In recent decades, experimental mathematics has emerged as a new branch of mathematics. This new branch is defined less by its subject matter, and more by its use of computer assisted reasoning. Experimental mathematics uses a variety of computer assisted approaches to verify or prove mathematical hypotheses. For example, there is “number crunching” such as searching for very large Mersenne primes, and showing that the Goldbach conjecture holds for all even numbers less than 2 × (...) 1018. There are “verifications” of hypotheses which, while not definitive proofs, provide strong support for those hypotheses, and there are proofs involving an enormous amount of computer hours, which cannot be surveyed by any one mathematician in a lifetime. There have been several attempts to argue that one or another aspect of experimental mathematics shows that mathematics now accepts empirical or inductive methods, and hence shows mathematical apriorism to be false. Assessing this argument is complicated by the fact that there is no agreed definition of what precisely experimental mathematics is. However, I argue that on any plausible account of ’experiment’ these arguments do not succeed. (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 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)

The rise of the field of “ experimental mathematics” poses an apparent challenge to traditional philosophical accounts of mathematics as an a priori, non-empirical endeavor. This paper surveys different attempts to characterize experimental mathematics. One suggestion is that experimental mathematics makes essential use of electronic computers. A second suggestion is that experimental mathematics involves support being gathered for an hypothesis which is inductive rather than deductive. Each of these options turns out (...) to be inadequate, and instead a third suggestion is considered according to which experimental mathematics involves calculating instances of some general hypothesis. The paper concludes with the examination of some philosophical implications of this characterization. (shrink)

In this paper, I describe some aspects of the phenomenon of "experimental mathematics" in order to discuss whether it constitutes a subdiscipline or a particular style of mathematics. My conclusion is that neither of these notions accurately capture the complex culture of experimental mathematics.

In the present paper, I go beyond these examples by bringing into play an example that I nd more experimental in nature, namely that of the use of the so-called PSLQ algorithm in researching integer relations between numerical constants. It is the purpose of this paper to combine a historical presentation with a preliminary exploration of some philosophical aspects of the notion of experiment in experimental mathematics. This dual goal will be sought by analysing these aspects as (...) they are presented by some of the protagonists of the eld and discussing them using notions from contemporary philosophy of science. (shrink)

This chapter presents an overview of the contributions to the section on Experimental Mathematics by focusing in particular on how they characterize the phenomenon of “experimental mathematics” and its origins. The second part presents two case studies illustrating how experimental mathematics is understood in contemporary analysis. The third section offers a systematic presentation of the contributions to the section.

This chapter is devoted to discussions about the nature of experiments in mathematics during the formative period of experimental mathematics and to a quantitative and qualitative analysis of the journal Experimental Mathematics founded in 1992. Particular attention is paid to interpret the roles of experiment and observation in the mathematical practices surrounding Experimental Mathematics.

This book explores the results of applying empirical methods to the philosophy of logic and mathematics. Much of the work that has earned experimental philosophy a prominent place in twenty-first century philosophy is concerned with ethics or epistemology. But, as this book shows, empirical methods are just as much at home in logic and the philosophy of mathematics. -/- Chapters demonstrate and discuss the applicability of a wide range of empirical methods including experiments, surveys, interviews, and data-mining. (...) Distinct themes emerge that reflect recent developments in the field, such as issues concerning the logic of conditionals and the role played by visual elements in some mathematical proofs. -/- Featuring leading figures from experimental philosophy and the fields of philosophy of logic and mathematics, this collection reveals that empirical work in these disciplines has been quietly thriving for some time and stresses the importance of collaboration between philosophers and researchers in mathematics education and mathematical cognition. (shrink)

The ArgumentThis paper is a study of the role of language in scientific activity. It recommends that language be viewed as a community's means of patterning its affairs. Language represents where the boundaries of the community are and who is entitled to speak within it, and it displays the structures of authority in the community. Moreover, language precipitates the community's view of what the world is like, such that linguistic usages can be taken as referring to that world. Thus, language (...) connects on the one side to a community's practical life, and, on the other, to the reality on which it “reports.”The example treated is Robert Boyle's view of the language proper for experimental practice, and his arguments against the value and legitimacy of mathematical representations within the experimental community. Boyle argued that the use of mathematics was inappropriate because (i) it would restrict the number of potential participants; (ii) it would give rise to unwarranted expectations of the certainty and accuracy to be expected of real physical inquiries; (iii) it would suggest views of natural law and God's relations with the natural world which were incorrect and possibly dangerous; and because (iv) the world upon which the experimentalist operated was not such as was represented by mathematical idealizations. I try to establish that decision about the appropriate language for the experimental community was a moral choice. (shrink)

