In this paper, following the claims made by various mathematicians, I try to construct a theory of levels of abstraction. I first try to clarify the basic components of the abstract method as it developed in the first quarter of the 20th century. I then submit an explication of the notion of levels of abstraction. In the final section, I briefly explore some of main philosophical consequences of the theory.
A Study of the History and Philosophy of Category Theory Jean-Pierre Marquis. to say that objects are dispensable in geometry. What is claimed is that the specific nature of the objects used is irrelevant. To use the terminology already ...
In this paper, I introduce the idea that some important parts of contemporary pure mathematics are moving away from what I call the extensional point of view. More specifically, these fields are based on criteria of identity that are not extensional. After presenting a few cases, I concentrate on homotopy theory where the situation is particularly clear. Moreover, homotopy types are arguably fundamental entities of geometry, thus of a large portion of mathematics, and potentially to all mathematics, at least according (...) to some speculative research programs. (shrink)
The aim of this paper is to clarify the role of category theory in the foundations of mathematics. There is a good deal of confusion surrounding this issue. A standard philosophical strategy in the face of a situation of this kind is to draw various distinctions and in this way show that the confusion rests on divergent conceptions of what the foundations of mathematics ought to be. This is the strategy adopted in the present paper. It is divided into 5 (...) sections. We first show that already in the set theoretical framework, there are different dimensions to the expression foundations of. We then explore these dimensions more thoroughly. After a very short discussion of the links between these dimensions, we move to some of the arguments presented for and against category theory in the foundational landscape. We end up on a more speculative note by examining the relationships between category theory and set theory. (shrink)
Le réalisme scientifique occupe une place centrale dans le système philosophique de Mario Bunge. Au cœur de cette thèse, on trouve l’affirmation selon laquelle nous pouvons connaître le monde partiellement. Il s’ensuit que les théories scientifiques ne sont pas totalement vraies ou totalement fausses, mais plutôt partiellement vraies et partiellement fausses. Ces énoncés sur la connaissance scientifique, à première vue plausible pour quiconque est familier avec la pratique scientifique, demandent néanmoins à être clarifiés, précisés et, ultimement, à être inclus dans (...) un cadre théorique plus large et rigoureux. Depuis ses toutes premières publications sur ces questions et jusqu’à récemment, Mario Bunge n’a cessé d’interpeller les philosophes afin qu’ils développent une théorie, au sens propre du terme, de la vérité partielle afin de clarifier les enjeux épistémologiques liés au réalisme scientifique. Bunge a lui-même proposé plusieurs parties de cette théorie au fil des années, mais aucune de ces propositions ne l’a satisfait pleinement et la construction de cette théorie demeure un problème entier. Dans ce texte, nous passerons rapidement en revue certaines des approches proposées par Bunge dans ses publications et nous esquisserons certaines pistes qui devraient servir à tout le moins de desiderata pour la construction d’une théorie de la vérité partielle. (shrink)
Structuralism has recently moved center stage in philosophy of mathematics. One of the issues discussed is the underlying logic of mathematical structuralism. In this paper, I want to look at the dual question, namely the underlying structures of logic. Indeed, from a mathematical structuralist standpoint, it makes perfect sense to try to identify the abstract structures underlying logic. We claim that one answer to this question is provided by categorical logic. In fact, we claim that the latter can be seen—and (...) probably should be seen—as being a structuralist approach to logic and it is from this angle that categorical logic is best understood. (shrink)
In this paper, we look at Bourbaki’s work as a case study for the notion of mathematical style. We argue that indeed Bourbaki exemplifies a mathematical style, namely the structuralist style.
In this paper, we try to establish that some mathematical theories, like K-theory, homology, cohomology, homotopy theories, spectral sequences, modern Galois theory (in its various applications), representation theory and character theory, etc., should be thought of as (abstract) machines in the same way that there are (concrete) machines in the natural sciences. If this is correct, then many epistemological and ontological issues in the philosophy of mathematics are seen in a different light. We concentrate on one problem which immediately follows (...) the recognition of the particular status of these theories: the demarcation problem between ‘natural kinds’ and ‘artefacts’. (shrink)
Approximations form an essential part of scientific activity and they come in different forms: conceptual approximations (simplifications in models), mathematical approximations of various types (e.g. linear equations instead of non-linear ones, computational approximations), experimental approximations due to limitations of the instruments and so on and so forth. In this paper, we will consider one type of approximation, namely numerical approximations involved in the comparison of two results, be they experimental or theoretical. Our goal is to lay down the conceptual and (...) formal foundations of a local theory of partial truth. This is done by introducing and exploring the concept of truth space. (shrink)
One of the key features of modern mathematics is the adoption of the abstract method. Our goal in this paper is to propose an explication of that method that is rooted in the history of the subject.
