Debates between contemporary platonist and nominalist conceptions of the metaphysical status of mathematical objects have recently included discussions of explanations of physical phenomena in which mathematics plays an indispensable role, termed mathematical explanations in science (MES). I will argue that MES requires an ontology that can (1) ground claims about mathematical necessity as distinct from physical necessity and (2) explain how that mathematical necessity applies to the physical world. I contend that nominalism fails to meet the first criterion and (...) platonism the second. I then articulate an alternative, Aristotelian approach to mathematical objects and defend such a view as meeting both criteria. (shrink)

This paper has two components. The first, longer component (sec. 1-6) is a critical exposition of Armstrong’s views about the metaphysics of mathematics, as they are presented in Truth and Truthmakers and Sketch for a Systematic Metaphysics. In particular, I discuss Armstrong’s views about the nature of the cardinal numbers, and his account of how modal truths are made true. In the second component of the paper (sec. 7), which is shorter and more tentative, I sketch an (...) alternative account of the metaphysics of mathematics. I suggest we insist that mathematical truths have physical truthmakers, without insisting that mathematical objects themselves are part of the physical world. (shrink)

This paper has two components. The first, longer component (sec. 1-6) is a critical exposition of Armstrong's views about the metaphysics of mathematics, as they are presented in Truth and Truthmakers and Sketch for a Systematic Metaphysics. In particular, I discuss Armstrong's views about the nature of the cardinal numbers, and his account of how modal truths are made true. In the second component of the paper (sec. 7), which is shorter and more tentative, I sketch an (...) alternative account of the metaphysics of mathematics. I suggest we insist that mathematical truths have physical truthmakers, without insisting that mathematical objects themselves are part of the physical world. (shrink)

In a previous paper, Thomas V. Morris and I sketched a view on which abstract objects, in particular, properties, relations, and propositions , are created by God no less than contingent, concrete objects. In this paper r suggest a way of extending this account to cover mathematical objects as well. Drawing on some recent work in logic and metaphysics, I also develop a more detailed account of the structure of PRPs in answer to the paradoxes that arise on a (...) naive understanding of the structure ofthe abstract universe. (shrink)

The instability inherent in the historical inventory of mathematical objects challenges philosophers. Naturalism suggests we can construct enduring answers to ontological questions through an investigation of the processes whereby mathematical objects come into existence. Patterns of historical development suggest that mathematical objects undergo an intelligible process of reification in tandem with notational innovation. Investigating changes in mathematical languages is a necessary first step towards a viable ontology. For this reason, scholars should not modernize historical texts without caution, as the use (...) of anachronistic notation tends to impede, rather than enhance, our ability to recognize the emergent nature of mathematical objects. (shrink)

Although Brouwer is well known for his Intuitionistic philosophy of mathematics, a constructivist philosophy which calls for restricted use of certain logical principles, there is much less awareness of the well-developed metaphysical basis which underlies those restrictions. In the first half of this paper I outline a basic interpretation of Brouwer's metaphysics, and then in the second half consider the compatibility of that metaphysics with Dummett's argument for a principled non-metaphysical approach to intuitionism. I conclude that once (...) the variously misleading accretions of the central concepts - metaphysics and logic - are set aside, Dummett and Brouwer's accounts can be seen to be at the very least compatible, if not complementar. (shrink)

Although Brouwer is well known for his Intuitionistic philosophy of mathematics, a constructivist philosophy which calls for restricted use of certain logical principles, there is much less awareness of the well-developed metaphysical basis which underlies those restrictions. In the first half of this paper I outline a basic interpretation of Brouwer's metaphysics, and then in the second half consider the compatibility of that metaphysics with Dummett's argument for a principled non-metaphysical approach to intuitionism. I conclude that once (...) the variously misleading accretions of the central concepts - metaphysics and logic - are set aside, Dummett and Brouwer's accounts can be seen to be at the very least compatible, if not complementar. (shrink)

Building on the seminal work of Kit Fine in the 1980s, Leon Horsten here develops a new theory of arbitrary entities. He connects this theory to issues and debates in metaphysics, logic, and contemporary philosophy of mathematics, investigating the relation between specific and arbitrary objects and between specific and arbitrary systems of objects. His book shows how this innovative theory is highly applicable to problems in the philosophy of arithmetic, and explores in particular how arbitrary objects can engage (...) with the nineteenth-century concept of variable mathematical quantities, how they are relevant for debates around mathematical structuralism, and how they can help our understanding of the concept of random variables in statistics. This fully worked through theory will open up new avenues within philosophy of mathematics, bringing in the work of other philosophers such as Saul Kripke, and providing new insights into the development of the foundations of mathematics from the eighteenth century to the present day. (shrink)

