Euclid was the dude who gave us (Euclidean) geometry.
He included the postulate below.
Given any straight line and a point not on it, there “exists one and only one straight line which passes” through that point and never never intersects the first line, no matter how far they are extended.
Well, this was later replaced with the assumption that more than one parallel can be drawn to a given line through a given point. One could also make the assumption no parallels can be drawn thusly. This led to a new type of geometry.
It was after this shift in thought that mathematics was recognized to be much more abstract than traditionally supposed:
- Because math statements can be construed in principle to be about anything, rather than some inherently circumscribed set of objects or traits of objects.
- Because the validity of math statements is grounded in the structure of statements rather than in the nature of a particular subject matter.
- Because any special meaning that may be associated with the terms in the postulates plays no essential role in deriving the theorems.
*Clumsily articulated from readings by Douglas Hofstadter as well as Roger Penrose
WrittenCasey 