While there are satisfactory answers to the question “How should we teach children mathematics?”, there are no satisfactory answers to the question “What mathematics should we teach children?”. This paper provides an answer to the last question for preschool children (early childhood), although the answer is also applicable to older children. This answer, together with an appropriate methodology on how to teach mathematics, gives a clear conception of the place of mathematics in the children’s world (...) and our role in helping children develop their mathematical abilities. Briefly, children’s mathematics consists of the world of children’s internal activities that they eventually purposefully organize in order to understand and control the outside world and organize their overall activities in it. We need to support a child in mathematical activities that she does spontaneously and in which she shows interest, and we need to teach her mathematics that she is interested in developing through these activities. In doing so, we must be fully aware that the child’s mathematics is part of the child’s world of internal activities and is not outside of it. We help the child develop mathematical abilities by developing them in the context of her world and not outside of it. From the point of view of this conception, the standards established today are limiting and too focused on numbers and geometric figures: these topics are too prominent and elaborated, and other mathematical contents are subordinated to them. Adhering to the standards, we drastically limit the mathematics of the child’s world, hamper the correct mathematical development of a child, and we can turn her away from mathematics. (shrink)

In this handbook, I put into practice my philosophical views on children's mathematics. The handbook contains brief instructions and examples of mathematical activities. In the INSTRUCTIONS section, instructions are given on how, and in part why that way, to help preschool children in their mathematical development. In the ACTIVITIES section, there are examples of activities through which the child develops her mathematical abilities.

This project explored what constitutes “ethical practice of mathematics”. Thematic analysis of ethical practice standards from mathematics-adjacent disciplines (statistics and computing), were combined with two organizational codes of conduct and community input resulting in over 100 items. These analyses identified 29 of the 52 items in the 2018 American Statistical Association Ethical Guidelines for Statistical Practice, and 15 of the 24 additional (unique) items from the 2018 Association of Computing Machinery Code of Ethics for inclusion. Three of (...) the 29 items synthesized from the 2019 American Mathematical Society Code of Ethics, and zero of the Mathematical Association of America Code of Ethics, were identified as reflective of “ethical mathematical practice” beyond items already identified from the other two codes. The community contributed six unique items. Item stems were standardized to, “The ethical mathematics practitioner…”. Invitations to complete the 30-min online survey were shared nationally (US) via Mathematics organization listservs and other widespread emails and announcements. We received 142 individual responses to the national survey, 75% of whom endorsed 41/52 items, with 90–100% endorsing 20/52 items on the survey. Items from different sources were endorsed at both high and low rates. A final thematic analysis yielded 44 items, grouped into “General” (12 items), “Profession” (10 items) and “Scholarship” (11 items). Moreover, for the practitioner in a leader/mentor/supervisor/instructor role, there are an additional 11 items (4 General/7 Professional). These results suggest that the community perceives a much wider range of behaviors by mathematicians to be subject to ethical practice standards than had been previously included in professional organization codes. The results provide evidence against the argument that mathematics practitioners engaged in “pure” or “theoretical” work have minimal, small, or no ethical obligations. (shrink)

Haskell Curry’s philosophy of mathematics is really a form of “structuralism” rather than “formalism” despite Curry’s own description of it as formalist (Seldin 2011). This paper explains Curry’s actual view by a formal analysis of a simple example. This analysis is extended to solve Keränen’s (2001) identity problem for structuralism, confirming Leitgeb’s (2020a, b) solution, and further clarifies structural ontology. Curry’s methods answer philosophical questions by employing a standard mathematical method, which is a virtue of the “methodological autonomy” emphasized (...) by Curry (1951, 1963) and more recently with greater clarity by Maddy (1997, 2007). (shrink)

This paper discusses Michael Dummett’s attempt to base the use of intuitionistic logic in mathematics on a proof-conditional semantics. This project is shown to face significant obstacles resulting from the existence of variants of standard intuitionistic logic. In order to overcome these obstacles, Dummett and his followers must give an intuitionistically acceptable completeness proof for intuitionistic logic relative to the BHK interpretation of the logical constants, but there are reasons to doubt that such a proof is possible. The paper (...) concludes by proposing an alternative way of thinking about why one should use intuitionistic logic when doing mathematics. (shrink)

It is argued that the philosophical and epistemological beliefs about the nature of mathematics have a significant influence on the way mathematics is taught at school. In this paper, the philosophy of mathematics of the NCTM's Standards is investigated by examining is explicit assumptions regarding the teaching and learning of school mathematics. The main conceptual tool used for this purpose is the model of two dichotomous philosophies of mathematics-absolutist versus- fallibilist and their relation to (...) mathematics pedagogy. The main conclusion is that a fallibilist view of mathematics is assumed in the Standards and that most of its pedagogical assumptions and approaches are based on this philosophy. (shrink)

