In recent years, several remarkable results have shown that certain theorems of finite combinatorics are unprovable in certain logical systems. These developments have been instrumental in stimulating research in both areas, with the interface between logic and combinatorics being especially important because of its relation to crucial issues in the foundations of mathematics which were raised by the work of Kurt Godel. Because of the diversity of the lines of research that have begun to shed light on these issues, there (...) was a need for a comprehensive overview which would tie the lines together. This volume fills that need by presenting a balanced mixture of high quality expository and research articles that were presented at the August 1985 AMS-IMS-SIAM Joint Summer Research Conference, held at Humboldt State University in Arcata, California.With an introductory survey to put the works into an appropriate context, the collection consists of papers dealing with various aspects of 'unprovable theorems and fast-growing functions'. Among the topics addressed are: ordinal notations, the dynamical systems approach to Ramsey theory, Hindman's finite sums theorem and related ultrafilters, well quasiordering theory, uncountable combinatorics, nonstandard models of set theory, and a length-of-proof analysis of Godel's incompleteness theorem. Many of the articles bring the reader to the frontiers of research in this area, and most assume familiarity with combinatorics and/or mathematical logic only at the senior undergraduate or first-year graduate level. (shrink)

In this paper I develop a theory of 'function' and function as a deontic modal word and phenomenon. Kratzer’s account of the semantics for the deontic modals is invoked and using her approach a formal schema for the semantics of 'function'-sentences is proposed. My account of function is a modalized and extended version of Cummins’ systems-type account of function. In the biological and physical sciences, on this account, function is a complex empirical deontic modal property. It is built on the (...) property of X’s doing Y well enough to enable Z, which is implicitly deontic because of the evaluative but nonetheless empirical in its biological and physical applications. This account of function resolves many of the traditional puzzles about biological and physical function, and extends naturally to include the other types of function. With a variant treatment of the semantics, this account is argued also to apply to mathematical function, where it is shown to be interestingly related to Frege’s account of function. (shrink)

I explore physics implications of the External Reality Hypothesis (ERH) that there exists an external physical reality completely independent of us humans. I argue that with a sufficiently broad definition of mathematics, it implies the Mathematical Universe Hypothesis (MUH) that our physical world is an abstract mathematical structure. I discuss various implications of the ERH and MUH, ranging from standard physics topics like symmetries, irreducible representations, units, free parameters, randomness and initial conditions to broader issues like consciousness, (...) parallel universes and Gödel incompleteness. I hypothesize that only computable and decidable (in Gödel’s sense) structures exist, which alleviates the cosmological measure problem and may help explain why our physical laws appear so simple. I also comment on the intimate relation between mathematical structures, computations, simulations and physical systems. (shrink)

This is a discussion of some themes in Max Tegmark’s recent book, Our Mathematical Universe. It was written as a review for Plus Magazine, the online magazine of the UK’s national mathematics education and outreach project, the Mathematics Millennium Project. Since some of the discussion---about symmetry breaking, and Pythagoreanism in the philosophy of mathematics---went beyond reviewing Tegmark’s book, the material was divided into three online articles. This version combines those three articles, and adds some other material, in particular (...) a brief defence of quidditism about properties. It also adds some references, to other Plus articles as well as academic articles. But it retains the informal style of Plus. (shrink)

I discuss some problems related to extreme mathematical realism, focusing on a recently proposed “shut-up-and-calculate” approach to physics. I offer arguments for a moderate alternative, the essence of which lies in the acceptance that mathematics is a human construction, and discuss concrete consequences of this—at first sight purely philosophical—difference in point of view.

The maturation of the physical image has made apparent the limits of our scientific understanding of fundamental reality. These limitations serve as motivation for a new form of metaphysical inquiry that restricts itself to broadly scientific methods. Contributing towards this goal we combine the mathematical universe hypothesis as developed by Max Tegmark with the axioms of Stewart Shapiro’s structure theory. The result is a theory we call the Theory of the Structural Multiverse (TSM). The focus is on informal (...) theory development and constraint satisfaction. Some empirical consequences of the theory are worked out, in particular the feasibility of a predictive observer selection effect. The explanatory, unifying, and generative powers of the theory are found to substantially support the theory. The TSM is demonstrated to be an empirically significant scientific theory that is foundational to and continuous with the rest of the scientific image. (shrink)

"One of Michael Blay's many fine achievements in Reasoning with the Infinite is to make us realize how velocity, and later instantaneous velocity, came to play a vital part in the development of a rigorous mathematical science of motion. ...

