This paper defends a conceptualistic version of structuralism as the most convincing way of elaborating a philosophical understanding of structuralism in line with the classical tradition. The argument begins with a revision of the tradition of “conceptual mathematics”, incarnated in key figures of the period 1850 to 1940 like Riemann, Dedekind, Hilbert or Noether, showing how it led to a structuralist methodology. Then the tension between the ‘presuppositionless’ approach of those authors, and the platonism of some recent versions of philosophical (...) structuralism, is presented. In order to resolve this tension, we argue for the idea of ‘logical objects’ as a form of minimalist realism, again in the tradition of classical authors including Peirce and Cassirer, and we introduce the basic tenets of conceptual structuralism. The remainder of the paper is devoted to an open discussion of the assumptions behind conceptual structuralism, and—most importantly—an argument to show how the objectivity of mathematics can be explained from the adopted standpoint. This includes the idea that advanced mathematics builds on hypothetical assumptions (Riemann, Peirce, and others), which is presented and discussed in some detail. Finally, the ensuing notion of objectivity is interpreted as a form of particularly robust intersubjectivity, and it is distinguished from fictional or social ontology. (shrink)

Modal logic provides an elegant way to understand the notion of potential infinity. This raises the question of what the right modal logic is for reasoning about potential infinity. In this article I identify a choice point in determining the right modal logic: Can a potentially infinite collection ever be expanded in two mutually incompatible ways? If not, then the possible expansions are convergent; if so, then the possible expansions are branching. When possible expansions are convergent, the right modal logic (...) is S4.2, and a mirroring theorem due to Linnebo allows for a natural potentialist interpretation of mathematical discourse. When the possible expansions are branching, the right modal logic is S4. However, the usual box and diamond do not suffice to express everything the potentialist wants to express. I argue that the potentialist also needs an operator expressing that something will eventually happen in every possible expansion. I prove that the result of adding this operator to S4 makes the set of validities Pi-1-1 hard. This result makes it unlikely that there is any natural translation of ordinary mathematical discourse into the potentialist framework in the context of branching possibilities. (shrink)

Hilbert's program attempts to show that our mathematical knowledge can be certain because we are able to know for certain the truths of elementary arithmetic. I argue that, in the absence of a theory of mathematical truth, Hilbert does not have a complete theory of our arithmetical knowledge. Further, while his deployment of a Kantian notion of intuition seems to promise an answer to scepticism, there is no way to complete Hilbert's epistemology which would answer to his avowed aims.

This paper explores the relationship borne by the traditional paradoxes of set theory and semantics to formal incompleteness phenomena. A central tool is the application of the Arithmetized Completeness Theorem to systems of second-order arithmetic and set theory in which various “paradoxical notions” for first-order languages can be formalized. I will first discuss the setting in which this result was originally presented by Hilbert & Bernays (1939) and also how it was later adapted by Kreisel (1950) and Wang (1955) in (...) order to obtain formal undecidability results. A generalization of this method will then be presented whereby Russell’s paradox, a variant of Mirimanoff’s paradox, the Liar, and the Grelling–Nelson paradox may be uniformly transformed into incompleteness theorems. Some additional observations are then framed relating these results to the unification of the set theoretic and semantic paradoxes, the intensionality of arithmetization (in the sense of Feferman, 1960), and axiomatic theories of truth. (shrink)

We present a case study of how mathematicians write for mathematicians. We have conducted interviews with two research mathematicians, the talented PhD student Adam and his experienced supervisor Thomas, about a research paper they wrote together. Over the course of 2 years, Adam and Thomas revised Adam’s very detailed first draft. At the beginning of this collaboration, Adam was very knowledgeable about the subject of the paper and had good presentational skills but, as a new PhD student, did not yet (...) have experience writing research papers for mathematicians. Thus, one main purpose of revising the paper was to make it take into account the intended audience. For this reason, the changes made to the initial draft and the authors’ purpose in making them provide a window for viewing how mathematicians write for mathematicians. We examined how their paper attracts the interest of the reader and prepares their proofs for validation by the reader. Among other findings, we found that their paper prepares the proofs for two types of validation that the reader can easily switch between. (shrink)

