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A TRANSDICIPLINARY ITINERARY

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EUCLIDEAN

TO THE

NON-EUCLIDEANP A R A D I G M

A TRANSDICIPLINARY ITINERARY

SOLOMON MARCUS

From the Euclidean to the Non-Euclidean Paradigm

EUCLID’S ELEMENTS,THE MANIFESTO OF THEAXIOMATIC-DEDUCTIVE THINKING

The Axiomatic Method

EUDOXUS, ARISTOTLE:

The Deductive Method

THALES, PYTHAGORAS:

The Invention of THEOREM and of its PROOF

According to the Greek etymology:

THEOREM = SPECTACLE

From the Euclidean to the Non-Euclidean Paradigm

EUCLID’S ELEMENTS

● Include not only geometry, but the whole Greek Mathematics heritage until Euclid:

- infinity of primes,

- irrationality of ,

- sum of a geometric progression,

- method of exhausting preparing the way to integral calculus,

- etc.

● Points out of the main mathematical way of thinking in successive steps, where each step is explicitly based on the previous steps.

This procedure prepared the way for introducing the artificial component of the mathematical language.

From the Euclidean to the Non-Euclidean Paradigm

EUCLID’S ELEMENTS(Continued)

● Makes clear the fact that the mathematical enterprise takes place in a fictional universe,whose mediation leads to the possibility to bridge mathematics with the real world.

● Shows:

- the COHESION and the COHERENCY of mathematics,

- its expansion in time,

- its synchronic and diachronic unity.

● Proves the huge number of researchers participating in the mathematical game.

Their solidarity in doing mathematics is stronger than their heterogeneity in many respects.

From the Euclidean to the Non-Euclidean Paradigm

THE EUCLIDEAN PARADIGMAND ITS SUPREMACY DURING 2000 YEARS

MATHEMATICS: A huge long-distance enterprise

The EUCLIDEAN MODEL as a guide for:

- The physicists of the Middle Ages

- St. Thomaso d’Aquino

- Benedict De Spinoza

- F.L. Gottlob Frege

- J.H. Woodger(in Biology)

- Archimedes

- Saint Augustin

- John Duns Scotus

- Isaac Newton

- Leonard Bloomfield(in Linguistics)

- Tadeusz Batog(in Phonology)

From the Euclidean to the Non-Euclidean Paradigm

NON-EUCLIDEAN GEOMETRYFACED WITH MISUNDERSTANDING,DOUBTS AND ADVERSITY

BIRTH: 1830 – 1831 Bólyai - Lobachevsky

STEPS TOWARDS ACCEPTANCE:

- Riemann 1854: Parabolic (Euclidean), Hyperbolic (Bólyai-Lobachevsky), Spherical (Elliptic) Riemann
- Opening, after 1855, of Gauss manuscripts
- Beltrami 1868: A model of hyperbolic geometry within Euclidean geometry
- Felix Klein 1871: Unification of all types of geometry by means of the generalised definition of DISTANCE (Cayley 1859)
- Special Relativity 1905
- Generalised Relativity 1917

Realities beyond the macroscopic one

From the Euclidean to the Non-Euclidean Paradigm

FROM FEAR AND REJECTIONTO A NEW UNDERSTANDING OF REALITY

- Geometry is no longer an exclusively axiomatic-deductive enterprise, it depends on physical aspects too.
- A challenge to Kant’s philosophy leading to doubt, mainly in Germany.
- A stimulus for art, mainly in Russia and France.
- A challenge to ethics and religion.
- In England, where people used to say the existence of God has the same degree of certainty as the statement that the sum of the angles of a triangle is two right angles.
- Euclidean geometry, as well as Galileo-Newtonian science, fit with the macroscopic Universe and, in most cases, with human sensorial-intuitive perception of the world.
- The reality considered in relativity theoryis that of non-Euclidean geometry, while the geometry of quantum reality involves Hilbert space and it is still under investigation.

From the Euclidean to the Non-Euclidean Paradigm

WE NEED BOTH THE EUCLIDEANAND THE NON-EUCLIDEAN PARADIGM

None of them aims to replace the other.

Sometimes the former is the framework permitting the allegory of the cave, where some shadows come from the Universe beyond the macroscopic world.

The following entities are equivalent:

Euclidean being.

Macroscopic being.

Galileo-Newtonian being.

Native speaker of human language.

Representation of the world according to our sensorial-intuitive perception and by using various prostheses to increase our muscular, sensorial and intellectual (cerebral) capacities (the most important being provided by the today computational tools).