In this paper, we argue that, contrary to the view held by most philosophers of mathematics, Bourbaki’s technical conception of mathematical structuralism is relevant to philosophy of mathematics. In fact, we believe that Bourbaki has captured the core of any mathematical structuralism.
Categorical foundations and set-theoretical foundations are sometimes presented as alternative foundational schemes. So far, the literature has mostly focused on the weaknesses of the categorical foundations. We want here to concentrate on what we take to be one of its strengths: the explicit identification of so-called canonical maps and their role in mathematics. Canonical maps play a central role in contemporary mathematics and although some are easily defined by set-theoretical tools, they all appear systematically in a categorical framework. The key (...) element here is the systematic nature of these maps in a categorical framework and I suggest that, from that point of view, one can see an architectonic of mathematics emerging clearly. Moreover, they force us to reconsider the nature of mathematical knowledge itself. Thus, to understand certain fundamental aspects of mathematics, category theory is necessary (at least, in the present state of mathematics). (shrink)
In this paper, I explore Bunge’s fictionism in philosophy of mathematics. After an overview of Bunge’s views, in particular his mathematical structuralism, I argue that the comparison between mathematical objects and fictions ultimately fails. I then sketch a different ontology for mathematics, based on Thomasson’s metaphysical work. I conclude that mathematics deserves its own ontology, and that, in the end, much work remains to be done to clarify the various forms of dependence that are involved in mathematical knowledge, in particular (...) its dependence on mental/brain states and material objects. (shrink)
Scientists use models to know the world. It i susually assumed that mathematicians doing pure mathematics do not. Mathematicians doing pure mathematics prove theorems about mathematical entities like sets, numbers, geometric figures, spaces, etc., they compute various functions and solve equations. In this paper, I want to exhibit models build by mathematicians to study the fundamental components of spaces and, more generally, of mathematical forms. I focus on one area of mathematics where models occupy a central role, namely homotopy theory. (...) I argue that mathematicians introduce genuine models and I offer a rough classification of these models. (shrink)
_Erich Reck* * and Georg Schiemer.** ** The Prehistory of Mathematical Structuralism. _Oxford University Press, 2020. Pp. 454. ISBN: 978-0-19-064122-1 ; 978-0-19-064123-8. doi: 10.1093/oso/9780190641221.001.0001.
Some concepts that are now part and parcel of mathematics used to be, at least until the beginning of the twentieth century, a central preoccupation of mathematicians and philosophers. The concept of continuity, or the continuous, is one of them. Nowadays, many philosophers of mathematics take it for granted that mathematicians of the last quarter of the nineteenth century found an adequate conceptual analysis of the continuous in terms of limits and that serious philosophical thinking is no longer required, except (...) perhaps when the question of the continuum is transferred to the arena of set theory where it takes the form of the infamous continuum hypothesis. As Philip Ehrlich has recently shown, this conviction goes back to the early writings of Russell who, in 1903 and then again in later writings, forcefully and eloquently pushed the view that mathematicians had given the final answer to immemorial conundrums arising from the continuous and infinitesimals . This proclamation of victory came with what was announced as the necessary defeat of the notion of the infinitesimal, despite the fact that mathematicians like Thomae, Du Bois-Reymond, Stolz, Bettazi, Veronese, Levi-Civita, and Hahn were investigating mathematical structures containing infinitesimals in a mathematically rigorous and logically consistent manner. In this respect Russell was merely walking in the footsteps of Cantor, and many of Russell's contemporaries were only too keen to keep infinitesimals out of Cantor's paradise. However, although Cantor certainly wanted to send the notion of infinitesimal to Hell, the notion kept a low profile in the mathematical purgatory, making its way in the study of non-Archimedean ordered algebraic systems. Of course, nowadays everyone has heard of Robinson's attempt at resurrecting infinitesimals in analysis in the form of non-standard analysis, but Robinson's work, despite the fact that it had, in …. (shrink)
The nature of truth has occupied philosophers since the very beginning of the field. Our goal is to clarify the notion of scientific truth, in particular the notion of partial truth of facts. Our strategy consists to brake the problem into smaller, more manageable, questions. Thus, we distinguish the truth of a scientific theory, what we call the "global" truth value of a theory, from the truth of a particular scientific proposition, what we call the "local" truth values of a (...) theory. We will present a new local theory of partial truth and will have few things to say about the global level. Moreover, we will also introduce some purely formal results, the most important being the introduction of a new class of algebraic structures which have some interesting connections with classical logic. (shrink)
Suppose we were to meet with extraterrestrials and that we were able to have a discussion about our respective cultures. At some point, they start asking questions about that something which we call “mathematics”. “What is it?”, they ask. Tough question. How should we answer them?
