[FOM] 739: Philosophical Geometry/17

Harvey Friedman hmflogic at gmail.com
Mon Jan 2 23:50:42 EST 2017

We now, for the first time, get into the guts of the Innate Geometric
Plane in the mind's eye.
http://www.cs.nyu.edu/pipermail/fom/2016-December/020210.html sits on
top of the usual ordered ring of reals (or ordered group in some
versions) with the plane viewed as ordered pairs of reals.

with links


   1.8. DIRECTED LINE (1 dimension)
   1.9 DIRECTED CIRCLES, TRIANGLES (1 dimension)
   1.10. EQUIDISTANCE REVISITED (1 dimension)
      2.1.1. Multiplicative Equidistance.
      2.1.2. Additive Equidistance.
      2.1.3. Triangle Inequality.
       2.2.1. Lines, Betweenness, Distance Comparison.
       2.2.2. Lines, Betweenness, Equidistance.
       2.2.3. Lines, Distance Comparison.
       2.2.4. Lines, Equidistance.
       2.2.4. Lines, Betweenness.
       2.2.5. Lines.
      2.3.1. Distance Comparison, Betweenness.
      2.3.2. Equidistance, Betweenness.
      2.3.3. Distance Comparison.
      2.3.4. Equidistance.



We now take up two dimensional Philosophical Geometry. In section 2.1,
we get far away from any kind of Geometric Plane to what we call the
Analytic Plane, directly reducing the geometry to analysis. There we
treat the plane as a set of ordered pairs of real numbers, and take
THE real number system for granted as an ordered ring - sometimes only
as an ordered group. There does remain the idea of what can be vividly
seen in the mind's eye, as the mind grasps and sees interaction
between the analytic and the geometric.

In sections 2.2-2.4, we focus on the Geometric Plane, involving no
direct use of quantities (real numbers).

We make a distinction between the Public and the Private Plane. This
is reflected in what relations are taken as primitive. For the Public
Plane, we use only relations that all minds agree on when looking at
the same (copy, instance) Public Plane.

DIGRESSION: Clear cases of Public Planes that readily come to mind
include blackboards, pieces of paper, desktops, computer screens,
square tables, football fields, etcetera. But these are bounded, not
like a genuine plane. So this suggests that everything that we do in
section 2 needs to be revisited for, say, closed rectangles with
interior. Recall that in section 1 we generally worked with a closed
interval as an edge of a piece of paper. However, it would be wrong to
be doing a tiny bit of philosophy like this and have this constrain
the exploratory mathematical investigations prematurely. It is better
to start with the unbounded plane from any mathematical point of view.

There are a number of issues concerning just what primitives we should
use for the Public and Private Planes. The Philosophy as expected is
not yet developed enough to fully decide this. Thus we take our usual
exploratory approaches always with the idea that the information
obtained is fully relevant to any future Philosophy.

We work with the following primitives:

i. Lines with Epsilon (membership of points in lines).
ii. 3-ary Betweenness. Distinct points x,y,z lie on a line with y between x,z.
iii. 4-ary Equidistance. The distance from x to y is the same as the
distance from z to w.
iv. 4-ary Distance Comparison. The distance from x to y is at most the
distance form z to w.

These are chosen partly because of how commonplace they are in
Euclidean geometry. Of course, they are all vivid in the mind's
geometric eye. There are some other familiar concepts but we will
limit ourselves to these. Is there a well defined criteria for
narrowing down to a reasonably short list like this of concepts from
Euclidean geometry, perhaps a bit longer but still reasonably short?
We have not addressed this question.

Now consider the structures (R^2,?), where ? is any of the 15 nonempty
subsets of i-iv.

There can be issues regarding which of i-iv should be part of the
Public Plane. The philosophical side of what we are doing is not
refined enough to really address this, and so we explore all of the
seemingly philosophically relevant possibilities.

THEOREM 2.1. All concepts are the standard ones in R^2. lines,epsilon
and betweenness are mutually first order definable in each other
without parameters. equidistance and distance comparison are mutually
first order definable in each other without parameters. betweenness is
first order definable in equidistance without parameters. equidistance
is not definable in betweenness without parameters. On the other hand,
lines,epsilon, betweenness, equidistance, distance comparison are
mutually first order definable with parameters. Any three parameters
will suffice provided that they do not all lie on a line.
(R^2,lines,epsilon,betweenness,equidistance,distance comparison) has
an automorphism other than the identity. However, for any of the four
above structures, if we add any three constants, not all lying on a
line, then we obtain a rigid structure (all automorphisms are the
identity), and furthermore in each of the four cases, the resulting
structures with such constants added are isomorphic. The definable
relations in (R^2,lines,epsilon,betweenness,equidistance, distance
comparison, parameters allowed, are exactly the semi algebraic
relations on R^2. This is the case with any three point parameters
that do not all lie on a line.

Thus, according to Theorem 2.1, we can argue that use of any one of
the primitives i-iv encompasses the whole of Euclidean Plane Geometry.
But from another point of view, Theorem 2.1 can be cited to say that
even all of the primitives i-iv is not encompassing the whole of
Euclidean Plane Geometry. An intermediate view is that historically
Euclidean Plane Geometry concerns lines, epsilon, betweenness,
equidistance, distance comparison, angles, and angle comparison, and
the latter two are definable from equidistance alone. So from this
point of view, (R^2,equidistance) encompasses the whole of Euclidean
Plane Geometry but (R^2,betweenness) does not.