This article aims at reducing the gap between mathematics and physics from a Wittgensteinian point of view. This gap is usually characterized by two discriminating features. The propositions of physics assert something which might be false; they have a hypothetical character. On the contrary, since mathematical propositions are rules that condition the form of assertions, they remain immune from falsification. The propositions of physics refer to facts that may confirm or refute them. On the contrary, since mathematical propositions have (...) no meaning independently of the demonstration procedures, it cannot be said that they refer to “facts” that pre-exist demonstrations. If we take a closer look, however, these two differences fade away. On the one hand, the propositions of physics are more resistant to experimental tests than has been said in the wake of logical positivism. On the other hand, the “factual” empirical material is defined and co-constituted by instruments whose arrangement is determined by the theory to be tested. I conclude by discussing the possibility of a Wittgensteinian philosophy of contemporary physics. (shrink)

In studies of scientific methodology, surprisingly little attention has been given to tests of hypotheses. Such testing constitutes a methodology common to various scientific disciplines and is an essential factor in the development of science since it determines which theories are retained. The classical theory of tests is a major accomplishment but requires modification in order to produce a theory that accounts for the success of science. The revised theory is an analysis of the nondeductive aspect of scientific reasoning. It (...) results in a new definition of probability and a nonclassical point of view with regard to the foundations of probability. (shrink)

In studies of scientific methodology, surprisingly little attention has been given to tests of hypotheses. Such testing constitutes a methodology common to various scientific disciplines and is an essential factor in the development of science since it determines which theories are retained. The classical theory of tests is a major accomplishment but requires modification in order to produce a theory that accounts for the success of science. The revised theory is an analysis of the nondeductive aspect of scientific reasoning. It (...) results in a new definition of probability and a nonclassical point of view with regard to the foundations of probability. (shrink)

In studies of scientific methodology, surprisingly little attention has been given to tests of hypotheses. Such testing constitutes a methodology common to various scientific disciplines and is an essential factor in the development of science since it determines which theories are retained. The classical theory of tests is a major accomplishment but requires modification in order to produce a theory that accounts for the success of science. The revised theory is an analysis of the nondeductive aspect of scientific reasoning. It (...) results in a new definition of probability and a nonclassical point of view with regard to the foundations of probability. (shrink)

This book is a collection of articles on research that attempts to connect logic and mathematics with empirical and cognition. There have been various such a...

The drastically increasing availability ofmodern computers coupled with the equally drasticallylower cost of a given amount of computer power inrecent years has resulted in the evolution of thetraditional experimental/theoretical dichotomy inchemistry into anexperimental/theoretical/computational trichotomy. This trichotomy can be schematically represented by atriangle with experimental,theoretical, and computational chemistry at the threevertices. The ET and EC edges of the ETC triangledepict the uses of theoretical and computationalchemistry, respectively, to predict and interpretexperimental results. The TC edge depicts therelationship between theoretical and (...) computationalchemistry. Mathematics plays an increasing role in allaspects of chemistry, particularly theoreticalchemistry, and has led to the evolution of thediscipline of mathematical chemistry. Research inmathematical chemistry can be considered to lie on achemistry-mathematics continuum depending on therelative depths of the underlying chemistry andmathematics. Examples of the author's own researchlying near each end of the chemistry-mathematicscontinuum include his work on applications of graphtheory and topology in inorganic coordination andcluster chemistry lying near the chemistry end and hiswork on chirality algebra lying near the mathematicsend. The general points in this essay are illustratedby an analysis of the roles of computational andtheoretical chemistry in developing an understandingof structure and bonding in deltahedral boranes andrelated carboranes. This work has allowed extensionof the concept of aromaticity from two dimensions asin benzene and other planar hydrocarbons to the thirddimension in deltahedral boranes. (shrink)