This essay discusses ways in which contemporary mathematical logic may be reconciled with John Dewey’s logical theory. Standard formal techniques drawn from dynamic modal logic, situation theory, generative grammar, generalized quantifier theory, category theory, lambda calculi, game theoretic semantics, network exchange theory, etc., are accommodated within a framework consistent with Dewey’s Logic: The Theory of Inquiry (1938). This essay outlines some basic features of Dewey’s logical theory, working in a top-down fashion through various technical notions pertaining to existential and ideational (...) aspects of inquiry, from contextual notions of situations and ideologies, to primitive operational and empirical notions of modes of action and particular qualities. The aim here is to display the scope of Dewey’s logical theory and thus establish a framework relative to which details may be explored without losing sight of their larger significance. (shrink)

When reassessing the role of Debreu's axiomatic method in economics, one has to explain both its success and unpopularity; one has to explain the "bright shadow" Debreu cast on the discipline: sheltering, threatening, and difficult to pin down. Debreu himself did not expect to have such an influence. Before he received the Bank of Sweden Prize in 1983 he had never openly engaged with the methodology or politics of mathematical economics. When in several speeches he later rigorously distinguished mathematical form (...) from economic content and claimed this as the virtue of mathematical economics, he did both: he defended mathematical reasoning against the theoretical innovations since the 1970s and expressed remorse for having promised too much because it cannot support claims about economic content. The analysis of this twofold role of Debreu's axiomatic method raises issues of the social and political responsibility of economists over and above standard epistemic issues. (shrink)

Summary The National Curriculum was introduced into British primary schools in 1989 to raise standards of attainment, especially in the basic skills of English and mathematics. Has this expensive innovation succeeded? This paper analyses the mathematics standards of eight cohorts of Year 6 children from five randomly selected primary schools within one Local Education Authority (n=1503) who had all done Mathematics 11 from 1989 to 1996. Examination of the means of the standardised mathematics scores (...) for each cohort reveals no evidence of rising standards in attainment over the eight years. There are fewer over? and underachievers than one would expect with a tendency for compression of scores around the mean. This paper questions the efficacy of the National Curriculum, assessment procedures and the publication of school performance tables in raising standards in mathematics at Key Stage 2. The need for an effective system of monitoring educational standards throughout the UK is reiterated. (shrink)

Mathematical proof is the primary form of justification for mathematical knowledge, but in order to count as a proper justification for a piece of mathematical knowl- edge, a mathematical proof must be rigorous. What does it mean then for a mathematical proof to be rigorous? According to what I shall call the standard view, a mathematical proof is rigorous if and only if it can be routinely translated into a formal proof. The standard view is almost an orthodoxy among contemporary (...) mathematicians, and is endorsed by many logicians and philosophers, but it has also been heavily criticized in the philosophy of mathematics literature. Progress on the debate between the proponents and opponents of the standard view is, however, currently blocked by a major obstacle, namely the absence of a precise formulation of it. To remedy this deficiency, I undertake in this paper to provide a precise formulation and a thorough evaluation of the standard view of mathematical rigor. The upshot of this study is that the standard view is more robust to criticisms than it transpires from the various arguments advanced against it, but that it also requires a certain conception of how mathematical proofs are judged to be rigorous in mathematical practice, a conception that can be challenged on empirical grounds by exhibiting rigor judgments of mathematical proofs in mathematical practice conflicting with it. (shrink)

Abstract.ΠMTL is a schematic extension of the monoidal t-norm based logic (MTL) by the characteristic axioms of product logic. In this paper we prove that ΠMTL satisfies the standard completeness theorem. From the algebraic point of view, we show that the class of ΠMTL-algebras (bounded commutative cancellative residuated l-monoids) in the real unit interval [0,1] generates the variety of all ΠMTL-algebras.

