Download FORMALIZATION OF HILBERT`S GEOMETRY OF INCIDENCE AND

JAN VON PLATO
FORMALIZATION OF HILBERT’S GEOMETRY OF INCIDENCE
AND PARALLELISM
ABSTRACT. Three things are presented: How Hilbert changed the original construction
postulates of his geometry into existential axioms; In what sense he formalized geometry;
How elementary geometry is formalized to present day’s standards.
1. INTRODUCTION
Hilbert’s axiomatization of geometry is one of the landmarks in the foundational debate on mathematics that began around the turn of the century.
This is not so much due to its subject matter as to the foundational viewpoint the work embodies. Indeed, even a glance in the -Bibliography
of Mathematical Logic, Vol. 1, Section B30, shows that the foundations
of geometry, particularly the synthetic kind as in Hilbert, has been just
a trickling side stream in this century’s logic and foundational study. We
remember Hilbert’s geometry for its idea of abstract axiomatization, and
for the dictum that existence is consistency. It need not be laid down what
the intended objects of the theory are, so that inference about them has to
be formalized. Hilbert’s geometry is the best known of the early attempts
at formalizing mathematics.
Three aspects of Hilbert’s geometry will be addressed here. The first
concerns construction and existence. A curious little finding testifies to
Hilbert’s changing of his mind about the order of the two notions. Secondly, we shall address the question, to what degree did Hilbert succeed in
formalizing his geometrical system. Finally, we show how part of Hilbert’s
original synthetic geometry is formalized by today’s standards. Consideration of plane incidence geometry will suffice for these purposes.
2. CONSTRUCTION AND EXISTENCE
Hilbert’s incidence geometry of the plane has just three axioms:
(I1)
For two points A, B there exists always a line a such that both
of the points A, B are incident with it.
Synthese 110: 127–141, 1997.
c 1997 Kluwer Academic Publishers. Printed in the Netherlands.
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(I2)
JAN VON PLATO
For two points A, B there exists not more than one line with
which both points A, B are incident.
Hilbert adds that one always intends expressions such as ‘two points’ in
the sense of two distinct points. The remaining axiom is:
(I3)
On a line there exist at least two points. There exist at least three
points such that they are not incident with one line.
Going through the subsequent development, one verifies that Hilbert never
refers to Axiom I1 in the proofs of his theorems. Yet Axiom I1 seems quite
like the other two axioms in appearance. The explanation of what seems
like a strange fact may be due to Hilbert’s unwillingness to revise the texts
of his proofs: Namely, the above formulations are those of the seventh
edition of 1930, and a comparison with the text of the first edition of 1899
reveals an essential difference. The original axioms read:
(I1)
(I2)
Two points A, B distinct from each other determine always a
line a; we shall set AB = a or BA = a.
Any two distinct points of a line determine this line; that is, if
AB = a and AC = a, and B =
6 C , then also BC = a.
These two axioms are followed by a similar one for planes:
(I3)
Three points A, B , C not on the same line determine always a
plane ; we set ABC = .
The existence of three noncollinear points is guaranteed by Axiom 7:
(I7)
On every line there are at least two points, on every plane at
least three points, not incident with one line, and in space there
are at least four points, not incident with one plane.
Thus, Hilbert’s original formulation of Axiom I1 was in the form of a
construction postulate, and in 1899 he even had a symbolic notation for the
condition required by the construction, the equality symbol crossed over
with a thick vertical bar. In the proofs of the theorems, he would simply
write AB whenever a line had to be constructed from two distinct points.
From 1903 on, the first two axioms read as follows:
A, B distinct from each other determine always a
(I1)
Two points
line a.
(I2)
Any two distinct points of a line determine this line.