In this paper, I introduce and examine the notion of “mathematical engineering” and its impact on mathematical change. Mathematical engineering is an important part of contemporary mathematics and it roughly consists of the “construction” and development of various machines, probes and instruments used in numerous mathematical fields. As an example of such constructions, I briefly present the basic steps and properties of homology theory. I then try to show that this aspect of contemporary mathematics has important consequences on our conception (...) of mathematical knowledge, in particular mathematical growth. (shrink)
The aim of this paper is to put into context the historical, foundational and philosophical significance of category theory. We use our historical investigation to inform the various category-theoretic foundational debates and to point to some common elements found among those who advocate adopting a foundational stance. We then use these elements to argue for the philosophical position that category theory provides a framework for an algebraic in re interpretation of mathematical structuralism. In each context, what we aim to show (...) is that, whatever the significance of category theory, it need not rely upon any set-theoretic underpinning. (shrink)
The aim of this paper is to put into context the historical, foundational and philosophical significance of category theory. We use our historical investigation to inform the various category-theoretic foundational debates and to point to some common elements found among those who advocate adopting a foundational stance. We then use these elements to argue for the philosophical position that category theory provides a framework for an algebraic _in re_ interpretation of mathematical structuralism. In each context, what we aim to show (...) is that, whatever the significance of category theory, it need not rely upon any set-theoretic underpinning. (shrink)
This paper looks at how the idea of pointless topology itself evolved during its pre-localic phase by analyzing the definitions of the concept of topological space of Menger and Nöbeling. Menger put forward a topology of lumps in order to generalize the definition of the real line. As to Nöbeling, he developed an abstract theory of posets so that a topological space becomes a particular case of topological poset. The analysis emphasizes two points. First, Menger's geometrical perspective was superseded by (...) an algebraic one, a lattice-theoretical one to be precise. Second, Menger's bottom–up approach was replaced by a top–down one. (shrink)
In this paper, we argue that Forrester’s paradox, as he presents it, is not a paradox of standard deontic logic. We show that the paradox fails since it is the result of a misuse of , a derived rule in the standard systems. Before presenting Forrester’s argument against standard deontic logic, we will briefly expose the principal characteristics of a standard system Δ. The modal system KD will be taken as a representative. We will then make some remarks regarding , (...) pointing out that its use is restricted to the standard system’s theorems, and cannot be applied to contingent conditionals. Finally, we will show that Forrester’s paradox is not a paradox of standard deontic logic, at least not in the sense he intended it to be. We show that the paradox cannot arise in KD since its semantical model is not rich enough to represent the intuitive validity of the conditional within Forrester’s paradox. We show that the paradox arises within a system that has a finer semantics. (shrink)
This volume has 41 chapters written to honor the 100th birthday of Mario Bunge. It celebrates the work of this influential Argentine/Canadian physicist and philosopher. Contributions show the value of Bunge’s science-informed philosophy and his systematic approach to philosophical problems. The chapters explore the exceptionally wide spectrum of Bunge’s contributions to: metaphysics, methodology and philosophy of science, philosophy of mathematics, philosophy of physics, philosophy of psychology, philosophy of social science, philosophy of biology, philosophy of technology, moral philosophy, social and political (...) philosophy, medical philosophy, and education. The contributors include scholars from 16 countries. Bunge combines ontological realism with epistemological fallibilism. He believes that science provides the best and most warranted knowledge of the natural and social world, and that such knowledge is the only sound basis for moral decision making and social and political reform. Bunge argues for the unity of knowledge. In his eyes, science and philosophy constitute a fruitful and necessary partnership. Readers will discover the wisdom of this approach and will gain insight into the utility of cross-disciplinary scholarship. This anthology will appeal to researchers, students, and teachers in philosophy of science, social science, and liberal education programmes. 1. Introduction Section I. An Academic Vocation Section II. Philosophy Section III. Physics and Philosophy of Physics Section IV. Cognitive Science and Philosophy of Mind Section V. Sociology and Social Theory Section VI. Ethics and Political Philosophy Section VII. Biology and Philosophy of Biology Section VIII. Mathematics Section IX. Education Section X. Varia Section XI. Bibliography. (shrink)
