This volume announces a new era in the philosophy of God. Many of its contributions work to create stronger links between the philosophy of God, on the one hand, and mathematics or metamathematics, on the other hand. It is about not only the possibilities of applying mathematics or metamathematics to questions about God, but also the reverse question: Does the philosophy of God have anything to offer mathematics or metamathematics? The remaining contributions tackle stereotypes in the philosophy of religion. The (...) volume includes 35 contributions. It is divided into nine parts: 1. Who Created the Concept of God; 2. Omniscience, Omnipotence, Timelessness and Spacelessness of God; 3. God and Perfect Goodness, Perfect Beauty, Perfect Freedom; 4. God, Fundamentality and Creation of All Else; 5. Simplicity and Ineffability of God; 6. God, Necessity and Abstract Objects; 7. God, Infinity, and Pascal's Wager; 8. God and (Meta-)Mathematics; and 9. God and Mind. (shrink)

The use of models of scientific theories should not be done without qualifications about the mathematics being used to build the models. This looks obvious, at least for logicians, but generally, it is not to the philosopher of science. Thus, some details about this point seem useful for both. Since any quick revision in the literature shows that in most cases, mainly after the raising of the semantic approach, the models are taken to be set-theoretical structures, in discussing the issue (...) we shall be concerned more with set theories, the locus where the play is usually developed. (shrink)

The Introduction to "Knowledge, Number and Reality. Encounters with the Work of Keith Hossack" provides an overview over Hossack's work and the contributions to the volume.

In describing numerosity as “a kind of ersatz number,” Clarke and Beck fail to consider a familiar and compelling definition of numerosity, which conceptualizes numerosity as the cognitive counterpart of the mathematical concept of cardinality; numerosity is the magnitude, whereas number is a scale through which numerosity/cardinality is measured. We argue that these distinctions should be considered.

The Forrest-Armstrong argument, as reconfigured by David Lewis, is a reductio against an unrestricted principle of recombination. There is a gap in the argument which Lewis thought could be bridged by an appeal to recombination. After presenting the argument, I show that no plausible principle of recombination can bridge the gap. But other plausible principles of plenitude can bridge the gap, both principles of plenitude for world contents and principles of plenitude for world structures. I conclude that the Forrest-Armstrong argument, (...) when fortified in one of these ways, demands that unrestricted recombination be rejected. The appropriate restriction comes from a consideration of what world structures are possible. I argue that, although there are too many worlds to form a set, for any world, the individuals at that world do form a set. To defend it I invoke a principle of Limitation of Size together with an iterative conception of structure. (shrink)

Is identity fundamental to every conceptual systems? In this article I intend to present reasons against the claim that every conceptual system presupposes the notion of identity. To address this debate I analyze the positions of Bueno (2014; 2016) and Krause and Arenhart (2015). While Bueno argues that identity is necessary for all conceptual systems, Krause and Arenhart present a series of objections against such position, thus defending that identity is not fundamental. I intend to show that the main objections (...) presented by Krause and Arenhart do not affect Bueno’s theory because of a misinterpretation of his arguments. Finally, I will present some objections against Bueno that, hopefully, does not miss the target. (shrink)

Approaching Infinity addresses seventeen paradoxes of the infinite, most of which have no generally accepted solutions. The book addresses these paradoxes using a new theory of infinity, which entails that an infinite series is uncompletable when it requires something to possess an infinite intensive magnitude. Along the way, the author addresses the nature of numbers, sets, geometric points, and related matters. The book addresses the need for a theory of infinity, and reviews both old and new theories of infinity. It (...) discussing the purposes of studying infinity and the troubles with traditional approaches to the problem, and concludes by offering a solution to some existing paradoxes. (shrink)