HOWEVER: There is no sharp frontier between the macroscopic world, on the one hand, and the infinitely small and the infinitely large worlds, on the other hand. Some signs from the latter ones are captured in the former.

From the Euclidean to the Non-Euclidean Paradigm

NON-EUCLIDEAN GEOMETRY AND GÖDEL INCOMPLETENESS THEOREM FACE TO FACE

At a distance of hundred years (1830 / 31 – 1930 / 31), they are the most important shocks of the 19th and 20th century mathematics.

Conflict with common sense.

Conflict withtraditional (Aristotle’s) logic.

Both of them refer to the Axiomatic-Deductive Foundations of Mathematics (Geometry and Arithmetic, respectively).

Both of them had to wait long time before being understood; the understanding came step by step and it is still active.

The impact of non-Euclidean geometry is universal.

The first systematic attention paid to Gödel’s result came in 1957 with the treatise of Jean Ladrière “Les limitations internes des formalismes. Etude sur la signification du théorème de Gödel et des théorèmes apparentés dans la théorie des fondements des mathématiques” (Paris), but part of the work of Turing in the ’30s was convergent with that of Gödel.

The huge Bourbakienterprise ignores both non-Euclidean geometry and Gödeltheorem.

From the Euclidean to the Non-Euclidean Paradigm

Both Non-Euclidean geometries

and Gödel Incompleteness Theorem

are missing from the Mathematical curriculum in high-school and in higher education.

MORE GENERALLY:

Most important ideas in the science of the 20th century and even some of those occurring in the 19th century are ignored by the existing curriculum and text books of mathematical education, of scientific education in general.

From the Euclidean to the Non-Euclidean Paradigm

As Euclidean beings, our language cannot account for phenomena in universes considered in Relativity Theory of in Quantum Mechanics.

See the arguments in this respect proposed by Niels Bohr:

The competence of human language, of human semiosis, is limited to the macroscopic universe.

Jean Dieudonné, expressing the Bourbaki view, proclaimed:

À BAS EUCLIDE!

in connection with the teaching of geometry in school.

Edsger W. Dijkstra (1988) rejects Euclidean geometry as a prototype of axiomatic-deductive thinking, referring to the large number of mistakes resulting from the use of “geometric intuition.”

Such mistakes were indicated by Morris Kline (1972). He requires from Euclid the standard of rigour met by Hilbert, Birkhoff and Tarski in their axiomatisation of geometry.

From the Euclidean to the Non-Euclidean Paradigm

THE AXIOM OF CHOICE (ZERMELO, 1905) REPEATS THE SCENARIO OF THE EUCLIDEAN PARALLEL POSTULATE

All attempts to prove them failed.

Many of these attempts lead to statements equivalent to the initial ones.

One could make a list of statements equivalent to the Parallel Postulate.

While a similar list of statements equivalent to the Axiom of Choice was made by Waclaw Sierpinski in his book about the Axiom of Choice in the ‘40s of the past century. For its review, see J.E. Rubin (Journal of Symbolic Logic 35, 1970, 1, p.145).

A similar job for the Parallel Postulate was done, paradoxically, very late: Marvin J. Greenberg, Euclidean and Non-Euclidean Geometries: Development and History, New York, Freeman, 2007.

From the Euclidean to the Non-Euclidean Paradigm

THE AXIOM OF CHOICE (ZERMELO, 1905) REPEATS THE SCENARIO OF THE EUCLIDEAN PARALLEL POSTULATE(continued)

The most spectacular and the most significant statement equivalent to the Parallel Postulate belongs to John Wallis (1663):

There exist similar figures that are not congruent.

For instance: - All circles are similarand this fact points out the deep meaning of the number π

- All squares are similar

- All spheres are similar

- All cubes are similar

- etc.

Wallis’ statement explains why Euclidean geometry is strongly related to the idea of simplicity, order and symmetry of the Greek art and of the Renaissance art, practically of the whole millenary history of art.

The challenge of this attitude came with Mandelbrot’s Fractal Geometry.

From the Euclidean to the Non-Euclidean Paradigm

THE AXIOM OF CHOICE (ZERMELO, 1905) REPEATS THE SCENARIO OF THE EUCLIDEAN PARALLEL POSTULATE(continued)

d’) The most spectacular statement equivalent to the axiom of choice came as a combination of Hausdorff’s result telling that the axiom of choice implies the existence of Lebesgue non-measurable sets of reals and R.Solovay (1970): the axiom of choice is equivalent to the existence of Lebesgue non-measurable sets of reals.