HILBERT’S GEOMETRY OF INCIDENCE AND PARALLELISM
129
These are just like the original formulations, but with most of the notation
dropped out. The notation AB is used also for line segments and rays. The
above is how the axioms remained until the seventh edition of the year
1930, when finally ‘bestimmen’ gave way for ‘es gibt’. The construction
postulate for connecting lines, as we may call it, was abandoned and a
formulation in terms of the existential quantifier was given. But Hilbert did
not change the practice of justifying steps in proofs by simply writing AB
for a line whenever two distinct points were available. Even if the notation
of constructions remained, the conceptual change had occurred already
around 1900. In the geometry Hilbert still talks of proving geometrical
truths from a few simple axioms. The study of the properties of the axioms
is based on “the logical analysis of our spatial intuition” (1899, 89–90
and Introduction). In his famous ‘mathematical problems’ paper of 1900
instead, the doctrine of existence as consistency is very clear: “If we
succeed in proving that the properties given to our objects never can lead
to a contradiction in a finite number of logical inferences, I will say that
the mathematical existence of an object, say a number or function fulfilling
certain properties, has been demonstrated” (1900a, 301). In another paper
of the same year, on the concept of number, Hilbert compares the ‘genetic’
and ‘axiomatic’ methods: In the former one begins with the number 1,
then builds up the sequence of natural numbers and their arithmetic, goes
on to rational numbers, and so on. In the axiomatic method, instead, one
begins with the assumption of the existence of the sets of things one talks
about, like points, lines and planes in geometry. These are related to each
other through axioms which have to form a consistent and complete system
(1900b, 180–181).
Further particulars testify to Hilbert’s changing ideas of mathematical
existence. We shall first look at the problem of establishing a construction
for the intersection point of two distinct lines, and then remark on Hilbert’s
treatment of geometric construction problems from a more general point
of view.
Hilbert mentions as an immediate consequence of the incidence axioms
the following theorem (Satz 1, 6): Two (distinct) lines of a plane have one
or no points in common. No proof is given, but certainly Hilbert’s idea was
that if two distinct lines a and b had at least two distinct points A and B
incident, Axiom I2 would give a = AB = b which is impossible. (Actually,
I found an argument to this effect in Hilbert and Cohn-Vossen 1932, p.
103.) Note that the argument is indirect, it only shows that the lines a and
b have at most one point in common.
Chapter VII of Hilbert’s book is devoted to geometric constructions.
There he states that the axioms of group I make it possible to execute
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the following task 1: “To connect two points with a line, and to find the
intersection point of two lines in case they are not parallel” (1899, 78). Now,
the second part is quite problematic, as parallelism is only introduced in
Axiom III: “In a plane with a point A outside a line a one and only one
line can be drawn that does not intersect the line a; It is called the parallel
to a through point A” (p. 10). Even with this definition, the intersection
point construction still remains to be effected. For the concept of parallels
just refers to lines obtained by axiom III, so, actually by a rule of parallel
line construction. What is needed is a general definition of parallels: that
of distinct lines that don’t have a point in common. Indeed, if a and b
are distinct and not parallel, this definition gives that they have a point
in common. Its uniqueness is guaranteed by Hilbert’s first theorem that
we just enunciated. Hilbert soon changed the axiom of parallels, from a
construction postulate into an existential axiom: “Let a be an arbitrary line
and A a point outside a: There is in the plane determined by a and A only
one line b that is incident with A and that does not intersect a” (1903, 15).
But only in the 1930 edition do we find the missing definition of parallels:
“Explanation: We call two lines parallel if they are in the same plane and
don’t intersect each other” (p. 28). Intersection, in turn, must be a concept
that applies to two distinct lines that have a point in common. This can be
gathered from the explanations following Axioms I1 and I2 in the 1930edition (p. 3): “If A is incident with a line a and also with another line b,
we shall also say: the lines a and b intersect, have a point in common, etc”.
