This paper aims to provide an updated synthesis on the works of Archimedes and the fundamental impact these have had on subsequent mathematical practice. The influence his mathematical processes have had on modern mathematics and how these have helped develop the field is discussed in historical perspective. Some of the recent investigations into the Archimedes Palimpsest are discussed and synthesized, namely, how they alter our understanding of some of his earlier works, and how Archimedean principles are seen to (...) have laid the foundations of possible new branches of mathematics. (shrink)

This paper continues investigations of the monotone fixed point principle in the context of Feferman's explicit mathematics begun in [14]. Explicit mathematics is a versatile formal framework for representing Bishop-style constructive mathematics and generalized recursion theory. The object of investigation here is the theory of explicit mathematics augmented by the monotone fixed point principle, which asserts that any monotone operation on classifications (Feferman's notion of set) possesses a least fixed point. To be more precise, the new (...) axiom not merely postulates the existence of a least solution, but, by adjoining a new constant to the language, it is ensured that a fixed point is uniformly presentable as a function of the monotone operation. Let T 0 + UMID denote this extension of explicit mathematics. [14] gave lower bounds for the strength of two subtheories of T 0 + UMID in relating them to fragments of second order arithmetic based on Π 1 2 comprehension. [14] showed that T $_0 \upharpoonright$ + UMID and T $_0 \upharpoonright$ + IND N + UMID have at least the strength of (Π 1 2 - CA) $\upharpoonright$ and (Π 1 2 - CA), respectively. Here we are concerned with the exact reversals. Let UMID N be the monotone fixed-point principle for subclassifications of the natural numbers. Among other results, it is shown that T $_0 \upharpoonright$ + UMID N and T $_0 \upharpoonright$ + IND N + UMID N have the same strength as (Π 1 2 - CA) $\upharpoonright$ and (Π 1 2 - CA), respectively. The results are achieved by constructing set-theoretic models for the aforementioned systems of explicit mathematics in certain extensions of Kripke-Platek set theory and subsequently relating these set theories to subsystems of second arithmetic. (shrink)

One of the criticisms of standard teaching practices is that they support merely ‘ritual’ as opposed to ‘principled’ knowledge, that is, knowledge which is procedural rather than being founded on principled explanation. This paper addresses issues and assumptions in current debate concerning the nature of mathematical knowledge, focusing on the ritual/principle distinction. Taking a discussion of centralism in logic and mathematics as its start-point, it seeks to resolve these issues through an examination of mathematics as a community of (...) practice and the teacher's role as epistemological authority in inducting pupils into such practices. (shrink)

The context for this paper is Feferman's theory of explicit mathematics, a formal framework serving many purposes. It is suitable for representing Bishop-style constructive mathematics as well as generalized recursion, including direct expression of structural concepts which admit self-application. The object of investigation here is the theory of explicit mathematics augmented by the monotone fixed point principle, which asserts that any monotone operation on classifications (Feferman's notion of set) possesses a least fixed point. To be more precise, (...) the new axiom not merely postulates the existence of a least solution, but, by adjoining a new functional constant to the language, it is ensured that a fixed point is uniformly presentable as a function of the monotone operation. The upshot of the paper is that the latter extension of explicit mathematics (when based on classical logic) embodies considerable proof-theoretic strength. It is shown that it has at least the strength of the subsystem of second order arithmetic based on Π 1 2 comprehension. (shrink)

This books states, as clearly and concisely as possible, the most fundamental principles of set-theory and mathematical logic. Included is an original proof of the incompleteness of formal logic. Also included are clear and rigorous definitions of the primary arithmetical operations, as well as clear expositions of the arithmetic of transfinite cardinals.

According to Stewart Shapiro's coherence principle, structures exist whenever they can be coherently described. I argue that Shapiro's attempts to justify this principle are circular, as he relies on criticisms of modal nominalism which presuppose the coherence principle. I argue further that when the coherence principle is not presupposed, his reasoning more strongly supports modal nominalism than ante rem structuralism.

