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Perspectives on Science, Volume 22, Number 1, Spring 2014, pp. 35-55
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Ideal Elements in
Hilbert’s Geometry
John Stillwell
University of San Francisco
Hilbert ªrst mentioned ideal elements in his 1898–99 lectures on geometry.
He described them as important, fruitful, and of frequent occurrence in mathematics, pointing to the examples of negative, irrational, imaginary, ideal
and transªnite numbers. In geometry, he had in mind the examples of points,
lines, and planes at inªnity, whose introduction gives geometry a certain completeness, by making theorems such as those of Pappus and Desargues universally valid.
In this article I will discuss how Hilbert transformed our view of the
Pappus and Desargues theorems by showing that they express the underlying
algebraic structure of projective geometry. I will compare this result with another of Hilbert’s great contributions, his calculus of ends. By studying the
ideal elements of the hyperbolic plane, Hilbert similarly extracted algebraic
structure from the axioms of hyperbolic geometry.
Hilbert’s treatments of projective and hyperbolic geometry have another
important common element: construction of real numbers. To achieve this,
Hilbert has to add an axiom of continuity to the geometry axioms, but he
evidently wants to show that the real numbers can be put on a geometric
foundation.
1. Ideal Elements before Hilbert
Hilbert took to using ideal elements in the 1890’s, in both algebraic number theory and geometry. His Zahlbericht of 1897 popularized the concept of the ideal introduced by Dedekind in 1871 (which in turn formalized the concept of “ideal number” introduced by Kummer in the 1840’s).
His geometric work likewise followed a long history of ideal elements,
some that originated in geometry and others that originated elsewhere
and were applied to geometry. Important examples were:
Perspectives on Science 2014, vol. 22, no. 1
©2014 by The Massachusetts Institute of Technology
doi:10.1162/POSC_a_00117
35
36
Ideal Elements in Hilbert’s Geometry
Figure 1. Piero della Francesca’s Ideal City.
1. Points at inªnity in projective geometry, originating in the 1430s
from the “vanishing points” in perspective drawing.
2. Imaginary points in algebraic geometry, originating from the imaginary numbers used by Bombelli (1572) to solve cubic equations.
Imaginary points are needed to give the “right” number of intersections between algebraic curves.
3. The point that completes the plane C of complex numbers to a
sphere, used by Riemann (1857) to study complex functions.
4. The boundary at inªnity of the hyperbolic plane, discovered by
Beltrami (1868) when he constructed the ªrst models of nonEuclidean geometry.
We now look brieºy at each of these in turn.
1.1 Points at inªnity and perspective
Today, we barely notice when correct perspective is achieved, as in the
Renaissance painting Ideal City (Figure 1).
This is perhaps because of the ubiquity of computer graphics, which
has technology for perspective built in. But rules for perspective drawing
were developed only around 1430, and blatantly incorrect perspective remained common for several decades after that, even in Renaissance Italy.
Consider, for example, the tiled ºoor in Figure 2. This drawing is an illustration from Savonarola’s Art of Dying Well, published in Florence around
1490.
Points at inªnity are the key to correct perspective. The method used
by Italian Renaissance artists generally used only one such point ( a “vanishing point”) because they drew tiles in rows parallel to the bottom of the
picture. However, one can avoid drawing parallels, and place the tiles in
an arbitrary orientation, by using three points at inªnity to control the positioning of lines. Figure 3 shows how. Given a single tile, with parallel
sides, all other tiles fall into position one by one.
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Figure 2. Tiled ºoor in the Art of Dying Well.
The three ideal points are where the three families of parallels—of the
sides and diagonals of the tiles—meet on the horizon (the ideal line or line
at inªnity). The diagonal of the ªrst tile determines the diagonal of the
second tile, because they have the same point at inªnity. The diagonal of
the second tile in turn determines the missing side of the second tile,
which determines the diagonal of the third tile, and so on.
1.2 Imaginary points
Newton (1665) stated the following result about algebraic curves, which
he called “lines,” and their degrees, which he called their “dimensions.”
