My ultimate goal in this paper is to illuminate, from a naturalistic point of view, the significance of the application of mathematics in the natural sciences for the practice of contemporary set theory.

Juliette Kennedy’s new book brims with intriguing ideas. I don’t understand all of them, and I’m not convinced that the ones I do understand all fit together, b.

This ambitious book explores the relationship between time and history and shows how an appreciation of long-term time helps to make sense of the past. The book is devoted to a wide-ranging analysis of the way different societies have conceived and interpreted time, and it develops a theory of the threefold roles of continuity, gradual change, and revolution which together form a "braided" history. Linking the interpretative chapters are intriguing brief expositions on time travel, time cycles, time lines, and time (...) pieces, showing the different ways in which human history has been located in time. In its global approach the book is part of the new shift toward “big history,” in which traditional period divisions are challenged in favor of looking at the entire past of the world from start to end. The approach is thematic. The result is a view of world history in which outcomes are shown to be explicable, once they happen, but not necessarily predictable before they do. This book will inform the work of historians of all periods and at all levels, and contributes to the current reconsideration of traditional period divisions (such as Modernity and Postmodernity), which the author finds outmoded. (shrink)

Naturalism in Mathematics investigates how the most fundamental assumptions of mathematics can be justified. One prevalent philosophical approach to the problem--realism--is examined and rejected in favor of another approach--naturalism. Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be successfully applied in set theory. Her clear, original treatment of this fundamental issue is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both disciplines.

Mathematicians tend to think of themselves as scientists investigating the features of real mathematical things, and the wildly successful application of mathematics in the physical sciences reinforces this picture of mathematics as an objective study. For philosophers, however, this realism about mathematics raises serious questions: What are mathematical things? Where are they? How do we know about them? Offering a scrupulously fair treatment of both mathematical and philosophical concerns, Penelope Maddy here delineates and defends a novel version of (...) mathematical realism. She answers the traditional questions and poses a challenging new one, refocusing philosophical attention on the pressing foundational issues of contemporary mathematics. (shrink)

How did humans respond to the eighteenth-century discovery of countless new species of animals? This book explores the gamut of human-animal interactions: from love to cultural identifications, moral reflections, philosophical debates, classification systems, mechanical copies, insults and literary creativity.

This paper outlines a second-philosophical account of arithmetic that places it on a distinctive ground between those of logic and set theory.

This chapter compares and contrasts Quine’s naturalism with the versions of two post-Quineans on the nature of science, logic, and mathematics. The role of indispensability in the philosophy of mathematics is treated in detail.

Mathematics depends on proofs, and proofs must begin somewhere, from some fundamental assumptions. For nearly a century, the axioms of set theory have played this role, so the question of how these axioms are properly judged takes on a central importance. Approaching the question from a broadly naturalistic or second-philosophical point of view, Defending the Axioms isolates the appropriate methods for such evaluations and investigates the ontological and epistemological backdrop that makes them appropriate. In the end, a new account of (...) the objectivity of mathematics emerges, one refreshingly free of metaphysical commitments. (shrink)

Stimulation of airway myocytes by contractile agents such as acetylcholine (ACh) activates a Ca2+-activated Cl– current (IClCa) which may play a key role in calcium homeostasis of airway myocytes and hence in airway reactivity. The aim of the present study was to model IClCa in airway smooth muscle cells using a computerised model previously designed for simulation of cardiac myocyte functioning. Modelling was based on a simple resistor-battery permeation model combined with multiple binding site activation by calcium. In order to (...) validate the model, a combination of equations, used to mimic [Ca2+]i response to ACh stimulation, were incorporated into the model. The results indicate that the model developed in this article accounts for experimental recordings and electrophysiological characteristics of this current in airway smooth muscle cells, with parameter values consistent with those calculated from experimental data. Such a model may thus be used to predict IClCa functioning, though additional experimental data from airway myocytes would be useful to more accurately determine some parameter values of the model. (shrink)

