A Virtue-Based Defense of Mathematical Apriorism

Mathematical apriorists usually defend their view by contending that axioms are knowable a priori, and that the rules of inference in mathematics preserve this apriority for derived statements—so that by following the proof of a statement, we can trace the apriority being inherited. The empiricist Philip Kitcher attacked this claim by arguing there is no satisfactory theory that explains how mathematical axioms could be known a priori. I propose that in analyzing Ernest Sosa’s model of intuition as an intellectual virtue, we can construct an “intuition–virtue” that could supply the missing explanation for the apriority of axioms. I first argue that this intuition–virtue qualifies as an a priori warrant according to Kitcher’s account, and then show that it could produce beliefs about mathematical axioms independent of experience. If my argument stands, this paper could provide insight on how virtue epistemology could help defend mathematical apriorism on a larger scale.     

Intuitions, evidence and hopefulness

Experimental philosophers have recently conducted surveys of folk judgements about a range of phenomena of interest to philosophy including knowledge, reference, and free will. Some experimental philosophers take these results to undermine the philosophical practice of appealing to intuitions as evidence. I consider several different replies to the suggestion that these results undermine philosophical appeal to intuition, both piecemeal replies which raise concerns about particular surveys, and more general replies. The general replies include the suggestions that the surveys consider the wrong sort of judgement, or the wrong kind of judge, or that the results of the surveys do not generate scepticism about philosophical appeal to intuition in particular, but rather a more problematic and general scepticism. I argue that the last of these general replies is the most promising. To assess its merits, I consider the most developed account of how the survey results are supposed to raise sceptical problems specifically for philosophical appeal to intuition, that presented in Weinberg (Midwest Stud Philos XXXI:318–343, 2007). I argue that there are significant objections to Weinberg’s account. I conclude that, so far, experimental philosophers have failed to show how their survey results raise a sceptical challenge that applies to philosophical appeal to intuition in particular, rather than having a problematic, more general scope.     

Intuition,intelligence, data compression

The main goal of my paper is to argue that data compression is a necessary condition for intelligence. One key motivation for this proposal stems from a paradox about intuition and intelligence. For the purposes of this paper, it will be useful to consider playing board games—such as chess and Go—as a paradigm of problem solving and cognition, and computer programs as a model of human cognition. I first describe the basic components of computer programs that play board games, namely value functions and search functions. I then argue that value functions both play the same role as intuition in humans and work in essentially the same way. However, as will become apparent, using an ordinary value function is just a simpler and less accurate form of relying on a database or lookup table. This raises our paradox, since reliance on intuition is usually considered to manifest intelligence, whereas usage of a lookup table is not. I therefore introduce another condition for intelligence that is related to data compression. This proposal allows that even reliance on a perfectly accurate lookup table can be nonintelligent, while retaining the claim that reliance on intuition can be highly intelligent. My account is not just theoretically plausible, but it also captures a crucial empirical constraint. This is because all systems with limited resources that solve complex problems—and hence, all cognitive systems—need to compress data.

     

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.     

LAW NECESSITARIANISM AND THE IMPORTANCE OF BEING INTUITIVE

The counter-intuitive implications of law necessitarianism pose a far more serious threat than its proponents recognize. Law necessitarians are committed to scientific essentialism, the thesis that there are metaphysically necessary truths which can be known only a posteriori. The most frequently cited arguments for this position rely on modal intuitions. Rejection of intuition thus threatens to undermine it. I consider ways in which law necessitarians might try to defend scientific essentialism without invoking intuition. I then consider ways in which law necessitarians who accept the general reliability of intuition might try to explain away the intuitions which conflict with their theory.     

Is Intuition Necessary for Defending Platonism?

G?del asserts that his philosophy falls under the category of conceptual realism. This paper gives a general picture of G?del’s conceptual realism’s basic doctrines, and gives a way to understand conceptual realism in the background of Leibniz’s and Kant’s philosophies. Among philosophers of mathematics, there is a widespread view that Platonism encounters an epistemological difficulty because we do not have sensations of abstract objects. In his writings, G?del asserts that we have mathematical intuitions of mathematical objects. Some philosophers do not think it is necessary to resort to intuition to defend Platonism, and other philosophers think that the arguments resorting to intuition are too na?ve to be convincing. I argue that the epistemic difficulty is not particular to Platonism; when faced with skepticism, physicalists also need to give an answer concerning the relationship between our experience and reality. G?del and Kant both think that sensations or combinations of sensations are not ideas of physical objects, but that, to form ideas of physical objects, concepts must be added. However, unlike Kant, G?del thinks that concepts are not subjective but independent of our minds. Based on my analysis of G?del’s conceptual realism, I give an answer to the question in the title and show that arguments resorting to intuition are far from na?ve, despite what some philosophers have claimed.     

