Download Axiomatizing Metatheory: A Fregean Perspective on Independence

Axiomatizing Metatheory: A Fregean Perspective
on Independence Proofs
Günther Eder
University of Vienna
Logic Colloquium, Helsinki
5/08/2015
Table of Contents
1 Hilbert’s ‘Foundations’
2 Frege on Hilbert’s ‘Foundations’
3 Frege’s Call for a New Science
4 What is the New Science?
5 Concluding Remarks
1 Hilbert’s ‘Foundations’
2 Frege on Hilbert’s ‘Foundations’
3 Frege’s Call for a New Science
4 What is the New Science?
5 Concluding Remarks
Hilbert’s The Foundations of Geometry
Hilbert’s The Foundations of Geometry (1899) is famous for its
Hilbert’s The Foundations of Geometry
Hilbert’s The Foundations of Geometry (1899) is famous for its
(1) emphasis on metatheoretical questions: Hilbert’s axiomatization
not just a foundation for Euclidean geometry, but mainly
designed to study questions like consistency and independence
of particular axioms / theorems.
Hilbert’s The Foundations of Geometry
Hilbert’s The Foundations of Geometry (1899) is famous for its
(1) emphasis on metatheoretical questions: Hilbert’s axiomatization
not just a foundation for Euclidean geometry, but mainly
designed to study questions like consistency and independence
of particular axioms / theorems.
(2) methodology: Metatheoretical questions are studied by
systematically considering various interpretations / models.
Hilbert’s Foundations of Geometry
The conceptual presuppositions that enable Hilbert’s semantic
consistency and independence proofs are that
Hilbert’s Foundations of Geometry
The conceptual presuppositions that enable Hilbert’s semantic
consistency and independence proofs are that
(1) the basic terms of an axiomatic theory can be variously
reinterpreted, they are understood schematically and are not
tied to their intended interpretations
Hilbert’s Foundations of Geometry
The conceptual presuppositions that enable Hilbert’s semantic
consistency and independence proofs are that
(1) the basic terms of an axiomatic theory can be variously
reinterpreted, they are understood schematically and are not
tied to their intended interpretations
(2) what matters, from a logical point of view, are the formal
relations between the basic terms, independently of their
intended interpretations
Hilbert’s Foundations of Geometry
• Hilbert’s independence and consistency results are presented
only informally
Hilbert’s Foundations of Geometry
• Hilbert’s independence and consistency results are presented
only informally
• No serious effort is made to explain the key notions of his
method (axioms being ‘fulfilled’ or ‘valid’ with respect to a
certain ‘system of things’)
Hilbert’s Foundations of Geometry
• Hilbert’s independence and consistency results are presented
only informally
• No serious effort is made to explain the key notions of his
method (axioms being ‘fulfilled’ or ‘valid’ with respect to a
certain ‘system of things’)
• There is evidence that Hilbert’s method isn’t even genuinely
‘semantic’, but more accurately described by the modern
notion of (syntactic) relative interpretability (see e.g. Hilbert
and Bernays 1934, p. 3)
1 Hilbert’s ‘Foundations’
2 Frege on Hilbert’s ‘Foundations’
3 Frege’s Call for a New Science
4 What is the New Science?
5 Concluding Remarks
Frege on Hilbert’s Foundations
• Frege, in a number of letters to Hilbert and a series of articles
published between 1903 and 1906, criticizes Hilbert’s
Foundations mainly for the perceived lack of clarity on
Hilbert’s part.
Frege on Hilbert’s Foundations
• Frege, in a number of letters to Hilbert and a series of articles
published between 1903 and 1906, criticizes Hilbert’s
Foundations mainly for the perceived lack of clarity on
Hilbert’s part.
• According to Frege
. . . we remain completely in the dark as to what he
[Hilbert] really believes he has proved and which logical
and extralogical laws and expedients he needs for this.
(Kluge 1971, p. 111)
Frege on Hilbert’s Foundations
Frege does provide a reconstruction of what he thinks Hilbert has in
mind (in his own terms) . . .
• Hilbert’s axioms are ‘really’ just second-level concepts
(‘propositional functions’)
• The basic terms in Hilbert’s axioms are ‘really’ just
second-order variables in disguise
• What is meant by satisfaction is ‘really’ just instantiation by a
sequence of meaningful terms
Frege on Hilbert’s Foundations
Frege does provide a reconstruction of what he thinks Hilbert has in
mind (in his own terms) . . .
• Hilbert’s axioms are ‘really’ just second-level concepts
(‘propositional functions’)
• The basic terms in Hilbert’s axioms are ‘really’ just
second-order variables in disguise
• What is meant by satisfaction is ‘really’ just instantiation by a
sequence of meaningful terms
. . . but dismisses it nonetheless.
