We investigate how epistemic injustice can manifest itself in mathematical practices. We do this as both a social epistemological and virtue-theoretic investigation of mathematical practices. We delineate the concept both positively—we show that a certain type of folk theorem can be a source of epistemic injustice in mathematics—and negatively by exploring cases where the obstacles to participation in a mathematical practice do not amount to epistemic injustice. Having explored what epistemic injustice in mathematics can amount to, we use the concept (...) to highlight a potential danger of intellectual enculturation. (shrink)
The first part of this paper presents asympathetic and critical examination of the approachof Shahid Rahman and Walter Carnielli, as presented intheir paper The Dialogical Approach toParaconsistency. In the second part, possibleextensions are presented and evaluated: (a) top-downanalysis of a dialogue situation versus bottom-up, (b)the specific role of ambiguities and how to deal withthem, and (c) the problem of common knowledge andbackground knowledge in dialogues. In the third part,I claim that dialogue logic is the best-suitedinstrument to analyse paradoxes of the (...) Sorites type.All these considerations lead to philosophicallyrelevant observations concerning principles of charityon the one hand, and compactness on the other. (shrink)
We explore aspects of an experimental approach to mathematical proof, most notably number crunching, or the verification of subsequent particular cases of universal propositions. Since the rise of the computer age, this technique has indeed conquered practice, although it implies the abandonment of the ideal of absolute certainty. It seems that also in mathematical research, the qualitative criterion of effectiveness, i.e. to reach one’s goals, gets increasingly balanced against the quantitative one of efficiency, i.e. to minimize one’s means/ends ratio. Our (...) story will lead to the consideration of some limit cases, opening up the possibility of proofs of infinite length being surveyed in a finite time. By means of example, this should show that mathematical practice in vital aspects depends upon what the actual world is like. (shrink)
Except in very poor mathematical contexts, mathematical arguments do not stand in isolation of other mathematical arguments. Rather, they form trains of formal and informal arguments, adding up to interconnected theorems, theories and eventually entire fields. This paper critically comments on some common views on the relation between formal and informal mathematical arguments, most particularly applications of Toulmin’s argumentation model, and launches a number of alternative ideas of presentation inviting the contextualization of pieces of mathematical reasoning within encompassing bodies of (...) explicit and implicit, formal and informal background knowledge. (shrink)
A solution of the zeno paradoxes in terms of a discrete space is usually rejected on the basis of an argument formulated by hermann weyl, The so-Called tile argument. This note shows that, Given a set of reasonable assumptions for a discrete geometry, The weyl argument does not apply. The crucial step is to stress the importance of the nonzero width of a line. The pythagorean theorem is shown to hold for arbitrary right triangles.
Most philosophers still tend to believe that mathematics is basically about producing formal proofs. A consequence of this view is that some aspects of mathematical practice are entirely lost from view. My contention is that it is precisely in those aspects that similarities can be found between practices in the exact sciences and in mathematics. Hence, if we are looking for a (more) unified treatment of science and mathematics it is necessary to incorporate these elements into our view of what (...) mathematics is about. As a helpful tool I introduce the notion of a mathematical argument as a more liberalized version of the notion of mathematical proof. (shrink)
Philosophy of mathematics today has transformed into a very complex network of diverse ideas, viewpoints, and theories. Sometimes the emphasis is on the "classical" foundational work (often connected with the use of formal logical methods), sometimes on the sociological dimension of the mathematical research community and the "products" it produces, then again on the education of future mathematicians and the problem of how knowledge is or should be transmitted from one generation to the next. The editors of this book felt (...) the urge, first of all, to bring together the widest variety of authors from these different domains and, secondly, to show that this diversity does not exclude a sufficient number of common elements to be present. In the eyes of the editors, this book will be considered a success if it can convince its readers of the following: that it is warranted to dream of a realistic and full-fledged theory of mathematical practices, in the plural. If such a theory is possible, it would mean that a number of presently existing fierce oppositions between philosophers, sociologists, educators, and other parties involved, are in fact illusory. (shrink)
An international group of distinguished scholars brings a variety of resources to bear on the major issues in the study and teaching of mathematics, and on the problem of understanding mathematics as a cultural and social phenomenon. All are guided by the notion that our understanding of mathematical knowledge must be grounded in and reflect the realities of mathematical practice. Chapters on the philosophy of mathematics illustrate the growing influence of a pragmatic view in a field traditionally dominated by platonic (...) perspectives. In a section on mathematics, politics, and pedagogy, the emphasis is on politics and values in mathematics education. Issues addressed include gender and mathematics, applied mathematics and social concerns, and the reflective and dialogical nature of mathematical knowledge. The concluding section deals with the history and sociology of mathematics, and with mathematics and social change. Contributors include Philip J. Davis, Helga Jungwirth, Nel Noddings, Yehuda Rav, Michael D. Resnik, Ole Skovsmose, and Thomas Tymoczko. (shrink)
Is alternative mathematics possible? More specifically, is it possible to imagine that mathematics could have developed in any other than the actual direction? The answer defended in this paper is yes, and the proof consists of a direct demonstration. An alternative mathematics that uses vague concepts and predicatesis outlined, leading up to theorems such as "Small numbers have few prime factors''.
In previous papers (see Van Bendegem , , , , , , and jointly with Van Kerkhove ) we have proposed the idea that, if we look at what mathematicians do in their daily work, one will find that conceiving and writing down proofs does not fully capture their activity. In other words, it is of course true that mathematicians spend lots of time proving theorems, but at the same time they also spend lots of time preparing the ground, if (...) you like, to construct a proof. (shrink)
The heuristics and strategies presented in Lakatos' Proofs and Refutations are well-known. However they hardly present the whole story as many authors have shown. In this paper a recent, rather spectacular, event in the history of mathematics is examined to gather evidence for two new strategies. The first heuristic concerns the expectations mathematicians have that a statement will be proved using given methods. The second heuristic tries to make sense of the mathematicians' notion of the quality of a proof.
