This is the most difficult of all.
For years we have been taught that a statement can not be false and true at the same time. There is nothing to be moderately and mildly false true at once, or the statement is totally false and totally true. This "self-evident truth" is known in logic as "the law of excluded middle."
Euclidean geometry tells us that at a point outside a given line is only possible to draw a parallel to that line.
But the elliptical geometry, in which the spherical geometry is a special case, says that simply is not possible to draw even a parallel to another line by a point outside that line.
Meanwhile, hyperbolic geometry tells us that not only can draw a parallel to a line from a point outside it, but it is possible to draw even more of a straight line parallel to another given by a point outside it.
If the universe we live in only one of the geometries may be correct, thus making the other geometries false, then what is the geometry of all true? Or is it possible that all are equally true in the universe in which we live? Is it possible that three contradictory statements are true but at the same time?
Unfortunately, our everyday experiences will not do anything to quietly extend our knowledge to things having to do with the infinite. Again and again, starting with optical illusions which our view can be easily fooled, and passing by the sophistries of ancient Greece (including the famous paradox of Achilles and the tortoise) with which the basis of certain things can draw the wrong conclusions, our intuition has proven to be very bad advisor to discover the reality of things. And if anything can make us fail our intuition is precisely to try to jump to conclusions about something that has to do with non-Euclidean geometry. A comparative example of this can be present considering a situation in which two fleas walking on a surface begin to move at the same time following each route perfectly "parallel" with respect to the other. Suppose that our measuring instruments have fleas incredibly sophisticated with which to start walking as do starting their journeys at the same time and with accurate initial deviation from zero degrees relative to one another. And suppose that we shall march forward along a path perfectly "straight, straight as can be imagined. Each of them swear by what is most sacred is moving "forward" without deviating even one thousandth of a degree of what appears to be a straight line. If both start your day in a perfectly flat surface in a Euclidean surface on which marks the fifth postulate of Euclid, then fleas while walking and keep going his career as straight as possible and will approach or be separated but remain at all times the same distance the one over the other. But if the surface on which they move is a surface curve with a curvature that they fail to perceive because it is a three-dimensional curvature , then go apart or closer depending on the surface will be elliptical ( as in the case of a baseball) or hyperbolic:
In one case, eventually crossing paths with their inevitable but may not do anything about it, and it will not be because their incredibly accurate measuring instruments are defective, but the fact that the space is moving curved . In the other case, eventually separating more and more without being able to do anything about it, also by the fact that the space is moving curved . The same thing happen to us in the case of a curvature being located in the fourth dimension be completely invisible to our physical senses. Fleas can travel in our example so that, regardless of the curvature of the surface will always stay the same distance away from each other. But In this case no longer be moving in a straight line will have to deviate from its trajectory by continuously adjusting their instruments, which in this case entirely lose the value for which they were designed. So how can we speak of a geometry is less valid or less possible than the other? On these interrrogantes
, a trial of AS Smogorzhevski entitled "On the Geometry of Lobachevsky" reads: "The issue concerning the structure of real space, are the responsibility of physics and can not be resolved with the forces of pure geometry. Its uniqueness is, inter alia, any geometry extension reflects the relationship with absolute accuracy, so, for example, due to the molecular structure of matter, there are no bodies available to the appreciation of the dimensions that have geometric properties of the ideal sphere. Precisely because of this, the application of geometric rules to solve specific problems inevitably leads to approximate results. Thus, our notion about the geometrical structure of real space is reduced in fact to the conviction based scientism that best describes a given geometry other real relationships with the ".
Expanding the latter, the development of General Theory Relativity leads us to an interesting conclusion: the three geometries, both the Euclidean and the elliptical and the hyperbolic, can be equally valid . Relativity does not preclude any of these possibilities.
This, of course, runs counter to our intuition, runs counter to everything we were supposed to be true. But if one thing has accustomed us to all of us modern physics, quantum mechanics starting to head stopped Einstein himself until the end of his days believed in the probabilistic view of the universe given by quantum mechanics clutching his concept of a deterministic universe, is that what we call intuition is very bad barometer of what we assume as "real." One thing is what we want it to be real, according to what our senses tell us and what our logic tells us, and quite another thing is what can be real out there beyond our senses.
What we have seen goes beyond simply offering alternative replacements to the Euclidean geometry. Take for example the course of analytic geometry taught in high schools as a preliminary to the study of differential and integral calculus, which takes place the great synthesis that made the French philosopher and mathematician René Descartes combining geometry with algebra. This geometry is based on plane geometry developed by Euclid. But if our field is, for example, a hyperbolic geometry, then we have to start all over again study, with a subject that would come calling hyperbolic geometry. Which of course is taught at leading universities around the world.
Is there anyone out there among those reading this who want to return to start studying all over again, because it is the starting point with which it started since high school was not the only one?
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