Putting the above observations together, we arrive at the following
picture of Hilbert’s geometry: The basic concepts are equal points, equal
lines, and incidence of a point with a line. Two lines intersect if they are
not equal and there is a point incident with both. Two lines are parallel if
they are not equal and there is no point incident with both. The classical
disjunction, two distinct lines are parallel or intersecting, is precisely what
is needed in order to obtain the two cases concluded in Hilbert’s Satz 1.
To be precise, Hilbert changed in 1903 the parallel line construction
into an axiom expressing the uniqueness, but not existence, of the parallel
to a line through a point. The reason is as follows: His first axiomatization
had five groups of axioms: group I for incidence, group II for order, group
III for parallels, group IV for congruence, and group V for continuity. In
1903 he changed the order of groups III and IV, claiming he can do the
parallel line construction through angle congruence: Let a line a and a
point A outside a be given. “Let us draw a line c which goes through A
and intersects a, and then a line b through A so that line c intersects lines
a and b with the same angles. It follows . . . that the lines a, b don’t have a
point in common” (1930, 28). The trouble here is that no axiom says there
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131
is, for any line and point outside the line, another line through the point
and intersecting the given line.
Let us see what things would be needed to resolve the situation. We can
try to use Axiom I3 which says that on any line there are at least two points,
say, B and C for line a. Then, if we succeed in proving that point A is
distinct from B , we can construct the line AB : Assume A and B are equal.
Since B is incident with a and A is outside a, we have a contradiction.
Therefore A and B are distinct. Since lines a and AB have a point in
common, it remains to prove that they are distinct lines. Assume they are
equal. It follows that A is incident with a, which is impossible. Therefore
a and AB are distinct lines. Now, finally, we have proved the existence of
the parallel to a given line through a given point. Therefore, if two lines
are distinct and not parallel, we can infer the existence of an intersection
point. This inference relies essentially on the purely existential Axiom I3.
Striking evidence of Hilbert’s change of mind regarding mathematical
existence can be gathered from his notes on geometry from the 1890s
(quoted from Toepell 1987, all page references to that volume). Five years
before the Grundlagen der Geometrie, Hilbert put the axioms as follows
(p. 60):
Existence axioms. Better: Axioms of incidence
1. Any 2 points A, B determine always one and only one line a.
2. Any 2 points A, B on line a determine the line a or in formulas, from AC = a and BC
= a, A B follows AB = a.
6=
Hilbert defines what it means for two lines to have a point in common, but
remarks that “it remains undecided whether two lines of a plane have a
common point at all” (p. 61). Further on, he remarks (p. 65):
It escapes our experience whether one can always find a point of intersection for two lines.
We therefore leave the matter preliminarily undecided and state only: 2 lines of a plane
have either one or no points in common.
It is a troublesome feature of Hilbert’s ‘Grundlagen’ of 1899 that it
retains the axioms of 1894, but claims to solve the task of constructing an
intersection point. More generally speaking, Hilbert’s treatment of geometric construction problems in his book displays the oddity that the existence
of solutions to geometric construction problems is proved purely indirectly.
The above passages from 1894 are evidence of Hilbert’s awareness of the
problem of effectiveness of a geometric construction. The question of constructive or nonconstructive existence became a sharply defined issue only
later, after Brouwer’s criticism of the classical law of excluded middle, but
by that time Hilbert had decided his way.
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3. DIFFERENT SENSES OF FORMALIZATION
A very common idea about formalization is that it consists in the use of
symbols. Thus, one used to speak of ‘symbolic logic’. One use for symbols
is to identify the clearly formal parts. But other means are available for that,
for example, a type different from the main text. Formalistic philosophy
of mathematics introduced the idea of a mathematical system as a set of
finitary rules for the manipulation of concrete symbols. A second, related
idea is formalization as machine executability. It calls for an explicit syntax
that is equated with formalization. A third idea is formalization as a process
in which structure is made explicit. None of these corresponds precisely to
Hilbert’s sense of formalization in his geometry. As for the first sense, his
geometry in its later editions uses letters as identifiers for points, lines, and
planes. It has a symbol for congruence, and a mnemonic sign for angles.