With respect to the metaphysics of infinity, the tendency of standard debates is to either endorse or to deny the reality of ‘the infinite’. But how should we understand the notion of ‘reality’ employed in stating these options? Wittgenstein’s critical strategy shows that the notion is grounded in a confusion: talk of infinity naturally takes hold of one’s imagination due to the sway of verbal pictures and analogies suggested by our words. This is the source of various philosophical pictures that (...) in turn give rise to the standard metaphysical debates: that the mathematics of infinity corresponds to a special realm of infinite objects, that the infinite is profoundly huge or vast, or that the ability to think about infinity reveals mysterious powers in human beings. First, I explain Wittgenstein’s general strategy for undermining philosophical pictures of ‘the infinite’ – as he describes it in Zettel; and then show how that critical strategy is applied to Cantor’s diagonalization proof in Remarks on the Foundations of Mathematics II. (shrink)

In “Mathematical Truth,” Paul Benacerraf raises an epistemic challenge for mathematical platonists. In this paper, I examine the assumptions that motivate Benacerraf’s original challenge, and use them to construct a new causal challenge for the epistemology of mathematics. This new challenge, which I call ‘Gödel’s Gap’, appeals to intuitive insights into mathematical knowledge. Though it is a causal challenge, it does not rely on any obviously objectionable constraints on knowledge. As a result, it is more compelling than the original challenge. (...) It is also more general; the challenge applies equally to platonistic and non-platonistic accounts of mathematical truth. And it can be generalized beyond the case of mathematical knowledge to pose a challenge for, e.g., moral knowledge. (shrink)

In this paper, elementary but hitherto overlooked connections are established between Wittgenstein's remarks on mathematics, written during his transitional period, and free-variable finitism. After giving a brief description of theTractatus Logico-Philosophicus on quantifiers and generality, I present in the first section Wittgenstein's rejection of quantification theory and his account of general arithmetical propositions, to use modern jargon, as claims (as opposed to statements). As in Skolem's primitive recursive arithmetic and Goodstein's equational calculus, Wittgenstein represented generality by the use of free (...) variables. This has the effect that negation of unbounded universal and existential propositions cannot be expressed. This is claimed in the second section to be the basis for Wittgenstein's criticism of the universal validity of the law of excluded middle. In the last section, there is a brief discussion of Wittgenstein's remarks on real numbers. These show a preference, in line with finitism, for a recursive version of the continuum. (shrink)

The Bernays-Müller debate was a dispute in the early 1920s between Paul Bernays and Aloys Müller regarding various philosophical issues related to “Hilbert’s program.” The debate is sometimes mentioned as a sidenote in discussions of Hilbert’s program, but there is little or no discussion of the debate itself in the secondary literature. This article aims to fill this gap and to provide a detailed analysis of the background of the debate, its contents, and the impact on its protagonists.

The main topic of this essay is symbolic mathematics or the method of symbolic construction, which I trace to the end of the sixteenth century when Franciscus Vieta invented the algebraic symbolism and started to use the word ‘symbolic’ in the relevant, non-ontological sense. This approach has played an important role for many of the great inventions in modern mathematics such as the introduction of the decimal place-value system of numeration, Descartes’ analytic geometry, and Leibniz’s infinitesimal calculus. It was also (...) central for the rigorization movement in mathematics in the late nineteenth century, as well as for the mathematics of modern physics in the 20th century.However, the nature of symbolic mathematics has been concealed and confused due to the strong influence of the heritage from the Euclidean and Aristotelian traditions. This essay sheds some light on what has been concealed by approaching some of the crucial issues from a historical perspective. Furthermore, I argue that the conception of modern mathematics as symbolic mathematics was essential to Wittgenstein’s approach to the foundations and nature of mathematics. This connection between Wittgenstein’s thought and symbolic mathematics provides the resources for countering the still prevalent view that he defended an uttrely idiosyncratic conception, disconnected from the progress of serious science. Instead, his project can be seen as clarifying ideas that have been crucial to the development of mathematics since early modernity. (shrink)

