Shaping the Enemy: Foundational Labelling by L.E.J. Brouwer and A. Heyting

The use of the three labels (logicism, formalism, intuitionism) to denote the three foundational schools of the early twentieth century are now part of literature. Yet, neither their number nor their adoption has been stable over the twentieth century. They were not introduced by the founding fathers of each school: namely, neither Frege nor Russell spoke of ‘logicism’; and even Hilbert did not use the word ‘formalism’ to introduce his foundational programs. At a certain point, only Brouwer used the label ‘intuitionism’ in his scientific production to personify his philosophy of mathematics and he used the label ‘formalism’ for Hilbert’s foundational viewpoint. Starting with Brouwer, the origin of the use of the three labels to represent a foundational meaning, will be analysed in this paper. Thereafter, the role that Brouwer’s pupil Arend Heyting had in the production and use of foundational labels will be considered. On the basis of the comparison of the attitudes of these two scholars I will finally advance the thesis that not only the creation but also the use of labels, far from being a mere gesture of academic reference to literature, can be a sign of the cultural operation each scholar wanted to do.     

The influence of Platonism on mathematics research and theological beliefs

An age-old debate in the philosophy of mathematics is whether mathematics is discovered or invented. There are four popular viewpoints in this debate, namely Platonism, formalism, intuitionism, and logicism. A natural question that arises is whether belief in one of these viewpoints affects the mathematician’s research? In particular, does subscribing to a Platonist or a formalist viewpoint influence how a mathematician conducts research? Does the area of research influence a mathematician’s beliefs on the nature of mathematics? How are the beliefs regarding the nature of mathematics connected to theological beliefs? In order to investigate these questions, five professional research mathematicians were interviewed. The mathematicians worked in diverse areas within analysis, algebra, and within applied mathematics, and had a combined 160 years of research experience. Although none of the mathematicians wanted to be pigeonholed into any one category of beliefs, the study revealed that four of the mathematicians leaned towards Platonism, which runs contrary to the popular notion that Platonism is an exception today. This study revealed that beliefs regarding the nature of mathematics influenced how mathematicians’ conducted research and were deeply connected to their theological beliefs. The findings are presented in the form of vignettes that give an insight into the mathematical and theological belief structures of the mathematicians.     

Proof vs Provability: On Brouwer’s Time Problem

Is a mathematical theorem proved because provable, or provable because proved? If Brouwer’s intuitionism is accepted, we’re committed, it seems, to the latter, which is highly problematic. Or so I will argue. This and other consequences of Brouwer’s attempt to found mathematics on the intuition of a move of time have heretofore been insufficiently appreciated. Whereas the mathematical anomalies of intuitionism have received enormous attention, too little time, I’ll try to show, has been devoted to some of the temporal anomalies that Brouwer has invited us to introduce into mathematics.     

Intuitionism,Meaning Theory and Cognition

Michael Dummett has interpreted and expounded upon intuitionism under the influence of Wittgensteinian views on language, meaning and cognition. I argue against the application of some of these views to intuitionism and point to shortcomings in Dummett's approach. The alternative I propose makes use of recent, post-Wittgensteinian views in the philosophy of mind, meaning and language. These views are associated with the claim that human cognition exhibits intentionality and with related ideas in philosophical psychology. Intuitionism holds that mathematical constructions are mental processes or objects. Constructions are, in the first instance, forms of consciousness or possible experience of a particular type. As such, they must be understood in terms of the concept of intentionality. This view has a historical basis in the literature on intuitionism. In a famous 1931 lecture Heyting in fact identifies constructions with fulfilled or fulfillable mathematical intentions. I consider some of the consequences of this identification and contrast them with Dummett's views on intuitionism.     

What Anit-realist Intuitionism Could Not Be

One of the two major parts of Dummett’s defense of intuitionism is the rejection of classical in favor of intuitionistic reasoning in mathematics, given that mathematical discourse is anti-realist. While there have been illuminating discussions of what Dummett’s argument for this might be, no consensus seems to have emerged about its overall form. In this paper I give an account of this form, starting by investigating a fundamental, but little discussed question: to what view of the relation between deductive principles and meaning is anti-realism committed? The result of this investigation is a constraint on meaning theoretic assessments of logical laws. Given this constraint, I show that, surprisingly, a consistent anti-realist critique of classical logic could not rely on the rejection of bivalence. Moreover, a consistent anti-realist defense of intuitionism must begin with a radical rejection of the very conception of logical consequence that underlies realist classical logic. It follows from these conclusions that anti-realist intuitionism seems committed to proceeding by proof theoretic means.     

VOLKER PECKHAUS (ed.), Oskar Becker und die Philosophie der Mathematik.