I argue that absolutism, the view that absolutely unrestricted quantification is possible, is to blame for both the paradoxes that arise in naive set theory and variants of these paradoxes that arise in plural logic and in semantics. The solution is restrictivism, the view that absolutely unrestricted quantification is not possible. -/- It is generally thought that absolutism is true and that restrictivism is not only false, but inexpressible. As a result, the paradoxes are blamed, not on illicit quantification, but (...) on the logical conception of set which motivates naive set theory. The accepted solution is to replace this with the iterative conception of set. -/- I show that this picture is doubly mistaken. After a close examination of the paradoxes in chapters 2--3, I argue in chapters 4 and 5 that it is possible to rescue naive set theory by restricting quantification over sets and that the resulting restrictivist set theory is expressible. In chapters 6 and 7, I argue that it is the iterative conception of set and the thesis of absolutism that should be rejected. (shrink)

Esta enciclopédia abrange, de uma forma introdutória mas desejavelmente rigorosa, uma diversidade de conceitos, temas, problemas, argumentos e teorias localizados numa área relativamente recente de estudos, os quais tem sido habitual qualificar como «estudos lógico-filosóficos». De uma forma apropriadamente genérica, e apesar de o território teórico abrangido ser extenso e de contornos por vezes difusos, podemos dizer que na área se investiga um conjunto de questões fundamentais acerca da natureza da linguagem, da mente, da cognição e do raciocínio humanos, bem (...) como questões acerca das conexões destes com a realidade não mental e extralinguística. A razão daquela qualificação é a seguinte: por um lado, a investigação em questão é qualificada como filosófica em virtude do elevado grau de generalidade e abstracção das questões examinadas (entre outras coisas); por outro, a investigação é qualificada como lógica em virtude de ser uma investigação logicamente disciplinada, no sentido de nela se fazer um uso intenso de conceitos, técnicas e métodos provenientes da disciplina de lógica. O agregado de tópicos que constitui a área de estudos lógico-filosóficos é já visível, pelo menos em parte, no Tractatus Logico-Philosophicus de Ludwig Wittgenstein, uma obra publicada em 1921. E uma boa maneira de ter uma ideia sinóptica do território disciplinar abrangido por esta enciclopédia, ou pelo menos de uma porção substancial dele, é extrair do Tractatus uma lista dos tópicos mais salientes aí discutidos; a lista incluirá certamente tópicos do seguinte género, muitos dos quais se podem encontrar ao longo desta enciclopédia: factos e estados de coisas; objectos; representação; crenças e estados mentais; pensamentos; a proposição; nomes próprios; valores de verdade e bivalência; quantificação; funções de verdade; verdade lógica; identidade; tautologia; o raciocínio matemático; a natureza da inferência; o cepticismo e o solipsismo; a indução; as constantes lógicas; a negação; a forma lógica; as leis da ciência; o número. (shrink)

The foremost aim of this paper is to realize the fourth part of the Aufbau. This part, which provides an actual phenomenalistic constitution system, is interpretable from a Kantian perspective (§§1-4). But Carnap plotted to overcome Kant’s old style of philosophy as well. We review this aspect of his constitution, focusing on space (§§7-13) and time (§§5-6), especially.

In this paper, we study the union axiom of ZFC. After a brief introduction, we sketch a proof of the folklore result that union is independent of the other axioms of ZFC. In the third section, we prove some results in the theory T:= ZFC minus union. Finally, we show that the consistency of T plus the existence of an inaccessible cardinal proves the consistency of ZFC.

One of the important challenges in the philosophy of mathematics is to account for the semantics of sentences that express mathematical propositions while simultaneously explaining our access to their contents. This is Benacerraf’s Dilemma. In this dissertation, I argue that cognitive science furnishes new tools by means of which we can make progress on this problem. The foundation of the solution, I argue, must be an ontologically realist, albeit non-platonist, conception of mathematical reality. The semantic portion of the problem can (...) be addressed by accepting a Chomskyan conception of natural languages and a matching internalist, mentalist and nativist view of semantics. A helpful perspective on the epistemic aspect of the puzzle can be gained by translating Kurt G ̈odel’s neo-Kantian conception of the nature of mathematics and its objects into modern, cognitive terms. (shrink)

