This paper deals with meta-statistical questions concerning frequentist statistics. In Sections 2 to 4 I analyse the dispute between Fisher and Neyman on the so called logic of statistical inference, a polemic that has been concomitant of the development of mathematical statistics. My conclusion is that, whenever mathematical statistics makes it possible to draw inferences, it only uses deductive reasoning. Therefore I reject Fisher's inductive approach to the statistical estimation theory and adhere to Neyman's deductive one. On the (...) other hand, I assert that Neyman-Pearson's testing theory, as well as Fisher's tests of significance, properly belong to decision theory, not to logic, neither deductive nor inductive. I then also disagree with Costantini's view of Fisher's testing model as a theory of hypothetico-deductive inferences.In Section 5 I disapprove Hacking1's evidentialists criticisms of the Neyman-Pearson's theory of statistics (NPT), as well as Hacking2's interpretation of NPT as a theory of probable inference. In both cases Hacking misses the point. I conclude, by claiming that Mayo's conception of the Neyman-Pearson's testing theory, as a model of learning from experience, does not purport any advantages over Neyman's behavioristic model. (shrink)

The explanatory filter is a proposed method to detect design in nature with the aim of refuting Darwinian evolution. The explanatory filter borrows its logical structure from the theory of statistical hypothesis testing but we argue that, when viewed within this context, the filter runs into serious trouble in any interesting biological application. Although the explanatory filter has been extensively criticized from many angles, we present the first rigorous criticism based on the theory of mathematical statistics.

In this paper I want to discuss some basic problems of inductive logic, i.e. of the attempt to solve the problem of induction by means of a calculus of logical probability. I shall try to throw some light upon these problems by contrasting inductive logic, based on logical probability, and working with undefined samples of observations, with mathematical statistics, based on statistical probability, and working with representative random samples.

The “embeddedness” of economic life in social relations has become a productive analytical principle and the basis of a penetrating critique of economic orthodoxy. But this critique raises another important, social and historical question, of how the economy became “disembedded” in the first place – how the multitude of transactions designated (somewhat arbitrarily) as economic were abstracted from the rest of social life and reconstituted as an object, the economy, which behaves according to its own logic. This article investigates the (...) social sources of some innovations in economic thought that contributed to the emergence of the economy, particularly statistical indicators and mechanical models. By examining the redefinitions of the object of economic research developed by Irving Fisher and Wesley Mitchell in the 1890s and the first decades of the twentieth century, I argue that the abstraction of the economy from the remainder of social life was a strategy linked to the position of these innovators within the field of economics, conceived as a social structure. Possessing a specialized scientific cultural capital, but lacking upper class background, contacts, and dispositions that characterized the founders of academic economics, Fisher and Mitchell elaborated new definitions of their discipline's object of study, and a new type of economic expertise. (shrink)

In this essay, I consider the use of mathematical statistics in the study of biological systems in the field, using as case studies the work of Ruth Geyer Shaw and her colleagues at the University of Minnesota. To address practical issues, like how to enhance prairie restoration, and how to prepare for (and perhaps prevent) the effect of rapid climate change, she and her colleagues combine mathematical modeling and intensive data collection in the field. Using ANOVA and the (...) more versatile approach of Aster Models, they show the effects of inbreeding (which follows from the fragmentation of prairie) to lower the fitness of plant populations. They study the genetic variance of plants in stable populations, which is important when that population must deal with changing conditions. And they study the consequences of rapid climate change: even good genetic variance and initial fitness may not save a given population from extinction. This leads to a new alliance of science with local, state, federal and global politics. (shrink)