Gravitation is interpreted to be an “ontomathematical” force or interaction rather than an only physical one. That approach restores Newton’s original design of universal gravitation in the framework of “The Mathematical Principles of Natural Philosophy”, which allows for Einstein’s special and general relativity to be also reinterpreted ontomathematically. The entanglement theory of quantum gravitation is inherently involved also ontomathematically by virtue of the consideration of the qubit Hilbert space after entanglement as the Fourier counterpart of pseudo-Riemannian space. Gravitation can be (...) also interpreted as purely mathematical or logical “force” or “interaction” as a corollary from its ontomathematical (rather than physical) realization. The ontomathematical approach to gravitation is implicit in general relativity equating it to operators in pseudo-Riemannian space obeying the Einstein field equation and also well-known by the “geometrization of physics”. Quantum mechanics shares the same by the separable complex Hilbert space and defining “physical quantity” by the Hermitian operators on it. One can interpret special Minkowski space involved by special relativity and the qubit Hilbert space of quantum information as Fourier counterparts immediately noticing that general relativity means gravitation as the Fourier counterpart of non-Hermitian operators implying non-unitarity and the violation of energy conservation and thus destroying Pauli’s particle paradigm. Since the Standard model obeys it, this explains the impossibility of “quantum gravitation” in any framework conservatively generalizing the Standard model so that it would include gravitation along with electromagnetic, weak, and strong interactions. Einstein’s geometrization of gravitation can be continued into a purely mathematical theory of it following Euclid’s realization for geometry to be exhaustively built in a deductive and axiomatic way as well as Riemann’s parametrization of all the class of Euclidean and non-Euclidean geometries by “space curvature”, then being generalized to Minkowski space as the operators on pseudo-Riemannian space as the Einstein field equation means gravitation. The transition from mathematical gravitation to logical one can rely on the historical lesson of the pair of Lobachevski’s and Riemann’s approaches now “reversely”, i.e., from the latter to the former. Logical gravitation is linkable to Hegel’s dialectical logic and ontological dialectics abandoning their interpretations as a new zero logic substituting classical propionyl logic. The approach of ontomathematics generalizing that of ontology, traceable even to Aristotle’s reformation of Plato’s doctrine, needs Hegel’s doctrine to be formalized as a first-order logic naturally containing Boolean algebra, isomorphic to both classical propositional logic and set theory being the class of all first-order logics, as a sub-logic along with Peano arithmetic as another sub-logic. The first-order logic at issue is called Hilbert arithmetic and elaborated in detail in other papers. It allows for both self-foundation of mathematics to be internally proved as complete and furthermore, quantum mechanics reinterpreted as quantum information to be included by the qubit Hilbert space interpretable in turn as a dual and physical counterpart of Hilbert arithmetic in a narrow sense, that is, both counterparts constitute Hilbert arithmetic in a wide sense, being mathematical and physical simultaneously and thus overcoming the Cartesian dualism of “body” gapped from “mind” by an abyss. Then, the proper philosophical interpretation of gravitation to be the fundamental ontomathematical force or interaction overcomes the ridiculous belief of the Big Bang wrongly alleged to be a scientific theory. Ontomathematical gravitation suggests an omnipresent and omnitemporal medium of “God’s” creation “ex nihilo” following only the natural necessity of quantum-information conservation particularly and locally manifested as energy conservation. (shrink)

ObjectiveAs the coronavirus disease 2019 pandemic spread across Shanghai, China, in late February 2022 and protective measures to mitigate its impact were enacted, this study aimed to estimate how home quarantine affected the mental health of preschool children in Shanghai, China and explore the association between lifestyle factors and mental health during this special period.MethodsA cross-sectional online survey of 2,110 preschool students from Shanghai, China, was conducted during May 20–25,2022. Preschooler’ mental health and daily activities were reported by (...) parents.ResultsThe sample involved 2,110 children with a mean age of 4.65 years [standard deviation : 0.91, range: 3–6 years]. Boys and children whose mother’s education level were college and high school had higher rate of mental health problems. Boys had significantly higher rates of peer problems and prosocial behaviors than girls. The 3-year-old group had significantly higher rates of prosocial behaviors than other groups. As compared to the Shanghai norm and the SDQ results of preschool children in Shanghai in 2019, there were a significant decrease in emotional symptoms score, as well as a significant increase in conduct problems score. Additionally, peer problems score significantly increased compared to SH2019. Decreased time spent on daily sleep was associated with the increased risk for preschoolers’ mental health problems.ConclusionThere was an increase in the frequency of emotional and behavioral problems, especially regarding conduct problems and peer problems, in preschool children during the COVID-19 home quarantine in Shanghai, China. Boys, younger preschool children and children whose mother’s education level were college and high school may be especially vulnerable to emotional and behavioral problems. It was also found that decreased time spent on sleep may aggravate preschool children’s mental health problems. It may be beneficial to differentiate and focus on conducting psychoeducation and implementing psycho-behavioral interventions to solve these issues. (shrink)

