The Univalent Foundations of Mathematics provide not only an entirely non-Cantorian conception of the basic objects of mathematics but also a novel account of how foundations ought to relate to mathematical practice. In this paper, I intend to answer the question: In what way is UF a new foundation of mathematics? I will begin by connecting UF to a pragmatist reading of the structuralist thesis in the philosophy of mathematics, which I will use to define a criterion that a formal (...) system must satisfy if it is to be regarded as a “structuralist foundation.” I will then explain why both set-theoretic foundations like ZFC and category-theoretic foundations like ETCS satisfy this criterion only to a very limited extent. Then I will argue that UF is better-able to live up to the proposed criterion for a structuralist foundation than any currently available foundational proposal. First, by showing that most criteria of identity in the practice of mathematics can be formalized in terms of the preferred criterion of identity between the basic objects of UF. Second, by countering several objections that have been raised against UF’s capacity to serve as a foundation for the whole of mathematics. (shrink)

In set theory, a maximality principle is a principle that asserts some maximality property of the universe of sets or some part thereof. Set theorists have formulated a variety of maximality principles in order to settle statements left undecided by current standard set theory. In addition, philosophers of mathematics have explored maximality principles whilst attempting to prove categoricity theorems for set theory or providing criteria for selecting foundational theories. This article reviews recent work concerned with the formulation, investigation and justification (...) of maximality principles. (shrink)

First published in 1903, _Principles of Mathematics_ was Bertrand Russell’s first major work in print. It was this title which saw him begin his ascent towards eminence. In this groundbreaking and important work, Bertrand Russell argues that mathematics and logic are, in fact, identical and what is commonly called mathematics is simply later deductions from logical premises. Highly influential and engaging, this important work led to Russell’s dominance of analytical logic on western philosophy in the twentieth century.

Lean is a new open source theorem prover being developed at Microsoft Research and Carnegie Mellon University, with a small trusted kernel based on dependent type theory. It aims to bridge the gap between interactive and automated theorem proving, by situating automated tools and methods in a framework that supports user interaction and the construction of fully specified axiomatic proofs. Lean is an ongoing and long-term effort, but it already provides many useful components, integrated development environments, and a rich API (...) which can be used to embed it into other systems. It is currently being used to formalize category theory, homotopy type theory, and abstract algebra. We describe the project goals, system architecture, and main features, and we discuss applications and continuing work. (shrink)

Since the beginning of the 20th century, philosophers of science have asked, "what kind of thing is a scientific theory?" The logical positivists answered: a scientific theory is a mathematical theory, plus an empirical interpretation of that theory. Moreover, they assumed that a mathematical theory is specified by a set of axioms in a formal language. Later 20th century philosophers questioned this account, arguing instead that a scientific theory need not include a mathematical component; or that the mathematical component need (...) not be specified by a set of axioms in a formal language. We survey various accounts of scientific theories entertained in the 20th century -- removing some misconceptions, and clearing a path for future research. (shrink)

I consider two usages of the expression "gauge theory". On one, a gauge theory is a theory with excess structure; on the other, a gauge theory is any theory appropriately related to classical electromagnetism. I make precise one sense in which one formulation of electromagnetism, the paradigmatic gauge theory on both usages, may be understood to have excess structure, and then argue that gauge theories on the second usage, including Yang-Mills theory and general relativity, do not generally have excess structure (...) in this sense. (shrink)

We inquire into the question whether the Aristotelean or classical \emph{ideal} of science has been realised by the Model Revolution, initiated at Stanford University during the 1950ies and spread all around the world of philosophy of science --- \emph{salute} P.\ Suppes. The guiding principle of the Model Revolution is: \emph{a scientific theory is a set of structures in the domain of discourse of axiomatic set-theory}, characterised by a set-theoretical predicate. We expound some critical reflections on the Model Revolution; the conclusions (...) will be that the philosophical problem of what a \emph{scientific theory} is has \emph{not} been solved yet --- \emph{pace} P.\ Suppes. While reflecting critically on the Model Revolution, we also explore a proposal of how to complete the Revolution and briefly address the intertwined subject of \emph{scientific representation}, which has come to occupy center stage in philosophy of science over the past decade. (shrink)

Classical logic has been attacked by adherents of rival, anti-realist logical systems: Ian Rumfitt comes to its defence. He considers the nature of logic, and how to arbitrate between different logics. He argues that classical logic may dispense with the principle of bivalence, and may thus be liberated from the dead hand of classical semantics.