Over the years, mathematics and statistics have become increasingly important in the social sciences1 . A look at history quickly confirms this claim. At the beginning of the 20th century most theories in the social sciences were formulated in qualitative terms while quantitative methods did not play a substantial role in their formulation and establishment. Moreover, many practitioners considered mathematical methods to be inappropriate and simply unsuited to foster our understanding of the social domain. Notably, the famous Methodenstreit also (...) concerned the role of mathematics in the social sciences. Here, mathematics was considered to be the method of the natural sciences from which the social sciences had to be separated during the period of maturation of these disciplines. All this changed by the end of the century. By then, mathematical, and especially statistical, methods were standardly used, and their value in the social sciences became relatively uncontested. The use of mathematical and statistical methods is now ubiquitous: Almost all social sciences rely on statistical methods to analyze data and form hypotheses, and almost all of them use (to a greater or lesser extent) a range of mathematical methods to help us understand the social world. Additional indication for the increasing importance of mathematical and statistical methods in the social sciences is the formation of new subdisciplines, and the establishment of specialized journals and societies. Indeed, subdisciplines such as Mathematical Psychology and Mathematical Sociology emerged, and corresponding journals such as The Journal of Mathematical Psychology (since 1964), The Journal of Mathematical Sociology (since 1976), Mathematical Social Sciences (since 1980) as well as the online journals Journal of Artificial Societies and Social Simulation (since 1998) and Mathematical Anthropology and Cultural Theory (since 2000) were established. What is more, societies such as the Society for Mathematical Psychology (since 1976) and the Mathematical Sociology Section of the American Sociological Association (since 1996) were founded. Similar developments can be observed in other countries. The mathematization of economics set in somewhat earlier (Vazquez 1995; Weintraub 2002). However, the use of mathematical methods in economics started booming only in the second half of the last century (Debreu 1991). Contemporary economics is dominated by the mathematical approach, although a certain style of doing economics became more and more under attack in the last decade or so. Recent developments in behavioral economics and experimental economics can also be understood as a reaction against the dominance (and limitations) of an overly mathematical approach to economics. There are similar debates in other social sciences. It is, however, important to stress that problems of one method (such as axiomatization or the use of set theory) can hardly be taken as a sign of bankruptcy of mathematical methods in the social sciences tout court. This chapter surveys mathematical and statistical methods used in the social sciences and discusses some of the philosophical questions they raise. It is divided into two parts. Sections 1 and 2 are devoted to mathematical methods, and Sections 3 to 7 to statistical methods. As several other chapters in this handbook provide detailed accounts of various mathematical methods, our remarks about the latter will be rather short and general. Statistical methods, on the other hand, will be discussed in-depth. (shrink)

Statistical tests of the primality of some numbers look similar to statistical tests of many nonmathematical, clearly empirical propositions. Yet interpretations of probability prima facie appear to preclude the possibility of statistical tests of mathematical propositions. For example, it is hard to understand how the statement that n is prime could have a frequentist probability other than 0 or 1. On the other hand, subjectivist approaches appear to be saddled with ‘coherence’ constraints on rational probabilities that require rational agents (...) to assign extremal probabilities to logical and mathematical propositions. In the light of these problems, many philosophers have come to think that there must be some way to generalize a Bayesian statistical account. In this article I propose that a classical frequentist approach should be reconsidered. I conclude that we can give a conditional justification of statistical testing of at least some mathematical hypotheses: if statistical tests provide us with reasons to believe or bet on empirical hypotheses in the standard situations, then they also provide us with reasons to believe or bet on mathematical hypotheses in the structurally similar mathematical cases. (shrink)

A subarea of the debate over the nature of evolutionary theory addresses what the nature of the explanations yielded by evolutionary theory are. The statisticalist line is that the general principles of evolutionary theory are not only amenable to a mathematical interpretation but that they need not invoke causes to furnish explanations. Causalists object that construction of these general principles involves crucial causal assumptions. A recent view claims that some biological explanations are statistically autonomous explanations (SAEs) whereby phenomena are (...) accounted for statistically and which prescind from micro-causal details. I raise three major problems for this account and then advance a view which unifies SAEs as mathematical explanations: the MSAE view. The MSAE view not only resolves the issues bedeviling the original SAE account but serves to importantly broaden the class of non-causal explanations in population biology. (shrink)

