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The trouble with lines

It would seem that when we examine the structure of an object we must ultimately come to some irreducible components that have an essence or intrinsic nature. For example we might consider the lumber, nails, concrete foundation and roofing shingles as being among the fundamental components of a house. Contemporary mathematics has deliberately and systematically purged itself of any objects with an intrinsic nature. Mathematicians did this because starting with objects that had an intrinsic nature, like lines, led them to make false assumptions like the parallel postulate of Euclidean Geometry.

For centuries, the parallel postulate of Euclid was considered to be a self evident truth. Two lines are parallel if they are both perpendicular to a third line. For example the legs of a well made table are parallel because they are perpendicular to the table top. No matter how far one extends the legs they will never meet if one keeps them straight and parallel. This is the parallel postulate. It seems self evident.

Now consider the laws of geometry on the surface of the earth. Sailors determine their location in the ocean by latitude and longitude. These are imaginary lines that circle the earth. Lines of latitude are parallel to the equator. Lines of longitude are perpendicular to the lines of latitude. Thus all lines of longitude are parallel with each other. However, if you look at a globe with the major lines of latitude and longitude marked, you will see that all the lines of longitude intersect at the north and south poles.

The surface of a sphere does not conform to our intuitive notions about parallel lines. We call geometries that obey the parallel postulate Euclidean. Many important geometries are not Euclidean including the surface of our planet. General relativity defines the geometry of our universe in contemporary physics. It too is not Euclidean.

Mathematicians wanted to avoid making assumptions that are not universally true like the parallel postulate. To that end they removed any fundamental entities like lines, planes or points from the formulation of mathematics. They invented set theory. In set theory there is a single primitive entity, the empty set, and a single primitive relationship, set membership. The only objects are the empty set and things constructed from the empty set. For example the number one is the set containing the empty set. The number two is the set containing the number one and the empty set.

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