Quine, Analyticity and Philosophy of Mathematics

Quine correctly argues that Carnap's distinction between internal and external questions rests on a distinction between analytic and synthetic, which Quine rejects. I argue that Quine needs something like Carnap's distinction to enable him to explain the obviousness of elementary mathematics, while at the same time continuing to maintain as he does that the ultimate ground for holding mathematics to be a body of truths lies in the contribution that mathematics makes to our overall scientific theory of the world. Quine's arguments against the analytic/synthetic distinction, even if fully accepted, still leave room for a notion of pragmatic analyticity sufficient for the indicated purpose.     

Peirce’s Philosophy of Mathematical Education: Fostering Reasoning Abilities for Mathematical Inquiry

I articulate Charles S. Peirce’s philosophy of mathematical education as related to his conception of mathematics, the nature of its method of inquiry, and especially, the reasoning abilities required for mathematical inquiry. The main thesis is that Peirce’s philosophy of mathematical education primarily aims at fostering the development of the students’ semeiotic abilities of imagination, concentration, and generalization required for conducting mathematical inquiry by way of experimentation upon diagrams. This involves an emphasis on the relation between theory and practice and between mathematics and other fields including the arts and sciences. For achieving its goals, the article is divided in three sections. First, I expound Peirce’s philosophical account of mathematical reasoning. Second, I illustrate this account by way of a geometrical example, placing special emphasis on its relation to mathematical education. Finally, I put forth some Peircean philosophical principles for mathematical education.     

作为一种分析哲学的数理逻辑——以相对可计算性与随机性概念研究为例

对于现代逻辑的运用是分析哲学的一大特征。分析哲学与现代逻辑甚至具有共同的起源:不仅分析哲学与现代逻辑的奠基者是同一批人,他们曾面对的还是同样的问题,并为此开发出我们所熟知的研究形态。然而,分析哲学和现代逻辑(尤其是数理逻辑)后来的发展似乎走上了不同的道路。笔者则希望论证当代数理逻辑研究仍然保留了分析哲学解决问题的形态——语言分析——的主要特征。论证主要基于两则案例分析,一个是对埃米尔·波斯特(Emil Post)关于可计算性理论的开创性工作,另一个是佩尔·马丁–洛夫(Per Martin-L?f)等人在随机性概念方面的工作。笔者还将进一步提议,仍然存在着一些值得当代分析哲学家和数理逻辑共同关心的问题。     

Arithmetic,set theory,reduction and explanation

Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important and well-known case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In defense of this claim, I offer some evidence from mathematical practice, and I respond to contrary suggestions due to Steinhart, Maddy, Kitcher and Quine. I then show how, even if set-theoretic reductions are generally not explanatory, set theory can nevertheless serve as a legitimate and successful foundation for mathematics. Finally, some implications of my thesis for philosophy of mathematics and philosophy of science are discussed. In particular, I suggest that some reductions in mathematics are probably explanatory, and I propose that differing standards of theory acceptance might account for the apparent lack of unexplanatory reductions in the empirical sciences.     

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.     

Platonic Number in the Parmenides and Metaphysics XIII

I argue here that a properly Platonic theory of the nature of number is still viable today. By properly Platonic, I mean one consistent with Plato's own theory, with appropriate extensions to take into account subsequent developments in mathematics. At Parmenides 143a-4a the existence of numbers is proven from our capacity to count, whereby I establish as Plato's the theory that numbers are originally ordinal, a sequence of forms differentiated by position. I defend and interpret Aristotle's report of a Platonic distinction between form and mathematical numbers, arguing that mathematical numbers alone are cardinals, by reference to certain non-technical features of a set-theoretical approach and other considerations in philosophy of mathematics. Finally I respond to the objections that such a conception of number was unavailable in antiquity and that this theory is contradicted by Aristotle's report in Metaph . XIII that Platonic numbers are collections of units. I argue that Aristotle reveals his own misinterpretation of the terms in which Plato's theory was expressed.     