For ye number of points in wch two lines may intersect can never bee
greater yn ye rectangle of ye numbers of their dimensions. And they
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Ideal Elements in Hilbert’s Geometry
Figure 3. Using points at inªnity to draw a tiled ºoor in perspective.
always intersect in soe many points, excepting those wch are
imaginarie onely.
This later became known as Bézout’s theorem, and a more concise statement
is as follows.
Bézout’s theorem: A curve of degree m meets a curve of degree n in exactly mn
points (if the curves have no common component).
However, the theorem was not properly proved for a couple more centuries, because to make it right one has to admit three kinds of ideal points.
1. Points at inªnity. For example, so that parallel straight lines (which
are curves of degree 1) have one point in common.
2. Imaginary points. For example, so that the parabola y⫽x2⫹1 meets
the line y⫽0 in two points; namely, the points x⫽i and x⫽⫺i.
3. Multiple points. For example, so that the parabola y⫽x2 (degree 2)
meets its tangent line y⫽0 (degree 1) in two points.
Imaginary points are already needed for the special case of Bézout’s theorem where one curve is y⫽0 and the other is y⫽p(x), where p is a polynomial of degree n. This case is essentially the fundamental theorem of algebra, in which case the multiplicity of a root x⫽a is the number of times the
factor x⫺a occurs among the linear factors of p(x).
Thus imaginary points and multiple points have an algebraic origin.
Points at inªnity have a geometric origin, but they can have implications
for algebra, as we will see. For the rest of this article we will be concerned
with points at inªnity, and the different contexts in which they arise. The
case of projective geometry shows that it is natural and convenient to add
a whole line of points at inªnity to the plane. The next case is one where it
is convenient to add a single point at inªnity to the plane.
Perspectives on Science
Figure 4. The square grid.
39
Figure 5. The grid of perpendicular
circles.
1.3 Complex functions
Imaginary, or complex, numbers were devised to give us solutions to
equations (initially cubic equations, but also simpler equations such as
x2⫹1⫽0). But they give more than this—something that we did not ask
for—when we consider functions of a complex variable. Complex differentiable functions are conformal maps of the plane (that is, maps that are angle-preserving or “similar in the small”). Here is an example, the function
that sends z to 1/z. This function maps the square grid shown in Figure 4
to the “grid” of perpendicular circles shown in Figure 5:
Thus differentiability acquires a deeper meaning when we allow the
variable to be complex. We see a property (preservation of angles) that is
only visible in two or more dimensions. This leads us to consider functions
on surfaces, and it turns out that we should consider surfaces beyond just
the plane C of complex numbers.
The Riemann Sphere Since complex functions can take the value ⬁ (for example, 1/z does at its pole z⫽0) we should think of their values lying in
the set C∪{⬁}, where C is the plane of complex numbers. Following
Riemann (1857), we view C∪{⬁} as a sphere via stereographic projection
(see Figure 6, which is by Jean-Christof Benoist on Wikimedia Commons). The idea of a differentiable function on the plane as one that maps
conformally can be transferred to the sphere, because stereographic projection is itself a conformal map.
The idea of “completing,” or compactifying, the plane C to a sphere by
means of the ideal point ∞is a useful idea, not only because it includes all
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Ideal Elements in Hilbert’s Geometry
Figure 6. Stereographic projection.
the values that a complex variable can take, but also because it makes
complex functions easier to classify. The differentiable functions on C are
hard to classify, because they include a multitude of transcendental functions such as ez, cos z, and sin z. However, the latter functions are not
meaningful at ⬁, because they approach different values as z approaches ⬁
in different directions. On the Riemann sphere, the picture is dramatically
simpler, because we need consider only functions that are meaningful
at ⬁. With this restriction:
1. Any entire function (one that is differentiable everywhere) is constant. (Liouville’s theorem).
2. Any meromorphic function (one differentiable except at poles) is
rational.
Thus the meromorphic functions on the Riemann sphere are precisely the
rational functions. A rational function r(z) can be written as the quotient
p(z)/q(z) of polynomials and, by the Fundamental Theorem of Algebra, the
polynomials p(z) and q(z) split into linear factors. It follows that a meromorphic function on C∪{⬁} is determined, up to a constant multiple, by
its zeros ai (the zeros of p) and poles bj (the zeros of q), and their respective
multiplicities mi and nj.