Over the past century, private tutoring in many countries has increased dramatically. Moreover, the main disadvantage of PT is that has a byproduct and a characteristic on the educational system in developing countries in terms of contributing to conditions such as large class sizes, low public expenditures, and an inadequate number of universities. In Saudi Arabia, the spread of PT at school and university levels has yet to be addressed by researchers. One goal of this examination was to research the (...) spread of PT in relation to mathematics being taught at school and university levels. This investigation used quantitative data of the questionnaire to determine the reasons for the spread. A total of 1000 students responded from University of Ha’il. The edge plans given results that are associated and like the assessment that is performed with the whole data. The current study found that a good family financial situation and lack of parental supervision of the children were significant factors underlying the spread of private tutoring at school and university levels. (shrink)

Statistical learning refers to the ability to identify structure in the input based on its statistical properties. For many linguistic structures, the relevant statistical features are distributional: They are related to the frequency and variability of exemplars in the input. These distributional regularities have been suggested to play a role in many different aspects of language learning, including phonetic categories, using phonemic distinctions in word learning, and discovering non-adjacent relations. On the surface, these different aspects share few commonalities. Despite this, (...) we demonstrate that the same computational framework can account for learning in all of these tasks. These results support two conclusions. The first is that much, and perhaps all, of distributional statistical learning can be explained by the same underlying set of processes. The second is that some aspects of language can be learned due to domain-general characteristics of memory. (shrink)

On the mathematical foundations of theoretical statistics.

This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and (...) made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

Through his S–R model of statistical relevance, Wesley Salmon offers a solution to the scientific explanation of objectively improbable events.

The author finds no support for the claim that J. M. Keynes had severe reservations, in general, as opposed to particular, concerning the application of mathematical, logical and statistical methods in economics. These misinterpretations rest on the omission of important source material as well as a severe misconstrual ofThe Treatise on Probability (1921).

The fundamental problem on which Ilya Prigogine and the Brussels–Austin Group have focused can be stated briefly as follows. Our observations indicate that there is an arrow of time in our experience of the world (e.g., decay of unstable radioactive atoms like uranium, or the mixing of cream in coffee). Most of the fundamental equations of physics are time reversible, however, presenting an apparent conflict between our theoretical descriptions and experimental observations. Many have thought that the observed arrow of time (...) was either an artifact of our observations or due to very special initial conditions. An alternative approach, followed by the Brussels–Austin Group, is to consider the observed direction of time to be a basic physical phenomenon due to the dynamics of physical systems. This essay focuses mainly on recent developments in the Brussels–Austin Group after the mid-1980s. The fundamental concerns are the same as in their earlier approaches (subdynamics, similarity transformations), but the contemporary approach utilizes rigged Hilbert space (whereas the older approaches used Hilbert space). While the emphasis on nonequilibrium statistical mechanics remains the same, their more recent approach addresses the physical features of large Poincare systems, nonlinear dynamics and the mathematical tools necessary to analyze them. (shrink)

This paper argues that a notion of statistical explanation, based on Salmon’s statistical relevance model, can help us better understand deep neural networks. It is proved that homogeneous partitions, the core notion of Salmon’s model, are equivalent to minimal sufficient statistics, an important notion from statistical inference. This establishes a link to deep neural networks via the so-called Information Bottleneck method, an information-theoretic framework, according to which deep neural networks implicitly solve an optimization problem that generalizes minimal sufficient statistics. The (...) resulting notion of statistical explanation is general, mathematical, and subcausal. (shrink)

A serious flaw in Hartry Field’s instrumental account of applied mathematics, namely that Field must overestimate the extent to which many of the structures of our mathematical theories are reflected in the physical world, underlies much of the criticism of this account. After reviewing some of this criticism, I illustrate through an examination of the prospects for extending Field’s account to classical equilibrium statistical mechanics how this flaw will prevent any significant extension of this account beyond field theories. I (...) note in the conclusion that this diagnosis of Field’s program also points the way to modifications that may work. (shrink)