Chomsky's criticism of Bloomfieldian structuralism's conception of linguistic reality applies equally to his own conception of linguistic reality. There are too many sentences in a natural language for them to have either concrete acoustic reality or concrete psychological or neural reality. Sentences have to be types, which, by Peirce's generally accepted definition, means that they are abstract objects. Given that sentences are abstract objects, Chomsky's generativism as well as his psychologism have to be given up. Langendoen and Postal's argument in (...) The Vastness of Natural Languages to show that there are more than denumerably many sentences is flawed. But, with the view that sentences are abstract objects, the flaws can be corrected. Once psychologism and generativism are abandoned, the revolution against Bloomfieldian structuralism can be brought to completion and linguistics can be put on a sound philosophical basis. (shrink)

The dream of a community of philosophers engaged in inquiry with shared standards of evidence and justification has long been with us. It has led some thinkers puzzled by our mathematical experience to look to mathematics for adjudication between competing views. I am skeptical of this approach and consider Skolem's philosophical uses of the Löwenheim-Skolem Theorem to exemplify it. I argue that these uses invariably beg the questions at issue. I say ?uses?, because I claim further that Skolem shifted his (...) position on the philosophical significance of the theorem as a result of a shift in his background beliefs. The nature of this shift and possible explanations for it are investigated. Ironically, Skolem's own case provides a historical example of the philosophical flexibility of his theorem. Our suspicion ought always to be aroused when a proof proves more than its means allow it. Something of this sort might be called ?a puffed-up proof?. Ludwig Wittgenstein, Remarks on the foundations of mathematics (revised edition), vol. 2, 21. (shrink)

Accelerating environmental uncertainty and the need to cope with increasingly complex market and social demands, combine to create high value for the intuitive approach to decision-making at the strategic level. Research on intuition suffers from marked fragmentation, due to the existence of disciplinary silos based on diverse, apparently irreconcilable, ontological and epistemological assumptions. Not surprisingly, there is no integrated interdisciplinary framework suitable for a rich account of intuition, contemplating how affect and cognition intertwine in the intuitive process, and how intuition (...) scales up from the individual to collective decision-making. This study contributes to the construction of a broad conceptual framework, suitable for a multi-level account of intuition and for a fruitful dialogue with distant research areas. It critically discusses two mainstream conceptualizations of intuition which claim to be grounded in a cross-disciplinary consensus. Drawing on the complexity paradigm, it then proposes a conceptualization of intuition as emergence. Finally, it explores the theoretical and practical implications. (shrink)

Far from being unwelcome or impossible in a mathematical setting, indeterminacy in various forms can be seen as playing an important role in driving mathematical research forward by providing “sources of newness” in the sense of Hutter and Farías :434–449, 2017). I argue here that mathematical coincidences, phenomena recently under discussion in the philosophy of mathematics, are usefully seen as inducers of indeterminacy and as put to work in guiding mathematical research. I suggest that to call a pair of mathematical (...) facts a coincidence is roughly to suggest that the investigation of connections between these facts isn’t worthwhile. To say of this pair, “That’s no coincidence!” is to suggest just the opposite. I further argue that this perspective on mathematical coincidence, which pays special attention to what mathematical coincidences do, may provide us with a better view of what mathematical coincidences are than extant accounts. I close by reflecting on how understanding mathematical coincidences as generating indeterminacy accords with a conception of mathematical research as ultimately aiming to reduce indeterminacy and complexity to triviality as proposed in Rota Indiscrete thoughts, Birkhäuser, 1997). (shrink)

Over the last century, there have been considerable variations in the frequency of use and types of diagrams used in mathematical publications. In order to track these changes, we developed a method enabling large-scale quantitative analysis of mathematical publications to investigate the number and types of diagrams published in three leading mathematical journals in the period from 1885 to 2015. The results show that diagrams were relatively common at the beginning of the period under investigation. However, beginning in 1910, they (...) were almost completely unused for about four decades before reappearing in the 1950s. The diagrams from the 1950s, however, were of a different type than those used earlier in the century. We see this change in publication practice as a clear indication that the formalist ideology has influenced mathematicians’ choice of representations. Although this could be seen as a minor stylistic aspect of mathematics, we argue that mathematicians’ representational practice is deeply connected to their cognitive practice and to the contentual development of the discipline. These changes in publication style therefore indicate more fundamental changes in the period under consideration. (shrink)

We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higher-order reflection principles.