1 Hilbert’s ‘Foundations’
2 Frege on Hilbert’s ‘Foundations’
3 Frege’s Call for a New Science
4 What is the New Science?
5 Concluding Remarks
Frege on Independence Proofs
• Frege makes clear that his worries with Hilbert’s proofs are
‘methodological’, not ‘mathematical’:
Now the question may still be raised whether, taking
Hilbert’s result as a starting point, we might not arrive at
a proof of the independence of the real axioms. (Kluge
1971, 103)
What are the objects of independence?
What I understand by independence in the realm of
thoughts may be clear from the following. I use the word
‘thought’ instead of ‘proposition’, since surely it is only
the thought-content that is relevant, and the former is
always present in the case of real propositions—and it is
only with these that we are here concerned. (Kluge 1971,
p. 103)
What are the objects of independence?
What I understand by independence in the realm of
thoughts may be clear from the following. I use the word
‘thought’ instead of ‘proposition’, since surely it is only
the thought-content that is relevant, and the former is
always present in the case of real propositions—and it is
only with these that we are here concerned. (Kluge 1971,
p. 103)
⇒ The objects of the New Science are Fregean thoughts!
How is independence specified?
Let Ω be a group of thoughts. Let a thought G follow from one or
several thoughts of this group by means of a logical inference such
that apart from the laws of logic, no proposition not belonging to Ω
is used. Let us now form a new group of thoughts by adding the
thought G to the group Ω. Call what we have just performed a
logical step. Now if through a sequence of such steps, where every
step takes the result of the preceding one as its basis, we can reach
a group of thoughts that contains the thought A, then we call A
dependent upon the group Ω. If this is not possible, then we call A
independent of Ω. (Kluge 1971, p. 104)
How is independence specified?
Let Ω be a group of thoughts. Let a thought G follow from one or
several thoughts of this group by means of a logical inference such
that apart from the laws of logic, no proposition not belonging to Ω
is used. Let us now form a new group of thoughts by adding the
thought G to the group Ω. Call what we have just performed a
logical step. Now if through a sequence of such steps, where every
step takes the result of the preceding one as its basis, we can reach
a group of thoughts that contains the thought A, then we call A
dependent upon the group Ω. If this is not possible, then we call A
independent of Ω. (Kluge 1971, p. 104)
⇒ Independence of thoughts is understood proof-theoretically!
How do we prove independence of
thoughts?
We now return to our question: Is it possible to prove the
independence of a real axiom from a group of real axioms? This
leads to the further question: How can one prove the independence
of a thought from a group of thoughts? First of all, it may be noted
that with this question we enter into a realm that is otherwise
foreign to mathematics. For although like all other disciplines
mathematics, too, is carried out in thoughts, still, thoughts are
otherwise not the object of its investigations. Even the
independence of a thought from a group of thoughts is quite
distinct from the relations otherwise investigated in mathematics.
Now we may assume that this new realm has its own specific, basic
truths which are as essential to the proofs constructed in it as the
axioms of geometry are to the proofs of geometry, and that we also
need these basic truths especially to prove the independence of a
thought from a group of thoughts. (Kluge 1971, p. 106)
How do we prove independence of
thoughts?
We now return to our question: Is it possible to prove the
independence of a real axiom from a group of real axioms? This
leads to the further question: How can one prove the independence
of a thought from a group of thoughts? First of all, it may be noted
that with this question we enter into a realm that is otherwise
foreign to mathematics. For although like all other disciplines
mathematics, too, is carried out in thoughts, still, thoughts are
otherwise not the object of its investigations. Even the
independence of a thought from a group of thoughts is quite
distinct from the relations otherwise investigated in mathematics.
Now we may assume that this new realm has its own specific, basic
truths which are as essential to the proofs constructed in it as the
axioms of geometry are to the proofs of geometry, and that we also
need these basic truths especially to prove the independence of a
thought from a group of thoughts. (Kluge 1971, p. 106)
⇒ Independence is to be proved in an axiomatic setting!
1 Hilbert’s ‘Foundations’
2 Frege on Hilbert’s ‘Foundations’
3 Frege’s Call for a New Science
4 What is the New Science?
5 Concluding Remarks
Axioms of the New Science
• Frege mentions three ‘basic laws’ for the New Science, two of
them are stated explicitly and a third one is elucidated
informally (Kluge 1971, pp. 107–110).
• The first two laws can be summarized by a single axiom:
(NS 1) If the thought G is provable from a group of true thoughts
Ω, then G is true
Axioms of the New Science
• Frege mentions three ‘basic laws’ for the New Science, two of
them are stated explicitly and a third one is elucidated
informally (Kluge 1971, pp. 107–110).
• The first two laws can be summarized by a single axiom:
(NS 1) If the thought G is provable from a group of true thoughts
Ω, then G is true
• The axiom provides a link between truth and proof which
Frege thinks is necessary for independence proofs.