The value problem and the nature of knowledge

Jason Baehr has argued that the intuition that knowledge is more valuable than mere true belief is neither sufficiently general nor sufficiently formal to motivate the value problem in epistemology. What he calls the “guiding intuition” is not completely general: our intuition does not reveal that knowledge is always more valuable than true belief; and not strictly formal: the intuition is not merely the abstract claim that knowledge is more valuable than true belief. If he is right, the value problem (as we know it) is not a real problem. I will argue in this paper that he is wrong about the generality claim: knowledge is always more valuable than true belief; and yet he is right about the formality claim—there is more to the intuition than just the abstract claim that knowledge is more valuable than true belief. What this amounts to, I will argue, is that there is still a value problem but that the guiding intuition can tell us how to solve it.     

Are Mathematical Theories Reducible to Non-analytic Foundations?

In this article I intend to show that certain aspects of the axiomatical structure of mathematical theories can be, by a phenomenologically motivated approach, reduced to two distinct types of idealization, the first-level idealization associated with the concrete intuition of the objects of mathematical theories as discrete, finite sign-configurations and the second-level idealization associated with the intuition of infinite mathematical objects as extensions over constituted temporality. This is the main standpoint from which I review Cantor’s conception of infinite cardinalities and also the metatheoretical content of some later well-known theorems of mathematical foundations. These are, the Skolem-Löwenheim Theorem which, except for its importance as such, it is also chosen for an interpretation of the associated metatheoretical paradox (Skolem Paradox), and Gödel’s (first) incompleteness result which, notwithstanding its obvious influence in the mathematical foundations, is still open to philosophical inquiry. On the phenomenological level, first-level and second-level idealizations, as above, are associated respectively with intentional acts carried out in actual present and with certain modes of a temporal constitution process.     

How Mathematical Concepts Get Their Bodies

When the traditional distinction between a mathematical concept and a mathematical intuition is tested against examples taken from the real history of mathematics one can observe the following interesting phenomena. First, there are multiple examples where concepts and intuitions do not well fit together; some of these examples can be described as “poorly conceptualised intuitions” while some others can be described as “poorly intuited concepts”. Second, the historical development of mathematics involves two kinds of corresponding processes: poorly conceptualised intuitions are further conceptualised while poorly intuited concepts are further intuited. In this paper I study this latter process in mathematics during the twentieth century and, more specifically, show the role of set theory and category theory in this process. I use this material for defending the following claims: (1) mathematical intuitions are subject to historical development just like mathematical concepts; (2) mathematical intuitions continue to play their traditional role in today's mathematics and will plausibly do so in the foreseeable future. This second claim implies that the popular view, according to which modern mathematical concepts, unlike their more traditional predecessors, cannot be directly intuited, is not justified.     

Platonism and metaphor in the texts of mathematics: Gödel and Frege on mathematical knowledge

In this paper, I challenge those interpretations of Frege that reinforce the view that his talk of grasping thoughts about abstract objects is consistent with Russell's notion of acquaintance with universals and with Gödel's contention that we possess a faculty of mathematical perception capable of perceiving the objects of set theory. Here I argue the case that Frege is not an epistemological Platonist in the sense in which Gödel is one. The contention advanced is that Gödel bases his Platonism on a literal comparison between mathematical intuition and physical perception. He concludes that since we accept sense perception as a source of empirical knowledge, then we similarly should posit a faculty of mathematical intuition to serve as the source of mathematical knowledge. Unlike Gödel, Frege does not posit a faculty of mathematical intuition. Frege talks instead about grasping thoughts about abstract objects. However, despite his hostility to metaphor, he uses the notion of ‘grasping’ as a strategic metaphor to model his notion of thinking, i.e., to underscore that it is only by logically manipulating the cognitive content of mathematical propositions that we can obtain mathematical knowledge. Thus, he construes ‘grasping’ more as theoretical activity than as a kind of inner mental ‘seeing’.