Riemann’s great insight was that, by generalizing the idea of the Riemann sphere to surfaces covering the sphere—Riemann surfaces—one may
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Figure 7. Non-Euclidean lines meeting at inªnity.
obtain a similarly simple view of algebraic functions of z. Namely, an algebraic function on a Riemann surface is determined, up to a constant multiple, by its zeros and poles.
1.4 Ideal points in non-Euclidean geometry
In his Euclides ab Omni Naevo Vindicatus (Euclid cleared of every defect), of
1733, Girolamo Saccheri explored the non-Euclidean geometry that we
call hyperbolic, and which he called geometry with “the hypothesis of the
acute angle.” He used this term because in this geometry a quadrilateral
with three right angles has a fourth angle which is acute. Saccheri investigated the acute angle hypothesis in an attempt to refute it, because he believed that:
The hypothesis of the acute angle is absolutely false; because it is
repugnant to the nature of the straight line.
The consequence of the acute angle hypothesis that he found repugnant
was a pair of lines that meet at inªnity and have a common perpendicular
there. However, repugnance is in the eye of the beholder, and in 1868
Beltrami found models of non-Euclidean hyperbolic geometry in which
the “lines” have precisely the behavior that Saccheri rejected. Moreover,
Beltrami’s models are rather attractive. In them, a pair of lines meeting at
inªnity look like those in Figure 7. Each of Beltrami’s models has a natural line at inªnity, and the line at inªnity is a common perpendicular to
lines that meet at inªnity. For example, in the half-plane model
•
•
•
Each “point” is a point of the upper half plane {(x,y): y⬎0}.
Each “line” is either a vertical half line or a semicircle with center on the x-axis.
The “line at inªnity” is the x-axis (to which all proper “lines” are
perpendicular).
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Ideal Elements in Hilbert’s Geometry
Figure 8. Periodicity pattern of the modular function.
Two of Beltrami’s models are particularly noteworthy because they are
conformal, or angle-preserving. This means that non-Euclidean geometry
has a natural role in complex analysis, and indeed Poincaré (1882) noticed
that some results already known in complex analysis had a non-Euclidean
interpretation. For example, the modular function, which goes back to
Gauss, is a periodic function on the half-plane whose periodicity is described by Figure 8.
The picture is from Klein and Fricke (1890). The values of the modular
function repeat under the transformations that send z to z⫹1 and to ⫺1/z.
These are among the transformations of the linear fractional form
f(z)⫽(az⫹b)/(cz⫹d ),
for real numbers a,b,c,d with ad-bc nonzero. Poincaré was studying functions invariant under linear fractional transformations around 1880 when
he noticed that such transformations are isometries of non-Euclidean geometry. They preserve the hyperbolic distance between points, which is deªned
by the distance element 公(dx2⫹dy2)/y. In particular, the curvilinear triangles shown in Figure 8 are congruent in the hyperbolic sense, because any
one of them can be mapped onto any other by a map that is a composite of
the hyperbolic isometries sending z to z⫹1 and to ⫺1/z.
Moreover, if we take z in the line at inªnity, R∪{⬁}, rather than in the
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Figure 9. The Pappus conªguration.
upper half-plane itself, these are nothing but the projective transformations
of the real projective line. Thus, non-Euclidean plane geometry has the
same group of transformations as one-dimensional projective geometry.
Also, a non-Euclidean motion of the half-plane is completely determined
by the corresponding transformation of its line at inªnity, so the ideal
points control the motion of the actual points in the non-Euclidean plane.
The projective transformations of the line a R∪{⬁} are made visible, so
to speak, by interpreting this line as the ideal boundary of the hyperbolic
plane. Poincaré (1883) made a spectacular extension of this idea. To understand transformations of C of the form
f(z)⫽(az⫹b)/(cz⫹d ),
with complex a, b, c, d, he viewed C as the ideal boundary of the upper halfspace, which turns out to be a model of the 3-dimensional non-Euclidean
geometry.