One of the traditional ways mathematical ideas and even new areas of mathematics are created is from experiments. One of the best-known examples is that of the Fermat hypothesis, which was conjectured by Fermat in his attempts to find integer solutions for the famous Fermat equation. This hypothesis led to the creation of a whole field of knowledge, but it was proved only after several hundred years. This book, based on the author's lectures, presents several new directions of (...) class='Hi'>mathematical research. All of these directions are based on numerical experiments conducted by the author, which led to new hypotheses that currently remain open, i.e., are neither proved nor disproved. The hypotheses range from geometry and topology (statistics of plane curves and smooth functions) to combinatorics (combinatorial complexity and random permutations) to algebra and number theory (continuous fractions and Galois groups). For each subject, the author describes the problem and presents numerical results that led him to a particular conjecture. In the majority of cases there is an indication of how the readers can approach the formulated conjectures (at least by conducting more numerical experiments). Written in Arnold's unique style, the book is intended for a wide range of mathematicians, from high school students interested in exploring unusual areas of mathematics on their own, to college and graduate students, to researchers interested in gaining a new, somewhat nontraditional perspective on doing mathematics. In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, MSRI and the AMS are publishing books in the Mathematical Circles Library series as a service to young people, their parents and teachers, and the mathematics profession. Titles in this series are co-published with the Mathematical Sciences Research Institute (MSRI). (shrink)

This book presents an historical overview of the field--from its development to the present--at an accessible mathematical level. This edition features two new chapters--one on factor analysis and the other on the rise of ANOVA usage in psychological research. Written for psychology, as well as other social science students, this book introduces the major personalities and their roles in the development of the field. It provides insight into the disciplines of statistics and experimental design through the examination of the (...) character of its founders and the nature of their views, which were sometimes personal and ideological, rather than objective and scientific. It motivates further study by illustrating the human component of this field, adding dimension to an area that is typically very technical. Intended for advanced undergraduate and/or graduate students in psychology and other social sciences, this book will also be of interest to instructors and/or researchers interested in the origins of this omnipresent discipline. (shrink)

The twentieth century witnessed a dramatic increase in the use of statistics by biologists, including systematists. The modern synthesis and new systematics stimulated this development, particularly after World War II. The rise of "the statistical frame of mind " resulted in a rethinking of the relationship between biological and mathematical points of view, the roles of objectivity and subjectivity in systematic research, the implications of new computing technologies, and the place of systematics among the biological disciplines.

In this study the theory of risk factors is discussed. The risk-concept is essential in cardiology and is, furthermore, important not only in medicine in general, but also and particularly in ecology. Since environmental risk factors endanger our health, ecological risks have to be taken as medical problems. If a factor or a set of factors is a necessary but not a sufficient condition for a disease we speak of a risk factor or of risk factors. Statistical analysis of risk (...) factors can be performed by multivariate methods. A method which is particularly useful for finding the most important of a set of risk factors, is discriminant analysis. (shrink)

Theory of linear estimation; General structure of analysis of designs; Standard designs; Applications of galois fields and finite geometry in the construction of designs; Some selected topics in design of experiments.

1. Deductive and Inductive Inference. Within the traditional treatments of scientific method, e.g., in and, it was customary to divide scientific inference into two parts: deductive and inductive. Deductive inference was taken to mean the activity of deducing theorems from postulates and definitions, whereas inductive inference represented the activity of constructing a general statement from a set of particular “facts.” Deductive inference was relegated to the mathematical sciences, and inductive inference to the empirical sciences. As a consequence, the whole (...) of science was split into two quite distinct parts: deductive science was conceived to start with precise axioms, postulates, and definitions, and its method consisted in drawing inferences from these “givens” in such a way that if the original assumptions are taken as true, then the theorems must be taken as true also. The obligation behind this “must” of the mathematical sciences was an obligation based on strict rules of deductive inference; in the earlier analyses, it was supposed that these rules were all aspects of the general theory of the syllogism, as it was described in an almost complete form in but in later work the rules have been formalized in more general terms. Inductive inference, on the other hand, was supposed to start with observed facts; these facts, or “data,” were presumably given by the senses, and described particular aspects of the external world. If the facts were given in a certain way, then the empirical scientist could “infer” a generalized description; this generalization had the characteristic that if it were granted as true, then all the facts could be deduced as consequences. The inductive process admittedly does not lead to unique generalizations, and the philosopher of science searched for other defining characteristics of good inference, such as “simplicity” and “convenience.” These empirical generalizations were taken to be “descriptions of the phenomenal world of the scientist”; a further inductive step might be taken by constructing a set of very general principles out of the more specialized generalizations within specific research problems. These “higher” inductive inferences were regarded as “theories,” which could be studied by themselves within the method of deductive inference. The scientist was then described as “checking” theory by determining whether or not the predicted consequences deduced from theory actually hold in the phenomenal world. (shrink)

