The Non-Archimedean geometries after David Hilbert

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The Non-Archimedean geometries after David Hilbert
Quest-ce que la géométrie aux époques modernes
et contemporaines ?  Luminy 16 au 20 avril 2007

• Cinzia Cerroni
• Università degli Studi di Palermo

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Veronese and the debate before the G.d.G. of
Hilbert

• ... The question about the existence of a
  segment which is infinitesimal in respect to
  another is ancient but neither the supporters
  neither the opponents have proved the possibility
  or the impossibility of this idea, because they
  did not put the question in a right way,
  complicating it with philosophical considerations
  unrelated with it. It, instead, has to be put in
  the same way of those about the parallel axiom
  and the space dimensions that is, if all the
  axioms hold, is Archimedes axiom a consequence
  of the others? Or in other words, let A and B be
  two segments (AltB), does it exist a geometry in
  which in general is not true An gt B, where n is a
  positive integer 1, 2, ..n ?
• If one takes as the continuity axiom,
  Dedekinds axiom or if one maps the points of the
  line in to the real numbers, then the previous
  relation is a conceguence of it. But I gave a new
  definition of the continuity axiom that does not
  contain Archimedes axiom ... . Veronese.

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Veroneses non-archimedean line
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• The work of Veronese caused a national and
  international discussion which involved Rodolfo
  Bettazzi, George Cantor, Wilhelm Killing, Tullio
  Levi Civita, Giuseppe Peano, Arthur Moritz
  Schönflies, Otto Stolz, and Giulio Vivanti.
• In particular, in 1893 there was a
  turning-point in the discussion Tullio Levi
  Civita published his work
• Sugli infiniti ed infinitesimi attuali quali
  elementi analitici
• in which he constructed from the reals number,
  in an analytical way, a number system whose
  numbers (the monosemii) are the marks of
  Veroneses infinite and infinitesimal segments.

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• The monosemii are pairs of real numbers a? a
  is called characteristic and ? index. The
  monosemii with index zero are the real numbers.
  Two monosemii are equals if and only if they have
  the same characteristic and index. Let a? and bµ
  be two differents monosemii.
• ? µ if the characteristic are equals then
  the monosemii are equals if the characteristic
  are differents, then a? is greater (or smaller)
  than bµ if and only if a is greater (or smaller)
  than b, and conversely.
• ? µ if the characteristics are both
  positives then a? is greater (or smaller) than bµ
  if and only if ? is greater (or smaller) than µ,
  and conversely. If the characteristics are not
  both positives, one uses the previous rule and
  then one inverts the unequal direction like in
  the real numbers.
• Notice that, the monosemii a? , with positive
  or negative index ?, are greater or smaller than
  every real number, respectively.

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The non-archimedean number-system of Hilbert

• He considered the set ?(t) consisting of the
  algebraic functions of t obtained from t by the
  five operations of addition, subtraction,
  multiplication, division, and the operation
  
• where ? is a function derived
• from the previous five
  operations.
• Moreover, if c is a function in ?(t), it will
  vanish only on finitely many values of t.
  Therefore, c, for positive large values of t, is
  either always positive or always negative.

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• If a and b are two functions in ?(t), a will be
  greater than b (a gt b) or a less than b (a lt b)
  if a b is always positive or always negative
  for positive large values of t, respectively.
• Let n be a positive integer. Then n is less than
  t (n lt t) since n t is always negative for
  large positive values of t.
• Consider the numbers 1 and t in ?(t). Every
  multiple of 1 is always less than t, so ?(t) is a
  non-Archimedean number system.
• An analytic geometry on this number system is
  non-Archimedean

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Dehn and non-Archimedean geometries

• On Hilberts suggestion, Max Dehn, in his
  dissertation Die Legendreschen Sätze über die
  Winkelsumme im dreieck (1900), studied the
  relationship between Legendres theorems and
  Archimedes axiom.
• In particular, he asked
• Kann man die Legedreschen Sätze ohne irgend
  ein Steigkeitspostulat beweisen, d.h. ohne von
  Archimedischen Axiom Gebrauch zu machen?