It has been contended that it is unjustified to believe, as Weyl did, that formalism's victory against intuitionism entails a defeat of the phenomenological approach to mathematics. The reason for this contention, recently put forth by Paolo Mancosu and Thomas Ryckman, is that, unlike intuitionistic Anschauung, phenomenological intuition could ground classical mathematics. I argue that this indicates a misinterpretation of Weyl's view, for he did not take formalism to prevail over intuitionism with respect to grounding classical mathematics. I also point out that the contention is false: if intuitionism fails, in the way Weyl thought it did, i.e. with respect to supporting scientific objectivity, then one should also reject the phenomenological approach, in the same respect.     

Hintikka’s Logical Revolution

Hintikka thinks that second-order logic is not pure logic, and because of Gödel’s incompleteness theorems, he suggests that we should liberate ourselves from the mistaken idea that first-order logic is the foundational logic of mathematics. With this background he introduces his independence friendly logic (IFL). In this paper, I argue that approaches taking Hintikka’s IFL as a foundational logic of mathematics face serious challenges. First, the quantifiers in Hintikka’s IFL are not distinguishable from Linström’s general quantifiers, which means that the quantifiers in IFL involve higher order entities. Second, if we take Wright’s interpretation of quantifiers or if we take Hale’s criterion for the identity of concepts, Quine’s thesis that second-order logic is set theory will be rejected. Third, Hintikka’s definition of truth itself cannot be expressed in the extension of language of IFL. Since second-order logic can do what IFL does, the significance of IFL for the foundations of mathematics is weakened.     

Ethical intuitionism and the linguistic analogy

It is a central tenet of ethical intuitionism as defended by W. D. Ross and others that moral theory should re?ect the convictions of mature moral agents. Hence, intuitionism is plausible to the extent that it corresponds to our well-considered moral judgments. After arguing for this claim, I discuss whether intuitionists o?er an empirically adequate account of our moral obligations. I do this by applying recent empirical research by John Mikhail that is based on the idea of a universal moral grammar to a number of claims implicit in W. D. Ross’s normative theory. I argue that the results at least partly vindicate intuitionism.     

On the status of proofs by contradiction in the seventeenth century

In this paper I show that proofs by contradiction were a serious problem in seventeenth century mathematics and philosophy. Their status was put into question and positive mathematical developments emerged from such reflections. I analyse how mathematics, logic, and epistemology are intertwined in the issue at hand. The mathematical part describes Cavalieri's and Guldin's mathematical programmes of providing a development of parts of geometry free of proofs by contradiction. The logical part shows how the traditional Aristotelean doctrine that perfect demonstrations are causal demonstrations influenced the reflection on proofs by contradiction. The main protagonist of this part is Wallis. Finally, I analyse some epistemological developments arising from the Cartesian tradition. In particular, I look at Arnauld's programme of providing an epistemologically motivated reformulation of Geometry free of proofs by contradiction. The conclusion explains in which sense these epistemological reflections can be compared with those informing contemporary intuitionism.     

The Russellian Influence on Hilbert and his School

The aim of the paper is to discuss the influence exercised by Russell's thought inGöttingen in the period leading to the formulation of Hilbert's program in theearly twenties. I show that after a period of intense foundational work, culminatingwith the departure from Göttingen of Zermelo and Grelling in 1910 we witnessa reemergence of interest in foundations of mathematics towards the end of 1914. Itis this second period of foundational work that is my specific interest. Through theuse of unpublished archival sources I will describe how Hilbert, Behmann, and Bernays,among others, were influenced by and reacted to the technical and philosophical thesespresented in Principia Mathematica. I also argue that there are some elements of continuity between Russell's approach and Hilbert's program as it was presented inthe early twenties.     

On Certainty,Change, and “Mathematical Hinges”

Annalisa Coliva (Int J Study Skept 10(3–4):346–366, 2020) asks, “Are there mathematical hinges?” I argue here, against Coliva’s own conclusion, that there are. I further claim that this affirmative answer allows a case to be made for taking the concept of a hinge to be a useful and general-purpose tool for studying mathematical practice in its real complexity. Seeing how Wittgenstein can, and why he would, countenance mathematical hinges additionally gives us a deeper understanding of some of his latest thoughts on mathematics. For example, a view of how mathematical hinges relate to Wittgenstein’s well-known river-bed analogy enables us to see how his way of thinking about mathematics can account nicely for a “dynamics of change” within mathematical research—something his philosophy of mathematics has been accused of missing (e.g., by Robert Ackermann (Wittgenstein’s city, The University of Massachusetts Press, Amherst, 1988) and Mark Wilson (Wandering significance: an essay on conceptual behavior, Oxford University Press, Oxford, 2006). Finally, the perspective on mathematical hinges ultimately arrived at will be seen to provide us with illuminating examples of how our conceptual choices and theories can be ungrounded but nevertheless the right ones (in a sense to be explained).