János Bólyay defined Absolute Geometry as the geometry obtained from the Euclidean one, when we eliminate the Parallel Postulate

In the past century, various types of constructive mathematics (Brouwer’s intuitionism, Bishop’s constructive analysis, etc), have investigated what becomes mathematics via the Zermelo-Fraenkel axioms of set theory when we ignore the axiom of choice

From the Euclidean to the Non-Euclidean Paradigm

THE AXIOM OF CHOICE (ZERMELO, 1905) REPEATS THE SCENARIO OF THE EUCLIDEAN PARALLEL POSTULATE(continued)

The answer came step by step, for instance, in 1950, we learned that the Tychonov’s theorem (the product of compact topological spaces is compact), while in 1972, we learned that the Krein-Milman theorem (the unit ball of the dual of a real normed linear space has an extreme point) are equivalent to the axiom of choice.

There is no proof of the intrinsic consistency of Euclidean geometry and no proof of the intrinsic consistency of the Zermelo-Fraenkel axiom system of set theory.

There is no proof of the intrinsic consistency of non-Euclidean geometry (be it absolute, hyperbolic or elliptic) and no proof of the intrinsic consistency of the axioms of set theory, after eliminating the axiom of choice.

For all considered geometries and for all axiom systems of set theory there are only theorems of relative consistency.

From the Euclidean to the Non-Euclidean Paradigm

DOES THE CONTINUUM HYPOTHESIS FOLLOW THE SCENARIO OF THE PARALLEL POSTULATE?

No YES-NO answer can be given so far.

A large part of the scenario of the parallel postulate and of the choice axiom remains valid for the Continuum Hypothesis, but W. Hugh Woodin, in a pioneering book “The Axiom of Determinacy, Forcing Axioms, and the Non-Stationary Ideal” (Berlin: Walter de Gruyter, 1999), and in a series of papers (2000, 2001) claims (in a comment by Patrick Dehornoy) that:

“For the first time, there is a realistic perspective to decide the Continuum Hypothesis, namely in the negative.”

This is really against any expectations.

From the Euclidean to the Non-Euclidean Paradigm

TWO OF THE MILLENNIUM PROBLEMS ARE IN QUESTION! PARALLEL POSTULATE?

DOES THE “P vs NP” PROBLEM FOLLOW THE SCENARIO OF THE PARALLEL POSTULATE?

A similar question for the Riemann Hypothesis

Two other possible candidates for the scenario of the parallel postulate and of the choice axiom:

- Goldbach conjecture: Every even natural number larger than 2 is the sum of two prime numbers.
- The Twin Primes conjecture: There is an infinite number of twin primes.

From the Euclidean to the Non-Euclidean Paradigm

FROM NAÏVE TO RIGOROUS (FORMAL) EUCLIDEAN GEOMETRY IN 2000 YEARS

FROM NAÏVE TO AXIOMATIC SET THEORY IN ONLY A FEW DECADES

With (pseudo) definitions, absence of primitive term, and inferences impregnated with intuitive-sensorial elements, Euclid’s version of geometry came in conflict with the standards of rigour emerging towards the end of the 19th century.

The first version of rigorous Euclidean geometry has been proposed by Hilbert (1899) in set theoretic terms;

Then, in 1941, George D. Birkhoff proposed four postulates for Euclidean geometry, taking angle and distance as primitive concepts;

A third version has been proposed by A Tarski in terms of first order logic, proving its consistency and completeness (Gödel theorem is not relevant here).

Having four variants of Euclidean geometry (one naïve, three formal), we have for each of them three variants of non-Euclidean geometry (absolute, hyperbolic, and elliptic), and for each of them we are faced with the problem of its consistency.

From the Euclidean to the Non-Euclidean Paradigm

DOSTOEVSKY’S YEARS“THE BROTHERS KARAMAZOV”AND NON-EUCLIDEAN GEOMETRIES

Just what was the weak point for scientist became the attractive point for artists and writers, in the early development of non-Euclidean geometries.

The possibility to have alternative geometries, i.e., alternative worlds, was perceived as a freedom allowing our imagination to try all kinds of directions.

All these attempts were placed under the metaphor of a fourth dimension.

Fyodor Dostoevsky, Robert Musil and Yevgeny Zamyatin – all of whom had studied engineering, seized on non-Euclidean geometry and imaginary numbers as useful metaphors for their fictional accounts of the coming-to-terms with the modern human condition.

Taken together, the writings of these three implied that mathematics was at least an accessory to the break-down of the traditional belief in God, the loss of certainty in human knowledge and at the same time an extreme and false rationality.