In the thesis some combinatorial statements are analysed from the reverse mathematics point of view. Reverse mathematics is a research program, which dates back to the Seventies, interested in finding the exact strength, measured in terms of set-existence axioms, of theorems from ordinary non set-theoretic mathematics. After a brief introduction to the subject, an on-line (incremental) algorithm to transitively reorient infinite pseudo-transitive oriented graphs is defined. This implies that a theorem of Ghouila-Houri is provable in RCA_0 and (...) hence is computably true. Interval graphs and interval orders are the common theme of the second part of the thesis. A chapter is devoted to analyse the relative strength of different characterisations of countable interval graphs and to study the interplay between countable interval graphs and countable interval orders. In this context the theme of unique orderability of interval graphs arises, which is studied in the following chapter. The last chapter about interval orders inspects the strength of some statements involving the dimension of countable interval orders. The third part is devoted to the analysis of two theorems proved by Rival and Sands in 1980. The first principle states that each infinite graph contains an infinite subgraph such that each vertex of the graph is adjacent either to none, or to one or to infinitely many vertices of the subgraph. This statement, restricted to countable graphs, is proved to be equivalent to ACA_0 and hence to be stronger than Ramsey's theorem for pairs, despite the similarity of partial order with finite width contains an infinite chain such that each point of the poset is comparable either to none or to infinitely many points of the chain. For each k less than or equal to 3, the latter principle restricted to countable poset of width k is proved to be equivalent to ADS. Some complementary results are presented in the thesis. (shrink)

The larger project broached here is to look at the generally sentence “if X is well-ordered then f is well-ordered”, where f is a standard proof-theoretic function from ordinals to ordinals. It has turned out that a statement of this form is often equivalent to the existence of countable coded ω-models for a particular theory Tf whose consistency can be proved by means of a cut elimination theorem in infinitary logic which crucially involves the function f. To illustrate this theme, (...) we prove in this paper that the statement “if X is well-ordered then εX is well-ordered” is equivalent to . This was first proved by Marcone and Montalban [Alberto Marcone, Antonio Montalbán, The epsilon function for computability theorists, draft, 2007] using recursion-theoretic and combinatorial methods. The proof given here is principally proof-theoretic, the main techniques being Schütte’s method of proof search [Kurt Schütte, Proof Theory, Springer-Verlag, Berlin, Heidelberg, 1977] and cut elimination for a fragment of. (shrink)

The paper portrays the influence of major philosophical ideas on the 1935 debates on quantum theory that reached their climax in the paper by Einstein, Podosky and Rosen, and describes the relevance of these ideas to the vast impact of the paper. I claim that the focus on realism in many common descriptions of the debate misses important aspects both of Einstein's and Bohr's thinking. I suggest an alternative understanding of Einstein's criticism of quantum mechanics as a manifestation of the (...) same methodological principles that served him in the construction of the special and the general theories of relativity. These principles address, in a very specific way, the relation of the theoretical mathematical representations to the represented physical systems. These ideas, I claim, played a key role in the influence of the paper on later works that changed our understanding of quantum theory despite the rejection of EPR's central conclusion. (shrink)

The leaders of the Scientific Revolution were not Baconian in temperament, in trying to build up theories from data. Their project was that same as in Aristotle's Posterior Analytics: they hoped to find necessary principles that would show why the observations must be as they are. Their use of mathematics to do so expanded the Aristotelian project beyond the qualitative methods used by Aristotle and the scholastics. In many cases they succeeded.

Since Descartes, mathematics has been dominated by a reductionist tendency, whose success would seem to promise greater certainty: the fewer basic objects mathematics can be understood as dealing with, and the fewer principles one is forced to assume about these objects, the easier it will be to establish a secure foundation for it. But this tendency has had the effect of sharply limiting the expressive power of mathematics, in a way that is made especially apparent by (...) its disappointing applications to the social sciences. We should move in the opposite direction: toward a mathematics that deals in general with constructed objects, and whose scope includes fictional, poetic characters as much as numbers and sets. (shrink)