Some recent work by philosophers of mathematics has been aimed at showing that our knowledge of the existence of at least some mathematical objects and/or sets can be epistemically grounded by appealing to perceptual experience. The sensory capacity that they refer to in doing so is the ability to perceive numbers, mathematical properties and/or sets. The chief defense of this view as it applies to the perception of sets is found in Penelope Maddy’s Realism in Mathematics, but a (...) number of other philosophers have made similar, if more simple, appeals of this sort. For example, Jaegwon Kim (1981, 1982), John Bigelow (1988, 1990), and John Bigelow and Robert Pargetter (1990) have all defended such views. The main critical issue that will be raised here concerns the coherence of the notions of set perception and mathematical perception, and whether appeals to such perceptual faculties can really provide any justification for or explanation of belief in the existence of sets, mathematical properties and/or numbers. (shrink)

ome recent work by philosophers of mathematics has been aimed at showing that our knowledge of the existence of at least some mathematical objects and/or sets can be epistemically grounded by appealing to perceptual experience. The sensory capacity that they refer to in doing so is the ability to perceive numbers, mathematical properties and/or sets. The chief defense of this view as it applies to the perception of sets is found in Penelope Maddy’s Realism in Mathematics, but a (...) number of other philosophers have made similar, if more simple, appeals of this sort. For example, Jaegwon Kim (1981, 1982), John Bigelow (1988, 1990), and John Bigelow and Robert Pargetter (1990) have all defended such views. The main critical issue that will be raised here concerns the coherence of the notions of set perception and mathematical perception, and whether appeals to such perceptual faculties can really provide any justification for or explanation of belief in the existence of sets, mathematical properties and/or numbers. (shrink)

The effort to fit simple logical truths–like `if it's either red or green and it's not red, then it must be green'–into Kant's account of knowledge turns up a position more subtle and intriguing than might be expected at first glance.

Many philosophers these days consider themselves naturalists, but it's doubtful any two of them intend the same position by the term. In Second Philosophy, Penelope Maddy describes and practices a particularly austere form of naturalism called "Second Philosophy". Without a definitive criterion for what counts as "science" and what doesn't, Second Philosophy can't be specified directly ("trust only the methods of science" for example), so Maddy proceeds instead by illustrating the behaviors of an idealized inquirer she calls (...) the "Second Philosopher". mhis Second Philosopher begins from perceptual common sense experimentation, theory formation and testing, working all the while to asses, correct and improve her methods as she goes. Second Philosophy is then the result of the Second Philosopher's investigations. Maddy delineates the Second Philosopher's approach by tracing her reactions to various familiar skeptical and transcendental views (Descartes, Kant, Carnap, late Putnam, van Fraassen), comparing her methods to those of other self-described naturalists (especially Quine), and examining a prominent contemporary debate (between disquotationalists and correspondence theorists in the theory of truth) to extract a properly second-philosophical line of thought. She then undertakes to practice Second Philosophy in her reflections on the ground of logical truth, the methodology, ontology and epistemology of mathematics, and the general prospects for metaphysics naturalized. (shrink)

Many philosophers claim to be naturalists, but there is no common understanding of what naturalism is. Maddy proposes an austere form of naturalism called 'Second Philosophy', using the persona of an idealized inquirer, and she puts this method into practice in illuminating reflections on logical truth, philosophy of mathematics, and metaphysics.

Pursues the theoretical level of the two‐tiered epistemology of set theoretic realism, the level at which more abstract axioms can be justified by their consequences at more intuitive levels. I outline the pre‐axiomatic development of set theory out of Cantor's researches, describe how axiomatization arose in the course of Zermelo's efforts to prove Cantor's Well‐ordering Theorem, and review the controversy over the Axiom of Choice. Cantor's Continuum Hypothesis and various questions of descriptive set theory were eventually shown to be independent (...) of the standard axioms, and new axiom candidates—Gödel's Axiom of Constructibility, on the one hand, determinacy and large cardinal axioms, on the other—offer dramatically different solutions. This situation presents a new epistemic challenge to the set theoretic realist: on what grounds can we adjudicate between new axiom candidates? (shrink)

Outlines a physicalistic version of set theoretic realism, and compares and contrasts it with Field's nominalism, with structuralism, and with modalism. I conclude that, despite metaphysical differences, versions of all these views share the new challenge raised in Ch. 4: how are new axiom candidates to be rationally evaluated?

Begins with a review of Benacerraf's metaphysical challenge to mathematical realism based on sets: how, for example, can number theory be the study of particular sets when other sets with the same structural relations would seem to do just as well? The set theoretic realist gives the straightforward response that numbers are not particular sets, but properties of sets. I close with a digression on the prospects for ‘Frege numbers’—i.e. numbers construed as proper classes.

Mathematical axioms have traditionally been thought of as obvious or self-evident truths, but current set theoretic work in the search for new axioms belies this conception. This raises epistemological questions about what other forms of justification are possible, and how they should be judged.

