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EventsI will detail transitions in proof theory between 1930 and 1934 - transitions, in which Hilbert and Gentzen were involved. The starting-point is Gödel's announcement of a restricted form of his first incompleteness theorem in Königsberg on 7 September 1930; the end-point is the draft of the first consistency proof for full arithmetic Gentzen completed in December of 1934. In the development between these two points, Hilbert's broad considerations in his last publication, Beweis des tertium non datur, seem to have played a significant role. Hilbert presented novel directions and concrete problems that needed to be addressed. Gentzen did resolve the problems in surprising ways, but fully in the spirit of Hilbert's view that true contentual thinking consists in operations on proofs. This still tentative picture has been made plausible by the outline of Gentzen's "urdissertation" that was found in the Bernays Nachlass by Jan von Plato.
I consider the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, in relation to recent criticisms of the “algebraic” approach to mathematical structuralism [Shapiro, 2005]. I first consider the Frege-Hilbert debate with the aim of distinguishing between axioms as assertions, i.e., as statements that are used to express or assert truths about a unique subject matter, and an axiom system as a schema that is used to provide “a system of conditions for what might be called a relational structure” (Bernays [1967], p. 497) so that axioms, as implicit definitions, are about whatever satisfies the conditions set forth. I then use this inquiry to rationally reconstruct those aspects of Hilbert’s “foundational” programme that can be drawn on to reevaluate arguments against using category theory to frame a structuralist philosophy of mathematics. Against the criticisms Shapiro, my aim is to show that category theory has as much to say about a pure algebraic consideration of meta-mathematical analyses of logical structure as it does about an algebraic consideration of mathematical analyses of mathematical structure, without either requiring an assertory mathematical or meta-mathematical background theory as a “foundation”, or turning meta-mathematical analyses of logical concepts into “philosophical” ones. Thus, we can use category theory to frame an interpretation of mathematics according to which we can be algebraic structuralists all the way down.
Godel showed the connection between the Lewis Modal System S4 and Intuitionistic Logic. Many have written about an algebraic formulation using as models topological spaces with the interior operator as necessitation, and the open subsets as modeling intuitionistic propositional logic. Topological models were also used to model many first-order logics. After Cohen's proofs were recast using Boolean-valued models, topological models for modal higher-order logic have been studied. For Boolean-valued logic, the complete Boolean algebra M = Meas([0,1])/Null of measurable subsets of [0, 1] modulo sets of measure zero gives every proposition a probability. Note that the measure algebra also carries a nontrivial S4 modality defined via the sublattice Open([0,1])/Null of open sets modulo null sets. Working by analogy to the Boolean-valued models for ZF, we construct over M a model for a modal ZF (MZF) where membership and equality predicates have interesting modal properties, and where real numbers correspond to random variables. Some theorems of Ergodic Theory then become principles about the proposed MZF.
A historical account of the development of Polish logic from its beginnings to the present situation. There will be ample opportunity to ask questions about this fascinating logic tradition, part of which will be familiar to the audience by the Tarski biography of Anita and Solomon Feferman.
Conceptual structuralism is a view of the nature of mathematics according to which mathematics emerges from humanly constructed, intersubjectively established, basic structural conceptions. Some of these are so clear that every problem posed in the language of the structure conceived can be said to have a definite truth value, whether or not that can be determined. Other conceptions may contain an inherently vague aspect, so that truth is only partial in them and not every problem can be asserted to have a definite truth value. In particular, this is behind my view that the Continuum Hypothesis is not a definite mathematical problem.
In A Logical Journey: From Gödel to Philosophy (Wang 1996, p. 164), Hao Wang writes that “Before 1959 Gödel had studied Plato, Leibniz, and Kant with care: his sympathies were with Plato and Leibniz. Yet he felt he needed to take Kant's critique of Leibniz seriously and find a way to meet Kant's objections to rationalism. He was not satisfied with Kant's dualism or with his restriction of intuition to sense intuition, which ruled out the possibility of intellectual or categorial intuition. It seems likely that, in the process of working on his Carnap paper in the 1950s, Gödel had realized that his realism about the conceptual world called for a more solid foundation than he then possessed. At this juncture [in 1959] it was not surprising for him to turn to Husserl's phenomenology, which promises a general framework for justifying certain fundamental beliefs that Gödel shared: realism about the conceptual world, the analogy of concepts and mathematical objects to physical objects, the possibility and importance of categorial intuition or immediate conceptual knowledge, and the one-sidedness of what Husserl call ‘the naive or natural standpoint’.”
In my talk I will elaborate on and develop some of these comments in detail, based on material in Wang's books on Gödel, my discussions with Wang in the nineteen eighties about Gödel's philosophical interests, and various items from the Gödel Nachlass. The talk is drawn from the first chapter of my book manuscript After Gödel: Platonism and Reason in Mathematics and Logic.
This talk investigates the status of arithmetical truths from a point of view that conjoins a non-reductionist version of logicism with a deflationary conception of abstraction. The resulting account accordingly combines two distinct tools: a non-standard (but still first-order) cardinality quantifier and an extra-logical operator representing numerical abstraction. The natural numbers are then characterized as abstracta of the equinumerosity relation, their properties derived from those of the cardinality quantifier in conjunction with the abstraction operator. As a result, it is the structural properties of the natural numbers are viewed as supervenient over the cardinal ones, rather than (as is more customary) the other way around.
Abstract: Historically, proof theory has its origin in Hilbert's foundational program. Building upon pioneering ideas of G. Kreisel, goeing back to the 50's, a new applied form of proof theory emerged during the last 20 year. Here the emphasis is on applications of so-called proof interpretations to concrete mathematical proofs with the aim of extracting effective bounds as well as new uniformity results from prima facie ineffective proofs. This has led to new results in number theory, approximation theory, nonlinear analysis, geodesic geometry and ergodic theory as well as the development of logical metatheorems that explain these results as instances of general logical phenomena. Specialized to the examples discussed in T. Tao's recent essay "Soft analysis, hard analysis, and the finite convergence principle" the logical machinery yields very much the type of quantitative finitary versions of analytical theorems as considered by Tao. We will argue that these logical methods provide a systematic approach to Tao's program of "hard analysis".
Abstract: The talk discusses some philosophical aspects of work in applied proof theory that has been carried out during the last 15-20 years inspired by G. Kreisel's program of `unwinding of proofs'. We will argue that this work sheds new light on concepts such as `constructive reasoning', `finitism',`ideal elements', `predicativism', `intensionality versus extensionality' among others and can help to reshape some traditional views on these topics towards a more realistic approach to the philosophy of mathematics.
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