Neste artigo, a partir de tópicos presentes na obra de Newton C. A. da Costa, propomos uma fundamentação rigorosa para de uma possível formulação de teorias científicas através da abordagem semântica. Seguindo da Costa, primeiramente desenvolveremos uma teoria geral das estruturas; no contexto desta teoria de estruturas mostraremos como caracterizar linguagens formais como um tipo particular de estrutura, mais especificamente, como uma álgebra livre. Em seguida, discutiremos como associar uma linguagem a uma estrutura, com a qual poderemos formular axiomas que (...) buscam captar a teoria da estrutura. Por fim, mostraremos como podemos, utilizando este aparato conceitual, fundamentar a formalização de da Costa e Chuaqui do chamado predicado de Suppes, utilizado para caracterizar teorias científicas de modo rigoroso. DOI:10.5007/1808-1711.2010v14n1p15. (shrink)

According to the most popular non-skeptical views about intuition, intuitions justify beliefs because they are based on understanding. More precisely: if intuiting that p justifies you in believing that p it does so because your intuition is based on your understanding of the proposition that p. The aim of this paper is to raise some challenges for accounts of intuitive justification along these lines. I pursue this project from a non-skeptical perspective. I argue that there are cases in which intuiting (...) that p justifies you in believing that p, but such that there is no compelling reason to think this is because your intuition is based on your understanding of the proposition that p. (shrink)

In this paper, I explore a question about deductive reasoning: why am I in a position to immediately infer some deductive consequences of what I know, but not others? I show why the question cannot be answered in the most natural ways of answering it, in particular in Descartes’s way of answering it. I then go on to introduce a new approach to answering the question, an approach inspired by Hume’s view of inductive reasoning.

Many experiments with infants suggest that they possess quantitative abilities, and many experimentalists believe that these abilities set the stage for later mathematics: natural numbers and arithmetic. However, the connection between these early and later skills is far from obvious. We evaluate two possible routes to mathematics and argue that neither is sufficient: (1) We first sketch what we think is the most likely model for infant abilities in this domain, and we examine proposals for extrapolating the natural number concept (...) from these beginnings. Proposals for arriving at natural number by (empirical) induction presuppose the mathematical concepts they seek to explain. Moreover, standard experimental tests for children's understanding of number terms do not necessarily tap these concepts. (2) True concepts of number do appear, however, when children are able to understand generalizations over all numbers; for example, the principle of additive commutativity (a+b=b+a). Theories of how children learn such principles usually rely on a process of mapping from physical object groupings. But both experimental results and theoretical considerations imply that direct mapping is insufficient for acquiring these principles. We suggest instead that children may arrive at natural numbers and arithmetic in a more top-down way, by constructing mathematical schemas. (shrink)

[1] You have a crystal ball. Unfortunately, it’s defective. Rather than predicting the future, it gives you the chances of future events. Is it then of any use? It certainly seems so. You may not know for sure whether the stock market will crash next week; but if you know for sure that it has an 80% chance of crashing, then you should be 80% confident that it will—and you should plan accordingly. More generally, given that the chance of a (...) proposition A is x%, your conditional credence in A should be x%. This is a chance-credence principle: a principle relating chance (objective probability) with credence (subjective probability, degree of belief). Let’s call it the Minimal Principle (MP). (shrink)

I am a realist of a metaphysical stripe. I believe in an immense realm of "modal" and "abstract" entities, of entities that are neither part of, nor stand in any causal relation to, the actual, concrete world. For starters: I believe in possible worlds and individuals; in propositions, properties, and relations (both abundantly and sparsely conceived); in mathematical objects and structures; and in sets (or classes) of whatever I believe in. Call these sorts of entity, and the reality they comprise, (...) metaphysical. In contrast, call the actual, concrete entities, and the reality they comprise, physical. Physical and metaphysical reality together comprise all that there is. In this paper, it is not my aim to defend realism about any particular metaphysical sort of entity. Rather, I ask quite generally whether and how any brand of realism about metaphysical sorts of entity could be justified? (shrink)

David Lewis (1980) proposed the Principal Principle (PP) and a “reformulation” which later on he called ‘OP’ (Old Principle). Reacting to his belief that these principles run into trouble, Lewis (1994) concluded that they should be replaced with the New Principle (NP). This conclusion left Lewis uneasy, because he thought that an inverse form of NP is “quite messy”, whereas an inverse form of OP, namely the simple and intuitive PP, is “the key to our concept of chance”. I argue (...) that, even if OP should be discarded, PP need not be. Moreover, far from being messy, an inverse form of NP is a simple and intuitive Conditional Principle (CP). Finally, both PP and CP are special cases of a General Principle (GP); it follows that so are PP and NP, which are thus compatible rather than competing. (shrink)