Derivationists, those wishing to explain the correctness and rigour of informal proofs in terms of associated formal proofs, are generally held to be supported by the success of the project of translating informal proofs into computer-checkable formal counterparts. I argue, however, that this project is a false friend for the derivationists because there are too many different associated formal proofs for each informal proof, leading to a serious worry of overgeneration. I press this worry primarily against Azzouni's derivation-indicator account, but (...) conclude that overgeneration is a major obstacle to a successful account of informal proofs in this direction. (shrink)

Category Theory has developed rapidly. This book aims to present those ideas and methods which can now be effectively used by Mathe­ maticians working in a variety of other fields of Mathematical research. This occurs at several levels. On the first level, categories provide a convenient conceptual language, based on the notions of category, functor, natural transformation, contravariance, and functor category. These notions are presented, with appropriate examples, in Chapters I and II. Next comes the fundamental idea of an adjoint (...) pair of functors. This appears in many substantially equivalent forms: That of universal construction, that of direct and inverse limit, and that of pairs offunctors with a natural isomorphism between corresponding sets of arrows. All these forms, with their interrelations, are examined in Chapters III to V. The slogan is "Adjoint functors arise everywhere". Alternatively, the fundamental notion of category theory is that of a monoid -a set with a binary operation of multiplication which is associative and which has a unit; a category itself can be regarded as a sort of general­ ized monoid. Chapters VI and VII explore this notion and its generaliza­ tions. Its close connection to pairs of adjoint functors illuminates the ideas of universal algebra and culminates in Beck's theorem characterizing categories of algebras; on the other hand, categories with a monoidal structure lead inter alia to the study of more convenient categories of topological spaces. (shrink)

Some reasons to regard the cumulative hierarchy of sets as potential rather than actual are discussed. Motivated by this, a modal set theory is developed which encapsulates this potentialist conception. The resulting theory is equi-interpretable with Zermelo Fraenkel set theory but sheds new light on the set-theoretic paradoxes and the foundations of set theory.

Frederick Suppe has come to enjoy a position of undisputed leadership in the post-positivistic philosophy of science.

Logical pluralism is the view that different logics are equally appropriate, or equally correct. Logical relativism is a pluralism according to which validity and logical consequence are relative to something. Stewart Shapiro explores various such views. He argues that the question of meaning shift is itself context-sensitive and interest-relative.

Mereotopology is a theory of connected parts. The existence of boundaries, as parts of everyday objects, is basic to any such theory; but in classical mereotopology, there is a problem: if boundaries exist, then either distinct entities cannot be in contact, or else space is not topologically connected . In this paper we urge that this problem can be met with a paraconsistent mereotopology, and sketch the details of one such approach. The resulting theory focuses attention on the role of (...) empty parts, in delivering a balanced and bounded metaphysics of naive space. (shrink)

Hao Wang was one of the few confidants of the great mathematician and logician Kurt Gödel. _A Logical Journey_ is a continuation of Wang's _Reflections on Gödel_ and also elaborates on discussions contained in _From Mathematics to Philosophy_. A decade in preparation, it contains important and unfamiliar insights into Gödel's views on a wide range of issues, from Platonism and the nature of logic, to minds and machines, the existence of God, and positivism and phenomenology. The impact of Gödel's theorem (...) on twentieth-century thought is on par with that of Einstein's theory of relativity, Heisenberg's uncertainty principle, or Keynesian economics. These previously unpublished intimate and informal conversations, however, bring to light and amplify Gödel's other major contributions to logic and philosophy. They reveal that there is much more in Gödel's philosophy of mathematics than is commonly believed, and more in his philosophy than his philosophy of mathematics. Wang writes that "it is even possible that his quite informal and loosely structured conversations with me, which I am freely using in this book, will turn out to be the fullest existing expression of the diverse components of his inadequately articulated general philosophy." The first two chapters are devoted to Gödel's life and mental development. In the chapters that follow, Wang illustrates the quest for overarching solutions and grand unifications of knowledge and action in Gödel's written speculations on God and an afterlife. He gives the background and a chronological summary of the conversations, considers Gödel's comments on philosophies and philosophers, and his attempt to demonstrate the superiority of the mind's power over brains and machines. Three chapters are tied together by what Wang perceives to be Gödel's governing ideal of philosophy: an exact theory in which mathematics and Newtonian physics serve as a model for philosophy or metaphysics. Finally, in an epilog Wang sketches his own approach to philosophy in contrast to his interpretation of Gödel's outlook. (shrink)

Saunders Mac Lane has drawn attention many times, particularly in his book Mathematics: Form and Function, to the system of set theory of which the axioms are Extensionality, Null Set, Pairing, Union, Infinity, Power Set, Restricted Separation, Foundation, and Choice, to which system, afforced by the principle, , of Transitive Containment, we shall refer as . His system is naturally related to systems derived from topos-theoretic notions concerning the category of sets, and is, as Mac Lane emphasises, one that is (...) adequate for much of mathematics. In this paper we show that the consistency strength of Mac Lane's system is not increased by adding the axioms of Kripke–Platek set theory and even the Axiom of Constructibility to Mac Lane's axioms; our method requires a close study of Axiom H, which was proposed by Mitchell; we digress to apply these methods to subsystems of Zermelo set theory , and obtain an apparently new proof that is not finitely axiomatisable; we study Friedman's strengthening of , and the Forster–Kaye subsystem of , and use forcing over ill-founded models and forcing to establish independence results concerning and ; we show, again using ill-founded models, that proves the consistency of ; turning to systems that are type-theoretic in spirit or in fact, we show by arguments of Coret and Boffa that proves a weak form of Stratified Collection, and that is a conservative extension of for stratified sentences, from which we deduce that proves a strong stratified version of ; we analyse the known equiconsistency of with the simple theory of types and give Lake's proof that an instance of Mathematical Induction is unprovable in Mac Lane's system; we study a simple set theoretic assertion—namely that there exists an infinite set of infinite sets, no two of which have the same cardinal—and use it to establish the failure of the full schema of Stratified Collection in ; and we determine the point of failure of various other schemata in . The paper closes with some philosophical remarks. (shrink)

