How to be a structuralist all the way down

This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, in relation to the “algebraic” approach to mathematical structuralism. My aim is to show that category theory has as much to say about an algebraic consideration of meta-mathematical analyses of logical structure as it does about mathematical analyses of mathematical structure, without either requiring an assertory mathematical or meta-mathematical background theory as a “foundation”, or turning meta-mathematical analyses of logical concepts into “philosophical” ones. Thus, we can use category theory to frame an interpretation of mathematics according to which we can be structuralists all the way down.     

Mathematizing phenomenology

Husserl is well known for his critique of the “mathematizing tendencies” of modern science, and is particularly emphatic that mathematics and phenomenology are distinct and in some sense incompatible. But Husserl himself uses mathematical methods in phenomenology. In the first half of the paper I give a detailed analysis of this tension, showing how those Husserlian doctrines which seem to speak against application of mathematics to phenomenology do not in fact do so. In the second half of the paper I focus on a particular example of Husserl’s “mathematized phenomenology”: his use of concepts from what is today called dynamical systems theory.

Saving the intuitions: polylithic reference

My main aim in this paper is to clarify the concepts of referential success and of referential continuity that are so crucial to the scientific realism debate. I start by considering the three dominant theories of reference and the intuitions that motivate each of them. Since several intuitions cited in support of one theory conflict with intuitions cited in support of another something has to give way. The traditional policy has been to reject all intuitions that clash with a chosen theory. A more radical policy, tied to some experimental philosophers, has called for the rejection of any evidential role for intuitions. I explore a largely ignored third alternative, i.e. saving intuitions (and their evidential role) even when they are at odds. To accommodate conflicting intuitions different sets of internally consistent (yet externally inconsistent) intuitions are taken to lend credence to different concepts of reference. In the current context, this means that the concepts of referential success and referential continuity are not monolithic. They are what I call ‘polylithic’. This paper is as much about meta-philosophical concerns with the role of intuitions as it is about reference and the scientific realism debate. Regarding the former I hope that a blueprint will emerge for similar projects in other philosophical domains. Regarding the latter, I hope that polylithicity helps disentangle claims about referential success and continuity in the scientific realism debate by making perspicuous which concepts are best equipped to evaluate the realist’s epistemic claims against the historical record of science.     

What “Intuitions” are Linguistic Evidence?

In “Intuitions in Linguistics” (2006a) and Ignorance of Language (2006b) I took it to be Chomskian orthodoxy that a speaker’s metalinguistic intuitions are provided by her linguistic competence. I argued against this view in favor of the alternative that the intuitions are empirical theory-laden central-processor responses to linguistic phenomena. The concern about these linguistic intuitions arises from their apparent role as evidence for a grammar. Mark Textor, “Devitt on the Epistemic Authority of Linguistic Intuitions” (2009), argues that I have picked the wrong intuitions: I should have picked non-judgmental linguistic “seemings”. These reside between metalinguistic judgments and linguistic performances and have an epistemic authority that the orthodox view may well be able to explain. Textor seems to think that the metalinguistic intuitions are not evidence at all. I argue that he is wrong about that. More importantly, I argue that there are no “in-between” linguistic seemings with epistemic authority.     

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     

A New Mixed View of Virtue Ethics, Based on Daniel Doviak’s New Virtue Calculus

In A New Form of Agent-Based Virtue Ethics, Daniel Doviak develops a novel agent-based theory of right action that treats the rightness (or deontic status) of an action as a matter of the action’s net intrinsic virtue value (net-IVV)—that is, its balance of virtue over vice. This view is designed to accommodate three basic tenets of commonsense morality: (i) the maxim that “ought” implies “can,” (ii) the idea that a person can do the right thing for the wrong reason, and (iii) the idea that a virtuous person can have “mixed motives.” In this paper, I argue that Doviak’s account makes an important contribution to agent-based virtue ethics, but it needs to be supplemented with a consequentialist account of the efficacy of well-motivated actions—that is, it should be transformed into a mixed (motives-consequences) account, while retaining its net-IVV calculus. This is because I believe that there are right-making properties external to an agent’s psychology which it is important to take into account, especially when an agent’s actions negatively affect other people. To incorporate this intuition, I add to Doviak’s net-IVV calculus a scale for outcomes. The result is a mixed view which accommodates tenets (ii) and (iii) above, but allows for (i) to fail in certain cases. I argue that, rather than being a defect, this allowance is an asset because our intuitions about ought-implies-can break down in cases where an agent is grossly misguided, and our theory should track these intuitions.     

