Inference to the best explanation and mathematical realism

Arguing for mathematical realism on the basis of Field’s explanationist version of the Quine–Putnam Indispensability argument, Alan Baker has recently claimed to have found an instance of a genuine mathematical explanation of a physical phenomenon. While I agree that Baker presents a very interesting example in which mathematics plays an essential explanatory role, I show that this example, and the argument built upon it, begs the question against the mathematical nominalist.     

<Emphasis Type="Italic">Wissenschaftslogik</Emphasis>: The role of logic in the philosophy of science

Carl Hempel introduced what he called “Craig’s theorem” into the philosophy of science in a famous discussion of the “problem of theoretical terms.” Beginning with Hempel’s use of ‘Craig’s theorem,” I shall bring out some of the key differences between Hempel’s treatment of the “problem of theoretical terms” and Carnap’s in order to illuminate the peculiar function of Wissenschaftslogik in Carnap’s mature philosophy. Carnap’s treatment, in particular, is fundamentally anti-metaphysical—he aims to use the tools of mathematical logic to dissolve rather solve traditional philosophical problems—and it is precisely this point that is missed by his logically-minded contemporaries such as Hempel and Quine.     

How to Define a Number? A General Epistemological Account of Simon Stevin’s Art of Defining

This paper explores Simon Stevin’s l’Arithmétique of 1585, where we find a novel understanding of the concept of number. I will discuss the dynamics between his practice and philosophy of mathematics, and put it in the context of his general epistemological attitude. Subsequently, I will take a close look at his justificational concerns, and at how these are reflected in his inductive, a postiori and structuralist approach to investigating the numerical field. I will argue that Stevin’s renewed conceptualisation of the notion of number is a sort of “existential closure” of the numerical domain, founded upon the practice of his predecessors and contemporaries. Accordingly, I want to make clear that l’Aritmetique have to be read not as an ontological analysis or exploration of the numerical field, but as an explication of a mathematical ethos. In this sense, this article also intends to make a specific contribution to the broader issue of the “ethics of geometry.”     

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.     

How is God-talk logically possible? A sketch for an argument on the logic of ‘God’

In this paper I want to argue for the optimal way to characterise the logical and semantical behaviour of the singular term ‘God’ used in religious language. The relevance of this enterprise to logical theory is the main focus as well. Doing this presupposes to outline the two rivaling approaches of well-definition of singular terms: Kripke’s (“rigid designators”) and Hintikka’s (“world-lines”). ‘God’ as a “rigid designator” is purified from all real-life-language-games of identification and only spells out a metaphysical tag, which favours the view of “anything goes”. Instead, ‘God’ as a “world-line,” plus two ways of quantification, is much more flexible to theological traditions, teachings of the church, religious practices and personal feelings. Thus, it provides a sufficiently well-defined singular term for the purposes of logical theory. The whole sketch is based on Jaakko Hintikka’s logical ideas, mainly on his responses to different authors in PJH. I have systematically omitted direct references to his texts because I have modified considerably his ideas for my own purposes.     

Internalism and Hyperexternalism About Reasons

Alan Goldman’s Reasons from Within is one of the most thorough recent defenses of what might be called ‘orthodox internalism’ about practical reasons. Goldman’s main target is an opposing view that includes a commitment to the following two theses: (O) that there are such things as objective values, and (E) that these values give rise to external reasons. One version of this view, which we can call ‘orthodox externalism’, also includes a commitment to the thesis (I) that rational people will be motivated by any reason they have of which they are aware. Goldman himself embraces (I), and deploys it frequently in his criticisms of orthodox externalism. But there is logical space for an externalist view that includes a commitment to (O) and (E), but that denies (I). The resulting “hyperexternalist” view holds that some reasons need not motivate us, even if we are rational. In this paper I argue that Goldman’s criticisms of orthodox externalism leave hyperexternalism untouched, and that his specific criticisms of my own version of hyperexternalism do not work. In light of Goldman’s criticisms of orthodox externalism and my own criticisms of Goldman’s view, hyperexternalism emerges as the favored option.     