     

Słupecki's Generalized Mereology and Its Flaws

One of the streams in the early development of set theory was an attempt to use mereology, a formal theory of parthood, as a foundational tool. The first such attempt is due to a Polish logician, Stanis?aw Le?niewski (1886–1939). The attempt failed, but there is another, prima facie more promising attempt by Jerzy S?upecki (1904–1987), who employed his generalized mereology to build mereological foundations for type theory. In this paper I (1) situate Le?niewski's attempt in the development of set theory, (2) describe and evaluate Le?niewski's approach, (3) describe S?upecki's strategy without unnecessary technical details, and (4) evaluate it with a rather negative outcome. The issues discussed go beyond merely historical interests due to the current popularity of mereology and because they are related to nominalistic attempts to understand mathematics in general. The introduction describes very briefly the situation in which mereology entered the scene of foundations of mathematics — it can be safely skipped by anyone familiar with the early development of set theory. Section 2 describes and evaluates Le?niewski's attempt to use mereology as a foundational tool. In Section 3, I describe an attempt by S?upecki to improve on Le?niewski's work, which resulted in a system called generalized mereology. In Section 4, I point out the reasons why this attempt is still not successful. Section 5 contains an explanation of why Le?niewski's use of Ontology in developing arithmetic also is not nominalistically satisfactory.     

Gödel And The Intuition Of Concepts

Gödel has argued that we can cultivate the intuition or ‘perception’ of abstractconcepts in mathematics and logic. Gödel's ideas about the intuition of conceptsare not incidental to his later philosophical thinking but are related to many otherthemes in his work, and especially to his reflections on the incompleteness theorems.I describe how some of Gödel's claims about the intuition of abstract concepts are related to other themes in his philosophy of mathematics. In most of this paper, however,I focus on a central question that has been raised in the literature on Gödel: what kind of account could be given of the intuition of abstract concepts? I sketch an answer to this question that uses some ideas of a philosopher to whom Gödel also turned in this connection: Edmund Husserl. The answer depends on how we understand the conscious directedness toward ‘objects’ and the meaning of the term ‘abstract’ in the context of a theory of the intentionality of cognition.     

Conceptual engineering for mathematical concepts

In this paper I investigate how conceptual engineering applies to mathematical concepts in particular. I begin with a discussion of Waismann’s notion of open texture, and compare it to Shapiro’s modern usage of the term. Next I set out the position taken by Lakatos which sees mathematical concepts as dynamic and open to improvement and development, arguing that Waismann’s open texture applies to mathematical concepts too. With the perspective of mathematics as open-textured, I make the case that this allows us to deploy the tools of conceptual engineering in mathematics. I will examine Cappelen’s recent argument that there are no conceptual safe spaces and consider whether mathematics constitutes a counterexample. I argue that it does not, drawing on Haslanger’s distinction between manifest and operative concepts, and applying this in a novel way to set-theoretic foundations. I then set out some of the questions that need to be engaged with to establish mathematics as involving a kind of conceptual engineering. I finish with a case study of how the tools of conceptual engineering will give us a way to progress in the debate between advocates of the Universe view and the Multiverse view in set theory.     

Hermann Weyl on Minkowskian Space–Time and Riemannian Geometry

Hermann Weyl as a founding father of field theory in relativistic physics and quantum theory always stressed the internal logic of mathematical and physical theories. In line with his stance in the foundations of mathematics, Weyl advocated a constructivist approach in physics and geometry. An attempt is made here to present a unified picture of Weyl’s conception of space–time theories from Riemann to Minkowski. The emphasis is on the mathematical foundations of physics and the foundational significance of a constructivist philosophical point of view. I conclude with some remarks on Weyl’s broader philosophical views.     

Hilbert,logicism, and mathematical existence

David Hilbert’s early foundational views, especially those corresponding to the 1890s, are analysed here. I consider strong evidence for the fact that Hilbert was a logicist at that time, following upon Dedekind’s footsteps in his understanding of pure mathematics. This insight makes it possible to throw new light on the evolution of Hilbert’s foundational ideas, including his early contributions to the foundations of geometry and the real number system. The context of Dedekind-style logicism makes it possible to offer a new analysis of the emergence of Hilbert’s famous ideas on mathematical existence, now seen as a revision of basic principles of the “naive logic” of sets. At the same time, careful scrutiny of his published and unpublished work around the turn of the century uncovers deep differences between his ideas about consistency proofs before and after 1904. Along the way, we cover topics such as the role of sets and of the dichotomic conception of set theory in Hilbert’s early axiomatics, and offer detailed analyses of Hilbert’s paradox and of his completeness axiom (Vollständigkeitsaxiom).     