Analogical reasoning by comparison is considered a special case of inductive reasoning, which is fundamental to the scientific method. By reasoning analogically, learners can abstract the underlying commonalities of several entities, thereby ignoring single objects’ superficial features. We tested whether different task environments designed to trigger analogical reasoning by comparison would support preschoolers’ induction of the concept of material kind to predict and explain objects’ floating or sinking as a central aspect of scientific reasoning. Specifically, in two experiments, we investigated (...) whether the number of presented objects (one versus two standards), consisting of a specific material and the labeling of objects with the respective material name, would benefit preschoolers’ material-based inferences. For each item set used in both experiments, we asked the children ( N = 59 in Experiment 1, N = 99 in Experiment 2) to predict an object’s floating or sinking by matching it to the standards and to verbally explain their selections. As expected, we found a significant effect for the number of standards in both experiments on the prediction task, suggesting that children successfully induced the relevance of material kind by comparison. However, labels did not increase the effect of the standards. In Experiment 2, we found that the children could transfer their conceptual knowledge on material kind but that transfer performance did not differ among the task environments. Our findings suggest that tasks inviting analogical reasoning by comparison with two standards are useful for promoting young children’s scientific reasoning. (shrink)

The call for evidence-based practice in education emphasizes the need for research to provide evidence for particular fields of educational practice. With this systematic literature review we summarize and analyze aggregated effectiveness information from 41 meta-analyses published between 2004 and 2019 to inform evidence-based practice in a particular field. In line with target specifications in education that are provided for a certain school subject and educational level, we developed and adopted a selection heuristic for filtering aggregated effect sizes specific to (...) both science and mathematics education and the secondary student population. The results include 78 context-specific aggregated effect sizes based on data from over one million students. The findings encompass a multitude of different teaching strategies, most of which offer a measurable advantage to alternatives. Findings demonstrate that context-specific effect size information may often differ from more general effect size information on teaching effectiveness and adherence to quality standards varies in sampled meta-analyses. Thus, although meta-analytic research has strongly developed over the last few years, providing context-specific and high-quality evidence still needs to be a focus in the field of secondary mathematics and science teaching and beyond. (shrink)

The present paper concerns the Wittgenstein ontology project: an attempt to create a Semantic Web representation of Ludwig Wittgenstein’s philosophy. The project has been in development since 2006, and its current state enables users to search for information about Wittgenstein-related documents and the documents themselves. However, the developers have much more ambitious goals: they attempt to provide a philosophical subject matter knowledge base that would comprise the claims and concepts formulated by the philosopher. The current knowledge representation technology is not (...) well-suited for this task, and a non-standard approach is required. The creators of the Wittgenstein ontology project are aware of this fact; recently, they have been discussing conceptual devices adjusting the technology to their needs. The main goal of this paper is to present examples of a representation of philosophical content that make use of both the devices already proposed and some new inventions. The represented content comes from the Tractatus Logico-Philosophicus; more specifically, its theses concerning the problems in philosophy of mathematics. (shrink)

This book develops new forms of logic: Operator Logic, Probabilistic Operator Logic and Quantum Operator Logic. It then proceeds to create a new view of metaphysics, Relativistic Quantum Metaphysics, for physical Reality. It then derives the form of The Standard Model of Elementary Particles. In particular it derives the origin of parity violation, the origin of the Strong interactions, and the origin of its peculiar symmetry. Also developed are new formalisms for Logic that are of interest in themselves. While (...) class='Hi'>mathematics is essential in the latter stages of the book it is presented with sufficient text discussion to make what it is doing understandable to the non-mathematical reader. Generally the jargon of Philosophy, Logic and Physics is avoided as much as possible. But the second part of this book is very mathematical of necessity. In it the form of The Standard Model is derived in detail from a fundamental set of principles. (shrink)

The origin of the genetic code has been attributed in part to an accidental assignment of codons to amino acids. Although several lines of evidence indicate the subsequent expansion and improvement of the genetic code, the hypothesis of Francis Crick concerning a frozen accident occurring at the early stage of genetic code evolution is still widely accepted. Considering Crick’s hypothesis, mathematical descriptions of hypothetical scenarios involving a huge number of possible coexisting random genetic codes could be very important to explain (...) the origin and evolution of a selected genetic code. This work aims to contribute in this regard, that is, it provides a theoretical framework in which statistical parameters of error functions are calculated. Given a genetic code and an amino acid property, the functional code robustness is estimated by means of a known error function. In this work, using analytical calculations, general expressions for the average and standard deviation of the error function distributions of completely random codes with standard stop codons were obtained. As a possible biological application of these results, any set of amino acids and any pure or mixed amino acid properties can be used in the calculations, such that, in case of having to select a set of amino acids to create a genetic code, possible advantages of natural selection of the genetic codes could be discussed. (shrink)

We review some results in the theory of non-relativistic quantum unstable systems. We account for the most important definitions of quantum resonances that we identify with unstable quantum systems. Then, we recall the properties and construction of Gamow states as vectors in some extensions of Hilbert spaces, called Rigged Hilbert Spaces. Gamow states account for the purely exponential decaying part of a resonance; the experimental exponential decay for long periods of time physically characterizes a resonance. We briefly discuss one of (...) the most usual models for resonances: the Friedrichs model. Using an algebraic formalism for states and observables, we show that Gamow states cannot be pure states or mixtures from a standard view point. We discuss some additional properties of Gamow states, such as the possibility of obtaining mean values of certain observables on Gamow states. A modification of the time evolution law for the linear space spanned by Gamow shows that some non-commuting observables on this space become commuting for large values of time. We apply Gamow states for a possible explanation of the Loschmidt echo. (shrink)

