Paolo Mancosu

Visualization, Explanation and Reasoning Styles in Mathematics

This book contains groundbreaking contributions to the philosophical analysis of mathematical practice. Several philosophers of mathematics have recently called for an approach to philosophy of mathematics that pays more attention to mathematical practice. Questions concerning concept-formation, understanding, heuristics, changes in style of reasoning, the role of analogies and diagrams, etc. have become the subject of intense interest. The historians and philosophers in this book agree that there is more to understanding mathematics than a study of its logical structure. How are mathematical objects and concepts generated? How does the process tie up with justification? What role do visual images and diagrams play in mathematical activity? What are the different epistemic virtues (explanatoriness, understanding, visualizability, etc.) which are pursued and cherished by mathematicians in their work? The reader will find here systematic philosophical analyses as well as a wealth of philosophically informed case studies ranging from Babylonian, Greek, and Chinese mathematics to nineteenth century real and complex analysis.

Mathematical Explanation: Problems and Prospects

Since this issue is devoted to the interaction between philosophy of mathematics and mathematical practice, I would like to begin with an introductory reflection on this topic, before I enter the specifics of my contribution. In the past thirty years there has been a marked shift in philosophy of mathematics, due to the appearance of research on aspects of mathematics that were previously ignored by philosophers of mathematics. A short, and very incomplete, list includes work on the dynamics of mathematical growth, the debate on computer proofs, the role of diagrammatic reasoning in mathematics, induction and conjecture in mathematics, problems at the interface of theoretical physics and mathematics. In some cases, these contributions have been accompanied by much fanfare about the need to pay attention to mathematical practice and by an attack on philosophy of mathematics as “foundations of mathematics”, variously called, “formalism”, “foundationalism”, “justificationism.” Whereas some of the polemical tone might have served to bring attention to new and exciting developments, I find that overall it is unwarranted and tends to muddle the issues. First of all, the characterization of the foundational programs, which are being attacked, is often one-sided at best and patently false in the worst cases. But even leaving questions of historical accuracy aside, all the programs in foundations of mathematics in this century have, in my opinion, been concerned with mathematical practice. In the grand foundational programs, say Hilbert’s, attention to practice was necessary to insure that the consistency program be able to account for all of mathematics, as opposed to a small part of it. And setting up the formalisms does require a very good sense of how much you need for various parts of mathematical practice. In this sense, many programs in contemporary logical foundations, such as reverse mathematics or predicative mathematics, are extremely sensitive to issues of mathematical practice. Moreover, the distinction between elementary and non-elementary methods, which was one of the cornerstones of Hilbert’s program, is a typical issue emerging from mathematical practice. However, it is true that many of the classical foundational programs “filter out” many aspects of mathematical practice which are irrelevant to their goals. Hence, there is a kernel of truth in the above mentioned criticisms. There are many aspects of mathematical practice that are irrelevant for some of the classical foundational programs but nonetheless worthy of philosophical attention. Thus, for instance, while a study of mathematical heuristics is not relevant to Hilbert’s program, it has much to offer to the philosophers and mathematicians who are interested in aspects of mathematics which go beyond the specific aims set by Hilbert for his task. But this, contrary to some of the polemical claims I referred to above, in itself does not invalidate Hilbert’s program (other considerations do!). It only calls for a liberalization concerning what aspects of mathematics should be objects of philosophical interest. I think that much of the alleged opposition between these developments can be deflated if one keeps in mind that the aims of both traditions are legitimate and all provide essential information about the complex reality we are interested in, i.e. mathematics. The topic of my paper, mathematical explanation, also escapes traditional foundational work. Part of the reason is that the subject area is admittedly vague, and consequently difficult to treat with precise mathematical or logical tools. Moreover, it does not bear directly upon some of the traditional foundational concerns, such as certainty, which have dominated much of philosophy of mathematics. It is nonetheless a subject of great philosophical interest. Consider, for instance, the situation in philosophy of science. There the topic of scientific explanation has received much attention. In this Mathematical Explanation: Problems and Prospects Paolo Mancosu

Visualization in logic and mathematics

In the last two decades there has been renewed interest in visualization in logic and mathematics. Visualization is usually understood in different ways but for the purposes of this article I will take a rather broad conception of visualization to include both visualization by means of mental images as well as visualizations by means of computer generated images or images drawn on paper, e.g. diagrams etc. These different types of visualization can differ substantially but I am interested in offering a characterization of visualizationthat is as broad as possible. The article describes and explains (1) the way in which visual thinking fell into desrepute, (2) the renaissance of visual thinking in mathematics over recent decades, (3) the ways in which visual thinking has been rehabilitated in epistemology of mathematics and logic.

MEASURING THE SIZE OF INFINITE COLLECTIONS OF NATURAL NUMBERS: WAS CANTOR’S THEORY OF INFINITE NUMBER INEVITABLE?

Cantor’s theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one association between two sets. As is well known, all countable infinite sets have the same ‘size’ in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collection A is properly included in a collection B then the ‘size’ of A should be less than the ‘size’ of B (part–whole principle). This second intuition was not developed mathematically in a satisfactory way until quite recently. In this article I begin by reviewing the contributions of some thinkers who argued in favor of the assignment of different sizes to infinite collections of natural numbers (Thabit ibn Qurra, Grosseteste, Maignan, Bolzano). Then, I review some recent mathematical developments that generalize the part–whole principle to infinite sets in a coherent fashion (Katz, Benci, Di Nasso, Forti). Finally, I show how these new developments are important for a proper evaluation of a number of positions in philosophy of mathematics which argue either for the inevitability of the Cantorian notion of infinite number (Godel) or for the rational nature of the Cantorian generalization as opposed to that, based on the part–whole principle, envisaged by Bolzano (Kitcher).