We show that when certain statements are provable in subsystems of constructive analysis using intuitionistic predicate calculus, related sequential statements are provable in weak classical subsystems. In particular, if a $\Pi^1_2$ sentence of a certain form is provable using E-HA ${}^\omega$ along with the axiom of choice and an independence of premise principle, the sequential form of the statement is provable in the classical system RCA. We obtain this and similar results using applications of modified realizability and the Dialectica interpretation. (...) These results allow us to use techniques of classical reverse mathematics to demonstrate the unprovability of several mathematical principles in subsystems of constructive analysis. (shrink)

View more Abstract If logical truth is necessitated by sheer syntax, mathematics is categorially unlike logic even if all mathematics derives from definitions and logical principles. This contrast gets obscured by the plausibility of the Synonym Substitution Principle implicit in conceptions of analyticity: synonym substitution cannot alter sentence sense. The Principle obviously fails with intercepting: nonuniform term substitution in logical sentences. ‘Televisions are televisions’ and ‘TVs are televisions’ neither sound alike nor are used interchangeably. Interception synonymy gets (...) assumed because logical sentences and their synomic interceptions have identical factual content, which seems to exhaust semantic content. However, intercepting alters syntax by eliminating term recurrence, the sole strictly syntactic means of ensuring necessary term coextension, and thereby syntactically securing necessary truth. Interceptional necessity is lexical, a notational artifact. The denial of interception nonsynonymy and the disregard of term recurrence in logic link with many misconceptions about propositions, logical form, conventions, and metalanguages. Mathematics is distinct from logic: its truth is not syntactic; it is transmitted by synonym substitution; term recurrence has no essential role. The ‘=’ of mathematics is an objectual relation between numbers; the ‘=’ of logic marks a syntactic relation of coreferring terms. -/- . (shrink)

Plausibly, mathematical claims are true, but the fundamental furniture of the world does not include mathematical objects. This can be made sense of by providing mathematical claims with paraphrases, which make clear how the truth of such claims does not require the fundamental existence of mathematical objects. This paper explores the consequences of this type of position for explanatory structure. There is an apparently straightforward relationship between this sort of structure, and the logical sort: i.e. logically complex claims are explained (...) by logically simpler ones. For example, disjunctions are explained by their (true) disjuncts, while generalizations are explained by their (true) instances. This would seem as plausible in the case of mathematics as elsewhere. Also, it would seem to be something that the anti-realist approaches at issue would want to preserve. It will be argued, however, that these approaches cannot do this: they lead not merely to violations of the familiar principles relating logical and explanatory structure, but even to reversals of these. That is, there are cases where generalizations explain their instances, or disjunctions their disjuncts. (shrink)

This volume shows Russell in transition from a neo-Kantian and neo-Hegelian philosopher to an analytic philosopher of the first rank. During this period his research centred on writing The Principles of Mathematics where he drew together previously unpublished drafts. These shed light on Russell's paradox. This material will alter previous accounts of how he discovered his paradox and the related paradox of the largest cardinal. The volume also includes a previously unpublished draft of an early attempt to solve (...) his paradox, as well as the earliest known version of his generalised relation arithmetic. It contains three articles which have never previously been published in English. (shrink)

In this paper, we propose a weak regularity principle which is similar to both weak König's lemma and Ramsey's theorem. We begin by studying the computational strength of this principle in the context of reverse mathematics. We then analyze different ways of generalizing this principle.

Contemporary analytic philosophy recognizes few principled constraints on its subject matter. When other disciplines also lay claim to a particular topic, however, important questions arise concerning the relation between these other disciplines and philosophy. A case in point is mathematics: traditional philosophy of mathematics defines a set of problems and certain general answers to those problems. However, mathematics is a subject matter that can be studied in many other ways: historically, sociologically, or even aesthetically, for example. Given (...) this, we may ask what implications discoveries in these disciplines have for the philosophy of mathematics. (shrink)

This paper deals with universes in explicit mathematics. After introducing some basic definitions, the limit axiom and possible ordering principles for universes are discussed. Later, we turn to least universes, strictness and name induction. Special emphasis is put on theories for explicit mathematics with universes which are proof-theoretically equivalent to Feferman's.