Boltzmann on Mathematics

Boltzmann’s lectures on natural philosophy point out how the principles of mathematics are both an improvement on traditional philosophy and also serve as a necessary foundation of physics or what the English call “Natura Philosophy”, a title which he will retain for his own lectures. We start with lecture #3 and the mathematical contents of his lectures plus a few philosophical comments. Because of the length of the lectures as a whole we can only give the main points of each but organized into a coherent study. Behind his mathematics stands his support of Darwinian evolution interpreted in a partly Lamarckian way. He also supported non-Euclidean geometry. Much of Boltzmann’s analysis of mathematics is an attempt to refute Kant’s static a priori categories and his identification of space with “non-sensuous intuition”. Boltzmann’s strong attention toward discreteness in mathematics can be seen throughout the lectures. Part II of this paper will touch on the historical background of atomism and focus on the discrete way of thinking with which Boltzmann approaches problems in mathematics and beyond. Part III briefly points out how Boltzmann related mathematics and discreteness to music. This revised version was published online in June 2006 with corrections to the Cover Date.     

Heinrich Hertz’s Neo-Kantian Philosophy of Science, and its Development by Harald Høffding

Summary  This article is an investigation of parallel themes in Heinrich Hertz’s philosophy science and Kant’s theory of schemata, symbols and regulative ideas. It is argued that Hertz’s “pictures” bears close similarities to Kantian “schemata”, that is, they are rules linking concepts to intuitions and provide them with their meaning. Kant’s distinction between symbols and schemata is discussed and related to Hertz’s three pictures of mechanics. It is argued that Hertz considered his own picture of mechanics (the “hidden mass” picture) as symbolic in a different way than the force and energy pictures. In the final part of the article it is described how Harald H?ffding soon after the publication of Hertz’s Principles of Mechanics developed a general theory of analogical reasoning, relying on the ideas of Hertz and Kant.     

No Reservations Required? Defending Anti-Nominalism

In a 2005 paper, John Burgess and Gideon Rosen offer a new argument against nominalism in the philosophy of mathematics. The argument proceeds from the thesis that mathematics is part of science, and that core existence theorems in mathematics are both accepted by mathematicians and acceptable by mathematical standards. David Liggins (2007) criticizes the argument on the grounds that no adequate interpretation of “acceptable by mathematical standards” can be given which preserves the soundness of the overall argument. In this discussion I offer a defense of the Burgess-Rosen argument against Liggins’s objection. I show how plausible versions of the argument can be constructed based on either of two interpretations of mathematical acceptability, and I locate the argument in the space of contemporary anti-nominalist views.     

Popper’S Evolutionary Epistemology Revamped

In a paper entitled “Revolution in Permanence”, published in the collection “Karl Popper: Philosophy and Problems”, John Worrall (1995) severely criticised several aspects of Karl Popper’s work before commenting that “I have no doubt that, given suffi-cient motivation, a case could be constructed on the basis of such remarks that Popper had a more sophisticated version of theory production......” (p. 102). Part of Worrall’s criticism is directed at a “strawpopper”: in his “Darwinian Model” emphasising the similarities and differences between genetic mutation, variation in animal behaviour and the gestation of scientific theories, Popper (1975, 1981, 1994) never stated that tentative scientific conjec-tures “while more or less random, are not completely blind.” He was referring to variation in animal species behaviour, and about tentative scientific conjectures he said nothing, although common sense would indicate that presumably he regarded them as being less blind and less random. In Popper (1977, 1983), giving a summary of his “Darwinian Model”, he repaired this omission about tentative scientific conjectures by inserting the sentence “On a level of World 3 theory formation they are of the character of planned gropings into the unknown.” Recent developments in the field of genetics (see for example Raff (1996), Lewis (1999), Korn (2002)) indicate that Popper’s intuitions were along the modern lines while Worrall’s intuitions are old fashioned. Therefore Popper’s “Darwinian Model” remains both viable and fruitful.     

