Wittgenstein on the Foundations of Mathematics

In Part I, an attempt is made to survey the original source material on which any detailed assessment of Wittgenstein's remarks on the foundations of mathematics from his middle and later periods ought to be based. This survey is presented within the context of a sketch of Wittgenstein's biography, which also mentions some of the major developments in his thinking. In addition, certain main themes are emphasized; these have to do primarily with the Kantian aspects of Wittgenstein's thought and with his mysticism or the 'religious point of view'. In Part II, Kreisel's critique of Wittgenstein's remarks on the foundations of mathematics, which has been developed since 1958 in a series of published articles, receives close examination, and, in connection with this, different approaches to the philosophical investigation of mathematics are considered which represent genuine alternatives to Wittgenstein's approach. There are separate sections on Lakatos's Proofs and Refutations and Bourbaki's 'L'Architecture des Mathématiques'. Finally, besides a bibliography which surveys the reception of Wittgenstein's views on the foundations of mathematics, there are two substantial appendices, which are supplemental to Part I. The first of these gives the manuscript sources for typescripts 221 and 222-4, and the correspondences in both directions between these typescripts. The second appendix is part of a chronological version of von Wright's catalogue of Wittgenstein's papers, beginning in 1929

A source modelling system and its use for uncertainty management

Human agents have to deal with a considerable amount of information from their environment and are also continuously faced with the need to take actions. As that information is largely of an uncertain nature, human agents have to decide whether, or how much, to believe individual pieces of information. To enable a reasoning system to deal in general with the demands of a real environment, and with information from human sources in particular, requires tools for uncertainty management and belief formation. This thesis presents a model for the management of uncertain information from human sources. Dealing, more specifically, with information which has been pre-processed by a natural language processor and transformed into an event-based representation, the model assesses information, forms beliefs and resolves conflicts between them in order to maintain a consistent world model. The approach is built on the fundamental principle that the uncertainty of information from people can, in the majority of situations, successfully be assessed through source models which record factors concerning the source's abilities and trustworthiness. These models are adjusted to reflect changes in the behaviour of the source. A mechanism is presented together with the underlying principles to reproduce such a behaviour. A high-level design is also given to make the proposed model reconstructible, and the successful operation of the model is demonstrated on two detailed examples

Metalevel and reflexive extension in mechanical theorem proving

In spite of many years of research into mechanical assistance for mathematics it is still much more difficult to construct a proof on a machine than on paper. Of course this is partly because, unlike a proof on paper, a machine checked proof must be formal in the strictest sense of that word, but it is also because usually the ways of going about building proofs on a machine are limited compared to what a mathematician is used to. This thesis looks at some possible extensions to the range of tools available on a machine that might lend a user more flexibility in proving theorems, complementing whatever is already available.In particular, it examines what is possible in a framework theorem prover. Such a system, if it is configured to prove theorems in a particular logic T, must have a formal description of the proof theory of T written in the framework theory F of the system. So it should be possible to use whatever facilities are available in F not only to prove theorems of T, but also theorems about T that can then be used in their turn to aid the user in building theorems of T.The thesis is divided into three parts. The first describes the theory FS₀, which has been suggested by Feferman as a candidate for a framework theory suitable for doing meta-theory. The second describes some experiments with FS₀, proving meta-theorems. The third describes an experiment in extending the theory PRA, declared in FS₀, with a reflection facility.More precisely, in the second section three theories are formalised: propositional logic, sorted predicate logic, and the lambda calculus (with a deBruijn style binding). For the first two the deduction theorem and the prenex normal form theorem are respectively proven. For the third, a relational definition of beta-reduction is replaced with an explicit function.In the third section, a method is proposed for avoiding the work involved in building a full Godel style proof predicate for a theory. It is suggested that the language be extended with quotation and substitution facilities directly, instead of providing them as definitional extensions. With this, it is possible to exploit an observation of Solovay's that the Lob derivability conditions are sufficient to capture the schematic behaviour of a proof predicate. Combining this with a reflection schema is enough to produce a non-conservative extension of PRA, and this is demonstrated by some experiments

Symmetry and complexity in propositional reasoning

We establish computational complexity results for a number of simple problem formulations connecting group action and prepositional formulas. The results are discussed in the context of complexity results arising from established work in the area of automated reasoning techniques which exploit symmetry

Formal verification of mathematical software

Methods are investigated for formally specifying and verifying the correctness of mathematical software (software which uses floating point numbers and arithmetic). Previous work in the field was reviewed. A new model of floating point arithmetic called the asymptotic paradigm was developed and formalized. Two different conceptual approaches to program verification, the classical Verification Condition approach and the more recently developed Programming Logic approach, were adapted to use the asymptotic paradigm. These approaches were then used to verify several programs; the programs chosen were simplified versions of actual mathematical software

The Bayes mind: From the St. Petersburg paradox to the New York Stock Exchange.

The thesis is an exposition and defence of Bayesianism as the preferred methodology of reasoning under uncertainty in social contexts. Chapter 1 gives a general outline of the foundations of probabilistic reasoning, as well as a critical exposition of the main non- Bayesian approaches to probability. After a brief discussion of the formal theory of probability, the thesis examines some non-Bayesian interpretations of the probability calculus, and purports to show their insufficiency. Chapter 2 provides an outline of the Bayesian (subjectivist) research programme. The opening sections of the chapter contain a historical overview of Bayesianism, as well as a defence of the assumptions on which it rests. The concluding sections then examine some of the key issues of contention between Bayesians and their critics, such as the nature of empirical confirmation and learning from experience. Since it is the author's contention that any sound methodology should be applicable, if necessary with modifications, across a wide range of contexts, the concluding two chapters make a Bayesian case first in a theoretical, and next in a practical setting. In particular, Chapter 3 discusses the issue of simplicity as a theoretical virtue. It argues that a Bayesian can coherently and successfully account for the structural or formal simplicity of the hypotheses that he entertains, by using his assignments of subjective prior probabilities in the process known as 'Bayesian Conditionalisation'. It also argues that some of the recent criticisms voiced against the Bayesian account of simplicity are inconsistent and/or question-begging. The process known in statistics as 'curve-fitting' provides the material for the discussion. Finally, Chapter 4 presents an extension of the Bayesian methodology to practical decision-making, using the context of investing activity. It purports to show that the most convincing picture of economic agents' investing behaviour is best explained by assuming that in the course of such behaviour, the agents maximise their expected utility, as is stipulated by the Bayesian decision theory. The argument revolves around the 'Efficient Markets Hypothesis' in the theory of finance, and the conclusion hinges both on the empirical adequacy of the various versions of this hypothesis and on its behavioural underpinnings. The thesis contains two appendices, intended to illustrate certain points made in the main body. The first appendix is a critical appraisal of a popular non-Bayesian account of causal inference in statistical contexts, with a bearing on the discussion in Chapter 3, while the second appendix provides a real-life illustration of some of the issues raised in Chapter 4. The overall structure of the thesis is intended to show how, from highly plausible assumptions, one can derive a powerful theory of reasoning under uncertainty that faithfully and uniformly represents both the theoretical and the practical concerns of the human mind

Space programs summary no. 37-62, volume 3 for the period 1 February - 31 March 1970. Supporting research and advanced development

Spacecraft communication, equipment, guidance, construction materials, propulsion, and related researc