In this first of two companion papers to a logico-mathematic, structural methodology , a meta-level analysis of the non metric structure is presented in relation to critiques based on standard experimental, statistical, and computational methods of contemporary psychology and cognitive science. The concept of a non metric methodology is examined as it relates to the epistemological and scientific goals of experimental, statistical, and computational methods. While sharing in these goals, differences and similarities between the two methodological approaches are (...) outlined. It is argued that typical experimental methods are not sufficient to extract and validate semantic information in verbal narratives. It is further suggested that a logico-mathematic, structural methodology can yield invariant law-like cognitive processes by careful methodological control of the specific case — instead of those found by current methods that produce “laws” based on statistical frequency. Lastly, the issue of experimental manipulation in relation to the logico-mathematic, structural methodology is examined. (shrink)

Nach einer allgemeinen Diskussion des Zusammenhanges zwischen theoretischer und experimenteller Forschung, wird in Hinblick auf die vom Verfasser entwickelten physikalisch-mathematischen Grundlagen der Biologie, eine Reihe von Einzelproblemen betrachtet. Es wird an Hand von Kurvenmaterial gezeigt wie weit die mathematisch vorausgesagten Beziehungen mit den experimentellen Befunden übereinstimmen. Folgende Fragen werden besprochen: Zellatmung, Zellgrössen, deren Abhängigkeit von Stoffwechsel, Zellteilung, Protoplasmaströmungen, Nervenerregung, psychophysische Gesetze, Reaktion auf geometrische Gestalten.Après une mise au point générale de la relation entre les sciences théoriques et expérimentales, diverses questions (...) sont discutées, du point de vue des fondations physico-mathématiques de la biologie, développées récemment par l'auteur. On montre par la comparaison des courbes calculées à celles observées, comment les relations prédites mathématiquement se trouvent vérifiées. Les questions suivantes sont discutées: respiration cellulaire, dimensions cellulaires et leur rapport au métabolisme, division cellulaire, mouvements protoplasmiques, excitation nerveuse, les lois psychophysiques, perception des formes géométriques. (shrink)

In 1966 Richard Levins argued that applications of mathematics to population biology faced various constraints which forced mathematical modelers to trade-off at least one of realism, precision, or generality in their approach. Much traditional mathematical modeling in biology has prioritized generality and precision in the place of realism through strategies of idealization and simplification. This has at times created tensions with experimental biologists. The past 20 years however has seen an explosion in mathematical modeling of biological systems with (...) the rise of modern computational systems biology and many new collaborations between modelers and experimenters. In this paper I argue that many of these collaborations revolve around detail-driven modeling practices which in Levins’ terms trade-off generality for realism and precision. These practices apply mathematics by working from detailed accounts of biological systems, rather than from initially idealized or simplified representations. This is possible by virtue of modern computation. The form these practices take today suggest however Levins’ constraints on mathematical application no longer apply, transforming our understanding of what is possible with mathematics in biology. Further the engagement with realism and the ability to push realistic models in new directions aligns well with the epistemological and methodological views of experimenters, which helps explain their increased enthusiasm for biological modeling. (shrink)

Recent experimental evidence from developmental psychology and cognitive neuroscience indicates that humans are equipped with unlearned elementary mathematical skills. However, formal mathematics has properties that cannot be reduced to these elementary cognitive capacities. The question then arises how human beings cognitively deal with more advanced mathematical ideas. This paper draws on the extended mind thesis to suggest that mathematical symbols enable us to delegate some mathematical operations to the external environment. In this view, mathematical symbols are not only (...) used to express mathematical concepts—they are constitutive of the mathematical concepts themselves. Mathematical symbols are epistemic actions, because they enable us to represent concepts that are literally unthinkable with our bare brains. Using case-studies from the history of mathematics and from educational psychology, we argue for an intimate relationship between mathematical symbols and mathematical cognition. (shrink)

Multiple authorship is so common and pervasive in our world that it is tempting to take it for granted. Prior to the twentieth century, however, multiple authorship was exceedingly rare.

This article outlines a philosophy of science in practice that focuses on the engineering sciences. A methodological issue is that these practices seem to be divided by two different styles of scientific reasoning, namely, causal-mechanistic and mathematical reasoning. These styles are philosophically characterized by what Kuhn called ?disciplinary matrices?. Due to distinct metaphysical background pictures and/or distinct ideas of what counts as intelligible, they entail distinct ideas of the character of phenomena and what counts as a scientific explanation. It is (...) argued that the two styles cannot be reduced to each other. At the same time, although they are incompatible, they must not be regarded as competing. Instead, they produce different kinds of epistemic results, which serve different kinds of epistemic functions. Moreover, some scientific breakthroughs essentially result from relating them. This view of complementary styles of scientific reasoning is supported by pluralism about metaphysical background pictures. (shrink)

