2) -1 is not the sum of squares;
3) for all x, x or -x is a square;
4) every polynomial of odd degree has a root.

Tarski showed that these axioms are complete. Thus a sentence is true in
(R,0,1,+,x) if and only if it is derivable from these axioms.

Think of (R^2,0,1,i,E) as corresponding to the geometric approach to
Euclidean plane geometry, and (R,0,1,+,x) as corresponding to the algebraic
approach to Euclidean plane geometry.

The following question arises. Can we give a similarly elegant and basic
axiomatization of the first order theory of (R^2,0,1,i,E) involving only
R^2,0,1,i,E?

We also consider a related matter. According to Tarski, the real algebraic
numbers are the 0-definable elements of (R,0,1,+,x). But they have a very
algebraic definition (hence the name "real algebraic"): the solutions to
nontrivial polynomials in one variable with integer coefficients. In fact,
this is the usual definition of real algebraic numbers.

Similarly, the 0-definable elements of (R^2,0,1,i,E) are also the real
algebraic numbers. But do they have a very geometric definition? I.e., a
simple definition in (R^2,0,1,i,E)?

NOTE: Of course, the two coordinate functions on R^2 are NOT considered
part of the language of (R^2,0,1,i,E), but equality is taken for granted.
Of course, by Theorem 1, the coordinate functions on R^2 are definable in
(R^2,0,1,i,E).

We take the second matter up, which is considerably easier.

Fundamental to all aspects of this theory is the concept of **diagrammatic
condition**. This is a formula in the language of (R^2,0,1,i,E) of the
following special form: a conjunction of one or more atomic formulas
without equality.

We say that this is diagrammatic for the following reason. Suppose the
variables in the formula are x_1,...,x_k. Then the formula indicates that
we have points x_1,...,x_k where the distances between specified pairs,
including 0,1,i, are equal. This is like having a diagram with k labeled
points plus 0,1,i, with some line segments drawn in and marked indicating
certain equalities among the lengths. Of course, one is entirely
noncommittal about degeneracies; e.g., about which of the x's are equal or
nonequal, which line segments cross or don't cross, which triples of points
are or are not colinear, etcetera.

THEOREM 3. Let x in R^2. The following are equivalent:
a) x is 0-definable in (R^2,0,1,i,E);
b) x is 0-definable in (R,0,1,+,x);
c) x is real algebraic (i.e., its components are real algebraic numbers);
d) x is a coordinate in some solution of some diagrammatic condition in
(R^2,0,1,i,E) that has at most finitely many solutions;
e) x is a coordinate in some solution of some diagrammatic condition in
(R^2,0,1,i,E) all of whose solutions are permutations of each other.

We can alter the notion of diagrammatic condition to incorporate various
committments. E.g., we can consider modified diagrammatic conditions which
consist of a conjunction of one or more atomic formulas without equality
and the conjunction asserting that all points used (say including 0,1,i)
are distinct. This eliminates the most basic of degeneracies, although
there are other important degeneracies still allowed. Then Theorem 3 still
holds. Furthermore, various other modifications with regard to the
elimination of degeneracies can be made, with the same result.

We now come to the first, more delicate matter. It is very useful to
separate out what we call basic Euclidean plane geometry and quadratic
Euclidean plane geometry.

Basic Euclidean plane geometry consists of the set of all sentences of
(R^2,0,1,i,E) that become provable from the axioms of real fields (no
square roots) under the obvious translation into (R,0,1,+,x). The quadratic
real field axioms consist of just axioms 1) - 3) above. Quadratic Euclidean
plane geometry consists of the set of all sentences of (R^2,0,1,i,E) that
become provable from the quadratic real field axioms under the obvious
translation into (R,0,1,+,x). Note that E can be defined without square
root.

One can give reasonably elegant axiomatizations of basic Euclidean plane
geometry and quadratic Euclidean plane geometry staying within the language
of (R^2,0,1,i,E). The latter corresponds closely to ruler and compass
constructions.

We now come to the main issue of giving a geometric form of axiom scheme 4)
- that every polynomial of odd degree has a root - within the language of
(R^2,0,1,i,E).

We certainly don't want to simulate this axiom scheme 4) directly. E.g.,
odd degree appears to be geometrically meaningless. But we are looking for
a kind of geometric construction principle.

We discuss three alternative schemes. They correspond to the following
three theorems from calculus:

1. Continuous functions on a product of finitely many closed disks obey the
intermediate value theorem.
2. Continuous functions on a compact set attain a value of least magnitude.
3. Any nonempty set of magnitudes has a greatest lower bound.

Obviously theorem 1 is the one that most closely resembles a construction
principle.

For clarity, we state our schemes somewhat informally.

First of all, we want to appropriately define the midpoint between two
points x,y. Note that there are points z,w (or w,z), such that x,y,z,w
forms a square with diagonal x,y (degenerate if and only if x = y). This is
defined using equidistance - the sides are all equal and the two diagonals
are equal. The unique point equidistant to the four corners is the desired
midpoint.

Next we define d(x,y) >= d(z,w) as follows. Let p be the midpoint between
x,y. Then d(x,y) >= d(z,w) if and only if there exists u such that d(z,u) =
d(u,w) = d(x,p).

Now that we have d(x,y) >= d(z,w), we can informally use the concept of
(closed) disk. This is the set of all points of distance at most a given
amount from the center. Here an amount is simply given by an ordered pair
of points.