The origins of testing scientific models with statistical techniques go back to 18th century mathematics. However, the modern theory of statistical testing was primarily developed through the work of Sir R.A. Fisher, Jerzy Neyman, and Egon Pearson in the inter-war period. Some of Fisher's papers on testing were published in economics journals (Fisher, 1923, 1935) and exerted a notable influence on the discipline. The development of econometrics and the rise of quantitative economic models in the mid-20th century made statistical significance (...) testing a commonplace, albeit controversial tool within economics. -/- In the debate about significance testing, methodological controversies intertwine with epistemological issues and sociological developments. Our aim in this chapter is to expound these connections and to show how the use of, and the debate about, significance testing in economics differs from other social sciences, such as psychology. (shrink)

Sociology is split into two antagonistic or mutually oblivious wings: quantitative and nonquantitative. Statistics does not occupy a privileged methodological position vis-a-vis qualitative, verbal sociology. Probability is a theory like any other, and each statistical method contains its particular theoretical bias. Such biases should be brought into the open and tested. Statistics may continue to be useful, though, as a substantive theory of change processes in the social world. A reorientation in our views of statistics may bring mathematical and (...) antimathematical branches of sociology back into a common enterprise. (shrink)

This article presents a challenge that those philosophers who deny the causal interpretation of explanations provided by population genetics might have to address. Indeed, some philosophers, known as statisticalists, claim that the concept of natural selection is statistical in character and cannot be construed in causal terms. On the contrary, other philosophers, known as causalists, argue against the statistical view and support the causal interpretation of natural selection. The problem I am concerned with here arises for the statisticalists because the (...) debate on the nature of natural selection intersects the debate on whether mathematical explanations of empirical facts are genuine scientific explanations. I argue that if the explanations provided by population genetics are regarded by the statisticalists as non-causal explanations of that kind, then statisticalism risks being incompatible with a naturalist stance. The statisticalist faces a dilemma: either she maintains statisticalism but has to renounce naturalism; or she maintains naturalism but has to content herself with an account of the explanations provided by population genetics that she deems unsatisfactory. This challenge is relevant to the statisticalists because many of them see themselves as naturalists. (shrink)

Games of chance are developed in their physical consumer-ready form on the basis of mathematical models, which stand as the premises of their existence and represent their physical processes. There is a prevalence of statistical and probabilistic models in the interest of all parties involved in the study of gambling – researchers, game producers and operators, and players – while functional models are of interest more to math-inclined players than problem-gambling researchers. In this paper I present a structural analysis (...) of the knowledge attached to mathematical models of games of chance and the act of modeling, arguing that such knowledge holds potential in the prevention and cognitive treatment of excessive gambling, and I propose further research in this direction. (shrink)

Inductive logic generalizes the idea of logical entailment and provides standards for the evaluation of non-conclusive arguments. A main application of inductive logic is the generalization of observational data to theoretical models. In the empirical sciences, the mathematical theory of statistics addresses the same problem. This paper argues that there is no separable purely logical aspect of statistical inference in a variety of complex problems. Instead, statistical practice is often motivated by decision-theoretic considerations and resembles empirical science.

