Doom wrote:
dcs wrote:
Pro Ecclesia Dei wrote:
Reading Lobachevsky and his utterly false geometry.
What is "utterly false" about hyperbolic geometry?
Or maybe you were employing -- er, hyperbole?
If I remember correctly, PED is a radical constructivist, only Euclidean geometry is valid, everything else is 'false' and only positive integers 'exist', all other numbers 'don't exist' or can be conceived only in terms of the integers.
Even if I were not a constructionist, Lobachevsky is false, at least the treatise I am reading. Geometrical Investigations on the Theory of Parallels
It is certainly true that you can take many of the propositions and apply them to true geometry, but his imaginary geometry is false. He claims that he is doing plane, straight line geometry.
Straight lines are not asymptotic, and to posit a planar geometry that denies the 5th postulate denies something that is evidently true. The angle of parallelism is always 90 degrees in math when speaking of straight lines, but for Lobachevsky they are always less.
And so if you start from the principles that Lobachevsky does then his geomtry is false, at least starting with proposition 22. I am sorry but the angles in a triangle equal two rights, that is true; to state that they never do, but are always less than 2 rights is therefore false.
There are two takes one can have here. Either one claims we do not know which is true "Euclidean geometry" or "Non Euclidean geometry" because we do not know whether the 5th postulate is true. But that is absurd. Or one can reject the geometry as formulated by Lobachevsky, and apply most of his propositions to curved space geometry, in which case they are valid and just as true as Euclidean planar geometry. But even then not every single proposition can thus be tranferred. For instance propositions 32 and 33. These follow only through perverse thinking...denying the 5th postulate in the context of planar geometry, for the boundary line would, since the 5th postulate is true (and the law of non contradiction therefore says Lobachevsky is wrong), actually be a straight line, not curved and the circle would not approach it as you made you radius bigger and you would not get the monstrous idea of an infinite circle.
Though it is bizarre that if you take his circle with an infinite radius and make a sphere and do geometry on that sphere you get Euclidean geometry. Which perhaps should be a sign that Lobachevsky was wrong to treat his geometry as planar and should have seen it must apply only in curved space (like the inside of a trumpet)