Most philosophers of mathematics try to show either that the sort of knowledge mathematicians have is similar to the sort of knowledge specialists in the empirical sciences have or that the kind of knowledge mathematicians have, although apparently about objects such as numbers, sets, and so on, isn't really about those sorts of things as well. Jody Azzouni argues that mathematical knowledge really is a special kind of knowledge with its own special means of gathering evidence. He analyses the (...) linguistic pitfalls and misperceptions philosophers in this field are often prone to, and explores the misapplications of epistemic principles from the empirical sciences to the exact sciences. What emerges is a picture of mathematics both sensitive to mathematical practice, and to the ontological and epistemological issues that concern philosophers. (shrink)

I discuss a puzzle that shows there is a need to develop a new metaphysical interpretation of mathematical theories, because all well-known interpretations conflict with important aspects of mathematical activities. The new interpretation, I argue, must authenticate the ontological commitments of mathematical theories without curtailing mathematicians' freedom and authority to creatively introduce mathematical ontology during mathematical problem-solving. Further, I argue that these two constraints are best met by a metaphysical interpretation of mathematics that takes mathematical entities to be constitutively (...) constructed by human activity in a manner similar to the constitutive construction of the US Supreme Court by certain legal and political activities. Finally, I outline some of the philosophical merits of metaphysical interpretations of mathematical theories of this type. (shrink)

In recent years, several remarkable results have shown that certain theorems of finite combinatorics are unprovable in certain logical systems. These developments have been instrumental in stimulating research in both areas, with the interface between logic and combinatorics being especially important because of its relation to crucial issues in the foundations of mathematics which were raised by the work of Kurt Godel. Because of the diversity of the lines of research that have begun to shed light on these issues, (...) there was a need for a comprehensive overview which would tie the lines together. This volume fills that need by presenting a balanced mixture of high quality expository and research articles that were presented at the August 1985 AMS-IMS-SIAM Joint Summer Research Conference, held at Humboldt State University in Arcata, California.With an introductory survey to put the works into an appropriate context, the collection consists of papers dealing with various aspects of 'unprovable theorems and fast-growing functions'. Among the topics addressed are: ordinal notations, the dynamical systems approach to Ramsey theory, Hindman's finite sums theorem and related ultrafilters, well quasiordering theory, uncountable combinatorics, nonstandard models of set theory, and a length-of-proof analysis of Godel's incompleteness theorem. Many of the articles bring the reader to the frontiers of research in this area, and most assume familiarity with combinatorics and/or mathematical logic only at the senior undergraduate or first-year graduate level. (shrink)

Since virtually every mathematical theory can be interpreted in set theory, the latter is a foundation for mathematics. Whether set theory, as opposed to any of its rivals, is the right foundation for mathematics depends on what a foundation is for. One purpose is philosophical, to provide the metaphysical basis for mathematics. Another is epistemic, to provide the basis of all mathematical knowledge. Another is to serve mathematics, by lending insight into the various fields. Another is (...) to provide an arena for exploring relations and interactions between mathematical fields, their relative strengths, etc. Given the different goals, there is little point to determining a single foundation for all of mathematics. (shrink)

Kant famously distinguishes between the methods of mathematics and of metaphysics, holding that metaphysicians err when they avail themselves of the mathematical method. Nonetheless, in the Metaphysical Foundations of Natural Science, he insists that mathematics and metaphysics must jointly ground ‘proper natural science’. This article examines the distinctive contributions and unity of mathematics and metaphysics to the foundations of the science of body. I argue that the two are distinct insofar as they involve distinctive (...) sorts of grounding relations – mathematics pertains to formal grounding, while metaphysics concerns material grounding – while they are unified insofar as they treat motion, the fundamental determination of the science of body. (shrink)

We make some remarks on the mathematics and metaphysics of the hole argument, in response to a recent article in this journal by Weatherall ([2018]). Broadly speaking, we defend the mainstream philosophical literature from the claim that correct usage of the mathematics of general relativity `blocks' the argument.

HorstenLeon* * _ The Metaphysics and Mathematics of Ordinary Objects. _Cambridge University Press, 2019. Pp. xviii + 231. ISBN: 978-1-107-03941-4 ; 978-1-10860177-1. doi: 10.1017/9781139600293.

