NATURALIZING DISSENSION

Abstract:  Mathematical naturalism forbids philosophical interventions in mathematical practice. This principle, strictly construed, places severe constraints on legitimate philosophizing about mathematics; it is also arguably incompatible with mathematical realism. One argument for the latter conclusion charges the realist with inability to take a truly naturalistic view of the Gödel Program in set theory. This argument founders on the disagreement among mathematicians about that program's prospects for success. It also turns out that when disagreements run this deep it is counterproductive to take too narrow a view of how philosophers of mathematics may legitimately proceed.     

The Necessity of Mathematics

Some have argued for a division of epistemic labor in which mathematicians supply truths and philosophers supply their necessity. We argue that this is wrong: mathematics is committed to its own necessity. Counterfactuals play a starring role.     

On Representations of Intended Structures in Foundational Theories

Journal of Philosophical Logic - Often philosophers, logicians, and mathematicians employ a notion of intended structure when talking about a branch of mathematics. In addition, we know that there...     

Philosophical reflections on the foundations of mathematics

This article was written jointly by a philosopher and a mathematician. It has two aims: to acquaint mathematicians with some of the philosophical questions at the foundations of their subject and to familiarize philosophers with some of the answers to these questions which have recently been obtained by mathematicians. In particular, we argue that, if these recent findings are borne in mind, four different basic philosophical positions, logicism, formalism, platonism and intuitionism, if stated with some moderation, are in fact reconcilable, although with some reservations in the case of logicism, provided one adopts a nominalistic interpretation of Plato's ideal objects. This eclectic view has been asserted by Lambek and Scott (LS 1986) on fairly technical grounds, but the present argument is meant to be accessible to a wider audience and to provide some new insights.     

Confronting Ideals of Proof with the Ways of Proving of the Research Mathematician

In this paper, we discuss the prevailing view amongst philosophers and many mathematicians concerning mathematical proof. Following Cellucci, we call the prevailing view the “axiomatic conception” of proof. The conception includes the ideas that: a proof is finite, it proceeds from axioms and it is the final word on the matter of the conclusion. This received view can be traced back to Frege, Hilbert and Gentzen, amongst others, and is prevalent in both mathematical text books and logic text books.     

On three arguments against categorical structuralism

Some mathematicians and philosophers contend that set theory plays a foundational role in mathematics. However, the development of category theory during the second half of the twentieth century has encouraged the view that this theory can provide a structuralist alternative to set-theoretical foundations. Against this tendency, criticisms have been made that category theory depends on set-theoretical notions and, because of this, category theory fails to show that set-theoretical foundations are dispensable. The goal of this paper is to show that these criticisms are misguided by arguing that category theory is entirely autonomous from set theory.     

Kreisel,the continuum hypothesis and second order set theory

Summary The major point of contention among the philosophers and mathematicians who have written about the independence results for the continuum hypothesis (CH) and related questions in set theory has been the question of whether these results give reason to doubt that the independent statements have definite truth values. This paper concerns the views of G. Kreisel, who gives arguments based on second order logic that the CH does have a truth value. The view defended here is that although Kreisel's conclusion is correct, his arguments are unsatisfactory. Later sections of the paper advance a different argument that the independence results do not show lack of truth values.     

Logic for physical space

Since the early days of physics, space has called for means to represent, experiment, and reason about it. Apart from physicists, the concept of space has intrigued also philosophers, mathematicians and, more recently, computer scientists. This longstanding interest has left us with a plethora of mathematical tools developed to represent and work with space. Here we take a special look at this evolution by considering the perspective of Logic. From the initial axiomatic efforts of Euclid, we revisit the major milestones in the logical representation of space and investigate current trends. In doing so, we do not only consider classical logic, but we indulge ourselves with modal logics. These present themselves naturally by providing simple axiomatizations of different geometries, topologies, space-time causality, and vector spaces.     

Berkeley and the Primary Qualities: Idealization vs. Abstraction

In the First of the Three Dialogues, Berkeley’s Hylas, responding to Philonous’s question whether extension and motion are separable from secondary qualities, says:

What! Is it not an easy matter, to consider extension and motion by themselves, . . . Pray how do the mathematicians treat of them?

After some introductory comments I propose to contrast Philonous’s (Berkeley’s) answer to this question, with an alternative, arguing for the following. (1) A distinction, Berkeley would accept should be made between abstraction as Berkeley conceives it in The Introduction to the Principles of Human Knowledge and idealization, exemplified by Galileo’s ignoring friction in formulating the law of free-fall. (2) Idealizations, being neither sensible objects nor Platonic forms, illustrate the way mathematically inclined natural philosophers of the time would treat some sensible objects. (3) Therefore one puzzle Berkeley raises, whether extension can exist without color or tactile qualities, disappears. (4) So too can the resemblance puzzle be easily avoided, that is how ideas (taken here to be sensible objects) can resemble what’s in principle insensible. Lastly I suggest this way of developing Hylas’s above remark is consistent with, though not requiring, Berkeley’s idealist metaphysics.     

The sense of incredibility in ethics

Philosophical Studies - It is often said that normative properties are “just too different” to reduce to other kinds of properties. This suggests that many philosophers find it...     