Plato is the first scientist whose work we still possess. He is our first writer to interpret the natural world mathematically, and also the first theorist of mathematics in the natural sciences. As no one else before or after, he set out why we should suppose a link between nature and mathematics, a link that has never been stronger than it is today. Mathematical Plato examines how Plato organized and justified the principles, terms, and methods of our (...) mathematical, natural science. "Roger Sworder deserves our gratitude for drawing attention to the significance of mathematics in Plato's thought and writings. He lays the principal discussions out before us with clarity. He also presents Plato as a theorist of nature: of physics and not just metaphysics, to use Aristotle's distinction. Not all readers, we should admit, will be equally convinced of the usefulness of Plato's science for today, but they will all be led more deeply into Plato's vision of reality."--ANDREW DAVISON, Westcott House, Cambridge "Here is Plato for an anti-Platonic age. The author gives careful attention to some of the most important passages in the Platonic dialogues and offers new solutions to some of Plato's most famous mathematical puzzles. He then considers the implications of these penetrating studies for the philosophy of science, and the natural sciences especially. This is a book that revivifies the core themes of Platonism and restores science to worship. It shows Roger Sworder to be one of the foremost students of Plato writing today, and places him in the noble tradition of Thomas Taylor."--RODNEY BLACKHIRST, author of Primordial Alchemy and Modern Religion: Essays on Traditional Cosmology. (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)

In every introductory course on logic, students learn that expressions like ‘somebody’, ‘nothing’ or ‘every woman’ are not names or referring expressions, but quantifiers, and that, owing to this,...

The goal of this paper is to answer the following question: Does it make sense to speak of thought experiments not only in physics, but also in mathematics, to refer to an authentic type of activity? One may hesitate because mathematics as such is the exercise of reasoning par excellence, an activity where experience does not seem to play an important role. After reviewing some results of the research on thought experiments in the natural sciences, we turn our (...) attention to experiments and thought experiments in mathematics, especially in fundamental mathematics, and investigate what thought experiments can teach us there. If we accept the principle that mathematical practice can sometimes be best described by a pragmatist approach where the concept of experience in mathematics is considered from a new perspective, thought experiments can in some cases be a useful instrument different from both ‘mathematical experiments’ and ‘formal’ proofs: they are mathematical experiments using deviant methods. (shrink)

Despite a vast philosophical literature on the epistemology of mathematics and much speculation about how, in principle, knowledge of this domain is possible, little attention has been paid to the psychological findings and theories concerning the acquisition, comprehension and use of mathematical knowledge. This contrasts sharply with recent philosophical work on language where comparable issues and problems arise. One topic that is the center of debate in the study of mathematical cognition is the question of innateness. This paper critically (...) examines the controversy. (shrink)

This chapter provides an overview of Thomas Hobbes's materialistic philosophy of mathematics. Hobbes's mathematical ontology rejects the seventeenth century's received view of the subject and his proposed first principles departed quite significantly from the tradition. Hobbes's understanding of geometry as a generalized science of material bodies puts him at odds with the traditional notion that the objects of geometrical investigation are radically distinct from the realm of material things. Hobbes's methodology holds that demonstrative knowledge must be based on (...) definitions that identify the causes of things. Hobbes was quite hostile to the algebraic methods characteristic of Descartes's analytic geometry, and he found fault with some presentations of the “method of indivisibles” that is an important precursor to the calculus. Hobbes approved of Bonaventura Cavalieri's method while dismissing John Wallis's approach as incoherent. The chapter investigates Hobbes's controversy with Wallis. Any discussion of Hobbes's mathematics takes place against the background of his controversy with Wallis. (shrink)