Kurt Gdel was the most outstanding logician of the 20th century and a giant in the field. This book is part of a five volume set that makes available all of Gdel's writings. The first three volumes, already published, consist of the papers and essays of Gdel. The final two volumes of the set deal with Gdel's correspondence with his contemporary mathematicians, this fourth volume consists of material from correspondents from A-G.

Mathematics is everywhere and yet its objects are nowhere. There may be five apples on the table but the number five itself is not to be found in, on, beside or anywhere near the apples. So if not in space and time, where are numbers and other mathematical objects such as perfect circles and functions? And how do we humans discover facts about them, be it Pythagoras’ Theorem or Fermat’s Last Theorem? The metaphysical question of what numbers are and (...) the epistemological question of how we know about them are central to the philosophy of mathematics. These and related philosophical questions are of particular interest because of mathematics’ unusual status. Mathematics is exceptional in that, on the one hand, it appears unhesitatingly true—no one doubts that 2 + 3 = 5—but on the other, as just noted, it is not about the physical world. This ambivalent status is what gives the philosophy of mathematics its special interest. The philosophy of mathematics is also one of the oldest academic fields, more or less coeval with philosophy itself. But contemporary philosophy of mathematics is rather different from its pre-twentieth-century antecedents, largely for three reasons. The first is that since the seventeenth century, mathematics has become integral to science. Science has over the past few centuries become increasingly mathematical, and indeed the fundamental science of nature, physics, is today recognised as a branch of applied mathematics. The second is that mathematics underwent a transformation in the course of nineteenth century: having started the century as a rather traditional-looking science of quantity it emerged a hundred years later a radically transformed abstract theory of structure. The final factor in the transformation of the philosophy of mathematics is the rise of modern logic. Developed by Frege, Cantor and others in the late nineteenth century, modern logic pervades contemporary mathematics, philosophy and computer science, and has had an immeasurable effect on the philosophy of mathematics. These volumes will collect the major works in this major field, with a focus on the last few decades. The anthology will include technical work, which interprets philosophically significant mathematical results or subfields of mathematics, as well as purely philosophical writing, aimed at those without advanced mathematics. The collection should be of interest to both philosophers and mathematicians, as well as to anyone who is susceptible to wondering what the main intellectual tool used in science, economics and finance, and indeed everyday life is ultimately about. (shrink)

Metaphysics, geometry, and the problems with diagrams -- The Pythagorean theorem: Euclid I.47 and VI.31 -- Thales and geometry: Egypt, Miletus, and beyond -- Pythagoras and the famous theorems -- From the Pythagorean theorem to the construction of the cosmos out of right triangles.

One effect of W. V. Quine’s assault on the analytic-synthetic distinction is pressure on the boundaries between mathematics and empirical science. Assumptions about reference and knowledge that are natural in the context of the empirical sciences have been exported to the case of mathematics. Problems then arise when we ask how, given the abstractness of mathematical entities, we can refer to them or know anything about them. For if abstractness entails causal impotence, and if reference and knowledge require (...) causal intercourse, then it seems a mystery how reference to, or knowledge of, abstract entities is at all possible. (shrink)

Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. In this survey article, the view is clarified and distinguished from some related views, and arguments for and against the view are discussed.

A green philosopher's peripeteia.--Physics and metaphysics of music.--The roots of arithmetic.--Critique of new geometrical abstractions.--The philosophical value of science.

PLATONISM or platonic realism in logic and mathematics is probably the most widespread contemporary view in the philosophy of mathematics. It has become popularly identified with the acceptance of an ontology of sets and/or classes as fundamental among the building materials of the cosmos and of all that is therein. Usually, also, these entities are regarded as "abstract" rather than "concrete," but no one has given us a sufficiently detailed and acceptable theory as to how this dichotomy is (...) to be drawn. Sets and classes are taken somehow as the paradigm of abstract entities, as over and against something called 'individuals', which are then the paradigm of concrete entities. This is all very unsatisfactory and in need of much analysis and clarification. (shrink)

The author elucidates the ontological basis of elementary mathematical theories and thereby assesses their certainty as a foundation for the more complex theories of modern mathematics, such as mathematical analysis and set theory. He adduces arguments in favor of the position of Frege, who held that geometry can provide a sufficiently broad and certain foundation for mathematics.