On fineness of grain

A central job for propositions is to be the objects of the attitudes. Propositions are the things we doubt, believe and suppose. Some philosophers have thought that propositions are sets of possible worlds. But many have become convinced that such an account individuates propositions too coarsely. This raises the question of how finely propositions should be individuated. An account of how finely propositions should be individuated on which they are individuated very finely is sketched. Objections to the effect that the account individuates propositions too finely are raised and responses to the objections are provided. It is also shown that theories that try to individuate propositions less finely have serious problems.     

Forgiveness or Fairness?

Abstract

Several philosophers who argue that forgiveness is an important virtue also wish to maintain the moral value of retributive emotions that forgiveness is meant to overcome. As such, these accounts explicate forgiveness as an Aristotelian mean between too much resentment and too little resentment. I argue that such an account ends up making forgiveness superfluous: it turns out that the forgiving person is not praised for a greater willingness to let go of her resentment, but rather for her fairness or good judgment. I conclude by arguing that the virtue of fair-mindedness is more compatible with maintaining the value of the retributive emotions than the virtue of forgiveness.     

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.     

Nature’s drawing: problems and resolutions in the mathematization of motion

The mathematical nature of modern science is an outcome of a contingent historical process, whose most critical stages occurred in the seventeenth century. ‘The mathematization of nature’ (Koyré 1957, From the closed world to the infinite universe, 5) is commonly hailed as the great achievement of the ‘scientific revolution’, but for the agents affecting this development it was not a clear insight into the structure of the universe or into the proper way of studying it. Rather, it was a deliberate project of great intellectual promise, but fraught with excruciating technical challenges and unsettling epistemological conundrums. These required a radical change in the relations between mathematics, order and physical phenomena and the development of new practices of tracing and analyzing motion. This essay presents a series of discrete moments in this process. For mediaeval and Renaissance philosophers, mathematicians and painters, physical motion was the paradigm of change, hence of disorder, and ipso facto available to mathematical analysis only as idealized abstraction. Kepler and Galileo boldly reverted the traditional presumptions: for them, mathematical harmonies were embedded in creation; motion was the carrier of order; and the objects of mathematics were mathematical curves drawn by nature itself. Mathematics could thus be assigned an explanatory role in natural philosophy, capturing a new metaphysical entity: pure motion. Successive generations of natural philosophers from Descartes to Huygens and Hooke gradually relegated the need to legitimize the application of mathematics to natural phenomena and the blurring of natural and artificial this application relied on. Newton finally erased the distinction between nature’s and artificial mathematics altogether, equating all of geometry with mechanical practice.     

Empress vs. Spider-Man: Margaret Cavendish on pure and applied mathematics

Synthese - The empress of Margaret Cavendish’s The Blazing World dismisses pure mathematicians as a waste of her time, and declares of the applied mathematicians that “there [is]...     

Internationalism or Search for Roots: A Tension in Modern Indian Thought

The problems of internationalism often look different and require different solutions depending on one's particular cultural standpoint. Contemporary Indian philosophers find that their philosophical efforts have long been too much oriented to the West, and that what Indian philosophy now needs is less international influence and more rootedness in their own rich tradition. This is needed for a more balanced international dialogue. The goal of internationalism in philosophy is not to bring into being a world-philosophy, but a global conversation on philosophical problems that is not inhibited by differing approaches.     

Disagreement and Intellectual Scepticism

Several philosophers have recently argued that disagreement with others undermines or precludes epistemic justification for our opinions about controversial issues (e.g. political, religious, and philosophical issues). This amounts to a fascinating and disturbing kind of intellectual scepticism. A crucial piece of the sceptical argument, however, is that our opponents on such topics are epistemic peers. In this paper, I examine the reasons for why we might think that our opponents really are such peers, and I argue that those reasons are either (a) too weak or (b) too strong, implying absurd conclusions. Thus, there is not a compelling case for disagreement-based intellectual scepticism.     

REDUCTION WITHOUT REDUCTIONISM: A DEFENCE OF NAGEL ON CONNECTABILITY

Unlike the overall framework of Ernest Nagel's work on reduction, his theory of intertheoretic connection still has life in it. It handles aptly cases where reduction requires complex representation of a target domain. Abandoning his formulation as too liberal was a mistake. Arguments that it is too liberal at best touch only Nagel's deductivist theory of explanation, not his condition of connectability. Taking this condition seriously gives a powerful view of reduction, but one which requires us to index explanatory power to sciences as they are formulated at particular times. While we may thereby reduce more than philosophers have supposed, we must abandon hope (as Nagel did) of saying anything useful about reductionism.     

Reflection Without Regress

Regress arguments show that to do something for a reason, one does not have to have reflectively endorsed that reason. This might seem to establish that reflection does not play a fundamental role in agency. This paper argues that this conclusion rests on too narrow a conception of agency. If agents are not just creatures who act for reasons but also creatures who can take ownership of the reasons for which they act, then there is a central role for reflection to play in agency even if it's not the role that philosophers of action have typically envisioned for it.     

Don’t Blame the Idealizations

Idealizing conditions are scapegoats for scientific hypotheses, too often blamed for falsehood better attributed to less obvious sources. But while the tendency to blame idealizations is common among both philosophers of science and scientists themselves, the blame is misplaced. Attention to the nature of idealizing conditions, the content of idealized hypotheses, and scientists’ attitudes toward those hypotheses shows that idealizing conditions are blameless when hypotheses misrepresent. These conditions help to determine the content of idealized hypotheses, and they do so in a way that prevents those hypotheses from being false by virtue of their constituent idealizations.