New directions for nominalist philosophers of mathematics

The present paper will argue that, for too long, many nominalists have concentrated their researches on the question of whether one could make sense of applications of mathematics (especially in science) without presupposing the existence of mathematical objects. This was, no doubt, due to the enormous influence of Quine’s “Indispensability Argument”, which challenged the nominalist to come up with an explanation of how science could be done without referring to, or quantifying over, mathematical objects. I shall admonish nominalists to enlarge the target of their investigations to include the many uses mathematicians make of concepts such as structures and models to advance pure mathematics. I shall illustrate my reasons for admonishing nominalists to strike out in these new directions by using Hartry Field’s nominalistic view of mathematics as a model of a philosophy of mathematics that was developed in just the sort of way I argue one should guard against. I shall support my reasons by providing grounds for rejecting both Field’s fictionalism and also his deflationist account of mathematical knowledge—doctrines that were formed largely in response to the Indispensability Argument. I shall then give a refutation of Mark Balaguer’s argument for his thesis that fictionalism is “the best version of anti-realistic anti-platonism”.     

Understanding risk: A guide for the perplexed

Over the course of the past decade, neurobiologists have become increasingly interested in concepts and models imported from economics. Terms such as “risk,” “risk aversion,” and “utility” have become commonplace in the neuroscientific literature as single-unit physiologists and human cognitive neuroscientists search for the biological correlates of economic theories of value and choice. Among neuroscientists, an incomplete understanding of these concepts has, however, led to a growing confusion that threatens to check the rapid advances in this area. Adding to the confusion, notions of risk have more recently been imported from finance, which employs quite different, although formally related, mathematical tools. Of course, the mixing of economic, financial, and neuroscientific traditions can only be beneficial in the long run, but truly understanding the conceptual machinery of each area is a prerequisite for obtaining that benefit. With that in mind, I present here an overview of economic and financial notions of risk and decision. The article begins with an overview of the classical economic approach to risk, as developed by Bernoulli. It then explains the important differences between the classical tradition and modern neoclassical economic approaches to these same concepts. Finally, I present a very brief overview of the financial tradition and its relation to the economic tradition. For novices, this should provide a reasonable introduction to concepts ranging from “risk aversion” to “risk premiums.”     

Walter Pitts and “A Logical Calculus”

Many years after the publication of “A Logical Calculus of the Ideas Immanent in Nervous Activity,” Warren McCulloch gave Walter Pitts credit for contributing his knowledge of modular mathematics to their joint project.

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Bad company tamed

Neo-Fregean logicism seeks to base mathematics on abstraction principles. But the acceptable abstraction principles are surrounded by unacceptable (indeed often paradoxical) ones. This is the “bad company problem.” In this introduction I first provide a brief historical overview of the problem. Then I outline the main responses that are currently being debated. In the course of doing so I provide summaries of the contributions to this special issue.     

Epistemological Challenges to Mathematical Platonism

Since Benacerraf’s “Mathematical Truth” a number of epistemological challenges have been launched against mathematical platonism. I first argue that these challenges fail because they unduely assimilate mathematics to empirical science. Then I develop an improved challenge which is immune to this criticism. Very roughly, what I demand is an account of how people’s mathematical beliefs are responsive to the truth of these beliefs. Finally I argue that if we employ a semantic truth-predicate rather than just a deflationary one, there surprisingly turns out to be logical space for a response to the improved challenge where no such space appeared to exist.     

