Pluralist mathematical realism, the view that there exists more than one mathematical universe, has become an influential position in the philosophy of mathematics. I argue that, if mathematical pluralism is true (and we have good reason to believe that it is), then mathematical realism cannot (easily) be justified by arguments from the indispensability of mathematics to science. This is because any justificatory chain of inferences from mathematical applications in science to the total body of mathematical theorems can cover at most (...) one mathematical universe. Indispensability arguments may thus lose their central role in the debate about mathematical ontology. (shrink)

I argue that recent attempts to deflect Access Problems for realism about a priori domains such as mathematics, logic, morality, and modality using arguments from evolution result in two kinds of explanatory overkill: the Access Problem is eliminated for contentious domains, and realist belief becomes viciously immune to arguments from dispensability, and to non-rebutting counter-arguments more generally.

The existence of fundamental moral disagreements is a central problem for moral realism and has often been contrasted with an alleged absence of disagreement in mathematics. However, mathematicians do in fact disagree on fundamental questions, for example on which set-theoretic axioms are true, and some philosophers have argued that this increases the plausibility of moral vis-à-vis mathematical realism. I argue that the analogy between mathematical and moral disagreement is not as straightforward as those arguments present it. In particular, I argue (...) that pluralist accounts of mathematics render fundamental mathematical disagreements compatible with mathematical realism in a way in which moral disagreements and moral realism are not. 11. (shrink)

Can art, religion, or philosophy afford ineffable insights? If so, what are they? The idea of ineffability has puzzled philosophers from Laozi to Wittgenstein. In Ineffability and its Metaphysics: The Unspeakable in Art, Religion and Philosophy, Silvia Jonas examines different ways of thinking about what ineffable insights might involve metaphysically, and shows which of these are in fact incoherent. Jonas discusses the concepts of ineffable properties and objects, ineffable propositions, ineffable content, and ineffable knowledge, examining the metaphysical pitfalls involved in (...) these concepts. Ultimately, she defends the idea that ineffable insights as found in aesthetic, religious, and philosophical contexts are best understood in terms of self-acquaintance, a particular kind of non-propositional knowledge. Ineffability as a philosophical topic is as old as the history of philosophy itself, but contributions to the exploration of ineffability have been sparse. The theory developed by Jonas makes the concept tangible and usable in many different philosophical contexts. (shrink)

Maimonides famously holds that, while it is perfectly possible to know (and say) that God exists, it is impossible to know (and say) what God is like because any positive attri- bution contradicts God’s essential oneness. Consequently, pure equivocity obtains between descriptions of the divine and descriptions of any other being. But this raises a puzzle: Knowledge of God seems vacuous if we lack all comprehension of God’s nature - so how can we have any comprehension of the divine without (...) being able to say what it is that we know? In this essay, I defend the substantiality of the concept of knowledge of God on Maimonides’ apophatic picture. Drawing on the concept of non-propositional knowledge, I argue that knowledge and comprehension can come apart, which lends support to Maimonides’ thesis that knowledge of God, though substantial and objective, cannot be expressed. (shrink)

Drawing an analogy between modal structuralism about mathematics and theism, I o er a structuralist account that implicitly de nes theism in terms of three basic relations: logical and metaphysical priority, and epis- temic superiority. On this view, statements like `God is omniscient' have a hypothetical and a categorical component. The hypothetical component provides a translation pattern according to which statements in theistic language are converted into statements of second-order modal logic. The categorical component asserts the logical possibility of the (...) theism struc- ture on the basis of uncontroversial facts about the physical world. This structuralist reading of theism preserves objective truth-values for theistic statements while remaining neutral on the question of ontology. Thus, it o ers a way of understanding theism to which a naturalist cannot object, and it accommodates the fact that religious belief, for many theists, is an essentially relational matter. (shrink)

This essay provides an overview of the ways in which contemporary philosophers have tried to make sense of ineffability as encountered in aesthetic contexts. Section 1 sets up the problem of aesthetic ineffability by putting it into historical perspective. Section 2 specifies the kinds of questions that may be raised with regard to aesthetic ineffability, as well as the kinds of answer each one of those questions would require. Section 3 investigates arguments that seek to locate aesthetic ineffability within the (...) object of aesthetic experiences, i.e. within aesthetic content. Section 4 discusses arguments that seek to locate aesthetic ineffability within the subject of aesthetic experience. (shrink)

In this paper, I argue that religious belief is epistemically equivalent to mathematical belief. Abstract beliefs don't fall under ‘naive’, evidence-based analyses of rationality. Rather, their epistemic permissibility depends, I suggest, on four criteria: predictability, applicability, consistency, and immediate acceptability of the fundamental axioms. The paper examines to what extent mathematics meets these criteria, juxtaposing the results with the case of religion. My argument is directed against a widespread view according to which belief in mathematics is clearly rationally acceptable whereas (...) belief in religion is not. The paper also aims to make some of the implications of contemporary mathematics available to philosophers working in different fields. (shrink)

Adorno’s metaphysics as developed in his Negative Dialectics revolves around what he calls the ‘Non-identical’. The Non-identical is essentially ineffable and can only be understood negatively, through Adorno’s method of ‘negative dialectics’. Negative dialectics is Adorno’s answer to Hegelian metaphysics, which he criticises for its ‘consistent resolution of non-identity into pure identity’. While Adorno endorses Hegel’s critique of Kant’s distinction between the realm of noumena and the realm of phenomena, he argues that Hegel is wrong in believing that a speculative (...) identity between thought and being can be achieved in a positive fashion. Human reason, Adorno argues, imposes unity and identity on objects in the world, thus suppressing their uniqueness and making equivalent what is essentially non-equivalent. The way Adorno sees it, ‘dialectics means to break the compulsion to achieve identity’. This paper aims (1) to explain Adorno’s answer to Hegelian metaphysics, (2) to elucidate Adorno’s idea of the Non-identical as ineffable, and (3) to highlight the importance of acknowledging the ineffable in contemporary philosophy. The paper thus expounds Adorno’s answer to Hegelian metaphysics, according to which philosophy must be prepared to bear not only contradiction, but also the ultimate impossibility of resolving non-identity into identity. (shrink)

The recognition of striking regularities in the physical world plays a major role in the justification of hypotheses and the development of new theories both in the natural sciences and in philosophy. However, while scientists consider only strictly natural hypotheses as explanations for such regularities, philosophers also explore meta-natural hypotheses. One example is mathematical realism, which proposes the existence of abstract mathematical entities as an explanation for the applicability of mathematics in the sciences. Another example is theism, which offers the (...) existence of a supernatural being as an explanation for the design-like appearance of the physical cosmos. Although all meta-natural hypotheses defy empirical testing, there is a strong intuition that some of them are more warranted than others. The goal of this paper is to sharpen this intuition into a clear criterion for the admissibility of meta-natural explanations for empirical facts. Drawing on recent debates about the indispensability of mathematics and teleological arguments for the existence of God, I argue that a meta-natural explanation is admissible just in case the explanation refers to an entity that, though not itself causally efficacious, guarantees the instantiation of a causally efficacious entity that is an actual cause of the regularity. (shrink)