Stable Algebraic Topology and Stable Topological Algebra

Algebraic topology is a young subject, and its foundations arenot yet firmly in place. I shall give some history, examplesand modern developments in that part of the subject called stablealgebraic topology, or stable homotopy theory. This is by farthe most calculationally accessible part of algebraic topology,although it is also the least intuitively grounded in visualizablegeometric objects. It has a great many applications to othersubjects such as algebraic geometry and geometric topology.Time will not allow me to say as much as I would like aboutthat. Rather, I shall emphasize some foundational issues thathave been central to this part of algebraic topology since theearly 1960s, but that have been satisfactorily resolved onlyin the last few years. 1991 Mathematics Subject Classification55P42, 55N20.     

An Invitation to Infinitary Computability

In the last two decades, various machine models of computability have been generalized to work in the transfinite. I will give an overview of these models, some of the main results concerning them and some reasons why studying them is of interest outside of foundational considerations. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The product representation theorem for interlaced pre-bilattices: some historical remarks

This note aims to unravel the history of the Product Representation Theorem for Interlaced Pre-bilattices. We will see that it has its lattice-theoretic roots in early attempts to solve one of the problems in Birkhoff’s Lattice Theory. The theorem was presented in its full generality by Czédli, Huhn and Szabó at a conference in Szeged, Hungary in 1980 (and published in 1983). This was several years before Ginsberg introduced bilattices at a conference on artificial intelligence in 1986 and in his foundational paper in 1988.     

Anti-Substitution Intuitions and the Content of Belief Reports

Philosophers of language traditionally take it that anti-substitution intuitions teach us about the content of belief reports. Jennifer Saul [1997, 2002 (with David Braun), 2007] challenges this lesson. Here I offer a response to Saul’s challenge. In the first two sections of the article, I present a common sense justification for drawing conclusions about content from anti-substitution intuitions. Then, in Sect. 3, I outline Saul’s challenge—what she calls ‘the Enlightenment Problem’. Finally, in Sect. 4, I argue that Saul’s challenge does not undermine the common sense justification presented in Sects. 1 and 2. I avoid the challenge by arguing that anti-substitution intuitions are not directly sensitive to the content of the sentences that produce them, but rather to the possibility that one could have distinct ways of thinking about an object.     

I Can Trust You Now … But Not Later: An Explanation of Testimonial Knowledge in Children

Children learn and come to know things about the world at a very young age through the testimony of their caregivers. The challenge comes in explaining how children acquire such knowledge. Since children indiscriminately receive testimony, their testimony-based beliefs seem unreliable, and, consequently, should fail to qualify as knowledge. In this paper I discuss some attempted explanations by Sandy Goldberg and John Greco and argue that they fail. I go on to suggest that what generates the problem is a hidden assumption that the standards for testimonial knowledge are invariant between children and cognitively mature adults. I propose that in order to adequately explain how children acquire testimonial knowledge we should reject this hidden assumption. I then argue that understanding knowledge in terms of intellectual skills gives us a plausible framework to do so.     

Complex calculus via foundation theorems

It was shown recently that continuous differentiation theory can be founded on a natural isometric linear isomorphism. We develop in this paper an analogous theory for complex differentiation; however, this requires different foundational isomorphism.     

Macrocosmos/microcosmos: celestial mechanics and old quantum theory

The Bohr atom was a solar system in miniature. Despite many deep foundational questions related to the origin of quantized motion, rapid progress was made in its mathematical development and its apparently successful application to spectral line series. In United States, where celestial mechanics flourished throughout the 19th and well into the 20th century, mathematicians and physicists were well prepared for just this sort of problem and made it their own far faster than many areas of the new physics. This paper examines the link between classical problems of perturbation theory, three-body and N-body orbital trajectories, the Hamilton–Jacobi equation, and the old quantum theory. I discuss why it was comparatively easy for American applied mathematicians, astronomers, and mathematical physicists to make significant contributions quickly to quantum theory and why further progress toward quantum mechanics by the same cohort was, in contrast, so slow.     