The temporality of Merleau-Ponty’s intertwining

In his last work, The Visible and the Invisible, Merleau-Ponty explored the fact that we believe that perception occurs in our heads (“in the recesses of a body”) and, hence, assert that the perceptual world is “in” us, while also believing that we are “in” the world we perceive.  In this article, I examine how this intertwining of self and world justifies the faith we have in perception. I shall do so by considering a number of examples.  In each case, the object “in itself” will turn out to be neither within us nor outside of us, but rather at the intersection set by the intertwining. I will then turn to what this disclosure of this object reveals about human temporality and, indeed, about human being as a place (or “clearing”) that permits disclosure.     

Higher Education,Knowledge for Its Own Sake,and an African Moral Theory

I seek to answer the question of whether publicly funded higher education ought to aim intrinsically to promote certain kinds of “blue-sky” knowledge, knowledge that is unlikely to result in “tangible” or “concrete” social benefits such as health, wealth and liberty. I approach this question in light of an African moral theory, which contrasts with dominant Western philosophies and has not yet been applied to pedagogical issues. According to this communitarian theory, grounded on salient sub-Saharan beliefs and practices, actions are right insofar as they respect relationships in which people both share a way of life, or identify with one another, and care for others’ quality of life, or are in solidarity with each other. I argue that while considerations of identity and solidarity each provide some reason for a state university to pursue blue-sky knowledge as a final end, they do not provide conclusive reason for it to do so. I abstain from drawing any further conclusion about whether this provides reason to reject the Afro-communitarian moral theory or the intuition that blue-sky knowledge is a proper final end of public higher education. I do point out, however, that the dominant Western moral theories on the face of it do no better than the African one at accounting for this intuition.     

Das Ontologische Dilemma der Normativen Ethik

The Ontological Dilemma of Normative Ethics. This paper pursues two goals. The first is to show that normative ethics is confronted with the following dilemma: to be coherent, this discipline is ontologically committed to acknowledge the existence of objective values, but, to be scientifically respectable, it is committed to repudiate such values. The second goal is to assess the possible solutions to this dilemma. To this end, the following strategies are discussed: Kant’s constructive objectivism, Jürgen Habermas’ “epistemic ersatzism”, Franz von Kutschera’s “confirmation pragmatism”, and David Brink’s “objectivist tour de force”. The paper’s conclusion is that the dilemma cannot be solved because it rests on a clash of intuitions none of which can be given up. This revised version was published online in August 2006 with corrections to the Cover Date.     

Susan Myers-Shirk’s <Emphasis Type="Italic">Helping the Good Shepherd</Emphasis>: A Review Essay

In this essay, I review a recent book that deals with the history of pastoral counseling. I offer an overview of the book, some criticism of the book, and a discussion of how this book relates to my own work. I argue that what Susan Myers-Shirk has identified as a “liberal moral sensibility” among pastoral counselors seems to have certain affinities with Peter Homans’s “mourning religion” thesis. I suggest that this thesis can shed light on the divide between liberal and conservative pastoral counselors, a divide that Myers-Shirk identifies, and that this thesis can build on Myers-Shirk’s historical work by providing a rubric for understanding the relationship between private experience and public theory among liberal pastoral counselors. I also suggest that Myers-Shirk should write a sequel to this book.     

Habermas on Understanding: Virtual Participation,Dialogue and the Universality of Truth

Although the success of Habermas’s theory of communicative action depends on his dialogical model of understanding in which a theorist is supposed to participate in the debate with the actors as a ‘virtual participant’ and seek context-transcendent truth through the exchange of speech acts, current literature on the theory of communicative action rarely touches on the difficulties it entails. In the first part of this paper, I will examine Habermas’s argument that understanding other cultural practices requires the interpreter to virtually participate in the “dialogue” with the actors as to the rationality of their cultural practice and discuss why, according to Habermas,such dialogue leads to the “context-transcendent truth”. In the second part, by using a concrete historical example, I will reconstruct a “virtual dialogue” between Habermas and Michael Polanyi as to the rationality of scientific practice and indicate why Habermas’s dialogical model of understanding based on the methodology of virtual participation cannot achieve what it professes to do.