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U of m non euclidean geometry
U of m non euclidean geometry





u of m non euclidean geometry

This is the result of Greek abstract thinking over centuries and became a pillar of European mathematics. We cannot visualize anything unless embedded in 3-dimensional flat space.Įuclid's axioms are a formalization of our intuition of space. Our imagination is limited to flat space of dimension three. I feel that the philosophical consequences of the discovery of non-Euclidean geometry and later its use in Relativity are overstated. It was still limited in the fact we don't know the thing-in-itself per Kant, and this hanging problem of the thing-in-itself served as the irritant-stimulant to the next great round of German philosophy: Fichte, Schelling, Hegel, Schopenhauer. And I should mention that Kant was trying to scrape together what knowledge he could.

U OF M NON EUCLIDEAN GEOMETRY SOFTWARE

Kant 1.0, 2.0 etc like software updates, but this kind of thinking does not fit well with certain kinds of metaphysics which seek permanent truth, fixity, etc. We don't seem to mind such changes today. So it would have damaged Kant's project, but it would not have damaged his way.

u of m non euclidean geometry

Remember, the world of phenomena is already contingent, so can't something stay put and be true, permanent and pure? So non-Euclidean geometry would have been a bombshell for Kant, it might have shaken him to the core (at least for a while). If you were to plug in different modules for space and time in the Transcendental Aesthetic, or if you were to fiddle around with his categories, what would this to to his project of wanting at least something to be "fixed" in place? True for all, if you will. But since geometry is the most compelling example, it robs the theory of one main 'hook' that makes us pay attention to it. So it is not deeply damaging to the theory as a whole. may be part of the form that proceeds from us, while its 'flatness' is synthetic, and would be different if we lived at a different scale or speed. Notions like the continuity of space, the basic properties of metrics, etc. The rest of the underlying mathematics may still be a form of our intuition. If space itself teaches us something about our imagination, like the fact it is off a bit at high speeds, then he is just wrong on that count. Kan't position is that space and time are not real, but are imposed on reality by our perception.

u of m non euclidean geometry

There should be nothing out there on the basis of which to modify it, if it is itself an aspect of ourselves and not of nature. It should not, then, be modified by experience. The problem is that our model of space is meant to be a ' form of intuition' for Kant. Sometimes in mathematics all one needs is a hint or a cue, and Kant may, and more than likely to, have provided this for him.

u of m non euclidean geometry

Gauss was known to have read Kants first critique where this extract is taken from (at least five times, according to one source) then one could conjecture that this - which is talking about geometry, his speciality - opened up for him the possibility of making a definite mathematical model of non-Euclidean geometry. This is where Kant opens up the possibility for non-Euclidean geometry if we exchange the axiom he mentions with a similar one (that is easier to work with, and changes nothing in what Kant wrote): that the angles of a triangle need not add upto 180 degrees then, if they add up to less, we get hyperbolic geometry, and if they add upto more, we get elliptic geometry. Similarly, geometric propositions, that, for instance in a triangle two sides together are greater than the third, can never be derived from the general concepts of line and triangle, but only from intuition, and indeed a priori with apodictic certainty (A24-5/B39-40) He's elaborating here what he means by a pure intuition - it's an ' a priori intuition'. It follows from this an a priori intuition (which is not empirical) underlies all concepts of space. This is simply saying we shouldn't confuse the immediate experience of space with the concepts that we use to talk about it this actually has been important in both physics and geometry, especially because of the popularity of the Cartesian notion of describing space, where one imposes a system of axes and then gives the coordinates of space instead, when we look at space we see no cartesian grid, taking this cue leads to the notion of general covariance in physics, and describing geometry intrinsically. Space is not a discursive, or as one says, general concept of relations of things in general, but a pure intuition.







U of m non euclidean geometry