Rethinking geometrical exactness

A crucial concern of early modern geometry was fixing appropriate norms for deciding whether some objects, procedures, or arguments should or should not be allowed into it. According to Bos, this is the exactness concern. I argue that Descartes’s way of responding to this concern was to suggest an appropriate conservative extension of Euclid’s plane geometry (EPG). In Section 2, I outline the exactness concern as, I think, it appeared to Descartes. In Section 3, I account for Descartes’s views on exactness and for his attitude towards the most common sorts of constructions in classical geometry. I also explain in which sense his geometry can be conceived as a conservative extension of EPG. I conclude by briefly discussing some structural similarities and differences between Descartes’s geometry and EPG.     

The poset of hypergraph quasirandomness

Chung and Graham began the systematic study of k‐uniform hypergraph quasirandom properties soon after the foundational results of Thomason and Chung‐Graham‐Wilson on quasirandom graphs. One feature that became apparent in the early work on k‐uniform hypergraph quasirandomness is that properties that are equivalent for graphs are not equivalent for hypergraphs, and thus hypergraphs enjoy a variety of inequivalent quasirandom properties. In the past two decades, there has been an intensive study of these disparate notions of quasirandomness for hypergraphs, and an open problem that has emerged is to determine the relationship between them. Our main result is to determine the poset of implications between these quasirandom properties. This answers a recent question of Chung and continues a project begun by Chung and Graham in their first paper on hypergraph quasirandomness in the early 1990's. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46,762–800, 2015     

Not from Nowhere History and Philosophy behind Mathematical Education

In this paper I discuss some of the historical and philosophical aspects of mathematical education, especially in foundational subjects where many of the principal issues seem to lie. Historical examples are chosen from the calculus, mathematical analysis, set theory and logic, but the philosophical points discussed also apply to other branches of mathematics.

The paper falls into four parts, which are alternately mainly historical and philosophical. The first part is motivated partly by problems arising in school mathematics (although the subject matter is treated from a more advanced viewpoint), and the second attempts to develop some general methodological considerations. The third part is relevant mainly to university mathematics, while the final part contains a number of broad conclusions, including the question of the role of the history of mathematics in mathematical education.     

A combinatorial-probabilistic analysis of bitcoin attacks*

In 2008, Satoshi Nakamoto famously invented bitcoin, and in his (or her, or their, or its) white paper sketched an approximate formula for the probability of a successful double spending attack by a dishonest party. This was corrected by Meni Rosenfeld, who, under more realistic assumptions, gave the exact probability (missing a foundational proof); and another formula (along with foundational proof), in terms of the Incomplete Beta function, was given later by Cyril Grunspan and Ricardo Pérez-Marco, that enabled them to derive an asymptotic formula for that quantity. Using Wilf-Zeilberger algorithmic proof theory, we continue in this vein and present a recurrence equation for the above-mentioned probability of success, that enables a very fast compilation of these probabilities. We next use this recurrence to derive (in algorithmic fashion) higher-order asymptotic formulas, extending the formula of Grunspan and Pérez-Marco who did the leading term. We then study the statistical properties (expectation, variance, etc.) of the duration of a successful attack.     

Sulla struttura dello spazio-tempo

Summary The paper fits in the sphere of foundational researches on Physical Geometry. A methodological line is followed which aims at giving general mathematical models of space and time and capturing the foundational value of the corresponding chronogeometrical structures; along this line a canonical model of space-time is provided which forms the largest substratum underlying both the Newtonian and the relativistic models of space-time, whose peculiar features are then derived by means of the invariance conditions imposed upon the chronogeometrical structures by some additional physical hypotheses.

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Lavoro eseguito nell'ambito del Gruppo Nazionale per la Fisica Matematica del C.N.R.     

Theories in practice: mathematics teaching and mathematics teacher education

Whilst research on the teaching of mathematics and the preparation of teachers of mathematics has been of major concern in our field for some decades, one can see a proliferation of such studies and of theories in relation to that work in recent years. This article is a reaction to the other papers in this special issue but I attempt, at the same time, to offer a different perspective. I examine first the theories of learning that are either explicitly or implicitly presented, noting the need for such theories in relation to teacher learning, separating them into: socio-cultural theories; Piagetian theory; and learning from practice. I go on to discuss the role of social and individual perspectives in authors’ approach. In the final section I consider the nature of the knowledge labelled as mathematical knowledge for teaching (MKT). I suggest that there is an implied telos about ‘good teaching’ in much of our research and that perhaps the challenge is to study what happens in practice and offer multiple stories of that practice in the spirit of “wild profusion” (Lather in Getting lost: Feminist efforts towards a double(d) science. SUNY Press, New York, 2007).     