Until recently, discussion of virtues in the philosophy of mathematics has been fleeting and fragmentary at best. But in the last few years this has begun to change. As virtue theory has grown ever more influential, not just in ethics where virtues may seem most at home, but particularly in epistemology and the philosophy of science, some philosophers have sought to push virtues out into unexpected areas, including mathematics and its philosophy. But there are some mathematicians already there, (...) ready to meet them, who have explicitly invoked virtues in discussing what is necessary for a mathematician to succeed. In both ethics and epistemology, virtue theory tends to emphasize character virtues, the acquired excellences of people. But people are not the only sort of thing whose excellences may be identified as virtues. Theoretical virtues have attracted attention in the philosophy of science as components of an account of theory choice. Within the philosophy of mathematics, and mathematics itself, attention to virtues has emerged from a variety of disparate sources. Theoretical virtues have been put forward both to analyse the practice of proof and to justify axioms; intellectual virtues have found multiple applications in the epistemology of mathematics; and ethical virtues have been offered as a basis for understanding the social utility of mathematical practice. Indeed, some authors have advocated virtue epistemology as the correct epistemology for mathematics (and perhaps even as the basis for progress in the metaphysics of mathematics). This topical collection brings together several of the researchers who have begun to study mathematical practices from a virtue perspective with the intention of consolidating and encouraging this trend. (shrink)

What is quantum mechanics about? The most natural way to interpret quantum mechanics realistically as a theory about the world might seem to be what is called wave function ontology: the view according to which the wave function mathematically represents in a complete way fundamentally all there is in the world. Erwin Schroedinger was one of the first proponents of such a view, but he dismissed it after he realized it led to macroscopic superpositions (if the wave function evolves in (...) time according to the equations that has his name). The Many-Worlds interpretation1 accepts the existence of such macroscopic superpositions but takes it that they can never be observed. Superposed objects and superposed observers split together in different worlds of the type of the one we appear to live in. For these who, like Schroedinger, think that macroscopic superpositions are a problem, the common wisdom is that there are two alternative views: "Either the wave function, as given by the Schroedinger equation, is not everything, or is not right" [Bell 1987]. The deBroglie-Bohm theory, now commonly known as Bohmian Mechanics, takes the first option: the description provided by a Schroedinger-evolving wave function is supplemented by the information provided by the configuration of the particles. The second possibility consists in assuming that, while the wave function provides the complete description of the system, its temporal evolution is not given by the Schroedinger equation. Rather, the usual Schroedinger evolution is interrupted by random and sudden "collapses". The most promising theory of this kind is the GRW theory, named after the scientists that developed it: Gian Carlo Ghirardi, Alberto Rimini and Tullio Weber.. It seems tempting to think that in GRW we can take the wave function ontologically seriously and avoid the problem of macroscopic superpositions just allowing for quantum jumps. In this paper we will argue that such "bare" wave function ontology is not possible, neither for GRW nor for any other quantum theory: quantum mechanics cannot be about the wave function simpliciter. That is, we need more structure than the one provided by the wave function. As a response, quantum theories about the wave function can be supplemented with structure, without taking it as an additional ontology. We argue in reply that such "dressed-up" versions of wave function ontology are not sensible, since they compromise the acceptability of the theory as a satisfactory fundamental physical theory. Therefore we maintain that: 1- Strictly speaking, it is not possible to interpret quantum theories as theories about the wave function; 2- Even if the wave function is supplemented by additional non-ontological structures, there are reasons not to take the resulting theory seriously. Moreover, we will argue that any of the traditional responses to the measurement problem of quantum mechanics (Bohmian mechanics, GRW and Many-Worlds), contrarily to what commonly believed, share a common structure. That is, we maintain that: 3- All quantum theories should be regarded as theories in which physical objects are constituted by a primitive ontology. The primitive ontology is mathematically represented in the theory by a mathematical entity in three-dimensional space, or space-time. (shrink)

The Polish philosophy of mathematics in the 19th century is not a well-researched topic. For this period, only five philosophers are usually mentioned, namely Jan Śniadecki, Józef Maria Hoene-Wroński, Henryk Struve, Samuel Dickstein, and Edward Stamm. This limited and incomplete perspective does not allow us to develop a well-balanced picture of the Polish philosophy of mathematics and gauge its influence on 19th- and 20th-century Polish philosophy in general. To somewhat complete our picture of the history of the Polish (...) philosophy of mathematics in those times, we here present the profiles of some lesser-known Polish Romantic philosophers of the 19th century, namely Karol Libelt, Bronisław Trentowski, and Józef Kremer. We discuss their contributions to the philosophy of mathematics and their metaphysical perspectives, and we also show how their metaphysical ideas have found some continuity in the studies of some Catholic philosophers. (shrink)