The question that is the subject of this article is not intended to be a sociological or statistical question about the practice of today’s mathematicians, but a philosophical question about the nature of mathematics, and specifically the method of mathematics. Since antiquity, saying that mathematics is problem solving has been an expression of the view that the method of mathematics is the analytic method, while saying that mathematics is theorem proving has been an expression of the view that the method (...) of mathematics is the axiomatic method. In this article it is argued that these two views of the mathematical method are really opposed. In order to answer the question whether mathematics is problem solving or theorem proving, the article retraces the Greek origins of the question and Hilbert’s answer. Then it argues that, by Gödel’s incompleteness results and other reasons, only the view that mathematics is problem solving is tenable. (shrink)

The debates between Bayesian, frequentist, and other methodologies of statistics have tended to focus on conceptual justifications, sociological arguments, or mathematical proofs of their long run properties. Both Bayesian statistics and frequentist (“classical”) statistics have strong cases on these grounds. In this article, we instead approach the debates in the “Statistics Wars” from a largely unexplored angle: simulations of different methodologies’ performance in the short to medium run. We conducted a large number of simulations using a straightforward decision problem (...) based around tossing a coin with unknown bias and then placing bets. In this simulation, we programmed four players, inspired by Bayesian statistics, frequentist statistics, Jon Williamson’s version of Objective Bayesianism, and a player who simply extrapolates from observed frequencies to general frequencies. The last player functions as a benchmark: a statistical methodology should at least outperform a crude form of induction. We focused on the performance of these methodologies in guiding the players towards good decisions. Unlike an earlier simulation study of this type, we found no systematic difference in performance between the Bayesian and frequentist players, provided the Bayesian used a flat prior and the frequentist used a low confidence level. Unlike that study, we were able to use Big Data methods to mitigate problems of random error in the simulation results. The Williamsonian player, who is a novel element of our study, also had no systematic differences in their performance, provided that they use a low confidence level. These players performed similarly even in the very short run, when players were making different decisions. Our study indicates that all three methodologies should be taken seriously by philosophers and practitioners of statistics. However, the frequentist and Williamsonian performed poorly when their confidence levels were high, and the Bayesian was surprisingly harmed by biased priors, providing some unexpected lessons for these methodologies when facing this type of decision problem. (shrink)

The city of Kruja dates back to its existence in the 5th and 6th centuries. In the inner city are preserved great historical, cultural, and architectural values that are inherited from generation to generation. In the city interact and coexist three different typologies of dwellings: historic buildings that belong to the XIII, XIV, XV, XIII, XIX centuries (built using the foundations of previous buildings); socialist buildings dating back to the Second World War until 1990; and modern buildings which were built (...) from 1990 onwards. According to the questionnaires, the creation of mathematical models applied to each category will result in contradictory attitudes but also fairy ones based on different percentages. There are underlined 5 quality of life indicators(questions) from a total of 30 questions of the questionnaire, which are involved in mathematical regression and are statistically significant with a significance level p<5% (with a reliability of 95%), which interact for all three categories of buildings. The quality of life indicators: dwelling area, heating mode during winter, level of dwelling improvement, time spent in the dwelling, and monthly electricity payments, are the main actors who will be compared in order to draw conclusions. According to the statistical calculations trend, it is noted that the socialist buildings category does not directly participate in the debate between historic and modern buildings by means of the quality of life indicators (questions). (shrink)

We develop a mathematical and interpretative foundation for the enterprise of decision-theoretic (DT) statistical causality, which is a straightforward way of representing and addressing causal questions. DT reframes causal inference as “assisted decision-making” and aims to understand when, and how, I can make use of external data, typically observational, to help me solve a decision problem by taking advantage of assumed relationships between the data and my problem. The relationships embodied in any representation of a causal problem require deeper (...) justification, which is necessarily context-dependent. Here we clarify the considerations needed to support applications of the DT methodology. Exchangeability considerations are used to structure the required relationships, and a distinction drawn between intention to treat and intervention to treat forms the basis for the enabling condition of “ignorability.” We also show how the DT perspective unifies and sheds light on other popular formalisations of statistical causality, including potential responses and directed acyclic graphs. (shrink)