The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous range of set-theoretic possibilities, a phenomenon that challenges the universe (...) view. In particular, I argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for. (shrink)

In this paper, I introduce the idea that some important parts of contemporary pure mathematics are moving away from what I call the extensional point of view. More specifically, these fields are based on criteria of identity that are not extensional. After presenting a few cases, I concentrate on homotopy theory where the situation is particularly clear. Moreover, homotopy types are arguably fundamental entities of geometry, thus of a large portion of mathematics, and potentially to all mathematics, at least according (...) to some speculative research programs. (shrink)

When ordinary people - mathematicians among them - take something to follow from something else, they are exposing the backbone of our self-ascribed ability to reason. Jody Azzouni investigates the connection between that ordinary notion of consequence and the formal analogues invented by logicians. One claim of the book is that, despite our apparent intuitive grasp of consequence, we do not introspect rules by which we reason, nor do we grasp the scope and range of the domain, as it were, (...) of our reasoning. This point is illustrated with a close analysis of a paradigmatic case of ordinary reasoning: mathematical proof. (shrink)

In the following, I argue that L. E. J. Brouwer's notion of the construction of purely mathematical objects and Edmund Husserl's notion of their constitution coincide.

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)

The purpose of this article is to explain why I believe that the Continuum Hypothesis (CH) is not a definite mathematical problem. My reason for that is that the concept of arbitrary set essential to its formulation is vague or underdetermined and there is no way to sharpen it without violating what it is supposed to be about. In addition, there is considerable circumstantial evidence to support the view that CH is not definite.

According to the semantic view of scientific theories, theories are classes of models. I show that this view -- if taken seriously as a formal explication -- leads to absurdities. In particular, this view equates theories that are truly distinct, and it distinguishes theories that are truly equivalent. Furthermore, the semantic view lacks the resources to explicate interesting theoretical relations, such as embeddability of one theory into another. The untenability of the semantic view -- as currently formulated -- threatens to (...) undermine scientific structuralism. (shrink)

This paper explores the set theoretic assumptions used in the current published proof of Fermat's Last Theorem, how these assumptions figure in the methods Wiles uses, and the currently known prospects for a proof using weaker assumptions.

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.

It seems to me that the ingredients of most theories both in Artificial Intelligence and in Psychology have been on the whole too minute, local, and unstructured to account–either practically or phenomenologically–for the effectiveness of common-sense thought. The "chunks" of reasoning, language, memory, and "perception" ought to be larger and more structured; their factual and procedural contents must be more intimately connected in order to explain the apparent power and speed of mental activities.

Preface This book is about semantics and logic. More specifically, it is about the semantics and logic of natural language; and, even more specifically than ...

The prime number theorem, established by Hadamard and de la Vallée Poussin independently in 1896, asserts that the density of primes in the positive integers is asymptotic to 1/ln x. Whereas their proofs made serious use of the methods of complex analysis, elementary proofs were provided by Selberg and Erdos in 1948. We describe a formally verified version of Selberg's proof, obtained using the Isabelle proof assistant.

This book is intended to be used as a textbook by students of mathematics, and also within limitations as a reference work.

This paper begins an axiomatic development of naive set theoryin a paraconsistent logic. Results divide into two sorts. There is classical recapture, where the main theorems of ordinal and Peano arithmetic are proved, showing that naive set theory can provide a foundation for standard mathematics. Then there are major extensions, including proofs of the famous paradoxes and the axiom of choice (in the form of the well-ordering principle). At the end I indicate how later developments of cardinal numbers will lead (...) to Cantor’s theorem, the existence of large cardinals, and a counterexample to the continuum hypothesis. (shrink)

This exploration of a notorious mathematical problem is the work of the man who discovered the solution. Written by an award-winning professor at Stanford University, it employs intuitive explanations as well as detailed mathematical proofs in a self-contained treatment. This unique text and reference is suitable for students and professionals. 1966 edition. Copyright renewed 1994.

A scientific community cannot practice its trade without some set of received beliefs. These beliefs form the foundation of the "educational initiation that prepares and licenses the student for professional practice". The nature of the "rigorous and rigid" preparation helps ensure that the received beliefs are firmly fixed in the student's mind. Scientists take great pains to defend the assumption that scientists know what the world is like...To this end, "normal science" will often suppress novelties which undermine its foundations. Research (...) is therefore not about discovering the unknown, but rather "a strenuous and devoted attempt to force nature into the conceptual boxes supplied by professional education". (shrink)

In this manuscript, published here for the first time, Tarski explores the concept of logical notion. He draws on Klein's Erlanger Programm to locate the logical notions of ordinary geometry as those invariant under all transformations of space. Generalizing, he explicates the concept of logical notion of an arbitrary discipline.