Vollkommene Syllogismen und reine Vernunftschlüsse: Aristoteles und Kant. Eine Stellungnahme zu Theodor Eberts Gegeneinwänden. Teil 2

In an earlier article (see J Gen Philos Sci (2010) 41: 341–355) I have compared Aristotle’s syllogistic with Kant’s theory of “pure ratiocination”. “Ratiocinia pura” („reine Vernunftschlüsse“) is Kant’s designation for assertoric syllogisms Aristotle has called ‘perfect’. In Kant’s view they differ from non-pure ratiocinia precisely in that their validity rests only on the validity of the Dictum de omni et nullo (which, however, in Kant’s view can be further reduced to more fundamental principles) whereas the validity of non-pure ratiocinia additionally presupposes the validity of inferences which Kant calls consequentiae immediatae. I have argued that Kant’s view is in some (not in all) essential features in accordance with Aristotle’s view concerning perfect syllogisms and certainly leading to a tenable and interesting logical theory. As a result I have rejected not only the interpretation of Aristotle adopted by Theodor Ebert, but also the objections he has raised against Kant’s logical theory. As far as Aristotle is concerned, Ebert has attempted to defend his position in the first part of his reply to my article published in J Gen Philos Sci (2009) 40: 357–365, and I have argued against this defence in issue 1 of the J Gen Philos Sci (2010) 41: 199–213 (cf. Ebert’s answer in the same issue pp. 215–231). In the following discussion I deal with Eberts defence of his criticism of Kant published in the second part of his reply to my article (see J Gen Philos Sci (2009) 40: 365–372). I shall argue, that Kant’s principle ‘nota notae est nota rei ipsius’ and his use of technical vocabulary stand up to the objections raised by Ebert. His attempts to prove that Kant’s logical theory is defective are based on several misinterpretations.     

Carnap,Kuhn, and revisionism: on the publication of <Emphasis Type="Italic">Structure</Emphasis> in <Emphasis Type="Italic">Encyclopedia</Emphasis>

In recent years, a revisionist process focused on logical positivism can be observed, particularly regarding Carnap’s work. In this paper, I argue against the interpretation that Kuhn’s The Structure of Scientific Revolutions having been published in the International Encyclopedia of Unified Science, co-edited by Carnap, is evidence of the revisionist idea that Carnap “would have found Structure philosophically congenial”. I claim that Kuhn’s book, from Carnap’s point of view, is not in philosophy of science but rather in history of science (in the context of a sharp discovery–justification distinction). It could also explain the fact that, despite his sympathetic letters to Kuhn as editor, Carnap never refers to Kuhn’s book in his work in philosophy of science.     

Abolishing Morality

Moral anti-realism comes in two forms – noncognitivism and the error theory. The noncognitivist says that when we make moral judgments we aren’t even trying to state moral facts. The error theorist says that when we make moral judgments we are making statements about what is objectively good, bad, right, or wrong but, since there are no moral facts, our moral judgments are uniformly false. This development of moral anti-realism was first seriously defended by John Mackie. In this paper I explore a dispute among moral error theorists about how to deal with false moral judgments. The advice of the moral abolitionist is to stop making moral judgments, but the contrary advice of the moral fictionalist is to retain moral language and moral thinking. After clarifying the choice that arises for the moral error theorist, I argue that moral abolitionism has much to recommend it. I discuss Mackie’s defense of moral fictionalism as well as a recent version of the same position offered by Daniel Nolan, Greg Restall, and Caroline West. Then I second some remarks Ian Hinckfuss made in his defense of moral abolitionism and his criticism of “the moral society.” One of the worst things about moral fictionalism is that it undermines our epistemology by promoting a culture of deception. To deal with this problem Richard Joyce offers a “non-assertive” version of moral fictionalism as perhaps the last option for an error theorist who hopes to avoid moral abolitionism. I discuss some of the problems facing that form of moral fictionalism, offer some further reasons for adopting moral abolitionism in our personal lives, and conclude with reasons for thinking that abolishing morality may be an essential step in achieving the goals well-meaning moralists and moral fictionalists have always cherished.

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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Why modal fictionalism is not self-defeating

Gideon Rosen’s [1990 Modal fictionalism. Mind, 99, 327–354] Modal Fictionalist aims to secure the benefits of realism about possible-worlds, whilst avoiding commitment to the existence of any world other than our own. Rosen [1993 A problem for fictionalism about possible worlds. Analysis, 53, 71–81] and Stuart Brock [1993 Modal fictionalism: A response to Rosen. Mind, 102, 147–150] both argue that fictionalism is self-defeating since the fictionalist is tacitly committed to the existence of a plurality of worlds. In this paper, I develop a new strategy for the fictionalist to pursue in response to the Brock–Rosen objection. I begin by arguing that modal fictionalism is best understood as a paraphrase strategy that concerns the propositions that are expressed, in a given context, by modal sentences. I go on to argue that what is interesting about paraphrastic fictionalism is that it allows the fictionalist to accept that the sentence ‘there is a plurality of worlds’ is true without thereby committing her to the existence of a plurality of worlds. I then argue that the paraphrastic fictionalist can appeal to a form of semantic contextualism in order to communicate her status as an anti-realist. Finally, I generalise my conception of fictionalism and argue that Daniel Nolan and John O’Leary-Hawthorne [1996 Reflexive fictionalisms. Analysis, 56, 26–32] are wrong to suggest that the Brock-Rosen objection reveals a structural flaw with all species of fictionalism.