It is a generally accepted idea that strict finitism is a rather marginal view within the community of philosophers of mathematics. If one therefore wants to defend such a position (as the present author does), then it is useful to search for as many different arguments as possible in support of strict finitism. Sometimes, as will be the case in this paper, the argument consists of, what one might call, a “rearrangement” of known materials. The novelty lies precisely in the (...) rearrangement, hence on the formal-axiomatic level most of the results presented here are not new. In fact, the basic results are inspired by and based on Mycielski (1981). (shrink)
For the first time in history, scholars working on language and culture from within an evolutionary epistemological framework, and thereby emphasizing complementary or deviating theories of the Modern Synthesis, were brought together. Of course there have been excellent conferences on Evolutionary Epistemology in the past, as well as numerous conferences on the topics of Language and Culture. However, until now these disciplines had not been brought together into one all-encompassing conference. Moreover, previously there never had been such stress on alternative (...) and complementary theories of the Modern Synthesis. Today we know that natural selection and evolution are far from synonymous and that they do not explain isomorphic phenomena in the world. ‘Taking Darwin seriously’ is the way to go, but today the time has come to take alternative and complementary theories that developed after the Modern Synthesis, equally seriously, and, furthermore, to examine how language and culture can merit from these diverse disciplines. As this volume will make clear, a specific inter- and transdisciplinary approach is one of the next crucial steps that needs to be taken, if we ever want to unravel the secrets of phenomena such as language and culture. (shrink)
What is philosophy of mathematics and what is it about? The most popular answer, I suppose, to this question would be that philosophers should provide a justification for our presently most cherished mathematical theories and for the most important tool to develop such theories, namely logico-mathematical proof. In fact, it does cover a large part of the activity of philosophers that think about mathematics. Discussions about the merits and faults of classical logic versus one or other ‘deviant’ logics as the (...) logical basis for mathematical theories, ranging from intuitionist over modal logic to paraconsistent logic, typically belong to this area, as do debates about the natural-number concept, its ‘nature’, its properties, especially its uniqueness, and so on. No doubt sociologists of knowledge could explain why philosophers of mathematics came to select these particular problems and deal with them the way they do. What it does imply, however, is that the question is meaningful whether different kinds of philosophies of mathematics are possible, and—why not?—perhaps even desirable. To a certain extent, the answer is trivial: look at what, e.g., phenomenologists have to say about mathematics and you must notice it does not fit into the scheme sketched above.1 Rather, the question should be whether there are forms that, on the one hand, reinterpret the whole enterprise, and, on the other hand, somehow remain related to the work of ‘mainstream’ philosophers of mathematics today.The book under review here does precisely that. Let me be a bit more precise. The general proposal, present throughout all the contributions, is that even at a first, superficial, glance at what mathematicians do when they do mathematics, it is far more complex than ‘just thinking about and …. (shrink)
The development of the philosophy of science in the twentieth century has created a framework where issues concerning funding dynamics can be easily accommodated. It combines the historical-philosophical approach of Thomas Kuhn. The University of Chicago Press, Chicago,  ) with the sociological approach of Robert K. Merton The sociology of science. Theoretical and empirical investigations. The University of Chicago Press, Chicago, pp 267–278,  ), linking the ‘exact’ sciences to economy and politics. Out of this came a new domain, (...) namely the study of scientific practices : Science as practice and culture. The University of Chicago Press, Chicago, 1992). Given this broad theoretical framework, we will specify by looking at the case of STEM education and its variant Science, Technology, Engineering and Mathematics with Art education as an example par excellence. Without going into the technical details of the financial support of the projects, we prefer to open a philosophical debate on the way how policies on academic subjects influence a whole society and the personal life of both researchers and people/pupils involved in education. (shrink)
No one will dispute, looking at the history of mathematics, that there are plenty of moments where mathematics is “in trouble”, when paradoxes and inconsistencies crop up and anomalies multiply. This need not lead, however, to the view that mathematics is intrinsically inconsistent, as it is compatible with the view that these are just transient moments. Once the problems are resolved, consistency (in some sense or other) is restored. Even when one accepts this view, what remains is the question what (...) mathematicians do during such a transient moment? This requires some method or other to reason with inconsistencies. But there is more: what if one accepts the view that mathematics is always in a phase of transience? In short, that mathematics is basically inconsistent? Do we then not need a mathematics of inconsistency? This paper wants to explore these issues, using classic examples such as infinitesimals, complex numbers, and infinity. (shrink)
The aim of this commentary is to show that a new development in formal logic, namely paraconsistent logic, should be connected with the laws of form. This note also includes some personal history to serve as background.
It is a rather safe statement to claim that the social dimensions of the scientific process are accepted in a fair share of studies in the philosophy of science. It is a somewhat safe statement to claim that the social dimensions are now seen as an essential element in the understanding of what human cognition is and how it functions. But it would be a rather unsafe statement to claim that the social is fully accepted in the philosophy of mathematics. (...) And we are not quite sure what kind of statement it is to claim that the social dimensions in theories of mathematics education are becoming more prominent, compared to the psychological dimensions. In our contribution we will focus, after a brief presentation of the above claims, on this particular domain to understand the successes and failures of the development of theories of mathematics education that focus on the social and not primarily on the psychological. (shrink)