In this paper, we describe "metaphysical reductions", in which the well-defined terms and predicates of arbitrary mathematical theories are uniquely interpreted within an axiomatic, metaphysical theory of abstract objects. Once certain (constitutive) facts about a mathematical theory T have been added to the metaphysical theory of objects, theorems of the metaphysical theory yield both an analysis of the reference of the terms and predicates of T and an analysis of the truth of the sentences of T. The well-defined terms and (...) predicates of T are analyzed as denoting abstract objects and abstract relations, respectively, in the background metaphysics, and the sentences of T have a reading on which they are true. After the technical details are sketched, the paper concludes with some observations about the approach. One important observation concerns the fact that the proper axioms of the background theory abstract objects can be reformulated in a way that makes them sound more like logical axioms. Some philosophers have argued that we should accept (something like) them as being logical. (shrink)

The essence of number was regarded by the ancient Greeks as the root cause of the existence of the universe, but it was only towards the end of the 19th century that mathematicians initiated an in-depth study of the nature of numbers. The resulting unavoidable actuality of infinities in the number system led mathematicians to rigorously investigate the foundations of mathematics. The formalist approach to establish mathematical proof was found to be inconclusive: Gödel showed that there existed true propositions (...) that could not be proved to be true within the natural number universe. This result weighed heavily on proposals in the mid-20th century for digital models of the universe, inspired by the emergence of the programmable digital computer, giving rise to the branch of philosophy recognised as digital philosophy. In this article, the models of the universe presented by physicists, mathematicians and theoretical computer scientists are reviewed and their relation to the natural numbers is investigated. A quantum theory view that at the deepest level time and space may be discrete suggests a profound relation between natural numbers and reality of the cosmos. The conclusion is that our perception of reality may ultimately be traced to the ontology and epistemology of the natural numbers. (shrink)

Graham Priest presents a ground-breaking account of the semantics of intentional language--verbs such as "believes," "fears," "seeks," or "imagines." Towards Non-Being proceeds in terms of objects that may be either existent or non-existent, at worlds that may be either possible or impossible. The book will be of central interest to anyone who is concerned with intentionality in the philosophy of mind or philosophy of language, the metaphysics of existence and identity, the philosophy of fiction, the philosophy of mathematics, (...) or cognitive representation in AI. (shrink)

The late 1960s saw the emergence of new philosophical interest in Kant's philosophy of mathematics, and since then this interest has developed into a major and dynamic field of study. In this state-of-the-art survey of contemporary scholarship on Kant's mathematical thinking, Carl Posy and Ofra Rechter gather leading authors who approach it from multiple perspectives, engaging with topics including geometry, arithmetic, logic, and metaphysics. Their essays offer fine-grained analysis of Kant's philosophy of mathematics in the context of (...) his Critical philosophy, and also show sensitivity to its historical background. The volume will be important for readers seeking a comprehensive picture of the current scholarship about the development of Kant's philosophy of mathematics, its place in his overall philosophy, and the Kantian themes that influenced mathematics and its philosophy after Kant. (shrink)

Metaphysics is sensitive to the conceptual tools we choose to articulate metaphysical problems. Those tools are a lens through which we view metaphysical problems; the same problems look different when we change the lens. There has recently been a shift to "postmodal" conceptual tools: concepts of ground, essence, and fundamentality. This shift transforms the debate over structuralism in the metaphysics of science and philosophy of mathematics. Structuralist theses say that patterns are "prior" to the nodes in the (...) patterns. In modal terms this would mean that the nodes cannot vary independently of the pattern. But its postmodal meaning is unclear. Does it mean, e.g., that a fundamental account of reality somehow speaks only of patterns? What would that mean? Three structuralist positions are examined through a postmodal lens: nomic/causal/dispositional essentialism, structuralism about individuals (e.g., structural realism, e.g., ante rem mathematical structuralism), comparativism about quantities. The question of when theories are equivalent, and how that impacts the debate over structuralism, is also discussed. (shrink)