The scientific study of consciousness, also referred to as consciousness science, is a young scientific field devoted to understanding how conscious experiences and the brain relate. It comprises a host of theories, experiments, and analyses that aim to investigate the problem of consciousness empirically, theoretically, and conceptually. This thesis addresses some of the questions that arise in these investigations from a formal and mathematical perspective. These questions concern theories of consciousness, experimental paradigms, methodology, and artificial consciousness. -/- Regarding theories of (...) consciousness, the thesis contributes to the understanding of the mathematical structure that some of the formal theories in the field propose. The work presented here targets the theory of consciousness known as Integrated Information Theory (IIT) and the neuroscientific theory known as Predictive Processing Theory or Free Energy Principle in its Active Inference form (AI-PP). The thesis provides axiomatic definitions of the mathematical structures that constitute these theories, and uses these definitions to address some of the open questions surrounding the theories. For AI-PP, this includes a rigorous derivation of the formula for Active Inference via Free Energy minimisation and a proof of compositionality of Free Energy. For IIT, this includes resolutions of some of the criticisms of IIT's formal scope and applications, but also the identification of new issues that concern the formalism and its derivation. When possible, the definitions are provided in the mathematical framework of category theory. -/- Regarding experiments, the thesis addresses the main paradigm for testing and falsifying theories of consciousness currently applied in the field. This paradigm consists of comparing the conscious experience that a theory predicts with the conscious experience that is inferred from behavioural data or report by use of measures of consciousness. The thesis provides a formal model of this paradigm and shows that under a certain condition—if inference and prediction are independent—, any minimally informative theory of consciousness can always be falsified. This is deeply problematic since the field’s reliance on report or behaviour to infer conscious experiences, in conjunction with the general structure of most contemporary theories of consciousness, implies such independence. This observation provides the exact formal underpinning of the well-known unfolding argument. The thesis analyses the origin of the problem and identifies precisely which changes are required to avoid this problem in future research. The thesis furthermore shows that the problem of falsifying theories of consciousness, and of empirical comparisons of theories of consciousness more generally, follows from a pervasive closure paradigm in consciousness science, that consists of taking a neuroscientific account of the brain as input to a theory of consciousness, so as to explain what consciousness is, without allowing for modifications or adaptations of the neuroscientific account that would accommodate consciousness as part of the brain's functioning. As is shown in the thesis, this paradigm has implications that point to a fundamental need of revision. -/- Regarding methodological and conceptual questions, the thesis contributes to the foundations of structural research in consciousness science. Structural research aims to use mathematical structures or mathematical spaces, instead of verbal descriptions or simple categorisations, to represent conscious experiences scientifically, for example when building theories of consciousness, or when exploring new empirical avenues to measure consciousness. Despite considerable advances in this realm, there was, prior to this thesis, no explicit definition of what a mathematical structure of conscious experience should be; that is, how the attribution of mathematical structure to conscious experiences should be systematically understood. Perhaps the most important contribution of this thesis to the field is to propose such a definition. The definition, a structural concept, extends existing approaches wherever available, and provides a basis for developing a common formal language to study consciousness, bridging developments as far apart as psychophysics and phenomenology. In addition, and independently of this proposal, the thesis offers a critical analysis of which metaphysical premises need to be presumed in structural research, whether the use of particular formal tools (such as structure-preserving mappings or homomorphisms) is justified, and how structural theories of consciousness could otherwise be built in the first place. An attempt to expand the results from consciousness to more general problems in philosophy of science is made in the context of the well-known Newman problem. -/- Regarding the question of artificial consciousness—can AI feel?—, the thesis contributes two results that take the form of no-go theorems, as known from physics. The first no-go theorem shows that if consciousness is relevant for the temporal evolution of a system's states—if it is dynamically relevant—, then contemporary AI systems cannot be conscious. That is because AI systems run on CPUs, GPUs, TPUs or other processors which have been designed and verified to adhere to computational dynamics that systematically preclude or suppress deviations. The second no-go theorem is situated in the context of computational functionalism, a view which posits that consciousness is a computation. The theorem shows that if computational functionalism holds true, consciousness cannot be a Turing computation. Rather, it must be a novel type of computation that has recently been proposed by Geoffrey Hinton, called mortal computation. -/- This thesis is part of a global effort to pioneer a mathematical perspective in consciousness science, now called Mathematical Consciousness Science. The hope behind the research carried out in this PhD is to illustrate the power and usefulness of mathematical approaches in different areas of consciousness science, and in doing so, to lay the foundations for future mathematical work that complements and supports empirical and theoretical work in the further development of this exciting field. (shrink)

