A way is found to add axioms to sequent calculi that maintains the eliminability of cut, through the representation of axioms as rules of inference of a suitable form. By this method, the structural analysis of proofs is extended from pure logic to free-variable theories, covering all classical theories, and a wide class of constructive theories. All results are proved for systems in which also the rules of weakening and contraction can be eliminated. Applications include a system of predicate logic (...) with equality in which also cuts on the equality axioms are eliminated. (shrink)

This is the only book to chart the history and development of modern probability theory. It shows how in the first thirty years of this century probability theory became a mathematical science. The author also traces the development of probabilistic concepts and theories in statistical and quantum physics. There are chapters dealing with chance phenomena, as well as the main mathematical theories of today, together with their foundational and philosophical problems. Among the theorists whose work is treated at some length (...) are Kolmogorov, von Mises and de Finetti. The principal audience for the book comprises philosophers and historians of science, mathematicians concerned with probability and statistics, and physicists. The book will also interest anyone fascinated by twentieth-century scientific developments because the birth of modern probability is closely tied to the change from a determinist to an indeterminist world-view. (shrink)

This is the only book to chart the history and development of modern probability theory. It shows how in the first thirty years of this century probability theory became a mathematical science. The author also traces the development of probabilistic concepts and theories in statistical and quantum physics. There are chapters dealing with chance phenomena, as well as the main mathematical theories of today, together with their foundational and philosophical problems. Among the theorists whose work is treated at some length (...) are Kolmogorov, von Mises and de Finetti. The principal audience for the book comprises philosophers and historians of science, mathematicians concerned with probability and statistics, and physicists. The book will also interest anyone fascinated by twentieth-century scientific developments because the birth of modern probability is closely tied to the change from a determinist to an indeterminist world-view. (shrink)

De Finetti's representation theorem is a special case of the ergodic decomposition of stationary probability measures. The problems of the interpretation of probabilities centred around de Finetti's theorem are extended to this more general situation. The ergodic decomposition theorem has a physical background in the ergodic theory of dynamical systems. Thereby the interpretations of probabilities in the cases of de Finetti's theorem and its generalization and in ergodic theory are systematically connected to each other.

Some of our earliest experiences of the conclusive force of an argument come from school mathematics: faced with a mathematical proof, we cannot deny the conclusion once the premises have been accepted. Behind such arguments lies a more general pattern of 'demonstrative arguments' that is studied in the science of logic. Logical reasoning is applied at all levels, from everyday life to advanced sciences, and a remarkable level of complexity is achieved in everyday logical reasoning, even if the principles behind (...) it remain intuitive. Jan von Plato provides an accessible but rigorous introduction to an important aspect of contemporary logic: its deductive machinery. He shows that when the forms of logical reasoning are analysed, it turns out that a limited set of first principles can represent any logical argument. His book will be valuable for students of logic, mathematics and computer science. (shrink)

Bruno de Finetti's earliest works on the foundations of probability are reviewed. These include the notion of exchangeability and the theory of random processes with independent increments. The latter theory relates to de Finetti's ideas for a probabilistic science more generally. Different aspects of his work are united by his foundational programme for a theory of subjective probabilities.

Three things are presented: How Hilbert changed the original construction postulates of his geometry into existential axioms; In what sense he formalized geometry; How elementary geometry is formalized to present day's standards.

Frege explained the notion of generality by stating that each its instance is a fact, and added only later the crucial observation that a generality can be inferred from an arbitrary instance. The reception of Frege’s quantifiers was a fifty-year struggle over a conceptual priority: truth or provability. With the former as the basic notion, generality had to be faced as an infinite collection of facts, whereas with the latter, generality was based on a uniformity with a finitary sense: the (...) provability of an arbitrary instance. (shrink)

Husserl’s Philosophie der Arithmetik came out in 1891. In this short essay, its place at the crossroads of two traditions in the philosophy and foundations of arithmetic is described; what preceded it and could thus have influenced Husserl. A brief look at what came later is also taken; where the choices at the crossroads led to.