Can an epistemic conception of truth and an endorsement of the excluded middle (together with other principles of classical logic abandoned by the intuitionists) cohabit in a plausible philosophical view? In PART I I describe the general problem concerning the relation between the epistemic conception of truth and the principle of excluded middle. In PART II I give a historical overview of different attitudes regarding the problem. In PART III I sketch a possible holistic solution.

The Aristotelian origins of formal systems are outlined, together with Aristotle's use of causal terms in describing syllogisms. The precision and exactness of a formalism, based on the projection of logical forms into perceptive signs, is contrasted with foundational, abstract concepts, independent of any formalism, which are presupposed for the understanding of a formal language. The definition of a formal system by means of a Turing machine is put in the context of Wittgenstein's general considerations of a machine understood as (...) a sign. A modification of Łukasiewicz's logic Ł3 with the inclusion of justification terms is proposed in order to formally analyse some features of formalistic reasoning as a mechanical, causal affair within a wider context of "indeterminacy". (shrink)

Let be the fragment of elementary Peano arithmetic in which induction is restricted to -formulas. More than three decades ago, Parsons showed that the provably total functions of are exactly the primitive recursive functions. In this paper, we observe that Parsons' result is a consequence of Herbrand's theorem concerning the -consequences of universal theories. We give a self-contained proof requiring only basic knowledge of mathematical logic.

Julius König is famous for his mistaken attempt to demonstrate that the continuum hypothesis was false. It is also known that the only positive result that could have survived from his proof is the paradox which bears his name. Less famous is his 1914 book Neue Grundlagen der Logik, Arithmetik und Mengenlehre. Still, it contains original contributions to logic, like the concept of metatheory and the solution of paradoxes based on the refusal of the law of bivalence. We are going (...) to discover them by analysing the content of the book. (shrink)

A partir del concepto themata de Gerald Holton, sugiero la noción de “tensiones temáticas” en un intento por abordar asuntos relacionados con la necesidad de establecer criterios de identidad en la evolución de controversias científicas. Por “tensiones temáticas” entiendo una variedad de presiones de fondo que moldean el desarrollo de ciertas controversias. Aplico la noción a dos disputas distantes en el tiempo para esclarecer su parentesco: la controversia que sostuvieron platónicos y aristotélicos entre los siglos iii a. c. y iii (...) d. c. sobre el modo de existencia del infinito, y la controversia que sostuvieron Georg Cantor y Leopold Kronecker a finales del siglo xix sobre la legitimidad de los números transfinitos. (shrink)

Wittgenstein defiende que los problemas filosóficos son ilegítimos. Sin embargo, no resulta del todo claro qué se entiende en general por problema filosófico, cuál es, pues, su naturaleza. El presente artículo intenta elucidar la observación de Wittgenstein que sostiene que la forma de un problema filosófico se puede presentar con la expresión “Yo no sé salir del atolladero” (IF § 123). En el artículo se elige, a manera de ejemplo, el escándalo kantiano, que demanda que demostremos con el uso de (...) la razón que la proposición “hay objetos físicos” es verdadera. Sin renunciar al ejemplo, se explora en qué sentido la expresión de Wittgenstein caracteriza en buena medida la naturaleza misma de la dificultad filosófica. En el ejercicio se atienden recomendaciones formales del Tractatus, terapias desplegadas en Sobre la Certeza y el recurso de distinguir entre sentido primario y sentido secundario de una expresión insinuado en las Investigaciones Filosóficas. (shrink)

Formalism in the philosophy of mathematics has taken a variety of forms and has been advocated for widely divergent reasons. In Sects. 1 and 2, I briefly introduce the major formalist doctrines of the late nineteenth and early twentieth centuries. These are what I call empirico-semantic formalism, game formalism and instrumental formalism. After describing these views, I note some basic points of similarity and difference between them. In the remainder of the paper, I turn my attention to Hilbert’s instrumental formalism. (...) My primary aim there will be to develop its formalist elements more fully. These are, in the main, its rejection of the axiom-centric focus of traditional model-construction approaches to consistency problems, its departure from the traditional understanding of the basic nature of proof and its distinctively descriptive or observational orientation with regard to the consistency problem for arithmetic. More specifically, I will highlight what I see as the salient points of connection between Hilbert’s formalist attitude and his finitist standard for the consistency proof for arithmetic. I will also note what I see as a significant tension between Hilbert’s observational approach to the consistency problem for arithmetic and his expressed hope that his solution of that problem would dispense with certain epistemological concerns regarding arithmetic once and for all. (shrink)