2. Geometry without Coordinates
Now we come back to projective geometry, taking up the story in the
19th century, when points at inªnity had been thoroughly assimilated
into the subject. Indeed, it was customary at this time to use the so-called
homogeneous coordinates, which put ordinary points and points at inªnity on
the same footing, describing them both by triples of real numbers. This
also allowed the use of algebraic methods in projective geometry, much as
Descartes had introduced into Euclidean geometry.
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Ideal Elements in Hilbert’s Geometry
At the same time it was noticed that the projective plane could also be
given an axiomatic description: the points and lines of a projective plane
satisfy the following axioms.
1. Any two points determine a unique line.
2. Any two lines meet in a unique point.
3. There exist at least four points, no three of which are in a line.
These axioms do not sufªce to prove all the theorems one can prove with
the help of homogeneous coordinates, so it remained to determine what
axioms should be added to the “obvious” axioms above. Hopefully, one
could ªnd axioms from which the existence of coordinates could be deduced. The ªrst such investigation was in the book Geometrie der Lage of
von Staudt (1847).
Following von Staudt (1847) and Wiener (1891), Hilbert in the 1890’s
analyzed the axiomatic approach to projective geometry, and how it may
be used to introduce coordinates “from inside.” The keys to this project
are the so-called “theorems” of Pappus and Desargues. Pappus discovered
his theorem around 350 CE, at a time when it was still part of Euclidean
geometry; Desargues discovered his theorem around 1640, along with
other pioneering results in projective geometry, which he was the ªrst to
view as a new branch of geometry.
Theorem of Pappus. For any hexagon with vertices alternately on two lines,
the intersections of opposite sides lie on a line (at the top of the picture in the example
shown).
(In Figure 9 we have deliberately drawn the line where the pairs of opposite sides meet so that it looks like the horizon, in which case the pairs of
opposite sides would be pairs of parallels. The freedom to interpret any
line L as the horizon, and hence to interpret lines that meet on L as parallels, is something we will exploit below, when we deªne “addition of
points” in projective geometry.)
This theorem states a projective property of points and lines; that is,
one involving only points, lines, and their intersections. But its proof requires non-projective concepts, such as lengths, and the congruence axioms
that govern them. The theorem of Desargues is a second theorem whose
statement involves only projective concepts, illustrated in Figure 10.
Theorem of Desargues. For any two triangles in perspective, the intersections
of corresponding sides lie on a line.
(Again, the picture is arranged so that the line where three intersections
occur looks like the horizon.)
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Figure 10. The Desargues conªguration.
This result really is a projective theorem, because it can be proved using
only projective concepts, but only if we work in projective space (see Figure 11). As one can see from Figure 11, corresponding sides of the triangles in perspective must meet in a line; namely, the line of intersection of
the two planes in which the triangles lie.
However, the proof breaks down if the two triangles lie in the same
plane. Like the Pappus theorem, the Desargues theorem in the plane is not
provable without the help of congruence axioms.
In fact, Hilbert showed that the Desargues theorem takes the place of
spatial axioms, and the Pappus theorem takes the place of congruence axioms. More precisely, using points at inªnity, one can deªne addition and
multiplication of points on a line. With the additional assumption of
Desargues or Pappus, this system of ‘coordinates’ has additional algebraic
structure, described by the following ªeld axioms.
a ⫹ b ⫽ b ⫹ a,
a ⫹ (b ⫹ c) ⫽ (a ⫹ b) ⫹ c
a⫹0⫽a
a ⫹ (⫺a) ⫽ 0
a(b⫹c)⫽ab⫹ac
ab ⫽ ba
a(bc) ⫽ (ab)c
a1 ⫽ a
aa-1 ⫽ 1 when a ⫽0
(commutativity)
(associativity)
(identity
(inverse)
(distributivity)
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Ideal Elements in Hilbert’s Geometry
Figure 11. Why the Desargues theorem holds in projective space.
Figure 12. Projective addition of a to b.
•
•
Assuming Desargues, the coordinate system is a skew ªeld (that
is, it satisªes all ªeld axioms except possibly commutative
multiplication). Assuming the projective space axioms, we get
exactly the same result, so Desargues replaces the space axioms.
Assuming Pappus and Desargues, the coordinate system is a ªeld.
