History of Non-Euclidean Geometry - Lies - Extra History - #6
Added 2018-06-29 17:31:00 +0000 UTCYou asked for so long, and here it is! SUGGESTED READING for this Extra History series:
--Euclid's The Elements
--Newton's Philosophiæ Naturalis Principia Mathematica
--Lobachevsky's Geometrical Investigations on the Theory of Parallel Lines
Happy reading!
Comments
I'm not aware of Euclid formally defining distance as far as I'm aware he just kinda assumes the notion of distance is obvious and further that given A, B, C the distances btw A,B,C are all finite and there is a point B' on AC which is kind of his circle axiom although to be fully rigorous he should say that any line drawn through the center of a circle intersects the circle at 2 points Here I'm assuming as few axioms as possible that is there are these objects P called points and 2 sets of objects S_L, S_C called lines and circles respectively such that given a,b in P not equal there is a unique L in S_L that contains a,b ... etc as an exercise using this set theory framework define distance and prove (introducing new axioms if necessary ) the triangle inequality
Hao Sun
2018-07-31 20:43:28 +0000 UTCPerhaps projective geometry, pole to polar transformations, inversion mapping of the plane, geometry of Z_p (integers modulo p) algebraic geometry, elementary set theory other college lvl mathematics are good for intuition However you may not to read that much for an example try to rigorously define distance using only the 5 postulates and reword Euclids 5 postulates in set theoretic terms as rigorously as possible to show that your definition of distance works
Hao Sun
2018-07-31 20:19:40 +0000 UTCo_O I've been told I have a 160 IQ, and I tried to follow these two comments, but they were too dense for me, with too any unfamiliar terms. Any chance for pointers to illustrated discussions of these subjects?
Bill Lemmond
2018-07-31 15:19:33 +0000 UTCI understand to some people I sound a bit ridiculous but just from the 4 axioms we only have that there are elements of a set called points and a subset of the power set L that we call "lines" with the property that for any two points a,b there is exactly one line going through them where in the 4 axioms have we used the fact that our "lines" are straight or even continuous? Where have we used the fact that our points are all in a plane?
Hao Sun
2018-07-02 19:49:04 +0000 UTCwhat took them so long to figure out the 5th postulate was independent of the first 4??? in the axioms there really isn't anything that says a "line" is a line as we understand and not say a great circle on a sphere or that points cannot be lines and lines planes (projective geometry) Or what if we let the "points" of our geometry to be everything except a point p in the plane along with a point at infinity "lines" in our geometry all the circles and lines that go through p and the angle of two circles C_1 ,C_2 is the angle made by the lines f(C_1) f(C_2) where f(x)=p+ (x-p)/|x-p|^2 is the inversion fcn which one of the 4 postulates says a line is not a curve?
Hao Sun
2018-07-02 19:26:37 +0000 UTCwhen is the next topic vote ?
schuyler
2018-07-01 00:01:24 +0000 UTCshould not there be a one off ?
schuyler
2018-06-30 00:41:49 +0000 UTCYes it will be!
Extra History
2018-06-29 20:42:09 +0000 UTCSo, is the Flu Pandemic Series going to be the first one with Matt as narrator?
Jason Fox
2018-06-29 19:33:25 +0000 UTCKeeping it easy with those recommendations, eh? I recall Newton's book being written in a form "to avoid being baited by little Smatterers in Mathematicks". For those interested, I'd recommend Jeffrey Weeks' "The Shape of Space". It can be a bit tough going at times, but doesn't require much formal math and does a good job at explaining this material as well as introducing the field of topology.
Paul
2018-06-29 18:29:55 +0000 UTC