Begins with a presentation and elaboration of Benacerraf's epistemic challenge to realism: how can we gain knowledge of an acausal world of non‐spatio‐temporal abstracts? I then outline a theory of perception based in part on neurological theories of Hebb and developmental evidence from Piaget, and I argue in these terms that we can, in fact, perceive sets of medium‐sized physical objects. This account of perception is elaborated into an account of physical and mathematical intuition, faculties that produce various rudimentary beliefs (...) that underlie the simplest physical and set theoretic assumptions. I conclude by comparing and contrasting this epistemology with some controversial passages from Gödel's writings. (shrink)

The early sections describe the pre‐theoretic realism of the mathematician, survey the basic forms of realism in philosophy, and attempt to disentangle the issues of realism from debates over the nature of truth. The final section begins by laying out traditional Platonism, intuitionism, and formalism. Quine's famous critique of Carnap's conventionalism then leads to Quine's realism, and Putnam's developments thereof, while a different sort of mathematical realism is found in Gödel's writings. Set theoretic realism emerges as an effort to build (...) on the strengths of Quine's and Gödel's realisms, while avoiding their weaknesses and filling in their lacunae. (shrink)

The Logical Must is an examination of Wittgenstein's philosophy of logic, early and late, from an austere naturalistic perspective called "Second Philosophy.".

A plea for natural philosophy --On the question of realism --Hume and Reid --Moore's hands --Wittgenstein on hinges --A note on truth and reference --The philosophy of logic --A Second Philosophy of logic --Psychology and the a priori sciences --Do numbers exist? --Enhanced if-thenism.

It’s often said that set theory provides a foundation for classical mathematics because every classical mathematical object can be modeled as a set and every classical mathematical theorem can be proved from the axioms of set theory. This is obviously a remarkable mathematical fact, but it isn’t obvious what makes it ‘foundational’. This paper begins with a taxonomy of the jobs set theory does that might reasonably be regarded as foundational. It then moves on to category-theoretic and univalent foundations, exploring (...) to what extent they do these same jobs, and to what extent they might do other jobs also reasonably regarded as foundational. (shrink)

What Do Philosophers Do? takes up the leading arguments for radical skepticism from an everyday point of view. A range of philosophical methods are examined and employed, for a revealing portrait of what philosophers do, and perhaps a quiet suggestion for what they should do, for what they do best.

This talk surveys a range of positions on the fundamental metaphysical and epistemological questions about elementary logic, for example, as a starting point: what is the subject matter of logic—what makes its truths true? how do we come to know the truths of logic? A taxonomy is approached by beginning from well-known schools of thought in the philosophy of mathematics—Logicism, Intuitionism, Formalism, Realism—and sketching roughly corresponding views in the philosophy of logic. Kant, Mill, Frege, Wittgenstein, Carnap, Ayer, Quine, and Putnam (...) are among the philosophers considered along the way. (shrink)

My goal here is to explore the relationship between pure and applied mathematics and then, eventually, to draw a few morals for both. In particular, I hope to show that this relationship has not been static, that the historical rise of pure mathematics has coincided with a gradual shift in our understanding of how mathematics works in application to the world. In some circles today, it is held that historical developments of this sort simply represent changes in fashion, or in (...) social arrangements, governments, power structures, or some such thing, but I resist the full force of this way of thinking, clinging to the old school notion that we have gradually learned more about the world over time, that our opinions on these matters have improved, and that seeing how we reached the point we now occupy may help us avoid falling back into old philosophies that are now no longer viable. In that spirit, it seems to me that once we focus on the general question of how mathematics relates to science, one observation is immediate: the march of the centuries has produced an amusing reversal of philosophical fortunes. Let me begin there. In the beginning, that is, in Plato, mathematical knowledge was sharply distinguished from ordinary perceptual belief about the world. According to Plato’s metaphysics, mathematics is the study of eternal and unchanging abstract Forms,1 while science is uncertain and changeable opinion about the world of mere becoming. Indeed, in Plato’s lights, of the two, only mathematics deserves to be called ‘knowledge’ at all! Of course if sense perception cannot give us knowledge, if mathematics is not about perceivable things, then Plato owes us an account of how we ordinary humans achieve this wonderful insight into the properties of the abstract world of Forms. Plato’s answer is that we do not actually acquire mathematical knowledge at all; rather, we recollect it from a time before birth, when our souls, unencumbered by physical bodies, were free to commune with the Forms, and not just the mathematical ones, either – also Truth, Beauty, Justice, the Good, and so on.2 Whatever appeal this position may have held for the ancient Greeks, it will not begin to satisfy a contemporary, scientifically minded philosopher.. (shrink)