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Max Dehn

• Born in Hamburg in 1878
• he received his doctorate in Göttingen at the
• age of twenty-one, under Hilberts
  supervision
• he obtained his Habilitation in Munich in 1901,
• solving the third of Hilberts twenty-three
  problems
• was Privatdozent in Munich from 1901 until 1911
  and became Ordinarius in Breslau in 1913
• he moved to the University of Frankfurt in 1921
  where he lectured until 1935
• he was the driving force of the seminar on the
  History of Mathematics, founded in Frankfurt in
  1922
• in 1939, since he was a Jew, he emigrated from
  Germany to Copenhagen and later to Trondheim in
  Norway
• in early 1941, when German troops occupied
  Trondheim, he emigrated to the United States
• in 1945, he arrived at Black Mountain College in
  North Carolina, where he died in 1952.

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The Legendres theorems

• The sum of the inner angles of a triangle is
  equal to or less than two right angles.
• If in a triangle the sum of the inner angles is
  equal to two right angles, it is so in every
  triangle.

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• Dehn first showed that Legendres second
  theorem is only a consequence of the incidence,
  order and congruence axioms by proving, in a
  geometry where such axioms hold, the following
  more general theorem
• If the sum of the inner angles of one triangle
  is less, equal, or greater, respectevely, than
  two right angles then this is true for every
  triangle.
• He constructed a Non-Legendrian Geometry in
  which there are infinite lines parallel to a
  fixed line through a point, Archimedes axiom
  does not hold and the sum of the inner angles of
  a triangle is greater than two right-angles.
• He constructed a Semi-Euclidean Geometry in
  which there are infinite lines parallel to a
  fixed line through a point, Archimedes axiom
  does not hold but the sum of the inner angles of
  every triangle is still equal to two right
  angles.

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Die Winkel-summe im Dreieck Durch einen Punkt giebt es zu einer Geraden Durch einen Punkt giebt es zu einer Geraden Durch einen Punkt giebt es zu einer Geraden
Keine Parallele Eine Parallele Unendlich viele Parallelen
gt 2 R Elliptische Geometrie (Unmöglich) Nicht-Legendresche Geometrie
2 R (Unmöglich) Euklidische Geometrie Semi-Euklidische Geometrie
lt 2 R (Unmöglich) (Unmöglich) Hyperbolische Geometrie
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• Therefore, Dehn had shown that, if Archimede
  axiom is not valid, the relationships between the
  hypothesis on the existence and the number of
  parallel lines through a point and the sum of the
  inner angles of a triangle do not hold.
• In particular
• ... Dass die Hypothese des stumpfen Winkels,
  wie sie Saccheri nennt, sich nicht deckt mit der
  Hypothese der Endlichkeit der Geraden

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Non Legendrian Geometry

• Like Hilbert, Dehn considered the
  non-Archimedean number system ?(t) and
  constructed an analytic geometry over this set.
• He constructed, over this non-Archimedean
  plane, an elliptic or Riemaniann geometry, as
  follows. He took the imaginary conic
• x2 y2 1 0
• and considered as points and lines all the
  points and lines of the non-Archimedean plane,
  together with the line at infinity with its
  points, and, as congruences, the real
  transformations that establish the conic.

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• He considered, as points of the new geometry,
  the points of elliptic geometry (x, y)
  satisfying the following conditions
• -n/t lt x lt n/t
• -n/t lt y lt n/t
• where n is an integer, and as lines, the lines
  whose points satisfy the above condition.
• Then he showed that through one point there
  exist infinite lines parallel to a fixed line but
  where the sum of the inner angles of a triangle
  is greater than two right-angles.

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Semi-Euclidean Geometry

• He considered the above non-Archimedean plane
  and constructed over it a new geometry as
  followsthe new points are the points (x, y) of
  the satisfying the following condition
• -n lt x lt n
• -n lt y lt n
• where n is a positive integer and the lines
  are the lines of the non-Archimedean plane whose
  points satisfy the condition above.