According to a grand narrative that long ago ceased to be told, there was a seventeenth century Scientific Revolution, during which a few heroes conquered nature thanks to mathematics. This grand narrative began with the exhibition of quantitative laws that these heroes, Galileo and Newton for example, had disclosed: the law of falling bodies, according to which the speed of a falling body is proportional to the square of the time that has elapsed since the beginning of its fall; (...) the law of gravitation, according to which two bodies are attracted to one another in proportion to the sum of their masses and in inverse proportion to the square of the distance separating them -- according to his own preferences, each narrator added one or two quantitative laws of this kind. The essential feature was not so much the examples that were chosen, but, rather, the more or less explicit theses that accompanied them. First, mathematization would be taken as the criterion for distinguishing between a qualitative Aristotelian philosophy and the new quantitative physics. Secondly, mathematization was founded on the metaphysical conviction that the world was created pondere, numero et mensura, or that the ultimate components of natural things are triangles, circles, and other geometrical objects. This metaphysical conviction had two immediate consequences: that all the phenomena of nature can be in principle submitted to mathematics and that mathematical language is transparent; it is the language of nature itself and has simply to be picked up at the surface of phenomena. Finally, it goes without saying that, from a social point of view, the evolution of the sciences was apprehended through what has been aptly called the "relay runner model," according to which science progresses as a result of individual discoveries. Grand narratives such as this are perhaps simply fictions doomed to ruin as soon as they are clearly expressed. In any case, the very assumption on which this grand narrative relies can be brought into question: even in the canonical domain of mechanics, the relevant epistemological units crucial to understanding the dynamics of the Scientific Revolution are perhaps not a few laws of motion, but a complex set of problems embodied in mundane objects. Moreover, each of the theses just mentioned was actually challenged during the long period of historiographical reappraisal, out of which we have probably not yet stepped. Against the sharp distinction between a qualitative Aristotelian philosophy and the new quantitative physics, numerous studies insist that Rome wasn't built in a day, so to speak. Since Antiquity, there have always been mixed sciences; the emergence of pre-classical mechanics depends on both medieval treatises and the practical challenges met by Renaissance engineers. It is indeed true that, for Aristotle, mathematics merely captures the superficial properties of things, but the Aristotelianisms were many during the Renaissance and the Early Modern period, with some of them being compatible with the introduction of mathematics in natural philosophy. In addition, the gap between the alleged program of mathematizing nature and its effective realization was underlined as most natural phenomena actually escaped mathematization; at best they were enrolled in what Thomas Kuhn began to rehabilitate under the appellation of the "Baconian sciences," i.e., empirical investigations aiming at establishing isolated facts, without relating them to any overarching theory. Hence, mathematization of nature cannot pretend to capture a historical fact: at most, it expresses an indeterminate task for generations to come. On top of these first two considerations, and against the thesis of the neutrality of the mathematical language, it was urged that mathematics is not "only a language" and that, exactly as other symbolic means or cognitive tools, it has its own constraints. For example, it has been thoroughly explained that the Euclidean theory of proportions both guides and frustrates the Galilean analysis of motion; its shortages were particularly clear with respect to the expression of continuity, which is crucial in the case of motion. Consequently, when calculus was invented and applied to the analysis of motion, it was not a transposition that left things as they stood. Even more clearly than in the case of a translation from one natural language to another, the shift from one symbolic language to another entails that certain possibilities are opened while others are closed. The cognitive constraints imposed by established mathematical theories, as seen in the theory of proportions or calculus, were not the only ones to be studied in relation to mathematization. Certain schemes dependent on the grammar of natural languages, e.g., the scheme of contrariety, or certain symbolic means of representation, e.g. geometrical diagrams and numerical tables, were also subject to such scrutiny. Lastly, it was insisted that, even if we concede the existence of scientific geniuses, mathematics is largely produced by intellectual communities and embedded within social practices. More attention was consequently paid to the forms of communication in given mathematical networks, or to the teaching of the discipline in, for example, Jesuit colleges and universities. The set of mathematical practices specific to specialized craftsmen, highly-qualified experts and engineers began to be studied in its own right. All these reflections may have helped us change our perspectives on the question of mathematization. It seems, however, that they were instead set aside, both because of a general distrust towards sweeping narratives that are always subject to the suspicion that they overlook the unyielding complexity of real history, and because of a shift in our interests. The more obscure and idiosyncratic they are, an alchemist, a patron of the sciences or a lunatic collector is nowadays honored in journals of the history of sciences. As for the general issues involved in the question of mathematization, they are rejected as obsolete, or reserved for specialized journals in the history of mathematics. Consequently, before presenting the essays of this fascicle, I would like to say a few words in favor of a renewed study of the forms of mathematization in the history of the early sciences. (shrink)