Axioms of the New Science
• The final law, which Frege calls an ‘efflux of the formal nature
of logical laws’ (roughly) says that provability is invariant under
substitutions (‘translations’) of the non-logical vocabulary
Axioms of the New Science
• The final law, which Frege calls an ‘efflux of the formal nature
of logical laws’ (roughly) says that provability is invariant under
substitutions (‘translations’) of the non-logical vocabulary
• More specifically, suppose S and S t are propositions and Σ
and Σt conjunctions of sets of propositions such that S t (Σt )
results from S (Σ) by replacing non-logical terms by
non-logical terms of the same grammatical category, then:
(NS 2) If the thought expressed by S is provable from the group of
thoughts expressed by Σ, then the thought expressed by S t
is provable from the group of thoughts expressed by Σt
Independence Proofs in the New Science
• How do we show that the thought expressed by S is
independent of the group of thoughts expressed by the
conjunction Σ of a set of propositions?
Independence Proofs in the New Science
• How do we show that the thought expressed by S is
independent of the group of thoughts expressed by the
conjunction Σ of a set of propositions?
(1) Find a translation t such that the group of thoughts expressed by Σt
is true while the thought expressed by S t is false.
(2) Assume for reductio the thought expressed by S were provable from
the group of thoughts expressed by Σ.
(3) Then, by (NS 2), the thought expressed by S t would be provable
from the group of thoughts expressed by Σt .
(4) Then, by (NS 1) and the fact that the group of thoughts expressed
by Σt is true, the thought expressed by S t would be true as well.
(5) Therefore, the thought expressed by S is not provable from the group
of thoughts expressed by Σ
1 Hilbert’s ‘Foundations’
2 Frege on Hilbert’s ‘Foundations’
3 Frege’s Call for a New Science
4 What is the New Science?
5 Concluding Remarks
Problems with the New Science
There are obviously many problems related to Frege’s New Science:
Problems with the New Science
There are obviously many problems related to Frege’s New Science:
• What are Fregean thoughts (and senses more generally)? How
are thoughts individuated? How are they structured?
Problems with the New Science
There are obviously many problems related to Frege’s New Science:
• What are Fregean thoughts (and senses more generally)? How
are thoughts individuated? How are they structured?
• Frege’s approach seems to implicitly rely on axiomatic theories
of truth and provability, but what are these theories supposed
to look like?
Problems with the New Science
There are obviously many problems related to Frege’s New Science:
• What are Fregean thoughts (and senses more generally)? How
are thoughts individuated? How are they structured?
• Frege’s approach seems to implicitly rely on axiomatic theories
of truth and provability, but what are these theories supposed
to look like?
• In the case of truth there is a specific problem: How to avoid
the semantic paradoxes?
• ...
Frege’s Followers?
• Frege’s basic contention—that at least certain questions about
metatheoretical concepts are best studied in an axiomatic
setting—does seem to have some followers though:
• Axiomatic theories of truth have been mentioned already by
Tarski; axiomatic theories of (informal) provability by Gödel;
both are still studied extensively.
• An interesting question here would be to compare the
motivations and aims for axiomatizing such concepts with
Frege’s motivations and aims for his New Science
• ...
Does the New Science Refute the
‘No-Metatheory Reading’ ?
• Various scholars have claimed that Frege’s ‘universalist’
conception of logic somehow prevents him from engaging in
serious metatheoretical investigations.
Does the New Science Refute the
‘No-Metatheory Reading’ ?
• Various scholars have claimed that Frege’s ‘universalist’
conception of logic somehow prevents him from engaging in
serious metatheoretical investigations.
• So does the existence of Frege’s New Science simply refute
these interpretations?
Does the New Science Refute the
‘No-Metatheory Reading’ ?
• Various scholars have claimed that Frege’s ‘universalist’
conception of logic somehow prevents him from engaging in
serious metatheoretical investigations.
• So does the existence of Frege’s New Science simply refute
these interpretations?
• No. Frege’s New Science has little to do with what we today
usually understand by ‘metatheory’.
Does the New Science Refute the
‘No-Metatheory Reading’ ?
• Various scholars have claimed that Frege’s ‘universalist’
conception of logic somehow prevents him from engaging in
serious metatheoretical investigations.
• So does the existence of Frege’s New Science simply refute
these interpretations?
• No. Frege’s New Science has little to do with what we today
usually understand by ‘metatheory’.
• But that obviously does not show that Frege doesn’t, at other
occasions, engage in investigations we would subsume under
the label ‘metatheory’.
References
Hilbert, David 1899: Grundlagen der Geometrie. Leipzig:
Teubner Verlag (1923).
Hilbert, David and Bernays, Paul 1934: Die Grundlagen der
Mathematik. Berlin: Julius Springer Verlag.
Kluge, Eike-Henner W. (ed.) 1971: On the Foundations of
Geometry and Formal Theories of Arithmetic. New Haven and
London: Yale University Press.