     

What Does It Mean That “Space Can Be Transcendental Without the Axioms Being So”?

In 1870, Hermann von Helmholtz criticized the Kantian conception of geometrical axioms as a priori synthetic judgments grounded in spatial intuition. However, during his dispute with Albrecht Krause (Kant und Helmholtz über den Ursprung und die Bedeutung der Raumanschauung und der geometrischen Axiome. Lahr, Schauenburg, 1878), Helmholtz maintained that space can be transcendental without the axioms being so. In this paper, I will analyze Helmholtz’s claim in connection with his theory of measurement. Helmholtz uses a Kantian argument that can be summarized as follows: mathematical structures that can be defined independently of the objects we experience are necessary for judgments about magnitudes to be generally valid. I suggest that space is conceived by Helmholtz as one such structure. I will analyze his argument in its most detailed version, which is found in Helmholtz (Zählen und Messen, erkenntnistheoretisch betrachtet 1887. In: Schriften zur Erkenntnistheorie. Springer, Berlin, 1921, 70–97). In support of my view, I will consider alternative formulations of the same argument by Ernst Cassirer and Otto Hölder.     

A critique of Thad Metz’s African theory of moral status

Thad Metz defends what he considers to be a novel theory of moral status, i.e. an account about what beings are owed direct duties in virtue of their moral significance. Metz claims that his account is African, it is plausible and that it is worth taking seriously like other competing accounts in the Western philosophical tradition. In this article, I give four reasons why we should doubt, if not reject, these claims of plausibility. Firstly, I show how a theory that accounts for moral status by relying solely on some facet of human nature ultimately fails to grant intrinsic value to non-human components, and as such it will always prefer human interests over those of nonhuman components, and further it won’t have a moral-theoretical basis to assign intrinsic value to non-human components. Secondly, I hope to demonstrate that this theory will not be able to account for the moral status of Martians and in turn show that it does not secure the standing of animals from such beings. I also argue that his account does not give credible evidence for the intuition that severely injured human persons have greater moral status than animals with similar internal properties. Finally, I briefly indicate that this theory does not have the corpus to explain our duties to people who have died, or at least, their bodies.     

Close Calls and the Confident Agent: Free Will, Deliberation, and Alternative Possibilities

Two intuitions lie at the heart of our conception of free will. One intuition locates free will in our ability to deliberate effectively and control our actions accordingly: the ‘Deliberation and Control’ (DC) condition. The other intuition is that free will requires the existence of alternative possibilities for choice: the AP condition. These intuitions seem to conflict when, for instance, we deliberate well to decide what to do, and we do not want it to be possible to act in some other way. I suggest that intuitions about the AP condition arise when we face ‘close calls,’ situations in which, after deliberating, we still do not know what we really want to do. Indeed, several incompatibilists suggest such close calls are necessary for free will. I challenge this suggestion by describing a ‘confident agent’ who, after deliberating, always feels confident about what to do (and can then control her actions accordingly). Because she maximally satisfies the DC condition, she does not face close calls, and the intuition that the AP condition is essential for free will does not seem to apply to her. I conclude that intuitions about the importance of the AP condition rest on our experiences of close calls and arise precisely to the extent that our deliberations fail to arrive at a clear decision. I then raise and respond to several objections to this thought experiment and its relevance to the free will debate.     

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.     

The Ontology of Musical Works and the Role of Intuitions: An Experimental Study

Philosophers of music often appeal to intuition to defend ontological theories of musical works. This practice is worrisome as it is rather unclear just how widely shared are the intuitions that philosophers appeal to. In this paper, I will first offer a brief overview of the debate over the ontology of musical works. I will argue that this debate is driven by a conflict between two seemingly plausible intuitions—the repeatability intuition and the creatability intuition—both of which may be defended on the grounds that they are reflective of our actual musical practices. The problem facing philosophers within this debate is that there is no clear way to determine which of the two conflicting intuitions is more reflective of our musical practices. Finally, I offer discussion of an experimental study that was designed to test participants' intuitions regarding the repeatability of musical works. The evidence presented there suggests that the participants broadly accept the repeatability of musical works, but in a much narrower way than philosophers would likely accept.     