Assuming congruence axioms, we get exactly the same result, so
Pappus replaces the congruence axioms.
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Figure 13. Projective addition of b to a.
Figure 14. Comparing a⫹b with b⫹a.
It is surprising that the nine ªeld axioms are implied by just ªve projective plane axioms (the three projective plane axioms plus Pappus and
Desargues). Indeed it is surprising that Pappus and Desargues have any algebraic content at all. In the next section we will say more about why this
is so.
2.1 Projective addition and multiplication
To illustrate how the Pappus theorem leads to a coordinate system whose
elements form a ªeld, we ªrst show how to add points on a line, and explain why the Pappus theorem ensures that addition is commutative. To
show the construction as clearly as possible, we draw certain pairs of lines
as parallels—exploiting the freedom to call any line the “horizon,” and to
call lines “parallel” when they meet on the “horizon.” Figure 12 shows
how to form the sum a⫹b of points a and b, using parallel lines to simulate the process of translating a ªgure along a line.
Intuitively, the line sloping upwards out of O and the line sloping
downwards to a together form a ‘pair of dividers’ spanning the interval
from 0 to a. When we “translate” the dividers parallel to themselves so
that the left end moves to b, the right end moves to what we call a⫹b.
(The faint lines mark the initial position of the “dividers,” the dark lines
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Ideal Elements in Hilbert’s Geometry
Figure 15. Multiplying a by b.
Figure 16. Multiplying b by a.
indicate the ªnal position.) This is a simple and natural construction of
a⫹b, but notice that the construction of b⫹a is different. Indeed, b⫹a is
constructed as shown in Figure 13.
Thus it is not immediate that a⫹b⫽b⫹a. Luckily, the Pappus theorem
comes to the rescue. If we superimpose the two constructions, we get the
following Figure 14.
The ªgure contains a Pappus conªguration (in which the opposite sides
of the hexagon are parallel), which ensures that lines ending at a⫹b and
b⫹a end at the same point. Thus, a⫹b⫽b⫹a.
There is an equally easy construction of the product of points, using
parallel lines to simulate a process of magniªcation. One begins with a
line with points marked 0 and 1, and arbitrary points a and b, and
‘magniªes’ a by b using the construction shown in Figure 15. The key
is the second line out of 0, along which we slide the joint in the “divid-
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Figure 17. Why ab⫽ba.
ers.” The “dividers” initially have their left end on 1 and the right end
on a. Then we move the two lines in the “dividers” parallel to themselves
until the left end is on b.
This magniªes the interval from 1 to a by b, so that the right end of the
“dividers” lands on ab.
Again, it is not clear that the process is commutative, because the construction of ba is different from the construction of ab; namely, ba is constructed as shown in Figure 16.
But again the two constructions lead to the same point because of the
Pappus theorem, as the Pappus conªguration in Figure 17 makes clear.
These proofs show that a⫹b⫽b⫹a and ab⫽ba are very natural consequences of the Pappus theorem. In fact, with some ingenuity, it is possible
to prove all the ªeld axioms. The Desargues theorem is particularly helpful in proving the associative laws.
However, this is as much as we can prove with purely projective axioms, since there are ªnite projective planes, whose coordinates come from
ªnite ªelds. To extract the real numbers from geometry, Hilbert needed
extra geometric axioms of “betweenness” and “continuity.” With these
axioms, one recovers the intuitive projective plane used by artists. The
betweenness axioms guarantee that points on a projective line (minus its
point at inªnity) have a left-to-right order, and that they are densely ordered; that is, between any two of them there is a third. Continuity guarantees that the line has no gaps, which makes it isomorphic to the real
number line.
Hilbert’s work on extracting algebra from geometry with the help of
ideal elements was the culmination of the investigations that began with
von Staudt (1847) and Wiener (1891). The latter authors attempted to
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Ideal Elements in Hilbert’s Geometry
construct projective geometry without numbers, but it was Hilbert who
clearly identiªed the role of the Desargues theorem (implying the structure of a skew ªeld, that is, a structure satisfying all the ªeld axioms except commutative multiplication) and the Pappus theorem (implying
commutative multiplication). Thus if both the Desargues and Pappus
theorems hold, we have the structure of a ªeld. In a surprising late development (more than 250 years after the discovery of the Desargues theorem!), Hessenberg (1905) discovered that Pappus implies Desargues, so
the Pappus theorem alone implies all the ªeld properties.
