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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?

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  • But which is "Chihara's book" mentioned ? – Mauro ALLEGRANZA Oct 24 '15 at 13:59
  • @MauroALLEGRANZA: A Structural Account of Mathematics. – user13627 Oct 25 '15 at 14:53
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For Frege, like in Russell's view before the developmen of the Type thoery as a solution to the paradoxes, the "underlying" philosophy was that logic was a sort of "science of everything", i.e. a set of laws so general to be applicable to absolutely "everything".

In the formalized language, e.g. "individual" variables of what we called today predicate logic must range on every object whatever : human beings, numbers, etc.

When we "instantiate" the variable with "names" of objects and predicates we get true or false statement regarding the "world" in a broad sense.

Thus, considering Frege's criticism of Hilbert's axiomatic approach, having produced an axiomatic theory that is consistent (the two-axioms theory of "God") and if Hilbert's view according to which every consistent theory must have a model, we are forced to conclude that "God exists".

The fault of Frege's counter-example is due to the misunderstanding of Hilbert's approach.

For Hilbert, the fact that e.g. being the Euclidean geometry consistent, it has a model, does not license us to conclude that the physical world we live in is a model of it, so if we agree on the fact that the above theory being consistent implies the existence of a model does not license us to assume that in the world we live in there is an "object" (call it : "God") that satisfy the axioms of the theory.

From a purely logical point of view, the two axioms amount to :

Ax.1 : for all x, if G(x), then O(x)

Ax.2 : there is at least one x such that G(x).

The axioms of Hilbert's geometry are expressed with terms of natural language with an "intended" interpretation, but we work with it in purely logical manner, avoidung to use what we know about the interpretation. I.e. Hilbert uses them as "variables"; when he proves the meta-theoretical properties of the theory, like consistency and independence, he intepret the terms and the axioms in a different "environment" (e.g. the real number of analytical geometry) and this is enough to conclude with e.g. consistency (provided that the math theory of real number is consistent).

For Frege, instead, a formal theory must be expressed in "symbolic" form, and when we use natural language terms we are already "instantiating" the formulae of the formal language interpreting them into our (unique existsing) "world", made of objects (women, men, animals), concepts, functions, numbers, truth-values but (maybe) not gods.

Simplifying the issue, we can say that both have a clear understanding of the syntax-semantics distinction.

Consistency is a syntactical property, holding of a formal theory when we cannor derive (by means of logic alone) from the axioms of the theory a formula like A and not-A.

To be true is a semantical property holding when we interpret the terms of the theory in a "world" and we get interpreted axioms and theorems that express true facts of that "world".

For Frege, there is only one world : "the world". The fact that a concept like round-square is inconsistent, it is enough to conclude with the non existence of a round-square in the world.

The fact that we can (presumibely) produce a consistent theory about the "King of U.S." does not imply that Pres.Obama is a king. This amounts to the denial from Frege to agree that a consistent theory must have a model, because he "read" it as : "a consistent theory, when correctly interpreted, must be true in our world" (i.e. in "the world").

But this is not the point of view of Hilbert; for him :

  • a theory, even if expressed in natural language, can have multiple interpretations; and

  • a consistent theory must be "instantiated" by some model, not necessarily the "intended" one, i.e. the existing model can be obtained interpreting the terms in a different way from what they means "originally" (see geometry).

For details, and for a (surely more clear) exposition, see :

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  • "For Hilbert, the fact that e.g. being the Euclidean geometry consistent, it has a model, does not license us to conclude that the physical world we live in is a model of it..."-isn't the situation same in Frege's argument? Frege only (if I have understood it right) claims the existence of a model, just like Hilbert. It seems to me that to both of them it is immaterial whether the model is a "physical" one or not. So, where is the difference? – user13627 Oct 24 '15 at 12:30
  • @The difference is the above axioms are satisfied by an interpretation of G as the predicate x=0 and O as the predicate x≥0. Thus the axioms : if x=0, then x≥0 and there is at least one x such that x=0 are clearly true in our world with the natural numbers, and this is enough for Hilbert to conclude that the theory is consistent. But the conclusion that 0 is God is not licensed neither for Hilbert nor for Frege. The difference between the two is that Hilbert (according to F) is not licensed to "reinterpret" the term God (nor the term : point). – Mauro ALLEGRANZA Oct 24 '15 at 12:38
  • But what actually Hilbert does is not proving the consistency of his system. He simply shows that a model exist satisfying his axioms and his system is "consistent" iff the later is. The same argument can be repeated to the later system and so on. So, does Hilbert actually prove the "consistency" of his system? – user13627 Oct 24 '15 at 12:44
  • Also, the "reinterpretation" technique works well in any "world" if you are indeed eligible to "reinterpret". But how do we know that we are able to "reinterpret"? – user13627 Oct 24 '15 at 12:51
  • But the reference you have given doesn't give an explanation of the questions I have asked in the previous comments (especially in the last two comments). – user13627 Oct 26 '15 at 3:36
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First question:

The axioms of geometry are not assertions about the real world

This is, on the whole correct; I take that formalism is associated with Platonism, so that mathematical truths exist in some ideal world; when we use the truths of geometry about the real world then we are adapting it there.

Second question:

the terms occurring in Hilbetts axioms ... are parameters

I'm familiar with Hilberts formalism on mathematics, and how it's been useful there to open a space for mathematical objects though they are tied organically to the whole body of mathematical thought.

It's never occurred to me that the objects that formalism could consider are outside of mathematics; but one person who did think that, apart from Frege, where he posited all logically consistent worlds - by a principle of plenitude - actually do exist; this is his notion of Plural Worlds; however it's worth noting when he announced this to his fellow philosophers there were many incredulous eyebrows raised but they let him get on with writing what he was thinking.

In his plural worlds, there could be a world a god actually exists, in so far the traditional attributes of a god - omniscience, omnipresence etc - are logically consistent; and one might ask if there is a god in one world, is then there a god of all the worlds ...

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  • Please note that my first question asks about the axioms of Frege and not of Hilbert. I wanted to know that if Frege's axioms are not necessarily about the real world (like Hilbert's axioms) then how can we say that Frege has misunderstood Hilbert's axioms from the examples that Frege has given? – user13627 Oct 24 '15 at 6:33

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