Beyond these few symbols, it is written in an informal language familiar to
anyone who reads German. Actually, any text can be thought of as a string
of symbols, as long as its characters come from some standard set such as
ASCII. Then, the idea of formalization as ‘the use of symbols’ is just like
saying that we do word processing. Thus, there is more to the formalizing
of the language of mathematics than writing a string of standard symbols,
namely the structure that is left unrevealed when treating text as a string.
We shall soon return to this matter.
Poincaré’s review of Hilbert’s book from 1902 is a wonderful illustration
of the idea of formalization as machine executability. Says Poincaré: “M.
Hilbert has tried to put the axioms in such a form that they could be applied
by one who doesn’t understand their meaning because he never sees a
point, line or plane. It must be possible to follow the reasonings by purely
mechanical rules”. Such formalization would be a “puerile exercise”, were
it not for the question of completeness: “Is the list of axioms complete, or
have some that we use unconsciously escaped us? . . . One has to find out
whether Geometry is a logical consequence of explicitly stated axioms,
or in other words, whether the axioms given to a reasoning machine will
make the sequence of all theorems appear” (1902, 252–253).
Hilbert himself states as the basic principle of his study of geometry,
“to express every question so that we could at the same time find out if
it is possible to answer it following some prescribed way and using given
restricted means”. The aim is “to decide which axioms, assumptions or
methods are needed in the proof of a geometrical truth” (1899, 89–90).
One motive for explicitness was to give a clear sense to the impossibility
proofs of geometry. We may describe this sense of formalization as the
quest for rigor, or ‘Strenge in der Beweisfuehrung’ as Hilbert liked to say.
The need for rigor is felt when one considers an axiom system as an implicit
HILBERT’S GEOMETRY OF INCIDENCE AND PARALLELISM
133
definition of the concepts therein. A purely symbolic development helps
keep apart unwarranted steps based on an intended interpretation. We have
seen that Hilbert’s axiomatizations of geometry until 1930 did not quite
reach to this standard. Even in the 1930 edition, many things remain quite
vague, for example, the treatment of the ordered plane. It has a concept of
‘given side’ of a line. Several results are mentioned, all without proof. To
get started with the proofs, one would need to make precise the concepts
‘same side’, ‘different side’, and a principle to the effect that two points on
different sides of a line are distinct, and so on. The axiom of Pasch that is
Hilbert’s only plane ordering axiom, will not suffice.
It is often said that Hilbert formalized the genuinely geometrical principles and left just the logical principles implicit. (As in Weyl 1944, 635,
640.) Hilbert’s first published formulation of the axioms in 1899 shows use
of the equality symbol for lines. General principles for equality would have
to be made explicit, unless these are considered part of the logical rules,
and the same goes for the rules of substitution. A related question concerns
the use of diagrams. In a truly formalized geometry, there should be no
place for diagrams, save as a practical aid. Diagrams are overdetermined
in comparison to the purely geometric assumptions language is able to
express. A typical consequence of their use is incomplete case analysis.
Hilbert’s degree of explicitness in formalization matters can be profitably compared to a forgotten section of an otherwise well-known paper
by Skolem, of the year 1920. We shall do that in the next section. Before
that, let us proceed to some newer ideas about formalization. There is a
second tradition of formalization, besides the one in mathematics, namely
the one of the formal grammars of natural language. There is the categorial grammar of Ajdukiewicz from 1935 and Chomsky’s phrase structure
grammar from 1957. In Montague grammar, predicate logic and simple
type theory are used to represent the structure of language. In continuation
of this tradition, Aarne Ranta (1994, 1995) has devised a grammar on the
basis of constructive type theory, and a generalization of Chomsky’s phrase
structure grammar that is exactly analogous to the way in which type theory
generalizes predicate logic. His grammarian’s experience is summarized
in the principle: formalization consists in the addition of structural information (1994, 171). Such addition is gained by making semantics formal.
