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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.