4,888 research outputs found
The Uses of Argument in Mathematics
Stephen Toulmin once observed that `it has never been customary for
philosophers to pay much attention to the rhetoric of mathematical debate'.
Might the application of Toulmin's layout of arguments to mathematics remedy
this oversight?
Toulmin's critics fault the layout as requiring so much abstraction as to
permit incompatible reconstructions. Mathematical proofs may indeed be
represented by fundamentally distinct layouts. However, cases of genuine
conflict characteristically reflect an underlying disagreement about the nature
of the proof in question.Comment: 10 pages, 5 figures. To be presented at the Ontario Society for the
Study of Argumentation Conference, McMaster University, May 2005 and LOGICA
2005, Hejnice, Czech Republic, June 200
Buying Logical Principles with Ontological Coin: The Metaphysical Lessons of Adding epsilon to Intuitionistic Logic
We discuss the philosophical implications of formal results showing the con-
sequences of adding the epsilon operator to intuitionistic predicate logic. These
results are related to Diaconescu’s theorem, a result originating in topos theory
that, translated to constructive set theory, says that the axiom of choice (an
“existence principle”) implies the law of excluded middle (which purports to be
a logical principle). As a logical choice principle, epsilon allows us to translate
that result to a logical setting, where one can get an analogue of Diaconescu’s
result, but also can disentangle the roles of certain other assumptions that are
hidden in mathematical presentations. It is our view that these results have not
received the attention they deserve: logicians are unlikely to read a discussion
because the results considered are “already well known,” while the results are
simultaneously unknown to philosophers who do not specialize in what most
philosophers will regard as esoteric logics. This is a problem, since these results
have important implications for and promise signif i cant illumination of contem-
porary debates in metaphysics. The point of this paper is to make the nature
of the results clear in a way accessible to philosophers who do not specialize in
logic, and in a way that makes clear their implications for contemporary philo-
sophical discussions. To make the latter point, we will focus on Dummettian discussions of realism and anti-realism.
Keywords: epsilon, axiom of choice, metaphysics, intuitionistic logic, Dummett,
realism, antirealis
Optimal network topologies: Expanders, Cages, Ramanujan graphs, Entangled networks and all that
We report on some recent developments in the search for optimal network
topologies. First we review some basic concepts on spectral graph theory,
including adjacency and Laplacian matrices, and paying special attention to the
topological implications of having large spectral gaps. We also introduce
related concepts as ``expanders'', Ramanujan, and Cage graphs. Afterwards, we
discuss two different dynamical feautures of networks: synchronizability and
flow of random walkers and so that they are optimized if the corresponding
Laplacian matrix have a large spectral gap. From this, we show, by developing a
numerical optimization algorithm that maximum synchronizability and fast random
walk spreading are obtained for a particular type of extremely homogeneous
regular networks, with long loops and poor modular structure, that we call
entangled networks. These turn out to be related to Ramanujan and Cage graphs.
We argue also that these graphs are very good finite-size approximations to
Bethe lattices, and provide almost or almost optimal solutions to many other
problems as, for instance, searchability in the presence of congestion or
performance of neural networks. Finally, we study how these results are
modified when studying dynamical processes controlled by a normalized (weighted
and directed) dynamics; much more heterogeneous graphs are optimal in this
case. Finally, a critical discussion of the limitations and possible extensions
of this work is presented.Comment: 17 pages. 11 figures. Small corrections and a new reference. Accepted
for pub. in JSTA
Mathematical proofs in practice: revisiting the reliability of published mathematical proofs
Mathematics seems to have a special status when compared to other areas of human knowledge. This special status is linked with the role of proof. Mathematicians often believe that this type of argumentation leaves no room for errors and unclarity. Philosophers of mathematics have differentiated between absolutist and fallibilist views on mathematical knowledge, and argued that these views are related to whether one looks at mathematics-in-the-making or finished mathematics. In this paper we take a closer look at mathematical practice, more precisely at the publication process in mathematics. We argue that the apparent view that mathematical literature, given the special status of mathematics, is highly reliable is too naive. We will discuss several problems in the publication process that threaten this view, and give several suggestions on how this could be countered
Simon's Bounded Rationality. Origins and use in economic theory
The paper aims to show how Simon's notion of bounded rationality should be interpreted in the light of its connection with artificial intelligence. This connection points out that bounded rationality is a highly structured concept, and sheds light on several implications of Simon's general views on rationality. Finally, offering three paradigmatic examples, the artic1e presents the view that recent approaches, which refer to Simon's heterodox theory, only partially accept the teachings of their inspirer, splitting bounded rationality from the context of artificl al intelligence.
Presuppositions in Context: Constructing Bridges
About the book: The First International and Interdisciplinary Conference on Modelling and Using Context, Rio de Janeiro, January 1997, gave rise to the present book, which contains a selection of the papers presented there, thoroughly refereed and revised. The treatment of contexts as bona fide objects of logical formalisation has gained wide acceptance, following the seminal impetus given by McCarthy in his Turing Award address. The field of natural language offers a particularly rich variety of examples and challenges to researchers concerned with the formal modelling of context, and several chapters in the volume deal with contextualisation in the setting of natural language. Others adopt a purely formal-logical viewpoint, seeking to develop general models of even wider applicability. The 12 chapters are organised in three groups: formalisation of contextual information in natural language understanding and generation, the application of context in mechanised reasoning domains, and novel non-classical logics for contextual application
The role of authorial context in mathematicians’ evaluations of proof
We report on a series of task-based interviews in which nine mathematicians were asked to evaluate a series of six mathematical arguments, purportedly produced either by fellow mathematicians or undergraduate students. In this paper, we attend to the role of context in mathematicians’ responses, leading to four themes in expectations when evaluating the proofs that they read. First, mathematicians’ evaluations of identical arguments were sensitive to researchers’ manipulation of authorship, with most accepting arguments purportedly produced by a colleague while taking a more critical view of that same argument if produced by an undergraduate student. Our thematic analysis of interview responses led to three context-based factors influencing mathematicians’ responses when evaluating student-produced texts: course goals, instructors’ expectations, and assessment type. In the final section, we consider implications for researchers focused on understanding common practice amongst mathematicians as well as the pedagogic consequences of our findings for practice in the classroom
The practice of arguing and the arguments: Examples from mathematics
In argumentation studies, almost all theoretical proposals are applied, in general, to the analysis and evaluation of written argumentative texts. I will consider mathematics to illustrate some differences between argumentative practice and the products of it, to emphasize the need to address the different types of argumentative discourse and argumentative situation. Argumentative practice should be encouraged when teaching technical subjects to convey a better understanding and to improve thought and creativity
Out of Many, One: Toward Rigorous Common Core Standards From the Ground Up
Analyzes high school standards for English in twelve states and math in sixteen states designed for college- and career-readiness. Examines their alignment with the American Diploma Project's benchmarks for common core standards. Discusses implications
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