On Mathematical Explanation

In the present paper I would like to present some reflections which occurred to me upon reading Grosholz’s paper “The Partial Unification of Domains, Hybrids, and the Growth of Mathematical Knowledge.” However, I should warn the reader that although Grosholz’s paper provided the original stimulus for mine, in the end I pursue a number of issues which were perhaps not the central ones Grosholz was addressing. Grosholz begins by claiming that, Relations among distinct areas of mathematical activity are most commonly discussed in terms of the reduction of axiomatized theories, where reduction is defined to be the deductive derivation of the axioms of the reduced theory as theorems of the reducing theory (Grosholz 1999, 81).

Between Vienna and Berlin: The Immediate Reception of Godel's Incompleteness Theorems

What were the earliest reactions to Godels incompleteness theorems? After a brief summary of previous work in this area I analyse, by means of unpublished archival material, the first reactions in Vienna and Berlin to Godels groundbreaking results. In particular, I look at how Carnap, Hempel, von Neumann, Kaufmann, and Chwistek, among others, dealt with the new results.

Harvard 1940–1941: Tarski, Carnap and Quine on a finitistic language of mathematics for science

Tarski, Carnap and Quine spent the academic year 1940–1941 together at Harvard. In their autobiographies, both Carnap and Quine highlight the importance of the conversations that took place among them during the year. These conversations centred around semantical issues related to the analytic/synthetic distinction and on the project of a finitist/nominalist construction of mathematics and science. Carnaps Nachlaß in Pittsburgh contains a set of detailed notes, amounting to more than 80 typescripted pages, taken by Carnap while these discussions were taking place. In my article, I present a survey of these notes with special emphasis on Tarskis rejection of the analytic/synthetic distinction, the passage from typed languages to first-order languages, Tarskis finitism/nominalism, and the construction of a finitist language for mathematics and science.

Between Russell and Hilbert: Behmann on the Foundations of Mathematics

After giving a brief overview of the renewal of interest in logic and the foundations of mathematics in Göttingen in the period 1914-1921, I give a detailed presentation of the approach to the foundations of mathematics found in Behmann’s doctoral dissertation of 1918, Die Antinomie der transfiniten Zahl und ihre Auflösung durch die Theorie von Russell und Whitehead. The dissertation was written under the guidance of David Hilbert and was primarily intended to give a clear exposition of the solution to the antinomies as found in Principia Mathematica. In the process of explaining the theory of Principia, Behmann also presented an original approach to the foundations of mathematics which saw in sense perception of concrete individuals the Archimedean point for a secure foundation of mathematical knowledge. The last part of the paper points out an important numbers of connections between Behmann’s work and Hilbert’s foundational thought. §1. Logic and Foundations ofMathematics in Göttingen from 1910 to 1921. Recent work on Hilbert’s program has focused, among other things, on the development of logic in Hilbert’s school and on the philosophical underpinnings of the program. Sieg [30] and Moore [26] have investigated the development of first-order logic in Hilbert’s 1917–18 lectures, Zach [37] has given an in-depth analysis of the propositional calculus in Hilbert’s school from 1918 to 1928, and Mancosu [25] has investigated the philosophical context of Hilbert’s approach to the foundations of mathematics. TheHabilitationsschrift by Bernays [8] and Hilbert’s 1917–1918 lectures [19] represent the starting point of these important developments. However, these lectures were not the product of a sudden reawakening of interest in logic and the Received December 2, 1998; revised July 20, 1999. I would like to thank Volker Peckhaus, Christian Thiel, Peter Bernhard and Richard Zach for comments and for making it possible to access and reproduce some of the materials contained in the Behmann Archive in Erlangen. I am grateful to an anonymous referee for his comments, which helped me sharpen a number of issues raised in the paper. I am also grateful to the curators of the following collections for their help: Russell archive at McMaster University, Hamilton; Bernays Nachlaß, ETH Zürich; Hugo Dingler-Nachlaß, Aschaffenburg; Hilbert Nachlaß, Göttingen. I would finally like to thank theWissenschaftskolleg zu Berlin for having provided ideal conditions for work on the first draft of this paper during the academic year 1997–98. c

ON THE RELATIONSHIP BETWEEN PLANE AND SOLID GEOMETRY

Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned areas. In this paper our major concern is with methodological issues of purity and thus we treat the connection to other areas of the planimetry/stereometry relation only to the extent necessary to articulate the problem area we are after. Our strategy will be as follows. In the first part of the paper we will give a rough sketch of some key episodes in mathematical practice that relate to the interaction between plane and solid geometry. The sketch is given in broad strokes and only with the intent of acquainting the reader with some of the mathematical context against which the problem emerges. In the second part, we will look at a debate (on “fusionism”) in which for the first time methodological and foundational issues related to aspects of the mathematical practice covered in the first part of the paper came to the fore. We conclude this part of the paper by remarking that only through a foundational and philosophical effort could the issues raised by the debate on “fusionism” be made precise. The third part of the paper focuses on a specific case study which has been the subject of such an effort, namely the foundational analysis of the plane version of Desargues’ theorem on homological triangles and its implications for the relationship between plane and solid geometry. Finally, building on the foundational case study analyzed in the third section, we begin

The metaphysics of the calculus: A foundational debate in the Paris Academy of Sciences, 1700–1706

Abstract The differential calculus faced a strong opposition within the Academy of Sciences of Paris at the very beginning of the 18th century. The opposition came from a group of mathematicians who criticized the new analysis both for what they considered to be its lack of rigor and for the results that it produced. A bitter debate raged for about 6 years until the proponents of the new calculus prevailed.