DIAGRAMMATIC INTERMEDIATE DISTANCE PRINCIPLE

Let V,W be disks and A(x_1,...,x_k,y_1,...,y_n,z_1,...,z_m) be a
diagrammatic condition with all variables shown. Let x_1,...,x_k be in R^2.
Suppose that for all y_1,...,y_n in V there are unique z_1,...,z_m in W
such that A(x_1,...,x_k,y_1,...,y_n,z_1,...,z_m). Then {d(z_m,0):z_m in W
and there exists y_1,...,y_n in V and z_1,...,z_m-1 in W such that
A(x_1,...,x_k,y_1,...,y_n,z_1,...,z_m)} is an interval. Note that this can
be naturally formulated without thinking of distances as objects. One only
needs the relation d(p,q) >= d(p',q') as discussed above.

DIAGRAMMATIC LEAST DISTANCE PRINCIPLE

Let V,W be disks and A(x_1,...,x_k,y_1,...,y_n,z_1,...,z_m) be a
diagrammatic condition with all variables shown. Let x_1,...,x_k be in R^2.
Suppose that for all y_1,...,y_n in V there are unique z_1,...,z_m in W
such that A(x_1,...,x_k,y_1,...,y_n,z_1,...,z_m). Then {d(z_m,0):z_m in W
and there exists y_1,...,y_n in V and z_1,...,z_m-1 in W such that
A(x_1,...,x_k,y_1,...,y_n,z_1,...,z_m)} has a least element. (The form with
"greatest" is equivalent). Again this can be naturally formulated without
thinking of distances as objects.

DIAGRAMMTIC CLOSED INTERVAL PRINCIPLE

We can combine the previous two axiom schemes into one stronger principle.

Let V,W be disks and A(x_1,...,x_k,y_1,...,y_n,z_1,...,z_m) be a
diagrammatic condition with all variables shown. Let x_1,...,x_k be in R^2.
Suppose that for all y_1,...,y_n in V there are unique z_1,...,z_m in W
such that A(x_1,...,x_k,y_1,...,y_n,z_1,...,z_m). Then {d(z_m,0):z_m in W
and there exists y_1,...,y_n in V and z_1,...,z_m-1 in W such that
A(x_1,...,x_k,y_1,...,y_n,z_1,...,z_m)} is a finite closed interval
(possibly degenerate). Again this can be naturally formulated without
thinking of distances as objects.

MODIFIED DIAGRAMMATIC LEAST DISTANCE PRINCIPLE

This principle is suitable for certain kinds of modified diagrammatic
conditions. Let us come back to the condition that all points are distinct,
as discussed above. The only principle that applies to this modification is
the greatest lower bound principle discussed below. But we can formulate an
appropriate least distance principle if we put a lower bound on the
distances between these distinct points. Here is the precise formulation.

Define mesh(x_1,...,x_t) as the minimum of d(x_i,x_j) for i notequal j.

Let V,W be disks and A(x_1,...,x_k,y_1,...,y_n,z_1,...,z_m) be a
diagrammatic condition with all variables shown. Let x_1,...,x_k,p,q,r,s be
elements of R^2, where p,q are distinct, and r,s are distinct, and
mesh(0,1,i,x_1,...,x_k) >= d(p,q). Suppose that for all y_1,...,y_n in V
such that mesh(0,1,i,x_1,...,x_k,y_1,...,y_n) >= d(p,q), there are unique
z_1,...,z_m in W such that mesh(0,1,i,x_1,...,x_k,y_1,...,y_n,z_1,...,z_m)
>= d(r,s) and A(x_1,...,x_k,y_1,...,y_n,z_1,...,z_m). Then {d(z_m,0):z_m in
W and there exists y_1,...,y_n in V, mesh(0,1,i,y_1,...,y_n) >= d(p,q), and
z_1,...,z_m-1 in W, mesh(0,1,i,x_1,...,x_k,y_1,...,y_n,z_1,...,z_m) >=
d(r,s), and A(x_1,...,x_k,y_1,...,y_n,z_1,...,z_m)} is empty or has a least
element. (The form with "greatest" is equivalent). Again this can be
naturally formulated without thinking of distances as objects.

MODIFIED DIAGRAMMATIC GLB DISTANCE PRINCIPLE

This principle is suitable for just about any reasonable kind of modified
diagrammatic condition. In particular, let us focus on the diagrammatic
conditions that require that all points are distinct.

Let V,W be disks and A(x_1,...,x_k,y_1,...,y_n,z_1,...,z_m) be a
diagrammatic condition with all variables shown. Let 0,1,i,x_1,...,x_k be
distinct elements of R^2. Suppose that for all y_1,...,y_n in V such that
0,1,i,x_1,...,x_k,y_1,...,y_n are distinct, there are unique z_1,...,z_m in
W such that 0,1,x_1,...,x_k,y_1,...,y_n,z_1,...,z_m are distinct and
A(x_1,...,x_k,y_1,...,y_n,z_1,...,z_m). Then {d(z_m,0):z_m in W and there
exists y_1,...,y_n in V, z_1,...,z_m-1 in W,
0,1,i,x_1,...,x_k,y_1,...,y_n,z_1,...,z_m are distinct, and
A(x_1,...,x_k,y_1,...,y_n,z_1,...,z_m)} has a greatest lower bound. (The
form with "greatest" is equivalent). Again this can be naturally formulated
without thinking of distances as objects.

THEOREM 4. The result of adding any of these five axiom schemes to any
complete axiomatization of quadratic Euclidean plane geometry is a complete
axiomatization of Euclidean plane geometry.

This probably holds if "quadratic" is replaced by "basic," but I haven't
looked some of the details for that yet.