The most minimalistic treatment of Euclidean Plane Geometry presented
here is in section 2.2.5 where the intended structure is
(R^2,lines,epsilon). As indicated above, we can define betweenness in
this structure without parameters. In order to address issues
concerning "the whole of Euclidean Plane Geometry" we also work with
lines,epsilon, augmented with three not collinear constants. In
particular, the constants (0,0),(1,0),(0,1) play a vivid role, with
the intended structure (R^2,lines,epsilon,(0,0),(1,0),(0,1)), which is

There is an issue regarding the suitability of just lines,epsilon, or
just equidistance. The issue arises even with
lines,epsilon,equidistance. Namely, how do we formulate Completeness?

Well, as indicated above, betweenness can be defined from any of these
primitives, and Completeness is vividly formulated in terms of
betweenness. But the definition of betweenness, which can even be done
without parameters, is a bit more awkward than what we would like. By
comparison, the definition of parallel lines is not eat all wkward. So
in sections 2.2 and 2.3 it is only in the last subsection that we
encounter this awkwardness in the formulation of Completeness.

It should be noted that in set theory, we do run into formulations of
axioms which are significantly facilitated by making definitions which
can involve awkwardness.In convenient axiomatizations of ZFC, we
define emptyset and {x|, {x,y}, and use it for Infinity. Of course,
purists know well that Infinity can instead be formulated as "there is
a set with an element and where every element is an element of an
element". However, for formulating ZC or Z, some well known issues
arise, as there are many inequivalent formulations of Infinity (but
not for ZFC). Also, I have never seen anyone (comfortably) formulate
any large cardinal hypothesis in primitive notation - except for what
is in my Primitive Independence Results in Journal of Mathematical
Logic 3 (1):67-83 (2003), 333. Primitive independence results, July
13, 2002, 23 pages,
Also I haven't seen any comfortable formulation of the abstract set
theoretically fundamental CH or GCH (or V = L) in primitive notation
(somewhat relevant is
http://www.cs.nyu.edu/pipermail/fom/2016-July/019980.html and its
predecessor postings).

Stepping back for the moment, how did this Innate Geometric Plane come
about? I am in no position to say anything deep about this now,. We
make two points.

1. It would be exciting to tie this up with evolution. We evolved to
hunt animals, and, as animals evolved to hunt us, we evolved to avoid
getting hunted down by animals. All of this suggests the evolution of
innate 3 dimensional geometry, with crucial focus on two dimensional
aspects. Obviously non Euclidean geometry is not a similar product of
evolutionary processes.

2. I would like to know if my Innate Geometric Plane is the "same" as
yours. If we agree on certain innate conditions, then we prove that we
have an isomorphism between yours and mine. This is a general THEME in
Philosophical Mathematics - Philosophical Geometry is a special thread
in Philosophical Mathematics. We want to show that yours is the same
as mine. We already have known, almost known, or new, results of this
kind. E.g., for the natural numbers, integers, rationals, reals,
finite set theory, the first omega levels of the cumulative hierarchy,
the first omega + omega levels of the cumulative hierarchy (associated
with Z,ZC), and in certain senses for other portions or even all of
the cumulative hierarchy, etc. There is generally some "cross
principle" needed to establish the isomorphisms. E.g., in the case of
the natural numbers under <, we need to be able to treat relations
between yours and mine as primitive. HOWEVER, we have found that in
order to prove that yours and my numbers <= 2^2^2^2^2^2^2^2 have a
unique isomorphism, we need only purely ordinary logical reasoning -
i.e., nothing. Also, that yours and my first 8 levels of the
cumulative hierarchy sets have a unique isomorphism, also need
nothing. Of course, 8 can be replaced by any positive integer, but
then of course the size of definition of the isomorphism goes up
proportionally to 8. This general matter of "yours is the same as
mine" will be taken up in other postings.

My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
This is the 739th 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

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
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
713: Foundations of Geometry/1  9/24/16  2:09PM
714: Foundations of Geometry/2  9/25/16  10:26PM
715: Foundations of Geometry/3  9/27/16  1:08AM
716: Foundations of Geometry/4  9/27/16  10:25PM
717: Foundations of Geometry/5  9/30/16  12:16AM
718: Foundations of Geometry/6  101/16  12:19PM
719: Large Cardinals and Emulations/22
720: Foundations of Geometry/7  10/2/16  1:59PM
721: Large Cardinals and Emulations//23  10/4/16  2:35AM
722: Large Cardinals and Emulations/24  10/616  1:59AM
723: Philosophical Geometry/8  10/816  1:47AM
724: Philosophical Geometry/9  10/10/16  9:36AM
725: Philosophical Geometry/10  10/14/16  10:16PM
726: Philosophical Geometry/11  Oct 17 16:04:26 EDT 2016
727: Large Cardinals and Emulations/25  10/20/16  1:37PM
728: Philosophical Geometry/12  10/24/16  3:35PM
729: Consistency of Mathematics/1  10/25/16  1:25PM
730: Consistency of Mathematics/2  11/17/16  9:50PM
731: Large Cardinals and Emulations/26  11/21/16  5:40PM
732: Large Cardinals and Emulations/27  11/28/16  1:31AM
733: Large Cardinals and Emulations/28  12/6/16  1AM
734: Large Cardinals and Emulations/29  12/8/16  2:53PM
735: Philosophical Geometry/13  12/19/16  4:24PM
736: Philosophical Geometry/14  12/20/16  12:43PM
737: Philosophical Geometry/15  12/22/16  3:24PM
738: Philosophical Geometry/16  12/27/16  6:54PM

Harvey Friedman

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