Mathematicians often speak of conjectures as being confirmed by evidence that falls short of proof. For their own conjectures, evidence justifies further work in looking for a proof. Those conjectures of mathematics that have long resisted proof, such as the Riemann hypothesis, have had to be considered in terms of the evidence for and against them. In recent decades, massive increases in computer power have permitted the gathering of huge amounts of numerical evidence, both for conjectures in pure mathematics and (...) for the behavior of complex applied mathematical models and statistical algorithms. Mathematics has therefore become (among other things) an experimental science (though that has not diminished the importance of proof in the traditional style). We examine how the evaluation of evidence for conjectures works in mathematical practice. We explain the (objective) Bayesian view of probability, which gives a theoretical framework for unifying evidence evaluation in science and law as well as in mathematics. Numerical evidence in mathematics is related to the problem of induction; the occurrence of straightforward inductive reasoning in the purely logical material of pure mathematics casts light on the nature of induction as well as of mathematical reasoning. (shrink)

This second part of a two-part essay discusses recent developments in the Brussels-Austin Group after the mid 1980s. The fundamental concerns are the same as in their similarity transformation approach (see Part I), but the contemporary approach utilizes rigged Hilbert space (whereas the older approach used Hilbert space). While the emphasis on nonequilibrium statistical mechanics remains the same, the use of similarity transformations shifts to the background. In its place arose an interest in the physical features of large Poincaré systems, (...) nonlinear dynamics and the mathematical tools necessary to analyze them. (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)

For the given logical calculus we investigate the proportion of the number of true formulas of a certain length n to the number of all formulas of such length. We are especially interested in asymptotic behavior of this fraction when n tends to infinity. If the limit exists it is represented by a real number between 0 and 1 which we may call the density of truth for the investigated logic. In this paper we apply this approach to the intuitionistic (...) logic of one variable with implication and negation. The result is obtained by reducing the problem to the same one of Dummett's intermediate linear logic of one variable (see [2]). Actually, this paper shows the exact density of intuitionistic logic and demonstrates that it covers a substantial part (more than 93%) of classical prepositional calculus. Despite using strictly mathematical means to solve all discussed problems, this paper in fact, may have a philosophical impact on understanding how much the phenomenon of truth is sporadic or frequent in random mathematics sentences. (shrink)

Degrees of belief; Dempster's rule of combination; Simple and separable support functions; The weights of evidence; Compatible frames of discernment; Support functions; The discernment of evidence; Quasi support functions; Consonance; Statistical evidence; The dual nature of probable reasoning.

An inductive logic is a system of inference that describes the relation between propositions on data, and propositions that extend beyond the data, such as predictions over future data, and general conclusions on all possible data. Statistics, on the other hand, is a mathematical discipline that describes procedures for deriving results about a population from sample data. These results include predictions on future samples, decisions on rejecting or accepting a hypothesis about the population, the determination of probability assignments over (...) such hypotheses, the selection of a statistical model for studying the population, and so on. Both inductive logic and statistics are calculi for getting from the given data to propositions or results that transcend the data. (shrink)

This book discusses in dialogue form the basic principles of mathematics and its applications including the question: What is mathematics? What does its specific method consist of? What is its relation to the sciences and humanities? What can it offer to specialists in different fields? How can it be applied in practice and in discovering the laws of nature? Dramatized by the dialogue form and shown in the historical movements in which they originated, these questions are discussed in their full (...) complexity, yet are easily comprehended. The first dialogue, whose chief actor is Socrates, leads the reader to the source of modern mathematics in Athens in the 5th Century BC. The second dialogue, featuring Archimedes, takes place during the siege of Syracuse in 212 BC and shows the birth of applied mathematics. The third dialogue occurs in the year 1633 in Rome, its chief character being Galileo Galilei who fully realized the central importance of the mathematical method in discovering the laws of nature. Intended as supplemental reading for philosophy of mathematics courses at the high school or college level it will be of interest to both specialists and non-specialists in mathematics. Alfréd Rényi was born in Budapest Hungary in 1921. He studied mathematics and physics at the University of Budapest and received his Ph. D. from the University of Szaged in 1945. Since 1950 he has been Director of the Mathematical Research Institute of the Hungarian Academy of Sciences and since 1952 a professor at the University of Budapest. Dr. Renyi was a visiting professor at Michigan State University in 1961, at the University of Michigan in 1964 and at Stanford University in 1966. His main fields of research are probability theory, mathematical statistics and information theory, and he has also worked in analytic number theory as well as in various branches of analysis, combinatorial analysis and geometry. (shrink)