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• The sum of inner angles is equal to two
  right-angles in every triangle but there are
  infinite lines parallel to a fixed line through a
  point.
• Consider the line through the points (t, 0)
  and (0, 1) this is a line of the new geometry,
  since it passes through the points (0, 1) and (1,
  t -1/t), but intersects the x axis in a point
  that is not a point of the new geometry.
• Consider the line through the points (- t, 0)
  and (0, 1) this line is a line of the new
  geometry which intersects the x axis in a point
  that is not a point of the new geometry.

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The two previous lines pass through the point
(1, 0) and are parallel to the x axis.
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Hilbert lectures on the Foundations of Geometry
(1902)

• These results were probably achieved by Dehn
  in 1899. Hilbert mentioned them in a letter to
  Hurtwitz written on 5 November 1899 quoted in
  Toepell 1986, p. 257.
• Hilbert gave some lectures on the Foundations
  of Geometry in the summer semester of 1902. There
  is an elaboration by August Adler (1863-1923) of
  these lectures.
• In these lectures, he constructed another
  model of Semi-Euclidean geometry, emphasizing
  that the theorem about the sum of the inner
  angles of a triangle is not equivalent to the
  parallel axiom. He was struck by this kind of
  geometry which he called Merkwürdige Geometrie.

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The Prinzipien der Geometrie of F. Enriques
(1907)

• The article was planned after 1892, and the
  author continued to work on it until its
  publication. The article contains the results on
  the Foundations of Geometry obtained until then
  it is divided in seven chapters and the last of
  them is on the non-Archimedean Geometry and is
  exposed Dehns results on the relationships
  between Legendres theorems and Archimedes
  axiom.

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• In this article the author distinguishes the
  elementary questions, such that the ones
  directly deducible by the geometric properties
  from the superior questions, such that the ones
  needed to study in depth, as the Theory of
  Continuum or the Projective Geometry and so
  on
• After the representation of these different
  concepts, we reported in the last chapter about
  the new developments, which, through the
  abstraction of the common concept of continuum,
  had given rise to the construction of the
  non-Archimedean Geometry

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• Remaining in the field of Geometry, it must
  not be forgotten that such science is science
  about physical or intuitive facts as they are
  intended to be considered. Logical formalism must
  be conceived not as an aim in se but as a tool to
  carry out and to advance the intuition. The same
  results, logically established, must not be
  considered a mature achievement until they can be
  intuitively understood. But in the principles,
  intuitive evidence must shine luminously
  
• 
  Enriques 1900

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• Enriques, in the elementary questions,
  dedicated a paragraph on the continuity and
  Archimedes axiom in which he explained the
  postulates of continuity of Dedekind, Cantor and
  Weierstrass and the relationships between these
  postulates and the Archimedes axiom. Moreover,
  he described the Veronese geometric model of
  non-Archimedean geometry and exposed, as
  Schoenflies 1906 did, the difference between
  Veronese and Dedekind definitions of continuity.

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Bonola research on Saccheris theorem

• Bonola was interested in understanding the role
  of Archimedes axiom in the proof of Saccheris
  theorem. In his work I teoremi del Padre
  Girolamo Saccheri sulla somma degli angoli di un
  triangolo e le ricerche di M. Dehn (1905), he
  demonstrated, in a direct way without the use of
  Archimedes axiom, Saccheris theorem on the sum
  of the inner angles of a triangle
• This note has as the goal a direct and
  elementary proof of the result by Dehn, that is
  of Saccheris theorem, without the use of
  Archimedes axiom.

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Roberto Bonola

• He was born in 1874 in Bologna
• he graduated in Mathematics in 1898 under the
  supervision of Enriques, who assumed him
  immediately as his assistant
• in 1900 he became teacher of mathematics in the
  Female School and he started teaching before in
  Petralia Sottana and after in Pavia, where he
  spent the best years of his short life
• in 1902 he became assistant to the course of
  Calculus at the University of Pavia and in 1904
  he gave lectures on the Foundations of Geometry
• in 1909 he obtained the Libera Docenza of
  Projective Geometry and in 1910 he became
  Ordinary Professor on the Regio Istituto
  Superiore di Magistero femminile in Rome
• he was seriously sick since 1900 and he died
  prematurely in 1911 while he was going to Rome.