In quantum mechanics it is usually assumed that mutually exclusives states of affairs must be represented by orthogonal vectors. Recent attempts to solve the measurement problem, most notably the GRW theory, require the relaxation of this assumption. It is shown that a consequence of relaxing this assumption is that arithmatic does not apply to ordinary macroscopic objects. It is argued that such a radical move is unwarranted given the current state of understanding of the foundations of quantum mechanics.

We continue with the study of the Katětov order on MAD families. We prove that Katětov maximal MAD families exist under $\mathfrak {b=c}$ and that there are no Katětov-top MAD families assuming $\mathfrak {s\leq b}.$ This improves previously known results from the literature. We also answer a problem form Arciga, Hrušák, and Martínez regarding Katětov maximal MAD families.

In the literature on enculturation—the thesis according to which higher cognitive capacities result from transformations in the brain driven by culture—numerical cognition is often cited as an example. A consequence of the enculturation account for numerical cognition is that individuals cannot acquire numerical competence if a symbolic system for numbers is not available in their cultural environment. This poses a problem for the explanation of the historical origins of numerical concepts and symbols. When a numeral system had not been created (...) yet, people did not have the opportunity to acquire number concepts. But, if people did not have number concepts, how could they ever create a symbolic system for numbers? Here I propose an account of the invention of symbolic systems for numbers by anumeric people in the remote past that is compatible with the enculturation thesis. I suggest that symbols for numbers and number concepts may have emerged at the same time through the re-semantification of words whose meanings were originally non-numerical. (shrink)

Beginning from some passages by Vācaspati Miśra and Bhāskararāya Makhin discussing the relationship between a crown and the gold of which it is made, this paper investigates the complex underlying connections among difference, non-difference, coreferentiality, and qualification qua relations. Methodologically, philological care is paired with formal logical analysis on the basis of ‘Navya-Nyāya Formal Language’ premises and an axiomatic set theory-based approach. This study is intended as the first step of a broader investigation dedicated to analysing causation and transformation in (...) non-difference. (shrink)

ABSTRACTIn this paper I investigate how conceptual engineering applies to mathematical concepts in particular. I begin with a discussion of Waismann’s notion of open texture, and compare it to Shapiro’s modern usage of the term. Next I set out the position taken by Lakatos which sees mathematical concepts as dynamic and open to improvement and development, arguing that Waismann’s open texture applies to mathematical concepts too. With the perspective of mathematics as open-textured, I make the case that this allows us (...) to deploy the tools of conceptual engineering in mathematics. I will examine Cappelen’s recent argument that there are no conceptual safe spaces and consider whether mathematics constitutes a counterexample. I argue that it does not, drawing on Haslanger’s distinction between manifest and operative concepts, and applying this in a novel way to set-theoretic foundations. I then set out some of the questions that need to be engaged with to establish mathematics as involving a kind of conceptual engineering. I finish with a case study of how the tools of conceptual engineering will give us a way to progress in the debate between advocates of the Universe view and the Multiverse view in set theory. (shrink)

Perichoresis, or “mutual indwelling,” is a crucial concept in Trinitarian theology. But the philosophical underpinnings of the concept are puzzling. According to ordinary conceptions of “indwelling” or “being in,” it is incoherent to think that two entities could be in each other. In this paper, I propose a mereological way of understanding “being in,” by analogy with standard examples in contemporary metaphysics. I argue that this proposal does not conflict with the doctrine of divine simplicity, but instead affirms it. I (...) conclude by discussing how mutual indwelling relates to the concepts of unity and identity. (shrink)