3. Extracting the Real Numbers from Non-Euclidean Geometry
One sees in retrospect that Hilbert’s approach to geometry has the aim of
extracting the ªeld of real numbers from geometry. If he had wanted only
to derive the theorems of Euclid, or of projective geometry, he need not
have included axioms of continuity. For Euclid’s geometry one needs only
a ªeld that includes the rationals and is closed under the square root (of
positive numbers), since these are the numbers that arise from ruler and
compass constructions.
In 1902–3, Hilbert carried out a similar program of extracting the real
numbers from axioms for the non-Euclidean hyperbolic plane. Again the
points at inªnity are crucial; they serve as real numbers, and Hilbert adds
and multiplies them in what is called his calculus of ends. A point ␣ at
inªnity is called an end because it is the common “end” of a family of asymptotic or “parallel” lines (Figure 18). Hilbert then uses the geometry of
asymptotic lines to deªne the sum and product of ends, and these turn out
to have the same behavior as the sum and product of real numbers.
The deªnitions are based on certain propositions that Hilbert proves
axiomatically, but we will look at them in the half-plane, where they are
easily seen to be true. Once it is seen why they are true, it is fairly easy to
construct axiomatic proofs. An example is the proposition that if the perpendicular bisectors of two sides of a triangle have the same end, so does the third
side. To see why this is true, let the end of two of the perpendicular bisectors be ⬁ in the half-plane model. Then all the perpendicular bisectors are
vertical lines, as Figure 19 shows, so they have the same end.
Given ends ␣ and ␤, view them as points on the x-axis, and draw the
vertical lines from 0, ␣, and ␤ to ⬁. Then the construction shown in Figure 20, which obviously produces ␣⫹␤, is expressible in the language of
hyperbolic geometry.
The construction is the following:
1. Choose a point X on the line from 0 to ⬁.
2. Find the reºection X␣ of X in the line from ␣ to ⬁.
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Figure 18. The end of a family of asymptotic lines.
3. Find the reºection X␤ of X in the line from ␤ to ⬁.
4. Construct the line through X␣ and X␤. The ends of its perpendicular bisector are ␣⫹␤ and ⬁.
3.1 Algebraic properties of sum and product
It is clear from the half-plane model that the sum construction is independent of the point X, and no surprise that this fact can be proved from
the axioms of hyperbolic geometry. It is also clear that ␣⫹␤⫽␤⫹␣, because we have the same triangle in both cases. Thus ␣⫹␤⫽␤⫹␣ is a theorem of hyperbolic geometry, which can be proved from the axioms. The same
applies to properties of the product of ends, which we deªne shortly. We
ªnd that sum and product satisfy all the ªeld properties, and that these
properties are theorems of hyperbolic geometry. (However, unlike the
derivation of ªeld properties from the Pappus and Desargues theorems, we
need to use axioms about length and angle.)
The product of ends can be expected to exist, because of the role of
multiplication in the half-plane model. Since the element of length in the
half-plane model is 公(dx2⫹dy2)/y, sending (x,y) to (␤x, ␤y) preserves
hyperbolic length for any ␤ ⬎0. This allows “multiplication by magniªcation,” not unlike the projective construction, except that instead
of magnifying by parallel displacement we magnify by displacement
through equal hyperbolic distances. Thus, in the situation shown in Figure 21, the line segments iA and BC have equal hyperbolic length, so we
can multiply the end ␣ by ␤ by displacing B upwards to C through a hyperbolic distance equal to iA.
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Ideal Elements in Hilbert’s Geometry
Figure 19. Perpendicular bisectors with a common end.
Figure 20. Constructing the sum of two ends.
In more detail, to construct ␣␤ from ends ␣ and ␤:
1. Draw the line from 1 to its reºection ⫺1 in the line 0⬁, meeting
0⬁ at i.