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• Bonola shares his masters vision of geometry,
  i.e. as a deeply intuitive discipline. Thus, only
  a direct proof could really satisfy our intuitive
  vision
• The way followed by Dehn to prove, without
  Archimedes axiom, Saccheris theorem is very
  elegant and logically complete. Geometrical
  intuition, however, needed a direct proof, that
  is a proof without formal systems, constructed
  over abstract concepts, that only formally
  satisfies the geometrical properties.

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• He started from the research of Father
  Saccheri Saccheri 1733 on Euclids V axiom. He
  considered the birectangular isosceles
  quadrilateral ABCD ( 1 right angle and AB CD),
  that is now called the Saccheri Quadrilateral and
  distinguished the three Hypothesis the one of
  the right angle, the one of the acute angle and
  the one of the obtuse angle.
• He then demonstrated Saccheris theorem If
  one of the three previous Hypotheses is valid in
  a Saccheri Quadrilateral, this hypothesis is
  valid in every Saccheri Quadrilateral without
  using Archimedes axiom.
• To prove Saccheris theorem, he considered a
  plane in which the axioms of connection, order,
  and congruence are satisfied, distinguishing two
  cases the closed line and the open line.

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Dehn Research Program

• The foundations of geometry represent a type
  of mathematical inquiry that highly suited some
  characteristic features of Dehns mind. However,
  he was not primarily interested in finding
  minimal sets of axioms or in separating the
  postulates of a given discipline into sets of
  weaker ones and then proving their independence
  and completeness. He was interested in finding
  solid and simple foundations for a theory, in
  particular for projective geometry
• the aim of the foundations of projective
  geometry, as well as of metric geometry, is to
  transform the projective relations
  (collineations) into algebraic relations. Dehn
  1926

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• This kind of approach was to become a real
  Research Program, for Dehn inspired many
  students.
• Ruth Moufang (1905-1977) in her main work
  Alternative Körper und der Satz von
  Vollstadingen Vierseit she constructs the
  non-desarguesian planes, that nowadays are called
  Moufang planes, exhibiting a delicate interplay
  of geometry and algebra.

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Hilbert had shown that a subset of his axioms
for plane geometry (essentially the incidence
axioms) together with the incidence theorem of
Desargues permits the introduction of coordinates
on a straight line which are elements of a skew
field. He proceeded as follows he had defined
the operations of addition and multiplication
and their inverse, using the incidence theorems.
So, the Desargues theorem and the incidence
axioms are sufficient to prove the calculus rules
and the commutative rule of product . Vice
versa, had constructed a geometry in which is
valid the Desargues theorem, by using the
elements of a skew field. We investigate in the
same way, by using a particular case of
Desargues theorem, called D9, which is
equivalence to the theorem of the complete
quadrilateral.
Moufang, 1933.
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Therefore, as Hilbert had shown that the
Desargues theorem together with the incidence
axioms of planes allows one to introduce
coordinates in a projective plane which are
elements of a skew field, and vice versa Ruth
Moufang proved that the configuration D9 (or
equivalently the theorem of complete quadrilater)
holds in a projective plane if, and only if can
it be coordinatizated by an Alternative division
ring (of characteristic not 2). M. Hall in
1943, introduced a general way to coordinatize
every projective planes with planary ternary
rings and made a classifications by using the
relationship between the algebraic properties and
the geometric properties.
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Conclusions

• We may note that Veroneses pioneering work
  did not give rise to a real mathematical school,
  but to a lasting debate on the subject of
  non-Archimedean geometry also within Italian
  geometers.
• Bonolas work was more influenced by Dehns
  and Hilberts approaches than by that of
  Veronese. In fact, since in his researches he was
  principally concerned about non-Euclidean
  geometries, he was above all interested in
  studying the relationships between Archimedes'
  axiom and the Saccheri theorem and besides, it
  was significant for him to obtain Dehns result
  under a more elementary point of view.

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• It follows from the previous considerations
  that Enriques (and his student Bonola) considered
  the foundations of geometry as a part of
  elementary mathematics, while the approach of
  Hilbert and Dehn is founded on the study of the
  interrelation between algebraic property and
  geometric property, and so considers foundations
  as a fundamental part of research in modern
  mathematics.