When did the concept of model begin to be used in mathematics? This question appears at first somewhat surprising since “model” is such a standard term now in the discourse on mathematics and “modelling” such a standard activity that it seems to be well established since long. The paper shows that the term— in the intended epistemological meaning—emerged rather recently and tries to reveal in which mathematical contexts it became established. The paper discusses various layers of argumentations and (...) reflections in order to unravel and reach the pertinent kernel of the issue. The specific points of this paper are the difference in the epistemological concept to the usually discussed notions of model and the difference between conceptions implied in mathematical practices and their becoming conscious in proper reflections of mathematicians. (shrink)

To explore the impact of academic burden on the physical and mental health of primary school students, combined with the results of the Programme for International Student Assessment report in 2018, the relationship among the development of mathematical literacy, mathematics academic burden, and the physical and mental health of primary school students is studied. First, the relationship between mathematical literacy and mathematics anxiety is analyzed, and related influencing factors and measurement methods of mathematics anxiety are introduced. A (...) questionnaire is then designed for primary school students’ mathematical stress, and the reliability and validity of the designed questionnaire are tested. Finally, a questionnaire survey is conducted on students, parents, and teachers in the third, fourth, and fifth grades of three standardized public primary schools. The results of the questionnaire survey show that students, teachers, and parents have a general understanding of the mathematics academic burden of primary school students at this stage. A total of 70% of teachers believe that primary school students have a heavy mathematics burden; 50% of parents think that primary school students are under heavy academic stress; 70% of primary school students believe that the heavy mathematics burden leads to reduced sleep time and extracurricular activities, which has a serious impact on the physical and mental health of primary school students. This research provides a reference for improving the current balance between education and students’ physical and mental health in China. (shrink)

Standard histories of mathematics and of analytic philosophy contend that work on the foundations of mathematics was motivated by a crisis such as the discovery of paradoxes in set theory or the discovery of non-Euclidean geometries. Recent scholarship, however, casts doubt on the standard histories, opening the way for consideration of an alternative motive for the study of the foundations of mathematics—unification. Work on foundations has shown that diverse mathematical practices could be integrated into a single framework (...) of axiomatic systems and that much of mathematics could be expressed in a single language. The new framework was the product of an interdisciplinary coalition whose ideas resemble those later adopted by the Vienna Circle and logical empiricists. (shrink)

What is mathematics about? Does the subject-matter of mathematics exist independently of the mind or are they mental constructions? How do we know mathematics? Is mathematical knowledge logical knowledge? And how is mathematics applied to the material world? In this introduction to the philosophy of mathematics, Michele Friend examines these and other ontological and epistemological problems raised by the content and practice of mathematics. Aimed at a readership with limited proficiency in mathematics but (...) with some experience of formal logic it seeks to strike a balance between conceptual accessibility and correct representation of the issues. Friend examines the standard theories of mathematics - Platonism, realism, logicism, formalism, constructivism and structuralism - as well as some less standard theories such as psychologism, fictionalism and Meinongian philosophy of mathematics. In each case Friend explains what characterises the position and where the divisions between them lie, including some of the arguments in favour and against each. This book also explores particular questions that occupy present-day philosophers and mathematicians such as the problem of infinity, mathematical intuition and the relationship, if any, between the philosophy of mathematics and the practice of mathematics. Taking in the canonical ideas of Aristotle, Kant, Frege and Whitehead and Russell as well as the challenging and innovative work of recent philosophers like Benacerraf, Hellman, Maddy and Shapiro, Friend provides a balanced and accessible introduction suitable for upper-level undergraduate courses and the non-specialist. (shrink)

The lives of few mathematicians offer the drama that is presented by the life of L. E. J. Brouwer, correctly identified on the cover of this book as a topologist, intuitionist, and philosopher, and before we go any further, it will be worth indicating why.It is not just that Brouwer would rank high among mathematicians for his work in topology alone: he set standards for rigour and created a theory of dimension for topological spaces, and his fixed-point theorem is (...) of great importance. Nor is it just that his philosophy of intuitionism created a new and vibrant branch of logic, that is, arguably, the only viable alternative to naïve, classical logic outside the enclave of professional logicians. Rather, it is that there is a tension, popularly taken to amount to a contradiction, in the fundamental ideas behind his topology and his intuitionism. Added to that, he took a highly principled stand on mathematical issues that led him into confrontations with major figures and a certain degre .. (shrink)

