Hilbert and Euclid
How to work with concepts, using an example from the history of mathematics.
| Euclid, Elements, around 300 BCE | David Hilbert, Foundations of Geometry, 1899 |
|---|---|
| A point is that which has no part. | Points are objects denoted by letters A, B, … |
| A line is breadthless length. | Lines are objects denoted by letters a, b, … There is a relation “point X lies on line x”; it can be true or false. |
| Through two different points one can always draw exactly one straight line. | If two points A and B are distinct, and lines a and b are such that A and B lie on both of them, then these are not two lines, but one line. |
Notice the difference in thinking.
“Whole - parts” and “length - breadth” are concepts of excursus. In Euclid, they are an essential part of the construction. In Hilbert, they are absent altogether.
Euclid can work with points without lines. Hilbert cannot.
I commit myself to building my course according to Hilbert.
This justifies the first and second rules of discourse: hold the term and introduce concepts in tuples.