This chapter discusses the relation between Christian Wolff's philosophy and the methodological views of early modern experimental philosophers. The chapter argues for three claims. First, Wolff's system relies on experience at every step and his views on experiments, observations, hypotheses, and the a priori are in line with those of experimental philosophers. Second, the study of Wolff's views demonstrates the influence of experimental philosophy in early eighteenth-century Germany. Third, references to Wolff's empiricism and rationalism are best identified (...) or replaced with references to his endorsement of the tenets of experimental philosophers and of a mathematical demonstrative method. (shrink)

We investigate experimentally the relationship between risk and incentives in a principal–agent setting. In contrast to the existing empirical literature that describes such relationship as ‘tenuous’ or inconclusive, we find a clear negative relationship—supporting the prediction of the standard theoretical model. Specifically, we find that principals reduce the size of the offered piece rates with an increase in risk and instead provide positive fixed wages. Furthermore, we find no relationship between the variance in the performance and the effort choice of (...) the agent, and a strong positive relationship between the effort choice of the agents and the offered piece rates as well as fixed wage, suggesting positive reciprocity. Finally, we find evidence of social projection by the principals regarding the agents’ degree of risk aversion. (shrink)

Both Carnap and Quine made significant contributions to the philosophy of mathematics despite their diversedviews. Carnap endorsed the dichotomy between analytic and synthetic knowledge and classified certainmathematical questions as internal questions appealing to logic and convention. On the contrary, Quine wasopposed to the analytic-synthetic distinction and promoted a holistic view of scientific inquiry. The purpose of thispaper is to argue that in light of the recent advancement of experimental mathematics such as Monte Carlosimulations, limiting mathematical inquiry to (...) the domain of logic is unjustified. Robustness studies implemented inMonte Carlo Studies demonstrate that mathematics is on par with other experimental-based sciences. (shrink)

Both Carnap and Quine made significant contributions to the philosophy of mathematics despite their diversed views. Carnap endorsed the dichotomy between analytic and synthetic knowledge and classified certain mathematical questions as internal questions appealing to logic and convention. On the contrary, Quine was opposed to the analytic-synthetic distinction and promoted a holistic view of scientific inquiry. The purpose of this paper is to argue that in light of the recent advancement of experimental mathematics such as Monte Carlo (...) simulations, limiting mathematical inquiry to the domain of logic is unjustified. Robustness studies implemented in Monte Carlo Studies demonstrate that mathematics is on par with other experimental-based sciences. (shrink)

To what extent was ordinary language philosophy a precursor to experimental philosophy? Since the conditions on pursuit of either project are at best unclear, and at worst protean, the general question is hard to address. I focus instead on particular cases, seeking to uncover some central aspects of J. L. Austin’s and John Cook Wilson’s ordinary language based approach to philosophical method. I make a start at addressing three questions. First, what distinguishes their approach from other more traditional approaches? (...) Second, is their approach a form of experimental philosophy? Third, given their aims, should it have been? I offer the following preliminary answers. First, their approach distinctively emphasizes attention to what we should say when. Second, their approach is closer to contemporary experimental mathematics than it is to some prominent forms of contemporary experimental philosophy. Third, some purported grounds for pursuing their aims by way of surveying what individual speakers would say when are not compelling. (shrink)

The important role of mathematical representations in scientific thinking has received little attention from cognitive scientists. This study argues that neglect of this issue is unwarranted, given existing cognitive theories and laws, together with promising results from the cognitive historical analysis of several important scientists. In particular, while the mathematical wizardry of James Clerk Maxwell differed dramatically from the experimental approaches favored by Michael Faraday, Maxwell himself recognized Faraday as “in reality a mathematician of a very high order,” and (...) his own work as in some respects a re‐representation of Faraday’s field theory in analytic terms. The implications of the similarities and differences between the two figures open new perspectives on the cognitive role of mathematics as a learned mode of representation in science. (shrink)