This is a concise introductory textbook for a one semester course in the history and philosophy of mathematics. It is written for mathematics majors, philosophy students, history of science students and secondary school mathematics teachers. The only prerequisite is a solid command of pre-calculus mathematics. It is shorter than the standard textbooks in that area and thus more accessible to students who have trouble coping with vast amounts of reading. Furthermore, there are many detailed explanations of (...) the important mathematical procedures actually used by famous mathematicians, giving more mathematically talented students a greater opportunity to learn the history and philosophy by way of problem solving. Several important philosophical topics are pursued throughout the text, giving the student an opportunity to come to a full and consistent knowledge of their development. These topics include infinity, the nature of motion, and Platonism. This book offers, in fewer pages, a deep penetration into the key mathematical and philosophical aspects of the history of mathematics. (shrink)

Let mathematical justification be the kind of justification obtained when a mathematician provides a proof of a theorem. Are Gettier cases possible for this kind of justification? At first sight we might think not: The standard for mathematical justification is proof and, since proof is bound at the hip with truth, there is no possibility of having an epistemically lucky justification of a true mathematical proposition. In this paper, I argue that Gettier cases are possible (and indeed actual) in mathematical (...) reasoning. By analysing these cases, I suggest that the Gettier phenomenon indicates some upshots for actual mathematical practice. (shrink)

The mathematical formalism of quantum theory has been established for nearly a century, yet its physical foundations remain elusive. In recent decades, connections between quantum theory and information theory have garnered increasing attention. This study presents a physical derivation of the mathematical formalism quantum theory based on information-theoretic considerations in physical systems. We postulate that quantum systems are characterized by single independent adjustable variables. Utilizing this physical postulate along with the conservation of total probability, we derive the standard Hilbert space (...) formalism of quantum theory, including the Born probability rule. This comprehensive derivation of quantum theory offers a clear and concise physical foundation for the mathematical formalism of quantum mechanics. (shrink)

Newton characterizes the reasoning of Principia Mathematica as geometrical. He emulates classical geometry by displaying, in diagrams, the objects of his reasoning and comparisons between them. Examination of Newton’s unpublished texts shows that Newton conceives geometry as the science of measurement. On this view, all measurement ultimately involves the literal juxtaposition—the putting-together in space—of the item to be measured with a measure, whose dimensions serve as the standard of reference, so that all quantity is ultimately related to spatial extension. I (...) use this conception of Newton’s project to explain the organization and proofs of the first theorems of mechanics to appear in the Principia. The placementof Kepler’s rule of areas as the first proposition, and the manner in which Newton proves it, appear natural on the supposition that Newton seeks a measure, in the sense of a moveable spatial quantity, of time. I argue that Newton proceeds in this way so that his reasoning can have the ostensive certainty of geometry. (shrink)

Although there are a few books available that give brief surveys of a variety of nonstandard logics, there is a growing need for a critical presentation providing both a greater depth and breadth of insight into these logics. This book assembles a wider and deeper view of the many potentially applicable logics. Three appendixes provide short tutorials on classical logic and modal logics, and give a brief introduction to the existing literature on the logical aspects of probability theory. These tutorials (...) and the bibliography included in the appendixes provide useful reference material for the reader. (shrink)

In the present paper we show that any at most countable linearly-ordered commutative residuated lattice can be embedded into a commutative residuated lattice on the real unit interval [0, 1]. We use this result to show that Esteva and Godo''s logic MTL is complete with respect to interpretations into commutative residuated lattices on [0, 1]. This solves an open problem raised in.

A complete philosophy of mathematics must address Paul Benacerraf’s dilemma. The requirements of a general semantics for the truth of mathematical theorems that coheres also with the meaning and truth conditions for non-mathematical sentences, according to Benacerraf, should ideally be coupled with an adequate epistemology for the discovery of mathematical knowledge. Standard approaches to the philosophy of mathematics are criticized against their own merits and against the background of Benacerraf’s dilemma, particularly with respect to the problem of understanding (...) the distinction between pure and applied mathematics and the effectiveness of applied mathematics in the natural sciences and engineering. The evaluation of these alternatives provides the basis for articulating a philosophically advantageous Aristotelian inherence concept of mathematical entities. An inherence account solves Benacerraf’s dilemma by interpreting mathematical entities as nominalizations of structural spatiotemporal properties inhering in existent spatiotemporal entities. (shrink)

The aim of this paper is to provide epistemic reasons for investigating the notions of informal rigour and informal provability. I argue that the standard view of mathematical proof and rigour yields an implausible account of mathematical knowledge, and falls short of explaining the success of mathematical practice. I conclude that careful consideration of mathematical practice urges us to pursue a theory of informal provability.