Until the Scientific Revolution, the nature and motions of heavenly objects were mysterious and unpredictable. The Scientific Revolution was revolutionary in part because it saw the advent of many mathematical tools—chief among them the calculus—that natural philosophers could use to explain and predict these cosmic motions. Michel Blay traces the origins of this mathematization of the world, from Galileo to Newton and Laplace, and considers the profound philosophical consequences of submitting the infinite to rational analysis. "One of Michael Blay's (...) many fine achievements in _Reasoning with the Infinite_ is to make us realize how velocity, and later instantaneous velocity, came to play a vital part in the development of a rigorous mathematical science of motion."—Margaret Wertheim, _New Scientist_. (shrink)

Until the Scientific Revolution, the nature and motions of heavenly objects were mysterious and unpredictable. The Scientific Revolution was revolutionary in part because it saw the advent of many mathematical tools—chief among them the calculus—that natural philosophers could use to explain and predict these cosmic motions. Michel Blay traces the origins of this mathematization of the world, from Galileo to Newton and Laplace, and considers the profound philosophical consequences of submitting the infinite to rational analysis. "One of Michael Blay's (...) many fine achievements in _Reasoning with the Infinite_ is to make us realize how velocity, and later instantaneous velocity, came to play a vital part in the development of a rigorous mathematical science of motion."—Margaret Wertheim, _New Scientist_. (shrink)

This paper deals with universes in explicit mathematics. After introducing some basic definitions, the limit axiom and possible ordering principles for universes are discussed. Later, we turn to least universes, strictness and name induction. Special emphasis is put on theories for explicit mathematics with universes which are proof-theoretically equivalent to Feferman's.

I present an argument that for any computer-simulated civilization we design, the mathematical knowledge recorded by that civilization has one of two limitations. It is untrustworthy, or it is weaker than our own mathematical knowledge. This is paradoxical because it seems that nothing prevents us from building in all sorts of advantages for the inhabitants of said simulation.

This paper is about two topics: 1. systems of explicit mathematics with universes and a non-constructive quantification operator $\mu$; 2. iterated fixed point theories with ordinals. We give a proof-theoretic treatment of both families of theories; in particular, ordinal theories are used to get upper bounds for explicit theories with finitely many universes.

Mathematical and philosophical Newton Content Type Journal Article Pages 1-10 DOI 10.1007/s11016-010-9520-2 Authors Steffen Ducheyne, Centre for Logic and Philosophy of Science, Ghent University, Blandijnberg 2, 9000 Ghent, Belgium Journal Metascience Online ISSN 1467-9981 Print ISSN 0815-0796.

Sequential decision theory formally solves the problem of rational agents in uncertain worlds if the true environmental prior probability distribution is known. Solomonoff's theory of universal induction formally solves the problem of sequence prediction for unknown prior distribution. We combine both ideas and get a parameter-free theory of universal Artificial Intelligence. We give strong arguments that the resulting AIXI model is the most intelligent unbiased agent possible. We outline how the AIXI model can formally solve a number of problem classes, (...) including sequence prediction, strategic games, function minimization, reinforcement and supervised learning. The major drawback of the AIXI model is that it is uncomputable. To overcome this problem, we construct a modified algorithm AIXItl that is still effectively more intelligent than any other time t and length l bounded agent. The computation time of AIXItl is of the order t x 2^l. The discussion includes formal definitions of intelligence order relations, the horizon problem and relations of the AIXI theory to other AI approaches. (shrink)

Andrew Marr has built this masterful study of Mutio Oddi on a set of ironies. He begins with a bitter blow of fortune: Oddi, in the middle of an apparently promising life as mathematician and architect in his native Urbino, had fallen afoul of his lord the Duke, accused of participating in a plot to depose him. After years of apparently unjust imprisonment, he was released in 1610, but into exile. Yet Oddi managed to recast his career in Milan and (...) then in Lucca, building upon a varied set of skills, and even returned eventually to Urbino. That varied set of skills had resulted from yet another, earlier, set of reversals and recoveries: he had turned to mathematics only after first training as an artist in the studio of Federico Barocci, a field he had been forced to abandon due to problems with his eyesight. Oddi was widely respected in his day, not only for his achievements themselves but also for his persistence and ingenuity in overcoming such obstacles. Yet to modern historians of .. (shrink)

In Plutarch’s Quæstiones Conviviales there is a discussion on the topic—π.

(I) MATHEMATICAL LECTURES. LECTURE I. Of the Name and general Division of the Mathematical Sciences. BEING about to treat upon the Mathematical Sciences, ...

In Metaphysics M.2, 1077a9-14, Aristotle appears to argue against the existence of Platonic Forms on the basis of there being certain universal mathematical proofs which are about things that are ‘beyond’ the ordinary objects of mathematics and that cannot be identified with any of these. It is a very effective argument against Platonism, because it provides a counter-example to the core Platonic idea that there are Forms in order to serve as the object of scientific knowledge: the universal of (...) which theorems of universal mathematics are proven in Greek mathematics is neither Quantity in general nor any of the specific quantities, but Quantity-of-type-x. This universal cannot be a Platonic Form, for it is dependent on the types of quantity over which the variable ranges. Since for both Plato and Aristotle the object of scientific knowledge is that F which explains why G holds, as shown in a ‘direct’ proof about an arbitrary F (they merely disagree about the ontological status of this arbitrary F, whether a Form or a particular used in so far as it is F), Plato cannot maintain that Forms must be there as objects of scientific knowledge - unless the mathematics is changed. (shrink)