From the Euclidean to the Non-Euclidean Paradigm

DOSTOEVSKY’S YEARS“THE BROTHERS KARAMAZOV”AND NON-EUCLIDEAN GEOMETRIES(Continued)

Both Ivan and his devil in Dostoevsky’s “The Brothers Karamazov” (1880) explore the relationship between changing ideas about geometry and changing ideas about God.

Paralleling the common ‘if, then’ arguments of mathematicians, Ivan declares to Alyosha that if there is a God and he created the world, he created it according to Euclidean geometry.

Dostoevsky however was acquainted with non-Euclidean geometries and so Ivan soon thereafter refers to

“geometricians and philosophers, and even some of the most distinguished who doubt whether the whole universe […] was only created in Euclid’s geometry; …[and who] dare to dream that two parallel lines …may meet somewhere in infinity”

(Fyodor Dostoevsky, The Brothers Karamazov, transl. C Garnett, ed. R.E. Matlaw 1976, New York: Norton, p.216.)

From the Euclidean to the Non-Euclidean Paradigm

DOSTOEVSKY’S YEARS“THE BROTHERS KARAMAZOV”AND NON-EUCLIDEAN GEOMETRIES(Continued)

Accepting the non-Euclidean geometries, even as he temporarily ignores their human creator, Ivan suggests furthermore that minds that can entertain only a 3-dimensional Euclidean Earth are incapable of deciding if God exists or not.

According to D.E.O. Thompson,

“What Ivan is really aiming at, is the idea that,since non-Euclidean geometry is not of this world, God is not of this world”

(1987:77 “Poetic transformation of scientific facts,” Dostoevsky Studies 8, 73-91)

From the Euclidean to the Non-Euclidean Paradigm

BEFORE THE SCIENTISTS, YEARSARTISTS AND WRITERS ENJOYED NON-EUCLIDEAN GEOMETRIES

Herbert G. Wells, Oscar Wilde, Joseph Conrad, Ford Madox Ford, Marcel Proust and Gertrude Stein are attracted by a possible fourth dimension.

Musicians such as Edgar Varèse, George Antheil, Alexander Skriabin see in the fourth dimension a higher reality.

Arnold Schönberg’s atonal music reflects one of the persistent topics related to the fourth dimension: the inadequacy of human language to the new reality of the higher dimensions.

It is also well known the involvement of hyperbolic geometry inMaurits C. Escher’s works (due partly to his collaboration with the great geometer H.S. MacDonald Coxeter).

This is another way to express Niels Bohr’s ideas about the crisis of the human semiosis when coping with phenomena beyond the macroscopic universe.

From the Euclidean to the Non-Euclidean Paradigm

BEFORE THE SCIENTISTS, YEARSARTISTS AND WRITERS ENJOYED NON-EUCLIDEAN GEOMETRIES(Continued)

Science Fiction took advantage of non-Euclidean geometry.

The horror fiction writer H. P. Lovecraft introduces many unnatural things which follow their specific laws of geometry.

The main character in Robert Pirsig’s Zen and the Art of Motorcycle Maintenance, refers several times to Riemannian geometry.

The Romanian poet Nichita Stănescu opens his cycle Laus Ptolemaei (1968) with a so-called Postulate:

“Two different things cannot stay in the same place”

and then he adds ironically that this Postulate does not belong to Euclid.

From the Euclidean to the Non-Euclidean Paradigm

BEFORE THE SCIENTISTS, YEARSARTISTS AND WRITERS ENJOYED NON-EUCLIDEAN GEOMETRIES(Continued)

Nichita Stănescu is addressing himself to the Geometer:

“Old, non-human Euclid,you have trusted a unique world,with non-human postulates”

Is there here the feeling of the need of freedom opened by the fourth dimension and by non-Euclidean geometries, pointing out the multiplicity of geometries fitting the multiplicity of realities (revealed by the Relativity Theory and by Quantum Mechanics)?

From the Euclidean to the Non-Euclidean Paradigm

WITH CUBISM, YEARSPAINTING MOVES FROM EUCLID TO THE NON-EUCLIDEAN WORLD

A guide in bridging visual arts and non-Euclidean geometry is Wylie Sypher’s Rococo to cubism in art and literature (Vintage Books, New York, 1960).

With cubism, we repeat the Renaissance’s mentality, when science was an aspect of art and the painter was mathematically educated.

“Rodin complicates the old 3-dimensional scenographic space […], there is with him a protest against the Euclidean space, but without having a richer, more advantageous structure. He should need the relativistic space-time.”