The Meaning of Category Theory for 21st Century Philosophy

Among the main concerns of 20th century philosophy was that of the foundations of mathematics. But usually not recognized is the relevance of the choice of a foundational approach to the other main problems of 20th century philosophy, i.e., the logical structure of language, the nature of scientific theories, and the architecture of the mind. The tools used to deal with the difficulties inherent in such problems have largely relied on set theory and its “received view”. There are specific issues, in philosophy of language, epistemology and philosophy of mind, where this dependence turns out to be misleading. The same issues suggest the gain in understanding coming from category theory, which is, therefore, more than just the source of a “non-standard” approach to the foundations of mathematics. But, even so conceived, it is the very notion of what a foundation has to be that is called into question. The philosophical meaning of mathematics is no longer confined to which first principles are assumed and which “ontological” interpretation is given to them in terms of some possibly updated version of logicism, formalism or intuitionism. What is central to any foundational project proper is the role of universal constructions that serve to unify the different branches of mathematics, as already made clear in 1969 by Lawvere. Such universal constructions are best expressed by means of adjoint functors and representability up to isomorphism. In this lies the relevance of a category-theoretic perspective, which leads to wide-ranging consequences. One such is the presence of functorial constraints on the syntax–semantics relationships; another is an intrinsic view of (constructive) logic, as arises in topoi and, subsequently, in more general fibrations. But as soon as theories and their models are described accordingly, a new look at the main problems of 20th century’s philosophy becomes possible. The lack of any satisfactory solution to these problems in a purely logical and set-theoretic setting is the result of too circumscribed an approach, such as a static and punctiform view of objects and their elements, and a misconception of geometry and its historical changes before, during, and after the foundational “crisis”, as if algebraic geometry and synthetic differential geometry – not to mention algebraic topology – were secondary sources for what concerns foundational issues. The objectivity of basic geometrical intuitions also acts against the recent version of structuralism proposed as ‘the’ philosophy of category theory. On the other hand, the need for a consistent and adequate conceptual framework in facing the difficulties met by pre-categorical theories of language and scientific knowledge not only provides the basic concepts of category theory with specific applications but also suggests further directions for their development (e.g., in approaching the foundations of physics or the mathematical models in the cognitive sciences). This ‘virtuous’ circle is by now largely admitted in theoretical computer science; the time is ripe to realise that the same holds for classical topics of philosophy. Text of a talk given at the Workshop and Symposium on the Ramifications of Category Theory, Florence, November 18–22, 2003. For further documentation on the conference, see     

A philosopher's view of values and ethics

Counseling is the practical art of making rational decisions about values; thus it is part science and part philosophy. As a professional activity it falls midway between science and philosophy and partakes of the characteristics of both. Counseling readily recognizes its dependence on professional science for empirical knowledge about fact and theory but tends to ignore the analytic contributions of professional philosophy for understanding the nature of value and value theory. Counseling will be a better art when counselors are as concerned with what philosophy says about values as they are with the contributions of the social sciences.     

The inquiring mind

There is nothing, either in the recent developments of philosophy or in the development of the sciences, which should prevent philosophy from continuing its role of mother‐science and the sciences from influencing methods and conclusions of philosophers. The inquiring mind respects no boundaries between disciplines except those which are imposed by differences in questions raised. But basic questions, whether raised by philosophers or by scientists, tend to have components requiring co‐ordination of research or analysis of highly different disciplines. Both Anglo‐Saxon and continental developments in philosophy justify, however, a distinction between cultivating philosophy and being engaged in solving or resolving a philosophical problem, the former comprising the latter.     

从本体-历史的观点看

一、科学哲学的岔路半个多世纪前问世的奎因的那本《从逻辑的观点看》(1953),至今还影响着我们:从整体主义的观点出发启迪了诸多后实证主义的灵感。与奎因的本体-逻辑的与认识-逻辑的观点不同,本文想要阐述的是一种本体-历史的观点。这是两类不同性质的理论。我认为,本体-历史的观点同样也适     

Logic of imagination. Echoes of Cartesian epistemology in contemporary philosophy of mathematics and beyond

Descartes’ Rules for the direction of the mind presents us with a theory of knowledge in which imagination, considered as an “aid” for the intellect, plays a key role. This function of schematization, which strongly resembles key features of Proclus’ philosophy of mathematics, is in full accordance with Descartes’ mathematical practice in later works such as La Géométrie from 1637. Although due to its reliance on a form of geometric intuition, it may sound obsolete, I would like to show that this has strong echoes in contemporary philosophy of mathematics, in particular in the trend of the so called “philosophy of mathematical practice”. Indeed Ken Manders’ study on the Euclidean practice, along with Reviel Netz’s historical studies on ancient Greek Geometry, indicate that mathematical imagination can play a central role in mathematical knowledge as bearing specific forms of inference. Moreover, this role can be formalized into sound logical systems. One question of general epistemology is thus to understand this mysterious role of the imagination in reasoning and to assess its relevance for other mathematical practices. Drawing from Edwin Hutchins’ study of “material anchors” in human reasoning, I would like to show that Descartes’ epistemology of mathematics may prove to be a helpful resource in the analysis of mathematical knowledge.     

