So, I was wondering this - I'm currently reading a couple of books on non-Euclidean geometry - and this tells the story of how mathematicians overcame the assumption that the 5th parallel postulate must be true to be able to create non-Euclidean geometries. All very interesting stuff. Anyway, even more basic than the 5th postulate are Euclid's first 2 axioms:
1. A point is that which has no part
2. A line is length without breadth
what's interesting is that both of these axioms immediately take geometry from the "real" to the "ideal" - points in real life do have size and lines do have breadth. So, what happens to Euclidean geometry when you remove these axioms? If you simply replace them as:
1. A point is that which is infinitesimally small
2. A line is length with infinitesimally small breadth
do you get exactly the same geometry left? And if you reject even these definitions, what happens?
Given that mathematicians spent huge energies on the 5th postulate, there doesn't seem much interests in these axioms.
Anyone know anything on this topic? Thanks :)
1. A point is that which has no part
2. A line is length without breadth
what's interesting is that both of these axioms immediately take geometry from the "real" to the "ideal" - points in real life do have size and lines do have breadth. So, what happens to Euclidean geometry when you remove these axioms? If you simply replace them as:
1. A point is that which is infinitesimally small
2. A line is length with infinitesimally small breadth
do you get exactly the same geometry left? And if you reject even these definitions, what happens?
Given that mathematicians spent huge energies on the 5th postulate, there doesn't seem much interests in these axioms.
Anyone know anything on this topic? Thanks :)
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