Although theoretical results for several algorithms in many application domains were presented during the last decades, not all algorithms can be analyzed fully theoretically. Experimentation is necessary. The analysis of algorithms should follow the same principles and standards of other empirical sciences. This article focuses on stochastic search algorithms, such as evolutionary algorithms or particle swarm optimization. Stochastic search algorithms tackle hard real-world optimization problems, e.g., problems from chemical engineering, airfoil optimization, or bioinformatics, where classical methods from mathematical optimization fail. (...) Nowadays statistical tools that are able to cope with problems like small sample sizes, non-normal distributions, noisy results, etc. are developed for the analysis of algorithms. Although there are adequate tools to discuss the statistical significance of experimental data, statistical significance is not scientifically meaningful per se. It is necessary to bridge the gap between the statistical significance of an experimental result and its scientific meaning. We will propose some ideas on how to accomplish this task based on Mayo's learning model. (shrink)

In this paper, we use the risk apportionment technique of Eeckhoudt, Rey and Schlesinger (2007) to study higher order risk preferences for others’ health as well as ex-ante and ex-post inequality preferences for social risky distributions, and their interaction. In an experiment on a sample of university students acting as impartial spectators, we observe risk aversion towards social health losses and a dislike of ex-ante inequality. In addition, evidence for ex-post inequality seeking is much weaker than evidence for ex-ante inequality (...) aversion. Because ex-ante inequality aversion is unrelated to risk aversion, we conclude that simple forms of utilitarianism are not relevant for individual judgment of social risk over health. Last, our investigation of precautionary distribution, which would occur when one particular group in the society suffers from background health risk, shows substantial polarization of preferences. (shrink)

Despite the importance of mathematics in our educational systems little is known about how abstract mathematical thinking emerges. Under the uniting thread of mathematical development, we hope to connect researchers from various backgrounds to provide an integrated view of abstract mathematical cognition. Much progress has been made in the last 20 years on how numeracy is acquired. Experimental psychology has brought to light the fact that numerical cognition stems from spatial cognition. The findings from neuroimaging and single cell (...) recording experiments converge to show that numerical representations take place in the intraparietal sulcus. Further research has demonstrated that supplementary neural networks might be recruited to carry out subtasks; for example, the retrieval of arithmetic facts is done by the angular gyrus. Now that the neural networks in charge of basic mathematical cognition are identified, we can move onto the stage where we seek to understand how these basics skills are used to support the acquisition and use of abstract mathematical concepts. (shrink)

The experimental (apparent) space-time transformations connect coordinates altered by length contraction and clock retardation. When clocks are synchronized by means of light signals (Einstein–Poincaré procedure) or by slow clock transport, the experimental space-time. transformations assume the mathematical form of the “Extended space-time transformations”.(4) These reduce to the Lorentz–Poincaré transformations when one of the frames they connect is the fundamental inertial frame. If the synchronization procedure were perfect, the experimental space-time transformations would assume the form of Selleri’s inertial (...) transformations.(5) The real space-time transformations are those which are disclosed when the systematic measurement distortions are corrected. (shrink)

We propose a characterization of mathematical experiments in terms of a setup, a process with an outcome, and an interpretation. Using a broad notion of process, this allows us to consider arithmetic calculations and geometric constructions as components of mathematical experiments. Moreover, we argue that mathematical experiments should be considered within a broader context of an experimental research project. Finally, we present a particular case study of the genesis of a geometric construction to illustrate the experimental use of (...) hand drawings and computers in mathematical practice. (shrink)

The foundation of Mathematics is both a logico-formal issue and an epistemological one. By the first, we mean the explicitation and analysis of formal proof principles, which, largely a posteriori, ground proof on general deduction rules and schemata. By the second, we mean the investigation of the constitutive genesis of concepts and structures, the aim of this paper. This “genealogy of concepts”, so dear to Riemann, Poincaré and Enriques among others, is necessary both in order to enrich the foundational (...) analysis with an often disregarded aspect (the cognitive and historical constitution of mathematical structures) and because of the provable incompleteness of proof principles also in the analysis of deduction. For the purposes of our investigation, we will hint here to a philosophical frame as well as to some recent experimental studies on numerical cognition that support our claim on the cognitive origin and the constitutive role of mathematical intuition. (shrink)

: This study examines the effectiveness of systematic desensitizationtreatment on mathematics anxiety and performance among year one collegestudents. This study employs a quasi-experimental research design. The samplefor this study is drawn based on convenience sampling. The sample consistsof 65 year one students of which 32 are under the experimental group andanother 33 are under to control group. The instruments used in collectingdata are The Adopt and Adapt Fennama-Sherman Mathematics Attitude Scale, Neo-Five-Factory Personality Inventory, MathematicsPerformance Test, and (...) The Systematic Desensitization Module. The study postulates: mathematics anxious students who receivesystematic desensitization treatment would report a reduction in the level ofmathematics anxiety as compared to those mathematics anxious students whodo not receive any treatment; mathematics anxious students who receivesystematic desensitization treatment would perform better as comparedto those mathematics anxious students who do not receive any treatment.Quantitative data is analysed using ANOVAs. (shrink)