Kant's Syntheticity Revisited by Peirce

This paper reconstructs the Peircean interpretation of Kant's doctrine on the syntheticity of mathematics. Peirce correctly locates Kant's distinction in two different sources: Kant's lack of access to polyadic logic and, more interestingly, Kant's insight into the role of ingenious experiments required in theorem-proving. In this second respect, Kant's analytic/synthetic distinction is identical with the distinction Peirce discovered among types of mathematical reasoning. I contrast this Peircean theory with two other prominent views on Kant's syntheticity, i.e. the Russellian and the Beckian views, and show how Peirce's interpretation of Kant solves the dilemma that each of these two views faces. I also show that Hintikka's criterion for Kant's synthetic judgments, i.e. a new individual introduced by the -instantiation rule, does not capture the most important characteristic of Peirce's theorematic reasoning, i.e. the process of choosing a correct individual.     

Towards a Philosophy of Chemistry. A short extract of this paper was first read at the 10th International Congress of Logic, Methodology and Philosophy of Science, Florence, August 19–25, 1995.

The paper shows epistemological, methodological and ontological peculiarities of chemistry taken as a classificatory science of materials using experimental methods. Without succumbing to standard interpretations of physical science, chemical methods of experimental investigation, classification, reference, theorizing, prediction and production of new entities are developed one by one as first steps towards a philosophy of chemistry. Chemistry challenges traditional concepts of empirical object, empirical predicate, reference frame and theory, but also the distinction commonly drawn between natural science and technology. Due to its many peculiarities, I propose to treat chemistry philosophically as a special type of science, apart from other sciences.     

Challenging epistemology: Interactive proofs and zero knowledge

This essay explores what (if anything) research on interactive zero knowledge proofs has to teach philosophers about the epistemology of mathematics and theoretical computer science. Though such proof systems initially appear ‘revolutionary’ and are a nonstandard conception of ‘proof’, I will argue that they do not have much philosophical import. Possible lessons from this work for the epistemology of mathematics—our models of mathematical proof should incorporate interaction, our theories of mathematical evidence must account for probabilistic evidence, our valuation of a mathematical proof should solely focus on its persuasive power—are either misguided or old hat. And while the differences between interactive and mathematical proofs suggest the need to develop a separate epistemology of theoretical computer science (or at least complexity theory) that differs from our theory of mathematical knowledge, a casual look at the actual practice of complexity theory indicates that such a distinct epistemology may not be necessary.     

Hobbes on the Order of Sciences: A Partial Defense of the Mathematization Thesis

Accounts of Hobbes's “system” of sciences oscillate between two extremes. On one extreme, the system is portrayed as wholly axiomatic‐deductive, with statecraft being deduced in an unbroken chain from the principles of logic and first philosophy. On the other, it is portrayed as rife with conceptual cracks and fissures, with Hobbes's statements about its deductive structure amounting to mere window‐dressing. This paper argues that a middle way is found by conceiving of Hobbes's Elements of Philosophy on the model of a mixed‐mathematical science, not the model provided by Euclid's Elements of Geometry. I suggest that Hobbes is a test case for understanding early‐modern system construction more generally, as inspired by the structure of the applied mathematical sciences. This approach has the additional virtue of bolstering in a novel way the thesis that the transformation of philosophy in the long seventeenth century was indebted to mathematics, a thesis that has come under increasing scrutiny in recent years.     

