I have recently read about the Frege-Hilbert controversy regarding the axioms of Euclidean Geometry. In Chihara's book it is written (under the subsection Hilbert's criterion of truth and existence) that,
Let us now consider Frege's objection to Hilbert's doctrine that, if a set of axioms is consistent, then the axioms are "true" and the things defined by the axioms exist. Frege submitted to Hilbert the following example of a set of axioms:
(A1) A is an intelligent being
(A2) A is omnipresent
(A3) A is omnipotent,
suggesting that if this set is consistent, then it should follow by Hilbert's doctrine that the axioms are true and that there exists a thing that is intelligent, omnipresent, and omnipotent. Frege clearly thought any such inference would be absurd, but he could not believe that Hilbert actually maintained any such implausible doctrine. So he asked Hilbert to clarify what he was espousing.
After Hilbert broke off the correspondence, Frege published a paper sharply criticizing Hilbert's views about the foundations of geometry, repeating many of the objections in his letters. In the essay, the gloves came off and Frege expressed his true attitude toward the above Hilbertian doctrine. This time, he set forth the following set of axioms:
EXPLANATION: We conceive of objects which we call gods.
AXIOM 1. Every god is omnipotent.
AXIOM 2. There is at least one god.
He then wrote: "If this were admissible, then the ontological proof for the existence of God would be brilliantly vindicated".
This objection again illustrates Frege's misunderstanding of Hilbert's views. Hilbert's axioms of geometry are not assertions about the real world. The terms occurring in Hilbert's axioms, such as 'point' and `line', are parameters, unlike the terms 'intelligent being', 'omnipresent', and 'omnipotent' occurring in Frege's examples. One would think that Hilbert could have pointed out such differences without much trouble, and in this way advanced the discussion considerably. But he didn't.
I have the following questions regarding the last paragraph.
"Hilbert's axioms of geometry are not assertions about the real world."-are the axioms of Frege "assertions about the real world"? If so, then how? If not then how does the rest of the paragraph make sense?
"The terms occurring in Hilbert's axioms, such as 'point' and `line', are parameters, unlike the terms 'intelligent being', 'omnipresent', and 'omnipotent' occurring in Frege's examples."-why the terms 'intelligent being', 'omnipresent', and 'omnipotent' occurring in Frege's examples are not parameters?