TWINNING, INORGANIC REPLACEMENT, AND THE ORGANISM VIEW

In explicating his version of the Organism View, Eric Olson argues that you begin to exist only after twinning is no longer possible and that you cannot survive a process of inorganic replacement. Assuming the correctness of the Organism View, but pace Olson, I argue in this paper that the Organism View does not require that you believe either proposition. The claim I shall make about twinning helps to advance a debate that currently divides defenders of the Organism View, while the claim I shall make about inorganic replacement will help to put the Organism View on a par with its rival views by allowing it to accommodate a plausible intuition that its rivals can accommodate, namely, the intuition that you can survive a process of inorganic replacement. Both claims, I shall also argue, are important for those who are interested in the identity condition of a human organism, even if they do not hold the view that you are essentially an organism.     

Wittgenstein and Brouwer

In this paper, I present a summary of the philosophical relationship betweenWittgenstein and Brouwer, taking as my point of departure Brouwer's lecture onMarch 10, 1928 in Vienna. I argue that Wittgenstein having at that stage not doneserious philosophical work for years, if one is to understand the impact of thatlecture on him, it is better to compare its content with the remarks on logics andmathematics in the Tractactus. I thus show that Wittgenstein's position, in theTractactus, was already quite close to Brouwer's and that the points of divergence are the basis to Wittgenstein's later criticisms of intuitionism. Among the topics of comparison are the role of intuition in mathematics, rule following, choice sequences, the Law of Excluded Middle, and the primacy of arithmetic over logic.     

Dispositional accounts of evil personhood

It is intuitively plausible that not every evildoer is an evil person. In order to make sense of this intuition we need to construct an account of evil personhood in addition to an account of evil action. Some philosophers have offered aggregative accounts of evil personhood, but these do not fit well with common intuitions about the explanatory power of evil personhood, the possibility of moral reform, and the relationship between evil and luck. In contrast, a dispositional account of evil personhood can allow that evil is explanatory, that an evil person can become good, and that luck might prevent evil persons from doing evil or cause non-evil persons to do evil. Yet the dispositional account of evil personhood implies that some evil persons are blameless, which seems to clash with the intuition that evil persons deserve our strongest moral condemnation. Moreover, since it is likely that a large proportion of us are disposed to perform evil actions in some environments, the dispositional account threatens to label a large proportion of people evil. In this paper I consider a range of possible modifications to the dispositional account that might bring it more closely into alignment with our intuitions about moral condemnation and the rarity of evil persons. According to the most plausible of these theories, S is an evil person if S is strongly disposed to perform evil actions when in conditions that favour S’s autonomy.     

When psychology undermines beliefs

This paper attempts to specify the conditions under which a psychological explanation can undermine or debunk a set of beliefs. The focus will be on moral and religious beliefs, where a growing debate has emerged about the epistemic implications of cognitive science. Recent proposals by Joshua Greene and Paul Bloom will be taken as paradigmatic attempts to undermine beliefs with psychology. I will argue that a belief p may be undermined whenever: (i) p is evidentially based on an intuition which (ii) can be explained by a psychological mechanism that is (iii) unreliable for the task of believing p; and (iv) any other evidence for belief p is based on rationalization. I will also consider and defend two equally valid arguments for establishing unreliability: the redundancy argument and the argument from irrelevant factors. With this more specific understanding of debunking arguments, it is possible to develop new replies to some objections to psychological debunking arguments from both ethics and philosophy of religion.     

Evidence,explanation and enhanced indispensability

In this paper I shall adopt a possible reading of the notions of ‘explanatory indispensability’ and ‘genuine mathematical explanation in science’ on which the Enhanced Indispensability Argument (EIA) proposed by Alan Baker is based. Furthermore, I shall propose two examples of mathematical explanation in science and I shall show that, whether the EIA-partisans accept the reading I suggest, they are easily caught in a dilemma. To escape this dilemma they need to adopt some account of explanation and offer a plausible answer to the following ‘question of evidence’: What is a genuine mathematical explanation in empirical science and on what basis do we consider it as such? Finally, I shall suggest how a possible answer to the question of evidence might be given through a specific account of mathematical explanation in science. Nevertheless, the price of adopting this standpoint is that the genuineness of mathematical explanations of scientific facts turns out to be dependent on pragmatic constraints and therefore cannot be plugged in EIA and used to establish existential claims about mathematical objects.