Ernst Cassirer's Neo-Kantian Philosophy of Geometry

One of the most important philosophical topics in the early twentieth century – and a topic that was seminal in the emergence of analytic philosophy – was the relationship between Kantian philosophy and modern geometry. This paper discusses how this question was tackled by the Neo-Kantian trained philosopher Ernst Cassirer. Surprisingly, Cassirer does not affirm the theses that contemporary philosophers often associate with Kantian philosophy of mathematics. He does not defend the necessary truth of Euclidean geometry but instead develops a kind of logicism modeled on Richard Dedekind's foundations of arithmetic. Further, because he shared with other Neo-Kantians an appreciation of the developmental and historical nature of mathematics, Cassirer developed a philosophical account of the unity and methodology of mathematics over time. With its impressive attention to the detail of contemporary mathematics and its exploration of philosophical questions to which other philosophers paid scant attention, Cassirer's philosophy of mathematics surely deserves a place among the classic works of twentieth century philosophy of mathematics. Though focused on Cassirer's philosophy of geometry, this paper also addresses both Cassirer's general philosophical orientation and his reading of Kant.     

BEALER ON THE AUTONOMY OF PHILOSOPHICAL AND SCIENTIFIC KNOWLEDGE

Abstract: In a series of influential articles, George Bealer argues for the autonomy of philosophical knowledge on the basis that philosophically known truths must be necessary truths. The main point of his argument is that the truths investigated by the sciences are contingent truths to be discovered a posteriori by observation, while the truths of philosophy are necessary truths to be discovered a priori by intuition. The project of assimilating philosophy to the sciences is supposed to be rendered illegitimate by the more or less sharp distinction in these characteristic methods and its modal basis. In this article Bealer's particular way of drawing the distinction between philosophy and science is challenged in a novel manner, and thereby philosophical naturalism is further defended.     

Logic as a Universal Medium or Logic as a Calculus? Husserl and the Presuppositions of “the Ultimate Presupposition of Twentieth Century Philosophy”

This paper discusses Jean van Heijenoort's (1967) and Jaakko and Merrill B. Hintikka's (1986, 1997) distinction between logic as a universal language and logic as a calculus, and its applicability to Edmund Husserl's phenomenology. Although it is argued that Husserl's phenomenology shares characteristics with both sides, his view of logic is closer to the model‐theoretical, logic‐as‐calculus view. However, Husserl's philosophy as transcendental philosophy is closer to the universalist view. This paper suggests that Husserl's position shows that holding a model‐theoretical view of logic does not necessarily imply a calculus view about the relations between language and the world. The situation calls for reflection about the distinction: It will be suggested that the applicability of the van Heijenoort and the Hintikkas distinction either has to be restricted to a particular philosopher's views about logic, in which case no implications about his or her more general philosophical views should be inferred from it; or the distinction turns into a question of whether our human predicament is inescapable or whether it is possible, presumably by means of model theory, to obtain neutral answers to philosophical questions. Thus the distinction ultimately turns into a question about the correct method for doing philosophy.     

自然主义集合实在论与数学哲学研究范式

In recent 50 years,the debate between mathematical realism and anti-realism has been dominating the mainstream development in the contemporary philosophy of mathematics. Penelope Maddy proposed a naturalistic set theoretic realism in 1990. This project brings the philosophy of mathematics a new research idea,that is,philosophy should attach importance to mathematical practice. This article will critically analyze Maddy's naturalistic set theoretic realism on the basis of research paradigm background belief....     

Two Dogmatists

Grice and Strawson's ‘In Defense of a Dogma’ is admired even by revisionist Quineans such as Putnam (1962) who should know better. The analytic/synthetic distinction they defend is distinct from that which Putnam successfully rehabilitates. Theirs is the post‐positivist distinction bounding a grossly enlarged analytic. It is not, as they claim, the sanctified product of a long philosophic tradition, but the cast‐off of a defunct philosophy ‐ logical positivism. The fact that the distinction can be communally drawn does not show that it is based on a real difference. Sub‐categories that can be grouped together by enumeration will do the trick. Quine's polemical tactic (against which Grice and Strawson protest) of questioning the intelligibility of the distinction is indeed objectionable, but his argument can be revived once it is realized that ‘analytic’ et al. are theoretic terms, and there is no extant theory to make sense of them. Grice and Strawson's paradigm of logical impossibility is, in fact, possible. Their attempt to define synonymy in Quinean terms is a failure, nor can they retain analyticity along with the Quinean thesis of universal revisability. The dogma, in short, is indefensible.