Consistency in Hilbert's Foundations of Geometry

Question

After reading the entry in the Stanford Encyclopedia of Philosophy on Hilbert and Frege’s correspondence regarding the former’s Foundations of Geometry, I am quite puzzled by a claim that is made by the author of the article on page 7. There, she writes that the following pair of sentences is “demonstrably consistent in Hilbert’s sense”:

• Point B lies on a line between points A and C
• Point B does not lie on a line between points C and A

I have not yet had the chance to read Hilbert’s original work on this subject, but it seems implausible to me that these two statements would not entail a contradiction in his formal system. It would be greatly appreciated if someone can clarify whether I am in fact mistaken about this.

Answer 1

One of the key points of the so-called Frege-Hilbert controversy was the use made by Hilbert of the method of "alternative interpretations" to prove consistency and independence results.

For Frege, a mathematical theory has a meaning: the intended interpretation, while for Hilbert a "formal system" must be developed without considering the intended interpretation.

"The central idea of the alternative interpretation is that for Frege, the question whether a given thought is logically entailed by a collection of thoughts is sensitive not just to the formal structure of the sentences used to express those thoughts, but also to the contents of the simple (e.g., geometric) terms that appear in those sentences."

In a nutshell: if "between" is not read as between, and it is treated as a binary relation whatever, without further axioms we are not entitled to assert that the relation is symmetric.

In formal terms, $R(A,C,B)$ and $¬R(A,C,B)$ are contradictory, while $R(A,C,B)$ and $¬R(C,A,B)$ are not.