Some of the consequences are:
It need not be the case that the use of symbols serves the aims of
formalization in the grammarian’s sense. Symbols appear in abbreviatory definitions, which is a pragmatic affair, and their use is to be
kept separate from formalization.
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Formalization admits of degrees: The more explicit the structure is
made, the more the text is formalized.
There is an opposite to formalization, called sugaring. It too admits of
degrees. It consists in the deletion of semantical type information and
of functional syntactic structure, which usually leads to overloading.
For example, we could present the functional structure of an incidence
predicate by the form
Coincident:(D:Set)(E:Set)(d:D)(e:E)Prop
Then, by applying the propositional function Coincident to the arguments
Point : Set, Point : Set, A : Point and B : Point, we get a value Coincident(Point, Point, A, B ) which belongs to the type Prop of propositions.
We get similarly Coincident(Line, Line, a, b) and Coincident(Point, Line,
A, a). Dropping the type information we get what seems to be the same
two place propositional function applied now to two points, now to two
lines, and now to a point and a line. Dropping the rest of the functional
structure we get the strings, A and B are coincident, a and b are coincident,
A and a are coincident.
Given the above, we need to say what ‘structure’ is. This was indicated
in connection with sugaring: it consists of semantical type information
and functional syntactic structure. Expressions of a language are, from
the grammatical point of view, built up by the application of rules. The
latter are given as functions that operate on linguistic objects of appropriate
categories, to produce other objects of determinate categories. We begin
with some given basic categories, and give ways of forming new categories,
and ways of forming objects of these categories. Semantical categories
combine with functional syntactic structure in the formalization of text,
the result of which is a syntax tree.
A mathematician would make nothing of the differences in grammatical
form, say, between the different constructions line a intersects with line b,
a and b are intersecting lines, and so on; Their logical content is the same.
On a formal level, this thought is realized by an interpretation function
from syntax trees to type theory. It will return the value Con(a,b) for the
syntax trees of the above examples. A sugaring function maps the syntax
trees into strings of words; This is the translation into natural language, or a
language identical to it in form and effect. Further tasks include the parsing
of expressions of informal language (or strings of words) into syntax trees.
Detailed illustrations of the grammarian’s approach to the formalization
of the language of mathematics, some geometrical ones included, can be
found in Ranta (1995).
HILBERT’S GEOMETRY OF INCIDENCE AND PARALLELISM
135
4. FORMALIZATION OF SKOLEM’S INCIDENCE GEOMETRY
Skolem’s basic notion in his axiomatization of plane incidence geometry
is ‘coincidence’, written as (AB ) for points, as (ab) for lines, and as (Aa)
for a point incident with a line. Equality of (AB ) with (BA), of (ab) with
(ba), and of (Aa) with (aA), is taken as part of the notation. The axioms
are (1920, 124–125):
(I)
For every A, (AA).
For every a, (aa).
(II)
(AB ) and (BC ) imply (AC ).
(ab) and (bc) imply (ac).
(III)
(AB ) and (Ac) imply (Bc).
(ab) and (Ca) imply (Cb).
(IV)
(Aa), (Ba), (Ab) and (Bb) imply (AB ) or (ab).
(V)
For every A and B there is a c such that (Ac) and (Bc).
For every a and b there is a C such that (aC ) and (bC ).
Here we have: Equality of geometric objects, in the sense of coincidence,
is an equivalence relation. Next, the third axiom permits substitution of
equal objects in the incidence relation. The fourth axiom is a classical
form of a general principle that gives the uniqueness of connecting lines
and intersection points as results (see our 1995, Section 12). Last we have
an axiom which implies the existence of connecting lines and intersection
points in projective geometry, and there is not much to add to Skolem at
his chosen level of notational accuracy.