Internal,External and Intra-Individual Relations

In this paper I argue that there are in fact external relations in Russell’s sense. The level at which we are forced to acknowledge them is, however, not the level of relations between concrete individual objects. All relations of this kind, which I will call “inter-individual” relations, can be construed as supervenient on the monadic properties of their terms. But if we pursue our ontological analysis a little bit deeper and consider the internal structure of a concrete individual, then we will inevitably find irreducible external relations. I mean for example the relation of instantiation (in the frame of a realist’s theory) or that of concurrence (in the frame of a trope theory). I will show that such “intra-individual” relations – the relations that make up the internal structure of a concrete individual out of more primitive metaphysical “building blocks” like universals or tropes – could not (even in principle) be construed as supervenient. I should like to thank David Frost for brushing up my English and the Austrian Foundation for the Promotion of Scientific Research (FWF) for the financial support.     

An Enlightened Revolt: on the Philosophy of Nicholas Maxwell

This paper is a reaction to the book “Science and the Pursuit of Wisdom”, whose central concern is the philosophy of Nicholas Maxwell. I distinguish and discuss three concerns in Maxwell’s philosophy. The first is his critique of standard empiricism (SE) in the philosophy of science, the second his defense of aim-oriented rationality (AOR), and the third his philosophy of mind. I point at some problematic aspects of Maxwell’s rebuttal of SE and of his philosophy of mind and argue in favor of AOR.     

Intrinsic Explanation and Field’s Dispensabilist Strategy

Abstract

Hartry Field defended the importance of his nominalist reformulation of Newtonian Gravitational Theory, as a response to the indispensability argument, on the basis of a general principle of intrinsic explanation. In this paper, I argue that this principle is not sufficiently defensible, and can not do the work for which Field uses it. I argue first that the model for Field’s reformulation, Hilbert’s axiomatization of Euclidean geometry, can be understood without appealing to the principle. Second, I argue that our desires to unify our theories and explanations undermines Field’s principle. Third, the claim that extrinsic theories seem like magic is, in this case, really just a demand for an account of the applications of mathematics in science. Finally, even if we were to accept the principle, it would not favor the fictionalism that motivates Field’s argument, since the indispensabilist’s mathematical objects are actually intrinsic to scientific theory.     

Christian materialism in a scientific age

Many Christians who argue against Christian materialism direct their arguments against what I call ‘Type-I materialism’, the thesis that I cannot exist without my organic body. I distinguish Type-I materialism from Type-II materialism, which entails only that I cannot exist without some body that supports certain mental functions. I set out a version of Type-II materialism, and argue for its superiority to Type-I materialism in an age of science. Moreover, I show that Type-II materialism can accommodate Christian doctrines like the Resurrection of the Body, the Incarnation, and the “intermediate state” (if there is one).     

Rigid Designators for Properties

Here I defend the position that some singular terms for properties are rigid designators, responding to Stephen P. Schwartz’s interesting criticisms of that position. First, I argue that my position does not depend on ontological parsimony with respect to properties – e.g., there is no need to claim that there are only natural properties – to get around the problem of “unusual properties.” Second, I argue that my position does not confuse sameness of meaning across possible worlds with sameness of designation, or rigid designation. Third, I argue that my position does not founder by way of failing to assign rigidity the work of grounding a posteriori necessity. I thank Steve Schwartz and Bernard Linsky for helpful feedback on this paper     

Hindriks on Rule-Following*

This paper is a reply to Frank Hindriks’ paper “A Modest Solution to the Problem of Rule-Following”. Hindriks claims to find room for what he calls a modest solution to the Kripkensteinian problem of rule-following, different from both straight and sceptical solutions. Hindriks criticises Philip Pettit’s “ethocentric” solution and goes on to provide his own, “modest” one. My paper is in two parts. In the first part, I argue that there is no room for a “modest” solution to sceptical problems: depending on how one reads Kripke, Hindriks’ “modest” solution is always going to turn out either straight or sceptical. In the second part, I defend the ethocentric solution against Hindriks’ arguments. In particular, I argue that the topic-neutral specifications of favourable conditions which Pettit uses are superior to Hindriks’ topic-specific ones. *I want to thank Frank Hindriks for comments on an earlier version, but I take full responsibility for any remaining errors or misunderstandings. This work has been financially supported by the Academy of Finland (project 202513).