I analyse the conceptual and mathematical foundations of Lagrangian quantum field theory (QFT) (that is, the ‘naive’ (QFT) used in mainstream physics, as opposed to algebraic quantum field theory). The objective is to see whether Lagrangian (QFT) has a sufficiently firm conceptual and mathematical basis to be a legitimate object of foundational study, or whether it is too ill-defined. The analysis covers renormalisation and infinities, inequivalent representations, and the concept of localised states; the conclusion is that Lagrangian QFT (at least (...) as described here) is a perfectly respectable physical theory, albeit somewhat different in certain respects from most of those studied in foundational work. (shrink)

Working in Z + KP, we give a new proof that the class of hereditarily finite sets cannot be proved to be a set in Zermelo set theory, extend the method to establish other failures of replacement, and exhibit a formula Φ(λ, a) such that for any sequence $\langle A_{\lambda} \mid \lambda \text{a limit ordinal} \rangle$ where for each $\lambda, A_{\lambda} \subseteq ^{\lambda}2$ , there is a supertransitive inner model of Zermelo containing all ordinals in which for every λ A (...) λ = {α ∣Φ(λ, a)}. (shrink)

Let $\varSigma $ be a signature without $0$-ary operation symbols and $\textsf{Sl}$ the category of semilattices. Then, after defining and investigating the categories $\int ^{\textsf{Sl}}\textrm{Isys}_{\varSigma }$, of inductive systems of $\varSigma $-algebras over all semilattices, which are ordered pairs $\boldsymbol{\mathscr{A}}= (\textbf{I},\mathscr{A})$ where $\textbf{I}$ is a semilattice and $\mathscr{A}$ an inductive system of $\varSigma $-algebras relative to $\textbf{I}$, and PłAlg$(\varSigma )$, of Płonka $\varSigma $-algebras, which are ordered pairs $(\textbf{A},D)$ where $\textbf{A}$ is a $\varSigma $-algebra and $D$ a Płonka operator for (...) $\textbf{A}$, i.e. in the traditional terminology, a partition function for $\textbf{A}$, we prove the main result of the paper: There exists an adjunction, which we call the Płonka adjunction, from $\int ^{\textsf{Sl}}\textrm{Isys}_{\varSigma }$ to PłAlg$(\varSigma )$. (shrink)

In 1939, the influential psychophysicist S. S. Stevens proposed definitional distinctions between the terms ‘number,’ ‘numerosity,’ and ‘numerousness.’ Although the definitions he proposed were adopted by syeveral psychophysicists and experimental psychologists in the 1940s and 1950s, they were almost forgotten in the subsequent decades, making room for what has been described as a “terminological chaos” in the field of numerical cognition. In this paper, I review Stevens’s distinctions to help bring order to this alleged chaos and to shed light on (...) two closely related questions: whether it is adequate to speak of a number sense and how philosophers can make sense of the claim by cognitive scientists that numbers are perceptual entities. Moreover, I offer further support to Stevens’s distinction between numerosity and numerousness by showing that they are relational properties that emerge through different relationships between agents and their environments. The final conclusion is that, by adopting Stevens’s distinctions, numbers do not need to be seen as perceptual entities, since the so-called number sense is better described as a sense of numerousness. (shrink)

In this paper, I argue from a metasemantic principle to the existence of analytic sentences. According to the metasemantic principle, an external feature is relevant to determining which concept one expresses with an expression only if one is disposed to treat this feature as relevant. This entails that if one isn’t disposed to treat external features as relevant to determining which concept one expresses, and one still expresses a given concept, then something other than external features must determine that one (...) does. I argue that, in such cases, what determines that one expresses the concept also puts one in a position to know that certain sentences are true—these sentences are thus analytic relative to this determination basis. Finally, I argue that there are such cases: some sentences are analytic relative to what determines that we express certain key concepts, and these sentences include ones that have always been thought to be the best candidates for being analytic, namely, stipulative truths, and first principles of mathematics. (shrink)

In the following work we propose to incorporate the main feature of quantum mechanics, i.e., the concept of indiscernibility. To achieve this goal, first we present two models of set theories: a quasi-set theory and a non-antisymmetric mereology. Next, we show how specific objects of QST—m-atoms—can be defined within NAM. Finally, we introduce a concept of a parthood of indiscernibles and discuss its features in respect to standard notions of indiscernibles and within NAM.