“Cézanne is more clever, but we should not follow his advice, requiring to represent nature by means of cylinder, sphere and cone […]. Although almost illiterate, he had a strong perception of Bradley’s philosophy and of Einstein’s, Riemann’s, Clifford’s and Gauss’s works.”

From the Euclidean to the Non-Euclidean Paradigm

WITH CUBISM, YEARSPAINTING MOVES FROM EUCLID TO THE NON-EUCLIDEAN WORLD(Continued)

In 1875, Clifford observes that geometry’s laws cannot be applied to small portions of the space, comparable with small hillocks; distortions spread as the waves.

Van Gogh had perhaps this move when he had to paint the extremely strong locally curved spaces in embankments and water.

As Francastel observes, cubism is a product of modern thinking, because it is based not only on new techniques, but also on a rich philosophical and scientific speculation.

Cubism realizes a mutation in the representation of the 3-dimensional space, because it is no longer reported to a fixed viewpoint. Things appear in multiple relations and change their appearance with respect to the adopted viewpoint. In their turn, the number of viewpoints increases, depending on time and light.

As in Renaissance, cubism opens new possibilities to see the world. Cubism may be ahead with respect to science’s modernity.

From the Euclidean to the Non-Euclidean Paradigm

WITH CUBISM, YEARSPAINTING MOVES FROM EUCLID TO THE NON-EUCLIDEAN WORLD(Continued)

The artist may be privileged with respect to the scientist, because he captures before the scientist the mentality of his time.

Picasso’s Guernica appears long time before the annihilation of Hiroshima, Dostoevsky penetrated the human unconscious before Freud, cubism is, in some sense, a preface to Relativity Theory.

By the revolution it produced in the way we represent the world, cubism has created a cinematic style. Cubism has rejected the Renaissance illusions of the 3-dimensional space.

In the same order of ideas, José Ortega y Gasset questions the existence of some thing more artificial than Euclidean geometry, which gave the foundation of painting before cubism.

Cubist painting solves the old conflict between primary, mathematical properties of objects and their secondary, material, sensorial properties. Accepting the plurality of the worlds, cubism is bridging perfectly the plurality of the geometries.

From the Euclidean to the Non-Euclidean Paradigm

NON-EUCLIDEAN GEOMETRIES EVERYWHERE YEARS

András Prékopa has already provided a very comprehensive historical and cultural account of non-Euclidean geometries in Non-Euclidean Geometries: János Bolyai Memorial Volume “The revolution of János Bolyai”, p. 3-60, Berlin: Springer, 2006.

Although not so easy, aspects that were not considered there were intended to be approached here.

As partly illustrated so far, in mathematics, the non-Euclidean ideas spread in all fields.

The universality of hyperbolic geometry is perhaps the most visible.

In the ’80s, of the past century, hyperbolic geometry became the protagonist of the program proposed by William Thurston (Fields Medal), aiming to assign a canonical structure to every 3-dimensional variety.

In computer science, a new problem appeared: computer’s dependence on the underlying space-time geometry; different geometries yield radically different computers.

From the Euclidean to the Non-Euclidean Paradigm

NON-EUCLIDEAN GEOMETRIES EVERYWHERE YEARS(Continued)

Hyperbolic geometry is involved in bio-computing and in molecular computing.

M. Margenstern, working in molecular computing and in cellular automata, considers various types of hyperbolic tilings (Journal of Universal Computer Science 9(5) and 10(9), Fundamenta Informaticae 56).

He proves the undecidability of the tiling problem for the hyperbolic plane.

A hot and controversial problem is the structure of the visual space.

Most arguments favor the non-Euclidean variant.

In this order of ideas, a more interesting result comes from the school of Jean Piaget, concerning the way children perceive the space and arguing that the topological and non-Euclidean properties are recognized by children before they learn to recognize the Euclidean properties (Piaget and Inhelder, The child’s concept of space, Routledge and Kegan Paul, London, 1956).

From the Euclidean to the Non-Euclidean Paradigm

NON-EUCLIDEAN GEOMETRIES EVERYWHERE YEARS(Continued)

In the same order of ideas, Patrick Heelan, in Space Perception in the Philosophy of Science, (University of California Press, Berkeley, 1983, 163-164) argues that a certain non-Euclidean (hyperbolic) visual capacity precedes the Euclidean perception.

In other words, the visual and the representation of space are, in their initial form, organized according to some non-Euclidean patterns and only later, by experience and education, an extension takes place to the Euclidean aspects.

The non-Euclidean, initially conceived as a paradoxical phenomenon, proves to be in the very nature of human being.

Together with the Euclidean, it accounts for the complexity of the reality.

From the Euclidean to the Non-Euclidean Paradigm

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