In this paper I detail three major mathematical developments that led to the emergence of Yang–Mills theories as the foundation for the standard model of particle physics. In less than 10 years, work on renormalizability, the renormalization group, and lattice quantum field theory highlighted the utility of Yang–Mills type models of quantum field theory by connecting poorly understood candidate dynamical models to emerging experimental results. I use this historical case study to provide lessons for theory construction in physics, and (...) touch on issues related to renormalization group realism from a more historical perspective. In particular, I highlight the fact that much of the hard work in theory construction comes when trying to understand the consequences and representational capacities of a theoretical framework. (shrink)

The role of mathematics in scientific practice is too readily relegated to that of formulating equations that model or describe what is being investigated, and then finding solutions to those equations. I survey the role of mathematics in: 1. Exact solutions of differential equations, especially conformal mapping; and 2. Simulations of solutions to differential equations via numerical methods and via agent-based models; and 3. The use of experimental models to solve equations (a) via physical analogies based on (...) similarity of the form of the equations, such as Prandtl’s soap-film method, and (b) the method of physically similar systems. Two major themes emerge: First, the role of mathematics in science is not well described by deduction from axioms, although it generally involves deductive reasoning. Creative leaps, the integration of experimental or observational evidence, synthesis of ideas from different areas of mathematics, and insight regarding analogous forms are required to find solutions to equations. Second, methods that involve mappings or transformations are in use in disparate contexts, from the purely mathematical context of conformal mapping where it is mathematical objects that are mapped, to the use of concrete physical experimental models, where one concrete thing is shown to correspond to another. (shrink)

Despite its long history and stunning experimental successes, the mathematical foundation of perturbative quantum field theory is still a subject of ongoing research. This book aims at presenting some of the most recent advances in the field, and at reflecting the diversity of approaches and tools invented and currently employed. Both leading experts and comparative newcomers to the field present their latest findings, helping readers to gain a better understanding of not only quantum but also classical field theories. Though (...) the book offers a valuable resource for mathematicians and physicists alike, the focus is more on mathematical developments. This volume consists of four parts: The first Part covers local aspects of perturbative quantum field theory, with an emphasis on the axiomatization of the algebra behind the operator product expansion. The second Part highlights Chern-Simons gauge theories, while the third examines (semi-)classical field theories. In closing, Part 4 addresses factorization homology and factorization algebras. (shrink)

To demarcate the limits of experimental knowledge, we probe the limits of what might be called an experiment. By appeal to examples of scientific practice from astrophysics and analogue gravity, we demonstrate that the reliability of knowledge regarding certain phenomena gained from an experiment is not circumscribed by the manipulability or accessibility of the target phenomena. Rather, the limits of experimental knowledge are set by the extent to which strategies for what we call ‘inductive triangulation’ are available: that (...) is, the validation of the mode of inductive reasoning involved in the source-target inference via appeal to one or more distinct and independent modes of inductive reasoning. When such strategies are able to partially mitigate reasonable doubt, we can take a theory regarding the phenomena to be well supported by experiment. When such strategies are able to fully mitigate reasonable doubt, we can take a theory regarding the phenomena to be established by experiment. There are good reasons to expect the next generation of analogue experiments to provide genuine knowledge of unmanipulable and inaccessible phenomena such that the relevant theories can be understood as well supported. This article is part of a discussion meeting issue ‘The next generation of analogue gravity experiments’. (shrink)

Analysis of online mathematics forums can help reveal how explanation is used by mathematicians; we contend that this use of explanation may help to provide an informal conceptualization of simplicity. We extracted six conjectures from recent philosophical work on the occurrence and characteristics of explanation in mathematics. We then tested these conjectures against a corpus derived from online mathematical discussions. To this end, we employed two techniques, one based on indicator terms, the other on a random sample of (...) comments lacking such indicators. Our findings suggest that explanation is widespread in mathematical practice and that it occurs not only in proofs but also in other mathematical contexts. Our work also provides further evidence for the utility of empirical methods in addressing philosophical problems. (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.