The main aim of this work is to evaluate whether Boolos’ semantics for second-order languages is model-theoretically equivalent to standard model-theoretic semantics. Such an equivalence result is, actually, directly proved in the “Appendix”. I argue that Boolos’ intent in developing such a semantics is not to avoid set-theoretic notions in favor of pluralities. It is, rather, to prevent that predicates, in the sense of functions, refer to classes of classes. Boolos’ formal semantics differs from a semantics of pluralities for Boolos’ (...) plural reading of second-order quantifiers, for the notion of plurality is much more general, not only of that set, but also of class. In fact, by showing that a plurality is equivalent to sub-sets of a power set, the notion of plurality comes to suffer a loss of generality. Despite of this equivalence result, I maintain that Boolos’ formal semantics does not committ second-order languages to second-order entities, contrary to standard semantics. Further, such an equivalence result provides a rationale for many criticisms to Boolos’ formal semantics, in particular those by Resnik and Parsons against its alleged ontological innocence and on its Platonistic presupposition. The key set-theoretic notion involved in the equivalence proof is that of many-valued function. But, first, I will provide a clarification of the philosophical context and theoretical grounds of the genesis of Boolos’ formal semantics. (shrink)

In recent philosophy of mathematics avariety of writers have presented ``structuralist''views and arguments. There are, however, a number ofsubstantive differences in what their proponents take``structuralism'' to be. In this paper we make explicitthese differences, as well as some underlyingsimilarities and common roots. We thus identifysystematically and in detail, several main variants ofstructuralism, including some not often recognized assuch. As a result the relations between thesevariants, and between the respective problems theyface, become manifest. Throughout our focus is onsemantic and metaphysical issues, (...) including what is orcould be meant by ``structure'' in this connection. (shrink)

Putnam’s model-theoretic arguments for the indeterminacy of reference have been taken to pose a special problem for mathematical languages. In this paper, I argue that if one accepts that there are theory-external constraints on the reference of at least some expressions of ordinary language, then Putnam’s model-theoretic arguments for mathematical languages don’t go through. In particular, I argue for a kind of descriptivism about mathematical expressions according to which their reference is “anchored” in the reference of expressions of ordinary language. (...) These anchors add enough to the content of mathematical expressions to forestall the radical kind of indeterminacy that model-theoretic arguments are purported to show, while still leaving room for a plausible, moderate kind of indeterminacy. (shrink)

In this paper, I explore an intriguing view of definable numbers proposed by a Cambridge mathematician Ernest Hobson, and his solution to the paradoxes of definability. Reflecting on König’s paradox and Richard’s paradox, Hobson argues that an unacceptable consequence of the paradoxes of definability is that there are numbers that are inherently incapable of finite definition. Contrast to other interpreters, Hobson analyses the problem of the paradoxes of definability lies in a dichotomy between finitely definable numbers and not finitely definable (...) numbers. To bypass this predicament, Hobson proposes a language dependent analysis of definable numbers, where the diagonal argument is employed as a means to generate more and more definable numbers. This paper examines Hobson’s work in its historical context, and articulates his argument in detail. It concludes with a remark on Hobson’s analysis of definability and Alan Turing’s analysis of computability. (shrink)

