DO WE REALLY NEED AXIOMS IN MATHEMATICS?

The third way: a realism with the human face

For a Philosophy of Mathematical Practice

It is a fact that of the logicist, the intuitionist, and Hilbert's programmes, the classical programmes set up to establish mathematics on safe foundations, some have failed, whereas others have long since lost their propulsive force. Two of the consequences of the crisis of the classical programmes in the foundations of mathematics were: a widespread scepticism towards the possibility of establishing the certainty of mathematical methods, and a new attention paid to mathematical practice. The main aim of this article is that of individuating some directions of research along which to develop a tenable philosophy of mathematical practice.

Carnap e il mito del sistema di riferimento in filosofia della matematica

STEFANO DONATI. I Fondamenti della Matematica nel Logicismo di Bertrand Russell [The Foundations of Mathematics in the Logicism of Bertrand Russell]

Productive Ambiguity in Mathematics

According to E. Grosholz, there is a phenomenon called `productive ambiguity' which plays a very important role in mathematics, and the sciences, because it is instrumental to the resolution of many open questions. The main task of this paper is that of assessing Grosholz's claim with regard to mathematics.

Review††Edited by Adrian Rice and Antoni MaletAll books, monographs, journal articles, and other publications (including films and other multisensory materials) relating to the history of mathematics are abstracted in the Abstracts Department. The Reviews Department prints extended reviews of selected publications.Materials for review, except books, should be sent to the Abstracts Editor, Sloan Despeaux, Western Carolina University, Cullowhee, NC 28723, USA. Books in English for review should be sent to Adrian Rice, Department of Mathematics, Randolph-Macon College, Ashland, VA 23005-5505, USA. Books in other languages for review should be sent to Antoni Malet, Universitat Pompeu Fabra, Department of Humanities, Ramon Trias Farga 25–27, Barcelona, 8005, Spain.Most reviews are solicited. However, colleagues wishing to review a book are invited to make their wishes known to the appropriate Book Review Editor. (Requests to review books written in the English language should be sent to Adrian Rice at the above address; requests to review books written in other languages should be sent to Antoni Malet at the above address.) We also welcome retrospective reviews of older books. Colleagues interested in writing such reviews should consult first with the appropriate Book Review Editor (as indicated above, according to the language in which the book is written) to avoid duplication.

Creativity in Conceptual Spaces

The main aim of this paper is contributing to what in the last few years has been known as computational creativity. This will be done by showing the relevance of a particular mathematical representation of G"ardenfors's conceptual spaces to the problem of modelling a phenomenon which plays a central role in producing novel and fruitful representations of perceptual patterns: analogy.

Object, Structure, and Form

The main task of this paper is to develop the non-Platonist view of mathematics as a science of structures I have called, borrowing the label from Putnam, `realism with the human face'. According to this view, if by `object' we mean what exists independently of whether we are thinking about it or not, mathematics is a science of patterns (structures), where patterns are neither objects nor properties of objects, but aspects (or aspects of aspects, etc.) of concrete objects which dawn on us when we represent objects (or aspects of... within a given system (of representation). Mathematical patterns, therefore, are real, because they ultimately depend on concrete objects, but are neither objec…

True V or not True V, That is the Question

In this paper we intend to argue that: (1) the question `True V or not True V' is central to both the philosophical and mathematical investigations of the foundations of mathematics; (2) when posed within a framework in which set theory is seen as a science of objects, the question `True V or not True V' generates a dilemma each horn of which turns out to be unacceptable; (3) a plausible way out of the dilemma mentioned at (2) is provided by an approach to set theory according to which this is considered to be a science of structures.

Book Review. 'I Fondamenti della Matematica nel Logicismo di Bertrand Russell'. Stefano Donati. Firenze (Firenze Atheneum). 2003. ISBN: 88-7255-204-4. 988 pages. €39.00

Bertrand Russell

Bertrand Russell has been one of the most influential philosophers of the 20th Century. The importance of his work can not be confined to a particular thematic area such as mathematical logic or philosophy of language. Extraordinary vital, extremely curious and with a very acute intellect, he has produced an immense body of work. Bertrand Russell è stato uno dei filosofi più influenti del '900. L'importanza del suo lavoro non può essere confinata ad un particolare ambito tematico quale la logica matematica o la filosofia del linguaggio. Dotato di una vitalità straordinaria, di una grande curiosità e di un intelletto acutissimo, la sua produzione scientifica è sterminata.

Some arguments against Field on the Indispensability Thesis

An algebra for the manipulation of conceptual spaces in cognitive agents

According to Gärdenfors, the theory of conceptual spaces describes a level of representation present in some cognitive agents between a sub-conceptual and a symbolic level of representation. In contrast to a large part of contemporary philosophical speculation on these matters for which concepts and conceptual content are propositional, conceptual spaces provide a geometric framework for the representation of concepts. In this paper we introduce an algebra for the manipulation of different conceptual spaces in order to formalise the process whereby an artificial agent rearranges its internal conceptual representations as a consequence of its perceptions, which are here rendered in terms of …

A Realist Philosophy of Mathematics

The realism/anti-realism debate is one of the traditional central themes in the philosophy of mathematics. The controversies about the existence of the irrational numbers, the complex numbers, the infinitesimals, etc. will be familiar to all who are acquainted with the history of mathematics. This book aims mainly at presenting and defending a non-Platonist form of mathematical structural realism which, in the respect of the history of mathematics, harmonizes with a plausible epistemology that naturally arises from it.

Book Review: Ferreiros,J. and Gray J. (eds.): 2006, The Architecture of Modern Mathematics: Essays in History and Philosophy. Oxford University Press, Oxford.

MATHEMATICS AS A QUASI-EMPIRICAL SCIENCE

The present paper aims at showing that there are times when set theoretical knowledge increases in a non-cumulative way. In other words, what we call ‘set theory’ is not one theory which grows by simple addition of a theorem after the other, but a finite sequence of theories T1, ..., T n in which Ti+1, for 1 ≤ i < n, supersedes T i . This thesis has a great philosophical significance because it implies that there is a sense in which mathematical theories, like the theories belonging to the empirical sciences, are fallible and that, consequently, mathematical knowledge has a quasi-empirical nature. The way I have chosen to provide evidence in favour of the correctness of the main thesis of t…