It would be possible to arrive at formalizations of geometry through
the program of parsing into syntax trees, followed by interpretation in
type theory. We shall instead take the less laborious way and formalize
directly Skolem’s and Hilbert’s geometries for the plane by writing down
their axioms in the notation of type theory, and by declaring functions
that prove the axioms. The notation of type theory can be gathered from
Chapter 8 of Ranta (1994). The formal is indicated through other means
than the use of symbols, namely, by a different type. Symbols are used for
making formalizations fit a line.
We shall begin by declaring the basic sets of plane geometry.
(4.1)
Point:Set, Line:Set
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We shall make the first symbolic abbreviations.
(4.2)
Pt=Point:Set, Ln=Line:Set
There will be three propositional functions.
(4.3)
Coincident_Points:(Pt)(Pt)Set
Coincident_Lines:(Ln)(Ln)Set
Incident:(Pt)(Ln)Set
We make the abbreviations
(4.4)
EqPt=Coincident_Points:(Pt)(Pt)Set
EqLn=Coincident_Lines:(Ln)(Ln)Set
Inc=Incident:(Pt)(Ln)Set
Next we declare objects that prove what corresponds to Skolem’s five
axiom groups. We shall use rightaway suggestive abbreviations as names
for these, for the pragmatic consideration that the formulas would become
too long to be read comfortably.
(4.5)
ref_eqpt:(A:Pt)EqPt(A,A)
ref_eqln:(a:Ln)EqLn(a,a)
trans_eqpt:(A,B,C:Pt)(EqPt(A,B))
(EqPt(A,C))EqPt(B,C)
trans_eqln:(a,b,c:Ln)(EqLn(a,b))
(EqLn(a,c))EqLn(b,c)
Thus, given three points A, B and C and proofs that EqPt(A, B ) and
EqPt(A, C ), the function trans_eqpt will return as value a proof of EqPt(B ,
C ). Note that our way of writing reflexivity and transitivity implies symmetry: Substituting A for C in transitivity, EqPt(A, B ) and EqPt(A, A)
give EqPt(B , A).
Next there are the substitution and uniqueness axioms
(4.6)
subst_pt_inc:(A,B:Pt)(a:Ln)(EqPt(A,B))
(Inc(A,a))Inc(B,a)
subst_ln_inc:(a,b:Ln)(C:Pt)(EqLn(a,b))
(Inc(C,a))Inc(C,b)
unique_inc:(A,B:Pt)(a,b:Ln)(Inc(A,a))
(Inc(B,a))(Inc(A,b))(Inc(B,b))
Or(EqPt(A,B),EqLn(a,b))
HILBERT’S GEOMETRY OF INCIDENCE AND PARALLELISM
137
Finally we have the two axioms with quantifiers
(4.7)
exist_ln:(A,B:Pt)
Exist(Ln,(c)And(Inc(A,c),Inc(B,c)))
exist_pt:(a,b:Ln)
Exist(Ln,(c)And(Inc(c,a),Inc(C,b)))
The existential axioms can be replaced by construction postulates that
require conditions: The connecting line construction requires that the points
to be connected are distinct, and the intersection point construction that
the lines are distinct. We formalize a propositional function expressing
the distinctness of two points, the corresponding construction, and the
properties of constructed objects:
(4.8)
Distinct_Points:(Pt)(Pt)Set
DiPt=Distinct_Points:(Pt)(Pt)Set
conn:(A,B:Pt)(w:DiPt(A,B))Ln
inc_connl:(A,B:Pt)(w:DiPt(A,B))
Inc(A,conn(A,B,w))
inc_conn2:(A,B:Pt)(w:DiPt(A,B))
Inc(B,conn(A,B,w))
The corresponding formalizations for intersection points of two distinct
lines are:
(4.9)
Distinct_Lines:(Ln)(Ln)Set
DiLn=Distinct_Lines:(Ln)(Ln)Set
int:(a,b:Ln)(w:DiLn(a,b))Pt
inc_intl:(a,b:Ln)(w:DiLn(a,b))
Inc(int(a,b,w),a)
inc_int2:(a,b:Ln)(w:DiLn(a,b))
Inc(int(a,b,w),b)
Type theory has been implemented on a computer in several forms,
such as the ALF proof editing system of Gothenburg (see Magnusson and
Nordström 1994 for a description of ALF). We can now do proof editing of
Skolem’s incidence geometry by just adding the above declarations and the
law of excluded middle, or the law of indirect proof, to the proof editor. The
axiom for the uniqueness of lines and points is essentially classical for the
reason that the point and line equalities are not decidable. A constructive
formulation is given in our (1995), Axiom III.