From Brahmagupta to Euler: on the formula for the area of a cyclic quadrilateral

A formula for the area of a cyclic quadrilateral in terms of its sides was first stated without proof by the early seventh-century Indian mathematician Brahmagupta. As early as the late tenth century, the Persian mathematician al-Shannī provided a proof of the Indian's claim. In this paper I discuss al-Shannī's derivation and compare it with two other derivations. The first of these is by the Kerala mathematician and astronomer Jye??hadeva (sixteenth century), while the second is a forgotten proof by the Dutch mathematical practitioner Abraham de Graaf that was published in 1706. I conclude with a discussion of Euler's much better known derivation of 1748.     

Foundational ways of thinking for understanding the idea of logarithm

This study explored two undergraduate precalculus students’ understandings of the idea of logarithm as they completed conceptually oriented exploratory lessons on exponential and logarithmic functions. The students participated in consecutive, individual teaching experiments that focused on Sparky – an animated mystical saguaro that doubled in height every week. The exponential lesson focused on developing students’ conceptions of growth factors and tupling (e.g., doubling) periods as a foundational understanding for conceptualizing logarithms and logarithmic properties meaningfully. This paper characterizes conceptions that are productive for students to acquire in introductory lessons on exponential and logarithmic growth, and discusses two understandings that were revealed to be foundational for students’ development of productive meanings for exponents, logarithms and logarithmic properties.     

Understanding and advancing graduate teaching assistants’ mathematical knowledge for teaching

Graduate student teaching assistants (GTAs) usually teach introductory level courses at the undergraduate level. Since GTAs constitute the majority of future mathematics faculty, their image of effective teaching and preparedness to lead instructional improvements will impact future directions in undergraduate mathematics curriculum and instruction. In this paper, we argue for the need to support GTAs in improving their mathematical meanings of foundational ideas and their ability to support productive student thinking. By investigating GTAs’ meanings for average rate of change, a key content area in precalculus and calculus, we found evidence that even mathematically sophisticated GTAs possess impoverished meanings of this key idea. We argue for the need, and highlight one approach, for supporting GTAs to improve their understanding of foundational mathematical ideas and how these ideas are learned.     

‘An inalienable prerogative of a liberated spirit’: postulating American mathematics

In the early twentieth century, researchers in the United States engaged with foundational studies in mathematics by building and evaluating postulate systems. At the same time, their contemporaries were evaluating the meaning and politics of knowledge more broadly. This article argues that the study of postulates in the United States was tied to important Progressive Era questions about the nature of knowledge, the status of the knower, and the development of American Pragmatism. While most investigations of postulate studies have considered their implications within mathematical research and education, this article looks instead to the role of postulate studies in the professionalization of mathematics in the United States and to its cultural status more broadly.     

Univalence as a principle of logic

It is sometimes convenient or useful in mathematics to treat isomorphic structures as the same. The recently proposed Univalence Axiom for the foundations of mathematics elevates this idea to a foundational principle in the setting of homotopy type theory. It provides a simple and precise way in which isomorphic structures can be identified. We explore the motivations and consequences, both mathematical and philosophical, of making such a new logical postulate.     

Black the libertarian

The most serious challenge to Frankfurt-type counterexamples to the Principle of Alternate Possibilities (PAP) comes in the form of a dilemma: either the counterexample presupposes determinism, in which case it begs the question; or it does not presuppose determinism, in which case it fails to deliver on its promise to eliminate all alternatives that might plausibly be thought to satisfy PAP. I respond to this challenge with a counterexample in whichconsidering an alternative course of action is anecessary condition fordeciding to act otherwise, and the agent does not in fact consider the alternative. I call this a “buffer case,” because the morally relevant alternative is “buffered” by the requirement that the agent first consider the alternative. Suppose further that the agent’s considering an alternative action—entering the buffer zone—is what would trigger the counterfactual intervener. Then it would appear that PAP-relevant alternatives are out of reach. I defend this counterexample to PAP against three objections: that considering an alternative isitself a morally relevant alternative; that buffer cases can be shown to containother alternatives that arguably satisfy PAP; and that even if the agent’spresent access to PAP-relevant alternatives were eliminated, PAP could still be satisfied in virtue ofearlier alternatives. I conclude that alternative possibilities are a normal symptom, but not an essential constituent, of moral agency.