The Role of Intersectionality in Marriage and Family Therapy Multicultural Supervision

This paper offers a framework for understanding intersectionality and its role within marriage and family therapy multicultural supervision. An explanation of intersectionality, social identity relationships, multicultural supervision foundations and the influence of power through supervision will be discussed. Further, this paper will describe foundational concepts in the multicultural supervisory dyad and challenges integrating intersectionality within marriage and family therapy supervision. Implications for clinical supervision and research directions are discussed through cross cultural models, a randomized control design and conversation analysis.     

Moral perception,inference, and intuition

Sarah McGrath argues that moral perception has an advantage over its rivals in its ability to explain ordinary moral knowledge. I disagree. After clarifying what the moral perceptualist is and is not committed to, I argue that rival views are both more numerous and more plausible than McGrath suggests: specifically, I argue that (a) inferentialism can be defended against McGrath’s objections; (b) if her arguments against inferentialism succeed, we should accept a different rival that she neglects, intuitionism; and (c), reductive epistemologists can appeal to non-naturalist commitments to avoid McGrath’s counterexamples.

     

The Meaning of Category Theory for 21st Century Philosophy

Among the main concerns of 20th century philosophy was that of the foundations of mathematics. But usually not recognized is the relevance of the choice of a foundational approach to the other main problems of 20th century philosophy, i.e., the logical structure of language, the nature of scientific theories, and the architecture of the mind. The tools used to deal with the difficulties inherent in such problems have largely relied on set theory and its “received view”. There are specific issues, in philosophy of language, epistemology and philosophy of mind, where this dependence turns out to be misleading. The same issues suggest the gain in understanding coming from category theory, which is, therefore, more than just the source of a “non-standard” approach to the foundations of mathematics. But, even so conceived, it is the very notion of what a foundation has to be that is called into question. The philosophical meaning of mathematics is no longer confined to which first principles are assumed and which “ontological” interpretation is given to them in terms of some possibly updated version of logicism, formalism or intuitionism. What is central to any foundational project proper is the role of universal constructions that serve to unify the different branches of mathematics, as already made clear in 1969 by Lawvere. Such universal constructions are best expressed by means of adjoint functors and representability up to isomorphism. In this lies the relevance of a category-theoretic perspective, which leads to wide-ranging consequences. One such is the presence of functorial constraints on the syntax–semantics relationships; another is an intrinsic view of (constructive) logic, as arises in topoi and, subsequently, in more general fibrations. But as soon as theories and their models are described accordingly, a new look at the main problems of 20th century’s philosophy becomes possible. The lack of any satisfactory solution to these problems in a purely logical and set-theoretic setting is the result of too circumscribed an approach, such as a static and punctiform view of objects and their elements, and a misconception of geometry and its historical changes before, during, and after the foundational “crisis”, as if algebraic geometry and synthetic differential geometry – not to mention algebraic topology – were secondary sources for what concerns foundational issues. The objectivity of basic geometrical intuitions also acts against the recent version of structuralism proposed as ‘the’ philosophy of category theory. On the other hand, the need for a consistent and adequate conceptual framework in facing the difficulties met by pre-categorical theories of language and scientific knowledge not only provides the basic concepts of category theory with specific applications but also suggests further directions for their development (e.g., in approaching the foundations of physics or the mathematical models in the cognitive sciences). This ‘virtuous’ circle is by now largely admitted in theoretical computer science; the time is ripe to realise that the same holds for classical topics of philosophy. Text of a talk given at the Workshop and Symposium on the Ramifications of Category Theory, Florence, November 18–22, 2003. For further documentation on the conference, see     

Overcoming Functional Fixedness in Naming Traditions: A Commentary on Pilcher’s Names and “Doing Gender”

Because people’s names are central to everyday life, their role in the gender system is often overlooked. In the target article, Pilcher (2017) brings novel attention to the ways in which naming traditions allow individuals to enact gender in their lives. In this commentary, I expand on Pilcher’s argument that naming traditions merit more attention than they currently receive. Specifically, I begin by discussing links between forenaming and the gender binary. I then describe the ways in which the marital surname tradition reflects gendered power dynamics. In an effort to spur additional scholarly attention to naming traditions, I also delineate fruitful areas for future research. These future directions primarily focus on identifying why and how individuals “do gender differently” through their naming practices.