Mathematical logic was first introduced to China in early 1920s. Although, the process of introduction was facilitated by the lectures of Bertrand Russel at Peking University in 1921 and continued by China’s most passionate adherents of Russell’s philosophy, the establishment of mathematical logic as an academic discipline occurred only in late 1920s, in the framework of a recently reorganised Qinghua University in Peking. The main aim of this paper is to shed some light on the process of establishment (...) of mathematical logic at the department of philosophy at Qinghua University, between the years 1926 and 1945. In its main line of discussion, the article highlights the curricular developments at the department, connecting them with concrete theoretical endeavours by the leading members of the department. Furthermore, in its later parts, the article summarises the main advances made in the context of studies of modern logic at Qinghua University, from the expositions on the quintessential work Principia Mathematica, to the subsequent integration of criticisms coming from circles of logicians at Harvard University and ground-breaking contributions of Kurt Gödel in the mid-1930s. (shrink)

La matematica viene generalmente considerata uno degli ambiti più affidabili dell’intera impresa scientifica. Il suo successo e la sua solidità sono testimoniati, ad esempio, dall’uso imprescindibile che ne fanno le scienze empiriche e dall’accordo pressoché unanime con cui la comunità dei matematici delibera sulla validità di un nuovo risultato. Tuttavia, dal punto di vista filosofico la matematica rappresenta un puzzle tanto intrigante quanto intricato. Philosophy of Mathematics di Ø. Linnebo si propone di presentare e discutere le concezioni filosofiche della matematica (...) che hanno dominato la scena da Frege ai giorni nostri. (shrink)

The paper describes the situation of teaching mathematics and its position at Prague University in the second half of the 18th century. In order to be able to adequately present the specific changes during this period, I first explain the development of the role of mathematics as a modern science among the Prague Jesuits in the two centuries before. It is pointed out that the Jesuits initially assigned only a very minor importance to mathematics. From the middle of the 17th (...) century, however, there was significant development. In the middle of the 18th century, under the influence of the Enlightenment, state reforms set in, which also significantly influenced the structure and content of education at Prague University. I describe the consequences of these reforms – that also led to the dissolution of the Jesuit Order – for mathematics. Finally, I deal with the life and work of mathematicians and astronomers at Prague University in the second half of the 18th century. (shrink)

The contradiction between the rapid mathematization of health care through the active introduction of modern technologies and methods based on mathematical achievements in the field of medicine and the lack of a system of training medical students corresponding to these scientific successes, which allows them to carry out mathematical modeling of complex physical, chemical and biological processes at the molecular level for the purpose of their analysis and subsequent forecasting,, problems that arise when teaching mathematical analysis in (...) medical Schools, and possible ways to solve them. The solution identified during the research of problematic issues of teaching of mathematical analysis in medical schools is offered by enhancing the quality of learning through the re-allocated for the development of the discipline course hours stimulate the motivation of students, adjusting content and methodical component of teaching and active introduction in educational process of modern information technologies, elimination of excessive mathematical formalism. (shrink)

This article presents Tarski's Address at the Princeton Bicentennial Conference on Problems of Mathematics, together with a separate summary. Two accounts of the discussion which followed are also included. The central topic of the Address and of the discussion is decision problems. The introductory note gives information about the Conference, about the background of the subjects discussed in the Address, and about subsequent developments to these subjects.

Recent advances in physics render a reconsideration of the Place of Mathematics in the Interpretation of the Universe particularly timely. On the one hand, we have the introduction of non-euclidian geometry, which has given rise to much controversy informed or otherwise; on the other hand, we find mysterious forms of mathematics, invented to cope with the quantum difficulties, which so far have escaped metaphysical investigation or criticism. It would seem most desirable that these modes of interpreting reality should be (...) examined from the broader philosophical point of view. (shrink)

The number of studies related to natural and artificial mechanisms of learning rapidly increases. However, there is no general theory of learning that could provide a unifying basis for exploring different directions in this growing field. For a long time the development of such a theory has been hindered by nativists' belief that the development of a biological organism during ontogeny should be viewed as parameterization of an innate, encoded in the genome structure by an innate algorithm, and nothing essentially (...) new is created during this process. Noam Chomsky has claimed, therefore, that the creation of a non-trivial general mathematical theory of learning is not feasible, since any algorithm cannot produce a more complex algorithm. This study refutes the above argumentation by developing a counter-example based on the mathematical theory of algorithms and computable functions. It introduces a novel concept of a Universal Learning System (ULS) capable of learning to control in an optimal way any given constructive system from a certain class. The necessary conditions for the existence of a ULS and its main functional properties are investigated. The impossibility of building an algorithmic ULS for a sufficiently complex class of controlled objects is shown, and a proof of the existence of a non-algorithmic ULS based on the axioms of classical mathematics is presented. It is argued that a non-algorithmic ULS is a legitimate object of not only mathematics, but also the world of nature. These results indicate that an algorithmic description of the organization and adaptive development of biological systems in general is not sufficient. At the same time, it is possible to create a rigorous non-algorithmic general theory of learning as a theory of ULS. The utilization of this framework for integrating learning-related studies is discussed. (shrink)