L. E. J. Brouwer, the founder of mathematical intuitionism, believed that mathematics and its objects must be humanly graspable. He initiated a program rebuilding modern mathematics according to that principle. This book introduces the reader to the mathematical core of intuitionism – from elementary number theory through to Brouwer's uniform continuity theorem – and to the two central topics of 'formalized intuitionism': formal intuitionistic logic, and formal systems for intuitionistic analysis. Building on that, the book proposes a systematic, (...) philosophical foundation for intuitionism that weaves together doctrines about human grasp, mathematical objects and mathematical truth. (shrink)

We examine the sense in which logic is a priori, and explain how mathematical theories can be dichotomized non-trivially into analytic and synthetic portions. We argue that Core Logic contains exactly the a-priori-because-analytically-valid deductive principles. We introduce the reader to Core Logic by explaining its relationship to other logical systems, and stating its rules of inference. Important metatheorems about Core Logic are reported, and its important features noted. Core Logic can serve as the basis for a foundational program that (...) could be called Natural Logicism, an exposition of which will build on the (meta)logical ideas explained here. (shrink)

The imperviousness of mathematical truth to anti-objectivist attacks has always heartened those who defend objectivism in other areas, such as ethics. It is argued that the parallel between mathematics and ethics is close and does support objectivist theories of ethics. The parallel depends on the foundational role of equality in both disciplines. Despite obvious differences in their subject matter, mathematics and ethics share a status as pure forms of knowledge, distinct from empirical sciences. A pure understanding of (...) class='Hi'>principles is possible because of the simplicity of the notion of equality, despite the different origins of our understanding of equality of objects in general and of the equality of the ethical worth of persons. (shrink)

"Proof" has been and remains one of the concepts which characterises mathematics. Covering basic propositional and predicate logic as well as discussing axiom systems and formal proofs, the book seeks to explain what mathematicians understand by proofs and how they are communicated. The authors explore the principle techniques of direct and indirect proof including induction, existence and uniqueness proofs, proof by contradiction, constructive and non-constructive proofs, etc. Many examples from analysis and modern algebra are included. The exceptionally clear style (...) and presentation ensures that the book will be useful and enjoyable to those studying and interested in the notion of mathematical "proof.". (shrink)

We give a derivation of exclusion principles for the elementary particles of the standard model, using simple mathematical principles arising from a set theory of identical particles. We apply the theory of permutation group actions, stating some theorems which are proven elsewhere, and interpreting the results as a heuristic derivation of Pauli's Exclusion Principle (PEP) which dictates the formation of elements in the periodic table and the stability of matter, and also a derivation of quark confinement. We arrive (...) at these properties by using a symmetry property of collections of the particles themselves as compared for example to the symmetry property of their wave function under interchange of two particles. (shrink)

This book discusses in dialogue form the basic principles of mathematics and its applications including the question: What is mathematics? What does its specific method consist of? What is its relation to the sciences and humanities? What can it offer to specialists in different fields? How can it be applied in practice and in discovering the laws of nature? Dramatized by the dialogue form and shown in the historical movements in which they originated, these questions are discussed (...) in their full complexity, yet are easily comprehended. The first dialogue, whose chief actor is Socrates, leads the reader to the source of modern mathematics in Athens in the 5th Century BC. The second dialogue, featuring Archimedes, takes place during the siege of Syracuse in 212 BC and shows the birth of applied mathematics. The third dialogue occurs in the year 1633 in Rome, its chief character being Galileo Galilei who fully realized the central importance of the mathematical method in discovering the laws of nature. Intended as supplemental reading for philosophy of mathematics courses at the high school or college level it will be of interest to both specialists and non-specialists in mathematics. Alfréd Rényi was born in Budapest Hungary in 1921. He studied mathematics and physics at the University of Budapest and received his Ph. D. from the University of Szaged in 1945. Since 1950 he has been Director of the Mathematical Research Institute of the Hungarian Academy of Sciences and since 1952 a professor at the University of Budapest. Dr. Renyi was a visiting professor at Michigan State University in 1961, at the University of Michigan in 1964 and at Stanford University in 1966. His main fields of research are probability theory, mathematical statistics and information theory, and he has also worked in analytic number theory as well as in various branches of analysis, combinatorial analysis and geometry. (shrink)

