This book offers an up-to-date overview of the research on philosophy of mathematics education, one of the most important and relevant areas of theory. The contributions analyse, question, challenge, and critique the claims of mathematics education practice, policy, theory and research, offering ways forward for new and better solutions. The book poses basic questions, including: What are our aims of teaching and learning mathematics? What is mathematics anyway? How is mathematics related to society in the 21st century? How do students (...) learn mathematics? What have we learnt about mathematics teaching? Applied philosophy can help to answer these and other fundamental questions, and only through an in-depth analysis can the practice of the teaching and learning of mathematics be improved. The book addresses important themes, such as critical mathematics education, the traditional role of mathematics in schools during the current unprecedented political, social, and environmental crises, and the way in which the teaching and learning of mathematics can better serve social justice and make the world a better place for the future. (shrink)

Provability, Computability and Reflection.

Survey results are in the first instance utterances, which require interpretation. Moreover, when the results seem to involve disagreement in intuitive responses to a thought experiment, the results are most directly responsive to the scenario as envisaged by the particular subject, where the text of the example can give rise to relevantly different scenarios, depending on how the scenario is shaped by the subjects involved, under the guidance of the text. All of this opens up a defense of intuitions against (...) results that ostensibly imply extensive intuitive disagreement based on cultural or socio-economic background. Critics of the armchair have replied to this defense in recent publications. This paper takes up some of those replies. (shrink)

“Rational beings pursue and value truth. Intellectual conduct is to be judged, accordingly, by how well it aids our pursuit of that ideal.” What does this mean, and is it true? Even if intelligent life had never evolved or otherwise existed, Venus would still have orbited the Sun, so it would still have been true that Venus orbited the Sun. It is not the being thus true of what is true that we value indiscriminately. Some truths are good, but not (...) all, far from it. In loving the truth, then, what we value is not the being true of the truths. What we value in pursuing truth is rather our grasping it, our having it. What does this mean? Only through believing it does one relevantly have a truth : we have the truth that snow is white by believing that snow is white. In pursuing the truth what we want is true beliefs. Suppose you enter your dentist’s waiting room and find all the magazines taken. Deprived of reading matter, you’re sure to doze off, but you need no sleep. Are you then rationally bound to reach for the telephone book in pursuit of truth? Were you not to do so, you would forfeit a chance to pluck some desired goods within easy reach. If random telephone numbers do not elicit a wide enough yawn, consider a randomly selected cubic foot of the Sahara. Here is a trove of facts, of the form grain x is so many millimeters in direction D from grain y, than which few can be of less interest. Or take some bit of trivia known to me at the moment: say, that it was sunny in Rhode Island at noon on October 21, 1999. I confess that I will not rue my loss of this information, nor do I care either that or how early it will be gone. As interpreted so far, the view that we rationally want truth as such reduces to absurdity, or is at best problematic. What then is it we want in pursuing truth? If it is not after all true beliefs indiscriminately, what then is it? A manageable number of true beliefs? Obviously not; it is not just a matter of numbers. (shrink)

First published in 1999. Routledge is an imprint of Taylor & Francis, an informa company.

The main thesis of this paper is that the most cogent demands of subjectivity, at least with respect to questions concerning the contents of our thoughts, can be accommodated within an objectivist framework. I begin with two theses: (1) Subjectivity: I can know (the contents of) my own thoughts without appeal to any knowledge of features external to my mind; (2) Environmentalism: (The contents of) my thoughts are determined by features external to my mind, at least in this sense: without (...) causal and/or social interaction between my internal states and various external features, these internal states would not have the particular contents they have and therefore would not be the mental states they are. Section I proceeds by elucidating various lines of environmentalism, an overtly objectivist thesis. Section II considers and refutes one purported environmentalist challenge to subjectivity, namely, that, according to it, thoughts are not in the head. Section III discusses a central subjectivist thesis, namely, thoughts do not admit of an appearance/reality distinction. Tyler Burge and Hilary Putnam, two environmentalists, seem to endorse such a distinction. It is argued that the two most reasonable routes of escape open to them require a retreat from environmentalism. This leaves us with an apparent dilemma: reject either subjectivity or environmentalism Section IV defends environmentalism. Section V considers and criticizes a maneuver Donald Davidson makes to preserve first?person authority over thought?contents. Section VI concludes with a brief defense of environmentalism which rests largely on rejecting a deeply entrenched epistemic model about how we access our own thoughts. It is argued that it is this bad epistemic model and not subjectivity per se, which has created whatever tension there appears to be between subjectivity and environmentalism. (shrink)

This paper was written for a panel session, in which I was asked to represent an analytic perspective. On reflection I found that there is no such thing, however, and that what best unifies the analytic traditions is not even a set of questions, much less a set of answers, but only agreement on certain standards of clarity and argumentation, and an interest in dialectic and debate. Certain issues have long dominated the analytic agenda, it is true, and I see (...) no better way to represent an analytic perspective than to enter into a bit of dialectic and debate on one of these, in a way that I hope will be clear and cogent. (shrink)

Preface.-- Introduction.--I. The problem of consciousness.-- II. Modern idealism.--III. The alogical and the logical as ultimate elements.--IV. The individual conciousness.--V. Reality and truth.--VI. The higher conciousness.--VII. The final goal of all things.--VIII. Problem of metaphysic.--IX. Survey of results.-- Index.