Putnam famously attempted to use model theory to draw metaphysical conclusions. His Skolemisation argument sought to show metaphysical realists that their favourite theories have countable models. His permutation argument sought to show that they have permuted models. His constructivisation argument sought to show that any empirical evidence is compatible with the Axiom of Constructibility. Here, I examine the metamathematics of all three model-theoretic arguments, and I argue against Bays (2001, 2007) that Putnam is largely immune to metamathematical challenges.
The metamathematical theorems of Gödel and Church are frequently applied to the philosophy of mind, typically as rational evidence against mechanism. Using methods of Post and Smullyan, these results are presented as purely mathematical theorems and various such applications are discussed critically. In particular, J. Lucas's use of Gödel's theorem to distinguish between conscious and unconscious beings is refuted, while more generally, attempts to extract philosophy from metamathematics are shown to involve only dramatizations of the constructivity problem in foundations. (...) More specifically, philosophical extrapolations from metamathematics are shown to involve premature extensions of Church's thesis. (shrink)
We discuss the development of metamathematics in the Hilbert school, and Hilbert’s proof-theoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we show how these considerations help frame our understanding of metamathematics and proof theory today.
This work is a sequel to the author's Godel's Incompleteness Theorems, though it can be read independently by anyone familiar with Godel's incompleteness theorem for Peano arithmetic. The book deals mainly with those aspects of recursion theory that have applications to the metamathematics of incompleteness, undecidability, and related topics. It is both an introduction to the theory and a presentation of new results in the field.
This paper is an attempt to bring together two separated areas of research: classical mathematics and metamathematics on the one side, non-monotonic reasoning on the other. This is done by simulating nonmonotonic logic through antitonic theory extensions. In the first half, the specific extension procedure proposed here is motivated informally, partly in comparison with some well-known non-monotonic formalisms. Operators V and, more generally, U are obtained which have some plausibility when viewed as giving nonmonotonic theory extensions. In the second (...) half, these operators are treated from a mathematical and metamathematical point of view. Here an important role is played by U -closed theories and U -fixed points. The last section contains results on V-closed theories which are specific for V. (shrink)
In this paper, we trace the conceptual history of the term ?metamathematics? in the nineteenth century. It is well known that Hilbert introduced the term for his proof-theoretic enterprise in about 1922. But he was verifiably inspired by an earlier usage of the phrase in the 1870s. After outlining Hilbert's understanding of the term, we will explore the lines of inducement and elucidate the different meanings of ?metamathematics? in the final decades of the nineteenth century. Finally, we will (...) investigate the earlier occurrences and come to the conclusion that the conceptual prehistory of the Hilbertian notion of metamathematics dates back to 1870, whereas the history of the word starts in 1799 at the latest. What is this: metamathematics? It is something amazing everybody, [?] since it makes the mind dizzy and withdraws thinking its sole fulcrum.1 1 Brunner 1898, p. 832: ?Was ist das: Metamathematik? Es ist etwas, was einen Jeden in das äusserste Staunen versetzt, [?] was das Gehirn schwindlig macht und dem Denken seinen einzigen Stütz- und Angelpunkt zu entziehen droht? (shrink)
Turing's programme, the idea that intelligence can be modelled computationally, is set in the context of a parallel between certain elements from metamathematics and Popper's schema for the evolution of knowledge. The parallel is developed at both the formal level, where it hinges on the recursive structuring of Popper's schema, and at the contentual level, where a few key issues common to both epistemology and metamathematics are briefly discussed. In light of this connection Popper's principle of transference, akin (...) to Turing's belief in the relevance of the theory of computation for modelling psychological functions, is widened into the extended principle of transference. Thus Turing's programme gains a solid epistemological footing. *I am grateful to Claude Lamontagne and Jean-Pierre Delage for their comments on this paper. (shrink)
Pursuing the proof-theoretic program of Friedman and Simpson, we begin the study of the metamathematics of countable linear orderings by proving two main results. Over the weak base system consisting of arithmetic comprehension, II 1 1 -CA0 is equivalent to Hausdorff's theorem concerning the canonical decomposition of countable linear orderings into a sum over a dense or singleton set of scattered linear orderings. Over the same base system, ATR0 is equivalent to a version of the Continuum Hypothesis for linear (...) orderings, which states that every countable linear ordering either has countably many or continuum many Dedekind cuts. (shrink)
The automatic verification of large parts of mathematics has been an aim of many mathematicians from Leibniz to Hilbert. While Gödel's first incompleteness theorem showed that no computer program could automatically prove certain true theorems in mathematics, the advent of electronic computers and sophisticated software means in practice there are many quite effective systems for automated reasoning that can be used for checking mathematical proofs. This book describes the use of a computer program to check the proofs of several celebrated (...) theorems in metamathematics including those of Gödel and Church-Rosser. The computer verification using the Boyer-Moore theorem prover yields precise and rigorous proofs of these difficult theorems. It also demonstrates the range and power of automated proof checking technology. The mechanization of metamathematics itself has important implications for automated reasoning, because metatheorems can be applied as labor-saving devices to simplify proof construction. (shrink)
The classical Ulm theory provides a complete set of invariants for countable abelian p-groups, and hence also for countable torsion abelian groups. These invariants involve countable ordinals. One can read off many simple structural properties of such groups directly from the Ulm theory. We carry out a reverse mathematics analysis of several such properties. In many cases, we reverse to ATR0, thereby demonstrating a kind of necessary use of Ulm theory.
A number of comparability theorems have been investigated from the viewpoint of reverse mathematics. Among these are various comparability theorems between countable well orderings (,), and between closed sets in metric spaces (,). Here we investigate the reverse mathematics of a comparability theorem for countable metric spaces, countable linear orderings, and sets of rationals. The previous work on closed sets used a strengthened notion of continuous embedding. The usual weaker notion of continuous embedding is used here. As a byproduct, we (...) sharpen previous results of ,. (shrink)
A prized property of theories of all kinds is that of generality, of applicability or least relevance to a wide range of circumstances and situations. The purpose of this article is to present a pair of distinctions that suggest that three kinds of generality are to be found in mathematics and logics, not only at some particular period but especially in developments that take place over time: ‘omnipresent’ and ‘multipresent’ theories, and ‘ubiquitous’ notions that form dependent parts, or moments, of (...) theories. The category of ‘facets’ is also introduced, primarily to assess the roles of diagrams and notations in these two disciplines. Various consequences are explored, starting with means of developing applied mathematics, and then reconsidering several established ways of elaborating or appraising theories, such as analogising, revolutions, abstraction, unification, reduction and axiomatisation. The influence of theories already in place upon theory-building is emphasised. The roles in both mathematics and logics of set theory, abstract algebras, metamathematics, and model theory are assessed, along with the different relationships between the two disciplines adopted in algebraic logic and in mathematical logic. Finally, the issue of monism versus pluralism in these two disciplines is rehearsed, and some suggestions are made about the special character of mathematical and logical knowledge, and also the differences between them. Since the article is basically an exercise in historiography, historical examples and case studies are described or noted throughout. (shrink)