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)

Assuming no previous study in logic, this informal yet rigorous text covers the material of a standard undergraduate first course in mathematical logic, using natural deduction and leading up to the completeness theorem for first-order logic. At each stage of the text, the reader is given an intuition based on standard mathematical practice, which is subsequently developed with clean formal mathematics. Alongside the practical examples, readers learn what can and can't be calculated; for example the correctness of a derivation (...) proving a given sequent can be tested mechanically, but there is no general mechanical test for the existence of a derivation proving the given sequent. The undecidability results are proved rigorously in an optional final chapter, assuming Matiyasevich's theorem characterising the computably enumerable relations. Rigorous proofs of the adequacy and completeness proofs of the relevant logics are provided, with careful attention to the languages involved. Optinal sections discuss the classification of mathematical structures by first-order theories; the required theory of cardinality is developed from scratch. Throughout the book there are notes on historical aspects of the material, and connections with linguistics and computer science, and the discussion of syntax and semantics is influenced by modern linguistic approaches. Two basic themes in recent cognitive science studies of actual human reasoning are also introduced. Including extensive exercises and selected solutions, this text is ideal for students in logic, mathematics, philosophy, and computer science. (shrink)

Mathematical rigor is commonly formulated by mathematicians and philosophers using the notion of proof gap: a mathematical proof is rig­orous when there is no gaps in the mathematical reasoning of the proof. Any philosophical approach to mathematical rigor along this line requires then an account of what a proof gap is. However, the notion of proof gap makes sense only relatively to a given conception of valid mathematical reasoning, i.e., to a given conception of the validity of mathematical inference. A (...) proof gap can in particular be conceived as a failure in drawing a valid mathematical inference. The aim of this paper is to discuss two possible views of the validity of math­ematical inference with respect to their capacity to yield a plausible account of the intuitive notion(s) of proof gap present in mathematical practice. The first view is the one provided by the contemporary standards of mathematical rigor: a mathematical inference is valid if and only if its conclusion can be formally derived from its premises. We will argue that this conception does not lead to a plausible account of the intuitive notion(s) of proof gap. The second view is based on a new account of the validity of inference proposed by Prawitz: an inference is valid if and only if it consists in an operation that provides a ground for its conclusion given (previously obtained) grounds for its premises. We will first specify Prawitz's account to mathematical inference and we will then argue that the resulting ground-based account is able to capture various intuitive notions of proof gap as different types of failure in drawing valid mathematical inferences. We conclude that the ground-based account ap­pears of particular interest for the philosophy of mathematical practice, and we finally raise several challenges facing a full development of a ground-based account of the notions of mathematical rigor, proof gap and the validity of mathematical inference. (shrink)

The foundations of mathematics include mathematical logic, set theory, recursion theory, model theory, and Gödel's incompleteness theorems. Professor Wolf provides here a guide that any interested reader with some post-calculus experience in mathematics can read, enjoy, and learn from. It could also serve as a textbook for courses in the foundations of mathematics, at the undergraduate or graduate level. The book is deliberately less structured and more user-friendly than standard texts on foundations, so will also be attractive (...) to those outside the classroom environment wanting to learn about the subject. (shrink)

The aim of this article is to give the proof-theoretic analysis of various subsystems of Feferman's theory T1 for explicit mathematics which contain the non-constructive μ-operator and join. We make use of standard proof-theoretic techniques such as cut-elimination of appropriate semiformal systems and asymmetrical interpretations in standard structures for explicit mathematics.

In this work we argue that there is no strong demarcation between pure and applied mathematics. We show this first by stressing non-deductive components within pure mathematics, like axiomatization and theory-building in general. We also stress the “purer” components of applied mathematics, like the theory of the models that are concerned with practical purposes. We further show that some mathematical theories can be viewed through either a pure or applied lens. These different lenses are tied to different (...) communities, which endorse different evaluative standards for theories. We evaluate the distinction between pure and applied mathematics from a late Wittgensteinian perspective. We note that the classical exegesis of the later Wittgenstein’s philosophy of mathematics, due to Maddy, leads to a clear-cut but misguided demarcation. We then turn our attention to a more niche interpretation of Wittgenstein by Dawson, which captures aspects of the aforementioned distinction more accurately. Building on this newer, maverick interpretation of the later Wittgenstein’s philosophy of mathematics, and endorsing an extended notion of meaning as use which includes social, mundane uses, we elaborate a fuzzy, but more realistic, demarcation. This demarcation, relying on family resemblance, is based on how direct and intended technical applications are, the kind of evaluative standards featured, and the range of rhetorical purposes at stake. (shrink)