This thesis provides an account of acquaintance with abstract objects. The notion of acquaintance is integral to theorising on reference and singular thought, since it is generally taken to be the relation that must exist between a subject and an object, in order for the subject to refer to, and entertain singular thoughts about the object. The most common way of understanding acquaintance is as a form of causal connection. However, this implies a problem. We seem to be able to (...) refer and have singular thoughts about abstract objects. But given that abstract objects are causally inert, this would mean that we are unable to become acquainted with them. This problem shall be the focus of this thesis. I first argue that these traditional causal interpretations of acquaintance are lacking. Instead, I show that acquaintance is dependent to some degree on factors internal to the subject, namely the skills that they possess. From doctors to sommeliers to mathematicians (and possibly even philosophers!) – these subjects seem to succeed in becoming acquainted with certain objects precisely in virtue of their respective skills. Thus, building off from Evans’ The Varieties of Reference, I present a novel account of acquaintance, which I term as Skill-based Acquaintance (SBA). On SBA, a subject is said to be acquainted with an object when they possess discriminating knowledge of that object, gained through the use of their capacities and skills. The SBA account is applied to virtual (Ch.3), fictional (Ch.4), and mathematical objects (Ch.5), as well as God (Ch.6). SBA is successful in explaining how subjects can indeed become acquainted with these problematic categories of objects - some of which are abstract – thus being able to refer and entertain singular thoughts about them. Overall, then, SBA is shown to have greater explanatory power than competing accounts and should thus be preferred. (shrink)

In “Function and Concept” and “On Concept and Object”, Frege argued that certain differences between dependent and independent meanings were inviolable and “founded deep in the nature of things” but, in those articles, he was not explicit about the actual consequences of violating such differences. However, since by creating a law that permitted one to pass from a concept to its extension, he himself mixed dependent and independent meanings, we are in a position to study some of the actual consequences (...) of his having done so. To make certain of Frege’s ideas about the inviolability of logical form more tangible, I describe a string of very interrelated consequences that his attempt to transform dependent meanings into independent meanings actually brought in its wake. (shrink)

Ignoring primitive terms leads to an infinite regress. The alternative is to account for an intuitive understanding into the meaning of such terms. The current investigation proceeds on the basis of an idea of the structure of the various modes of being within which concrete entities function. Examples of primtive terms are given from disciplines such as mathematics, physics and logic and they are related to the general idea of a modal aspect. It is argued that primitive terms are not (...) isolated but reveal their meaning only through their interconnections with other primitive terms that are embedded in other modal aspects. However, although primitive terms are found within the various aspects, the meaning of an aspect only comes to expression through its coherence with other aspects, evinced in modal analogies that are qualified by the core meaning of an aspect. There appears to be two options, either reduce what is irreducible or merely provide synonymous terms for given primitives. The former happens when other unique terms are used to define a specific one and the latter when the attempted ‘definitions’ revert to terms with which the original terms could be meaningfully replaced. It is been pointed out that the coherence between primitives invites every academic discipline to account for the meaning attached to the analogies of primitive terms it is employing, without exploring this additional theme any further. (shrink)

En este ensayo respondo negativamente a la pregunta del título al sostener que el enunciado “La suma de los ángulos internos de un triángulo es igual a 180°” es contingentemente verdadero. Para ello, intento refutar la tesis de Ramsey de que las verdades geométricas necesariamente son verdades necesarias, así como la tesis de Kripke de que no puede haber proposiciones matemáticas contingentemente verdaderas. Además, recurriendo a la concepción fregeana sobre lo a priori y lo a posteriori, sostengo que hay verdades (...) geométricas que pueden ser a priori sin tener que serlo. -/- . (shrink)