2. Draw the line from ␣ to its reºection ⫺␣ in 0⬁, meeting 0⬁ at A.
3. Draw the line from ␤ to its reºection ⫺␤ in 0⬁, meeting 0⬁ at B.
4. Make BC on 0⬁ with the same (hyperbolic) length as iA.
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Figure 21. Multiplication and hyperbolic length.
The perpendicular to BC at C then has ends ␣␤ and ⫺␣␤. With this construction it is not hard to prove ␣␤⫽␤␣, and in fact all the ªeld axioms
are provable as theorems of hyperbolic geometry.
Thus, Hilbert has again succeeded in extracting the real numbers from
geometry, with the help of ideal elements. Indeed, in hyperbolic geometry,
the real numbers are the ideal elements.
4. Summary of Hilbert’s Theory of Ends
1. The theory is probably motivated by the half-plane model,
which Hilbert used in all his early courses on the foundations of
geometry.
2. Ends themselves are motivated by the ideal points on the boundary
of the half-plane, that is, by real numbers.
3. By replacing ideal points by sets of objects inside the abstract
hyperbolic plane (families of asymptotic lines), Hilbert was able to
build the arithmetic of real numbers by pure geometry.
4. Thus numbers, and hence the half-plane model of hyperbolic geometry, is implicit in the geometry itself.
5. Therefore, what holds in hyperbolic geometry is what holds in the halfplane model. In particular, the Hilbert axioms of hyperbolic geometry are complete, because what follows from them is what is true in
the half-plane model.
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Ideal Elements in Hilbert’s Geometry
Hilbert’s experience with ideal elements in geometry clearly had an important inºuence on his later thought. In On the Inªnite (1925), he reºected as follows on ideal elements, revisiting many of the ideas we have
touched on above, and envisaging a continuing role for ideal elements in
all ªelds of mathematics that involve the concept of inªnity:
[. . .] as is well known, the introduction of ideal elements, namely,
points at inªnity and a line at inªnity, renders the proposition that
two straight lines always intersect in a unique point universally
valid [. . .]
The ordinary complex magnitudes of algebra likewise are [. . .]
ideal elements; they serve to simplify the theorems on the existence
and number of roots of an equation.
Just as in geometry inªnitely many straight lines, namely, a family of parallels, are used to deªne an ideal point, so in higher arithmetic certain systems of inªnitely many numbers deªne a number
ideal, and indeed probably no use of ideal elements is a greater
stroke of genius [. . .]
Now we come to analysis [. . .] in a sense mathematical analysis
is but a single symphony of the inªnite.
5. Conclusion
As Hilbert intimated in the above remarks from On the Inªnite, most of
the ideal elements in mathematics are related to inªnity, and indeed to
inªnite sets. The points at inªnity that occur in projective and nonEuclidean geometry are admittedly motivated by the idea of inªnite distance rather than inªnite sets. But, as Hilbert realized, each point at
inªnity corresponds to an inªnite set of lines (a set of parallels in the projective case, a set of asymptotic lines in the non-Euclidean case). Thus, as
in the case of ideals in algebraic number theory, one does not have to go
“outside” the domain of actual elements—an ideal element is simply a
certain inªnite set of actual elements.
Hilbert seems to be fascinated by the construction of the real numbers
in geometric systems. In part, this may be due to the ancient roots of real
numbers in Euclid’s Elements and earlier; from the discovery of irrational
quantities in Pythagorean times to the development of the “theory of proportions” by Eudoxus and its exposition in Euclid’s Book V. The Greeks
believed that geometric quantities were more general than numerical
quantities, since they did not consider 公2 to be a number. Hilbert of
course had no such qualms, but he evidently sympathized with the idea
that geometry is logically prior to the theory of real numbers. At any rate,
Perspectives on Science
55
he wanted to show how the real numbers could be constructed from geometric foundations.
In the long run, however, the real numbers are too important to be
taken as a special feature of geometric systems, and inªnity is too important to be left as some unexplained ideal element. As Hilbert well knew,
the “symphony of the inªnite” that is analysis leads to questions about
inªnity deeper than those posed by the existence of ideal elements. Nevertheless, ideal elements provide a model for the use of inªnity in mathematics, and they make a convincing case for its indispensability.
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