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5. FORMALIZATION OF HILBERT’S GEOMETRY OF INCIDENCE AND
PARALLELISM
Skolem axiomatized directly projective geometry where any two distinct
lines intersect. Hilbert instead used the concept of parallel lines. To formalize a geometry of incidence and parallelism that corresponds to Hilbert’s
original axiomatization of 1899, we add to the formalization of Skolem’s
first four axioms, 4.1–6 above, two concepts with their corresponding
constructions: first the concept of distinct points and the connecting line
construction as formalized above in 4.8. Second, we add the concept of
intersecting lines:
(5.1)
Intersecting_Lines:(Ln)(Ln)Set
Con=Intersecting_Lines:(Ln)(Ln)Set
The construction postulate is
(5.2)
pt:(a,b:Ln)(w:Con(a,b))Ln
Its properties are given by the axioms
(5.3)
inc_ptl:(a,b:Ln)(w:Con(a,b))
Inc(pt(a,b,w),a)
inc_pt2:(a,b:Ln)(w:Con(a,b))
Inc(pt(a,b,w),b)
To prove uniqueness, we need the principle that Con(a, b) implies DiLn(a,
b). This will make the previous uniqueness axiom applicable. Projective
geometry can be done by adding the converse, any two distinct lines
converge. The classical expression of this principle is, any two parallel
lines are equal. (For a discussion of the relations between the different
geometries, see our 1995, Section 6.)
Hilbert’s original construction postulate for the parallel line construction had the condition that the point had to be outside the line. Such a
condition leads to unnecessary complications. Its original purpose was to
prevent reflexivity of the parallelism relation. Perhaps it was felt that the
parallel to line a through point A ‘is there already’ if Inc(A, a). But the
parallel and the line a are only coincident, not identical. There are in fact
proofs in which one constructs a parallel to line a through point A even
if Inc(A, a), the proof of the uniqueness of the parallel line construction
being a case in point. The complication with an irreflexive parallelism
relation is also reflected in the transitivity axiom: There it would have to
be postulated separately that the three lines involved are pairwise distinct.
HILBERT’S GEOMETRY OF INCIDENCE AND PARALLELISM
139
The formalization of parallelism, with reflexivity, is:
(5.4)
Parallel:(Ln)(Ln)Set
Par=Parallel:(Ln)(Ln)Set
ref_par:(a:Ln)Par(a,a)
trans_par:(a,b,c:Ln)(Par(a,b))
(Par(a,c))Par(b,c)
subst_ln_par:(a,b,c:Ln)(EqLn(a,b))
(Par(b,c))Par(a,c)
par:(A:Pt)(a:Ln)Ln
ispar_par:(A:Pt)(a:Ln)Par(a,par(A,a))
inc_par:(A:Pt)(a:Ln)Inc(A,par(A,a))
unique_par:(A:Pt)(a,b:Ln)(Inc(A,a))
(Inc(A,b))(Par(a,b))EqLn(a,b)
Let us show, as claimed above, that the uniqueness of the parallel line
construction calls for the construction par(A, a) even if A is incident with
a. Let us assume Inc(A, a). By the properties of par, we have Inc(A, par(A,
a)) and Par(a, par(A, a)). The three conditions of the uniqueness of par are
fulfilled, and we can conclude that EqLn(a, par(A, a)).