I argue for the claim that there are instances of a priori justified belief – in particular, justified belief in moral principles – that are not analytic, i.e., that cannot be explained solely by the understanding we have of their propositions. §1–2 provides the background necessary for understanding this claim: in particular, it distinguishes between two ways a proposition can be analytic, Basis and Constitutive, and provides the general form of a moral principle. §§3–5 consider whether Hume's Law, properly (...) interpreted, can be established by Moore's Open Question Argument, and concludes that it cannot: while Moore's argument – appropriately modified – is effective against the idea that moral judgments are either reductively analyzable or Constitutive-analytic, a different argument is needed to show that they are not Basis-analytic. Such an argument is supplied in §6. §§7–8 conclude by considering how these considerations bear on recent discussions of “alternative normative concepts”, on the epistemology of intuitions, and on the differences between disagreement in moral domains and in other a priori domains such as logic and mathematics. (shrink)

La pregunta por la naturaleza de las relaciones es de gran importancia en los escritos tempranos de Bentrand Russell, ya que sus desacuerdos con el idealismo británico se centraban en las relaciones, y su filosofía de las matemáticas depende crucialmente de las relaciones. A pesar de esto, no hay una discusión sistemática y extendida sobre las relaciones en el Russell temprano. Después de examinar la definición de relación de Russell, el autor examina crítica y sistemáticamente los puntos de vista de (...) Russell en los Principia Mathematica sobre los siguientes problemas: si una relación existe aparte de sus términos; el carácter intensional de las relaciones; las dificultades en las relaciones reflexivas; y si las relaciones simples pueden relacionar más de dos relatos. (shrink)

In this article I examine the dispute between F. H. Bradley and Bertrand Russell concerning the reality of relations. I show that Bradley’s objections to Russell’s view, that there are such things as relations which serve to effect the unity of complex items, were rooted in a methodological approach which Russell did not share. On Bradley’s view, one must be able to offer reductive analyses of the items one postulates in order that commitment to those items be justified. I argue (...) that Russell expressly rejected this methodological principle of Bradley’s, and instead adopted the view that one may justifiably postulate entities if doing so aids in the illumination of mathematical truths. I show that the postulation of relations does, on Russell’s view, serve to provide that illumination. Russell held that the truths of mathematics constitute fixed data, and that philosophical positions may be judged as successful according to the extent that they possess explanatory power with respect to this data. I argue that Bradley’s and Russell’s exchanges in print, as well as in private correspondence, reflect Russell’s awareness of a fundamental difference in methodological approach. I conclude that Russell elected not to answer Bradley’s objections on their own terms, but rather rejected the methodological assumptions from which those objections emerged. (shrink)

In this paper we devise some technical tools for dealing with problems connected with the philosophical view usually called mathematical instrumentalism. These tools are interesting in their own right, independently of their philosophical consequences. For example, we show that even though the fragment of Peano's Arithmetic known as IΣ₁ is a conservative extension of the equational theory of Primitive Recursive Arithmetic (PRA). IΣ₁ has a super-exponential speed-up over PRA. On the other hand, theories studied in the Program of Reverse (...) class='Hi'>Mathematics that formalize powerful mathematical principles have only polynomial speed-up over IΣ₁. (shrink)

In recent years, there has been a substantial amount of work in reverse mathematics concerning natural mathematical principles that are provable from RT, Ramsey's Theorem for Pairs. These principles tend to fall outside of the "big five" systems of reverse mathematics and a complicated picture of subsystems below RT has emerged. In this paper, we answer two open questions concerning these subsystems, specifically that ADS is not equivalent to CAC and that EM is not equivalent to (...) RT. (shrink)