The goal of inquiry in this essay is to ascertain to what extent the Principle of Compositionality – the thesis that the meaning of a complex expression is determined by the meaning of its parts and its mode of composition – can be justifiably imposed as a constraint on semantic theories, and thereby provide information about what meanings are. Apart from the introduction and the concluding chapter the thesis is divided into five chapters addressing different questions pertaining to the overarching (...) goal of inquiry. Chapter Two is an attempt to determine whether the Principle of Compositionality is a trivial principle. It is argued that this is not the case in the context of providing the semantics of natural languages since it is an open question whether the best theories that respect available syntactic and semantic data are compositional. Chapter Three and Chapter Four ask whether there are reasons to think that the Principle of Compositionality is true. It is argued that the three most commonly cited reasons for thinking that correct semantic theories must be compositional – Linguistic Creativity, Productivity and Systematicity – in fact only give us reasons to think that the meaning of complex expressions depend on the meanings of their parts and their modes of composition, but not that the meaning of complex expressions depend ONLY on those factors. Chapter Five asks whether there are any reasons to think that the Principle of Compositionality is false. It is argued that all the phenomena that have been suggested entail non-compositionality are in fact compatible with some versions of compositionality. Even if this conclusion is reached in this chapter, the conclusion of the previous two chapters entails that we are not justified in imposing the Principle of Compositionality as an adequacy constraint on Semantic theories. Chapter Six explores the ways in which information about what meanings are can be derived by imposing constraints on semantic theories. It is argued that the distinction emphasized in Chapter Three and Chapter Four between depending on and depending ONLY on is of critical importance. It is demonstrated that the informative potential of the Principle of Compositionality goes far beyond that of the principle we are in fact justified in imposing on semantic theories. So not only is it the case that we cannot learn anything about meanings from the justified imposition of the Principle of Compositionality – since we cannot justifiably impose it as a constraint on semantic theories – what we could have learned from it had we been justified in imposing it is very different from what we can learn from the principle that we are justified in imposing. (shrink)

Penelope Maddy advances a purportedly naturalistic account of mathematical methodology which might be taken to answer the question 'What justifies axioms of set theory?' I argue that her account fails both to adequately answer this question and to be naturalistic. Further, the way in which it fails to answer the question deprives it of an analog to one of the chief attractions of naturalism. Naturalism is attractive to naturalists and nonnaturalists alike because it explains the reliability of scientific practice. Maddy's (...) account, on the other hand, appears to be unable to similarly explain the reliability of mathematical practice without violating one of its central tenets. (shrink)

According to the dialetheist argument from the inconsistency of informal mathematics, the informal version of the Gödelian argument leads us to a true contradiction. On one hand, the dialetheist argues, we can prove that there is a mathematical claim that is neither provable nor refutable in informal mathematics. On the other, the proof of its unprovability is given in informal mathematics and proves that very sentence. We argue that the argument fails, because it relies on the unjustified and unlikely assumption (...) that the informal Gödel sentence is informally provable. (shrink)

A standard formalization of a scientific theory is a system of axioms for that theory in a first-order language (possibly many-sorted; possibly with the membership primitive $$\in$$ ). Suppes (in: Carvallo M (ed) Nature, cognition and system II. Kluwer, Dordrecht, 1992) expressed skepticism about whether there is a “simple or elegant method” for presenting mathematicized scientific theories in such a standard formalization, because they “assume a great deal of mathematics as part of their substructure”. The major difficulties amount to these. (...) First, as the theories of interest are mathematicized, one must specify the underlying applied mathematics base theory, which the physical axioms live on top of. Second, such theories are typically geometric, concerning quantities or trajectories in space/time: so, one must specify the underlying physical geometry. Third, the differential equations involved generally refer to coordinate representations of these physical quantities with respect to some implicit coordinate chart, not to the original quantities. These issues may be resolved. Once this is done, constructing standard formalizations is not so difficult—at least for the theories where the mathematics has been worked out rigorously. Here we give what may be claimed to be a simple and elegant means of doing that. This is for mathematicized scientific theories comprising differential equations for $$\mathbb{R}$$ -valued quantities Q (that is, scalar fields), defined on n (“spatial” or “temporal”) dimensions, taken to be isomorphic to the usual Euclidean space $$\mathbb{R}^n$$. For illustration, I give standard (in a sense, “text-book”) formalizations: for the simple harmonic oscillator equation in one-dimension and for the Laplace equation in two dimensions. (shrink)

The terms of a mathematical problem become precise and concise if they are expressed in an appropriate notation, therefore notations are useful to mathematics. But are notations only useful, or also essential? According to prevailing view, they are not essential. Contrary to this view, this paper argues that notations are essential to mathematics, because they may play a crucial role in mathematical discovery. Specifically, since notations may consist of symbolic notations, diagrammatic notations, or a mix of symbolic and diagrammatic notations, (...) notations may play a crucial role in mathematical discovery in different ways. Some examples are given to illustrate these ways. (shrink)