On the Hypotheses which lie at the Bases of Geometry.

Bernhard Riemann

Translated by William Kingdon Clifford[Nature, Vol. VIII. Nos. 183, 184, pp. 14–17, 36, 37.]

Seminar on the History of Hyperbolic Geometry, by Greg Schreiber

We began with an exposition of Euclidean geometry, first from Euclid’s perspective (as given in his Elements) and then from a modern perspective due to Hilbert (in his Foundations of Geometry). Almost all criticisms of Euclid up to the 19th century were centered on his fifth postulate, the so-called Parallel Postulate.The first half of the course dealt with various attempts by ancient, medieval, and (relatively) modern mathematicians to prove this postulate from Euclid’s others. Some of the most noteworthy efforts were by the Roman mathematician Proclus, the Islamic mathematicians Omar Khayyam and Nasir al-Din al-Tusi, the Jesuit priest Girolamo Sacchieri, the Englishman John Wallis, and the Frenchmen Lambert and Legendre. Each one gave a flawed proof of the parallel postulate, containing some hidden assumption equivalent to that postulate. In this way properties of hyperbolic geometry were discovered, even though no one believed such a geometry to be possible.

There is some question here as to what signifies a liberation of a kind and how this may have affected your perceptions. How is it so easy for what you may have read of one page to come back to it later and see and read something different? So how had you changed?

Wigner’s friend is a thought experiment proposed by the physicist Eugene Wigner; it is an extension of the Schrödinger’s cat experiment designed as a point of departure for discussing the Quantum mind/body problem.See: WIGNER’S FRIEND

Conclusion: *The state of mind of the observer plays a crucial role in the perception of time.* On the Effects of External Sensory Input on Time Dilation.” A. Einstein, Institute for Advanced Study, Princeton, N.J.

Einstein:Since there exist in this four dimensional structure [space-time] no longer any sections which represent “now” objectively, the concepts of happening and becoming are indeed not completely suspended, but yet complicated. It appears therefore more natural to think of physical reality as a four dimensional existence, instead of, as hitherto, the evolution of a three dimensional existence.

The value of non-Euclidean geometry lies in its ability to liberate us from preconceived ideas in preparation for the time when exploration of physical laws might demand some geometry other than the Euclidean.Bernhard Riemann

Riemannian Geometry, also known as elliptical geometry, is the geometry of the surface of a sphere. It replaces Euclid’s Parallel Postulate with, “Through any point in the plane, there exists no line parallel to a given line.” A line in this geometry is a great circle. The sum of the angles of a triangle in Riemannian Geometry is > 180°.

While this may seem abstract in term of it’s mathematical underpinnings, it allows us to see in ways that we might ever have been privileged to see before. So you turn your head to everything you have observed before and a whole new light has been thrown on the world. By consensus, this new view allows you to see deeper into the universe in ways that we had only taken from a standpoint of a man looking into outer space.

The Binary Pulsar PSR 1913+16:

So while being lead through the circumstance of historical individual pursuers to solving the Parallel postulate, liberation was found in order to move a geometrical proposition forward in time. Some may say that time is a illusion then?

So as a new paradigmatic change that has been initiated it’s application and is pushed into the world so as to ascertain it’s functionality. Does it then become real?

“…underwriting the form languages of ever more domains of mathematics is a set of deep patterns which not only offer access to a kind of ideality that Plato claimed to see the universe as created with in the Timaeus; more than this, the realm of Platonic forms is itself subsumed in this new set of design elements– and their most general instances are not the regular solids, but crystallographic reflection groups. You know, those things the non-professionals call . . . kaleidoscopes! * (In the next exciting episode, we’ll see how Derrida claims mathematics is the key to freeing us from ‘logocentrism’– then ask him why, then, he jettisoned the deepest structures of mathematical patterning just to make his name…)* H. S. M. Coxeter, Regular Polytopes (New York: Dover, 1973) is the great classic text by a great creative force in this beautiful area of geometry (A polytope is an n-dimensional analog of a polygon or polyhedron. Chapter V of this book is entitled ‘The Kaleidoscope’….)”