It is common in the literature on classical electrodynamics (ED) and relativity theory that the transformation rules for the basic electrodynamical quantities are derived from the hypothesis that the relativity principle (RP) applies to Maxwell’s electrodynamics. As it will turn out from our analysis, these derivations raise several problems, and certain steps are logically questionable. This is, however, not our main concern in this paper. Even if these derivations were completely correct, they leave open the following questions: (1) Is the (...) RP a true law of nature for electrodynamical phenomena? (2) Are, at least, the transformation rules of the fundamental electrodynamical quantities, derived from the RP, true? (3) Is the RP consistent with the laws of ED in a single inertial frame of reference? (4) Are, at least, the derived transformation rules consistent with the laws of ED in a single frame of reference? Obviously, (1) and (2) are empirical questions. In this paper, we will investigate problems (3) and (4). First we will give a general mathematical formulation of the RP. In the second part, we will deal with the operational definitions of the fundamental electrodynamical quantities. As we will see, these semantic issues are not as trivial as one might think. In the third part of the paper, applying what J. S. Bell calls “Lorentzian pedagogy”—according to which the laws of physics in any one reference frame account for all physical phenomena— we will show that the transformation rules of the electrodynamical quantities are identical with the ones obtained by presuming the covariance of the equations of ED, and that the covariance is indeed satisfied. As to problem (3), the situation is much more complex. As we will see, the RP is actually not a matter of the covariance of the physical equations, but it is a matter of the details of the solutions of the equations, which describe the behavior of moving objects. This raises conceptual problems concerning the meaning of the notion “the same system in a collective motion”.. (shrink)

SummaryThe platonist, in affirming the principle of bivalence for sentences for which there is no decision procedure, disconnects their truth‐conditions from conditions that would enable us to prove them ‐ as if Goldbach's conjecture, say, might just happen to be true. According to Dummett, what has gone wrong here is that the meaning of the relevant sentences has been conceived so as to go beyond what could be learned in learning to use them, or displayed in using them competently. Dummett (...) draws the general conclusion that accounts of meaning must traffic only in decidable circumstances. I suggest that Dummett can be right about platonism but wrong in this general conclusion: the centrality of decidable circumstances in competent use of language is a special feature of mathematical language. This means that someone who recoils from the anti‐realism constituted by Dummett's generalized anti‐platonism, in the case of, say, statements about other minds, need not be recoiling into a close analogue of platonism, as Dummett suggests. We can reinstate the intuitive idea that platonism goes wrong by inappropriately modelling the epistemology and metaphysics of mathematics on the epistemology and metaphysics of the natural world. And we make room for the suggestion that anti‐realism makes a converse mistake; in this vein, I propose a picture of Dummettian anti‐realism as a novel expression of familiar and suspect epistemological and metaphysical thoughts. (shrink)

We investigate the complexity of various combinatorial theorems about linear and partial orders, from the points of view of computability theory and reverse mathematics. We focus in particular on the principles ADS (Ascending or Descending Sequence), which states that every infinite linear order has either an infinite descending sequence or an infinite ascending sequence, and CAC (Chain-AntiChain), which states that every infinite partial order has either an infinite chain or an infinite antichain. It is well-known that Ramsey's Theorem (...) for pairs (RT₂²) splits into a stable version (SRT₂²) and a cohesive principle (COH). We show that the same is true of ADS and CAC, and that in their cases the stable versions are strictly weaker than the full ones (which is not known to be the case for RT₂² and SRT₂²). We also analyze the relationships between these principles and other systems and principles previously studied by reverse mathematics, such as WKL₀, DNR, and BΣ₂. We show, for instance, that WKL₀ is incomparable with all of the systems we study. We also prove computability-theoretic and conservation results for them. Among these results are a strengthening of the fact, proved by Cholak, Jockusch, and Slaman, that COH is Π₁¹-conservative over the base system RCA₀. We also prove that CAC does not imply DNR which, combined with a recent result of Hirschfeldt, Jockusch, Kjos-Hanssen, Lempp, and Slaman, shows that CAC does not imply SRT₂² (and so does not imply RT₂²). This answers a question of Cholak, Jockusch, and Slaman. Our proofs suggest that the essential distinction between ADS and CAC on the one hand and RT₂² on the other is that the colorings needed for our analysis are in some way transitive. We formalize this intuition as the notions of transitive and semitransitive colorings and show that the existence of homogeneous sets for such colorings is equivalent to ADS and CAC, respectively. We finish with several open questions. (shrink)