The traditional way of presenting mathematical knowledge is logical deduction, which implies a monolithic structure with topics in a strict hierarchical relationship. Despite many recent developments and methodical inventions in mathematics education, many curricula are still close in spirit to this hierarchical structure. However, this organisation of mathematical ideas may not be the most conducive way for learning mathematics. In this paper, we suggest that flattening curricula by developing self-contained micro topics and by providing multiple entry points to (...) knowledge by making the dependency graph of notions and subfields as sparse as possible could improve the effectiveness of teaching mathematics. We argue that a less strictly hierarchical schedule in mathematics education can decrease mathematics anxiety and can prevent students from ‘losing the thread’ somewhere in the process. This proposal implies a radical re-evaluation of standard teaching methods. As such, it parallels philosophical deconstruction. We provide two examples of how the micro topics can be implemented and consider some possible criticisms of the method. A full-scale and instantaneous change in curricula is neither feasible nor desirable. Here, we aim to change the prevalent attitude of educators by starting a conversation about the flat curriculum alternative. (shrink)

Benjamin Callard has recently suggested that causation between Platonic objects—standardly understood as atemporal and non-spatial—and spatio-temporal objects is not ‘a priori’ unintelligible. He considers the reasons some have given for its purported unintelligibility: apparent impossibility of energy transference, absence of physical contact, etc. He suggests that these considerations fail to rule out a priori Platonic-object causation. However, he has overlooked one important issue. Platonic objects must causally affect different objects differently, and different Platonic objects must causally affect the same objects (...) differently. How are Platonic objects—ones outside space and time—supposed to do that? CiteULike Connotea Del.icio.us What's this? (shrink)

This thesis provides an introduction to the physics of the Standard Model and beyond, and to the methods used to analyse Large Hadron Collider (LHC) data. The 'hierarchy problem', astrophysical data and experiments on neutrinos indicate that new physics can be expected at the now accessible TeV scale. This work investigates extensions of the Standard Model with gravitons and gravitinos (in the context of supergravity). The production of these particles in association with jets is studied as one of the most (...) promising avenues for researching new physics at the LHC. Advanced simulation techniques and tools, such as algorithms allowing the computation of Feynman graphs and helicity amplitudes are first developed and then employed. (shrink)

David Hume is the most influential precursor of modern empiri cism. By modern empiricism, I intend a belief that all cognitive conflicts can be resolved, in principle, by either appeal to matters offact, via scientific procedure, or by appeal to some sets of natural or conventional standards, whether linguistic, mathematical, aes thetic or political. This belief itself is a consequent of an old appre hension that all synthetic knowledge is based on experience, and that the rest can be reduced (...) to a set of self-evident truths. In this broad sense, Modern Empiricism encompasses classes, such as Logi cal Empiricism, Logical Atomism and Philosophical Analysis, and unique individuals such as Russell and Moore. It excludes, thereby, the present day continental philosophies, such as Thomism, Exist entialism, and Dialectical Materialism. Modern empiricists, to be sure, are influenced by many other phi losophers. Locke, Berkeley, and Mill, among the classical empiri cists, and Leibniz and Kant, among the rationalists (the former especially on the logico-mathematical side) in one way or other are responsible for the appearance of empiricism in its new form. But none of them were as influential as Hume. This, by itself is not news. Weinberg, in his well-known book, An Examination of Logical Positivism, observes that: Many, if not all, of the principal doctrines of contemporary positivism derive from Hume. (shrink)

The present paper deals with the predicate version MTL of the logic MTL by Esteva and Godo. We introduce a Kripke semantics for it, along the lines of Ono''s Kripke semantics for the predicate version of FLew (cf. [O85]), and we prove a completeness theorem. Then we prove that every predicate logic between MTL and classical predicate logic is undecidable. Finally, we prove that MTL is complete with respect to the standard semantics, i.e., with respect to Kripke frames on the (...) real interval [0,1], or equivalently, with respect to MTL-algebras whose lattice reduct is [0,1] with the usual order. (shrink)

Considered by many to be Abraham Robinson's magnum opus, this book offers an explanation of the development and applications of non-standard analysis by the mathematician who founded the subject. Non-standard analysis grew out of Robinson's attempt to resolve the contradictions posed by infinitesimals within calculus. He introduced this new subject in a seminar at Princeton in 1960, and it remains as controversial today as it was then. This paperback reprint of the 1974 revised edition is indispensable reading for anyone interested (...) in non-standard analysis. It treats in rich detail many areas of application, including topology, functions of a real variable, functions of a complex variable, and normed linear spaces, together with problems of boundary layer flow of viscous fluids and rederivations of Saint-Venant's hypothesis concerning the distribution of stresses in an elastic body. (shrink)