It is easy to derive existential axioms from a formulation in terms of constructions: Let w : DiPt(A, B ). Apply the connecting line construction to
get conn(A, B , w) : Ln. By the properties of connecting lines, inc_conn1(A,
B , w) : Inc(A, conn(A, B , w)) and inc_conn2(A, B , w) : Inc(B , conn(A,
B , w)). It is routine to construct the proof of
(A, B : Pt)(DiPt(A, B ))Exist(Ln, (c)And(Inc(A, c),Inc(B , c)))
from these. The existential axioms for the intersection point of two distinct
lines, and of two convergent lines, are derived similarly.
Skolem’s formulation of the existential axioms does not require the two
points (resp. two lines) to be distinct. His axioms have the strong existential
consequence that for any line, there exists a point incident with that line,
and for any point, there exists a line with which that point is incident. The
construction postulates and their axioms, in turn, display a type-theoretical
dependency structure: A construction such as conn(A, B , w) can only be
executed if we have proved its condition, namely that w : DiPt(A, B ), and
a proposition such as Inc(A, conn(A, B , w)) can likewise only be formed
if w : DiPt(A, B ). The proof object w is the carrier of the computational
information that gets discarded in a purely existential formulation.
In a classical axiomatization, the propositional functions appearing in
the constructions conn, int and pt as conditions, would be defined as follows:
DiPt(A, B ) = EqPt(A, B ),
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JAN VON PLATO
DiLn(a, b) = EqLn(a, b),
Con(a, b) = Par(a, b).
Now the previous concepts of distinct points, distinct lines and intersecting
lines become defined in terms of equal points, equal lines and parallel lines.
Thus, to complete the formalization of Hilbert’s geometry of incidence and
parallelism, we add to the formalizations 4.1–6, 4.8 and 5.1–4 the following
definitions:
(5.5)
DiPt(A,B)=Not(EqPt(A,B)):Set,
Con(a,b)=Not(Par(a,b)):Set.
The above definitions will bring into clear light the “oddity” of classical
geometry that we mentioned in the end of Section 2, namely that one
can prove purely indirectly the existence of a solution to a geometric
construction problem. We know from constructive analysis that a negative
condition such as Par(a, b) is not sufficient for the computability of
solutions. The same is true here on an abstract level. As long as we have
just proved the impossibility of parallelism of two lines, we have not shown
them positively convergent; the latter, however, would be precisely the
computational information needed for the intersection point construction
to be effectively executable. In a constructive formalization, one starts from
positive concepts such as DiPt or Con and makes the definitions rather the
other way around: EqPt(A, B ) = DiPt(A, B ), Par(a, b) = Con(a, b),
and so on. (The axioms in terms of the constructive basic concepts can
be found in our 1995.) Since the principle of indirect proof, or the law of
double negation, is not available, we cannot infer the positive condition
from EqPt(A, B ) or Par(a, b). The effect of adopting the constructive
conditions in geometrical construction and demonstration is that solutions
to geometric problems are provably terminating algorithms.
REFERENCES
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Hilbert, D.: 1900a, ‘Mathematische Probleme’, as reprinted in Hilbert’s collected works,
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Hilbert, D.: 1900b, ‘Ueber den Zahlbegriff’, Jahresbericht der Deutschen MathematikerVereinigung 8, 180–184.
Hilbert, D. and S. Cohn-Vossen: 1932, Anschauliche Geometrie, Springer Verlag, Berlin.
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HILBERT’S GEOMETRY OF INCIDENCE AND PARALLELISM
141
von Plato, J.: 1995, ‘The Axioms of Constructive Geometry’, Annals of Pure and Applied
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Department of Philosophy
University of Helsinki
Helsinki
Finland
[email protected]