Opinionated state of the art paper on mathematical explanation. After a general introduction to the subject, the paper is divided into two parts. The first part is dedicated to intra-mathematical explanation and the second is dedicated to extra-mathematical explanation. Each of these parts begins to present a set of diverse problems regarding each type of explanation and, afterwards, it analyses relevant models of the literature. Regarding the intra-mathematical explanation, the models of deformable proofs, mathematical saliences and the demonstrative structure of (...) mathematical induction are addressed. Regarding the extra-mathematical explanation, modal, abstract and deductive models are addressed. (shrink)

What is the source of logical and mathematical truth? This book revitalizes conventionalism as an answer to this question. Conventionalism takes logical and mathematical truth to have their source in linguistic conventions. This was an extremely popular view in the early 20th century, but it was never worked out in detail and is now almost universally rejected in mainstream philosophical circles. Shadows of Syntax is the first book-length treatment and defense of a combined conventionalist theory of logic and mathematics. It (...) argues that our conventions, in the form of syntactic rules of language use, are perfectly suited to explain the truth, necessity, and a priority of logical and mathematical claims, as well as our logical and mathematical knowledge. (shrink)

I propose a deductive-nomological model for mathematical scientific explanation. In this regard, I modify Hempel’s deductive-nomological model and test it against some of the following recent paradigmatic examples of the mathematical explanation of empirical facts: the seven bridges of Königsberg, the North American synchronized cicadas, and Hénon-Heiles Hamiltonian systems. I argue that mathematical scientific explanations that invoke laws of nature are qualitative explanations, and ordinary scientific explanations that employ mathematics are quantitative explanations. I analyse the repercussions of this deductivenomological model (...) on causal explanations. (shrink)

A prominent objection against the logicality of second-order logic is the so-called Overgeneration Argument. However, it is far from clear how this argument is to be understood. In the first part of the article, we examine the argument and locate its main source, namely, the alleged entanglement of second-order logic and mathematics. We then identify various reasons why the entanglement may be thought to be problematic. In the second part of the article, we take a metatheoretic perspective on the matter. (...) We prove a number of results establishing that the entanglement is sensitive to the kind of semantics used for second-order logic. These results provide evidence that by moving from the standard set-theoretic semantics for second-order logic to a semantics which makes use of higher-order resources, the entanglement either disappears or may no longer be in conflict with the logicality of second-order logic. (shrink)

Duncan Pritchard recently proposed a Wittgensteinian solution to closure-based skepticism. According to Wittgenstein, all epistemic systems assume certain truths. The notions that we are not disembodied brains, that the Earth has existed for a long time and that one’s name is such-and-such all function as “hinge commitments.” Pritchard views a hinge commitment as a positive propositional attitude that is not a belief. Because closure principles concern only knowledge-apt beliefs, they do not apply to hinge commitments. Thus, from the fact that (...) a subject knows that he is sitting in a room, and the fact that the subject’s sitting in a room entails his bodily existence, it does not follow that the subject also knows that he is not an envatted brain. This paper rejects Pritchard’s non-belief reading of hinge commitments. I start by showing that the non-belief reading fails to solve the skeptical paradox because the reasons that Pritchard uses to support the non-belief reading do not exempt hinge propositions from closure principles. I then proceed to argue that the non-belief reading is false as it claims that hinge commitments, unlike ordinary beliefs, are rationally unresponsive—with the help of a scenario in which a subject’s experience is internally chaotic, we can safely conclude that the hinge commitment that one is not systematically mistaken about the world is equally responsive to one’s evidential situations. (shrink)

In order to graduate, Florida's high school students by law must learn that Communism is evil, dangerous, and fallacious. All students must learn that the U.S. produces the highest standard of living and more freedom than any other economic system on earth. State universities in Florida are creating a curriculum to implement the Americanism versus Communism Act of 1961 and the Free Enterprise and Consumer Education Act of 1975. ;The Florida Department of Education says that ideology, noncritical thinking, is superior (...) to critical thinking and that the superiority of the U.S. political economy and the dignity of the individual rest in part on peoples' being able to express themselves freely. ;Florida's A.V.C. and F.E.C.E.A. proponents have a special way of convincing audiences that they will develop loyalty while at the same time imposing a set of ideas not open to question. ;This work argues that, intentionally or not, Florida's legislature and Department of Education have set up an official ideology and the mechanism to purvey it. This ideological approach defeats the aim of comparing systems objectively. While the stated aim is to promote critical thinking, the D.O.E.'s special philosophy underscoring words like "democracy" results in indoctrinating students with a questionable description of the U.S. system. (shrink)

I argue that three main interpretations of the aim of Russell’s early logicism in The Principles of Mathematics (1903) are mistaken, and propose a new interpretation. According to this new interpretation, the aim of Russell’s logicism is to show, in opposition to Kant, that mathematical propositions have a certain sort of complete generality which entails that their truth is independent of space and time. I argue that on this interpretation two often-heard objections to Russell’s logicism, deriving from Gödel’s incompleteness theorem (...) and from the non-logical character of some of the axioms of Principia Mathematica respectively, can be seen to be inconclusive. I then proceed to identify two challenges that Russell’s logicism, as presently construed, faces, but argue that these challenges do not appear unanswerable. (shrink)

"A theoretical juice extractor: The role of mathematics in scientific explanation". There have recently been proposed cases where, supposedly, mathematics would play a genuinely explanatory role in science. These have been divided into those situations where the explanatory role would be played by mathematical operations, and those where it would be played by mathematical entities. In this article, I analyze some of these purported cases and argue that claims that mathematics can be genuinely explanatory are unfounded. Throughout my discussion, I (...) emphasize the representational role of mathematics, as opposed to its supposed explanatory role: the role of mathematics, even in the cases that I discuss, is to represent physical facts and help draw inferences about those facts. (shrink)

Résumé. En trente ans, le rapport d’Internet à l’entreprise scientifique a changé. Nous sommes passés d’un instrument de collaboration scientifique à un dispositif de réseaux sociaux qui assure la plus large diffusion à l’irrationalisme et à l’alt-factualisme. Pour comprendre ce changement, il convient de réexaminer les mécanismes de la convergence des opinions. La première source de cette convergence est l’existence d’un monde commun, qui nous expose aux mêmes faits et qui détermine par révision successive des croyances de chacun, sans que (...) la moindre communication soit requise, une convergence asymptotique des opinions, et ce quelles que soient les croyances de départ. Naturellement, si les faits connus par les uns sont communiqués aux autres, cette convergence s’accélère considérablement, si bien que nous devons considérer la connaissance comme le résultat d’une activité collective, et l’échange d’informations comme l’une de ses sources nécessaires. D’une part, cette coopération épistémique est naturelle et facilement implémentable : en vertu des propriétés caractéristiques de l’information, qui est un bien que nous conservons lorsque nous le donnons, le partage informationnel est exempt des difficultés qui affectent généralement la coopération. D’autre part, cette coopération est intrinsèquement productive : la collaboration entre celui qui sait que φ et celui qui sait que φ implique ψ met chacun des deux agents en possession de la connaissance de ψ, qui n’était pourtant détenue par personne avant l’échange. L’internet des débuts donne à cette coopération informationnelle une ampleur inégalée. La situation actuelle est différente. Un certain nombre de facteurs, dont la porosité croissante entre les scientifiques et le large public, ont mis au premier plan l’échange, non des informations factuelles, mais celui des opinions. Les biais notoires de la psychologie humaine, au premier rang desquels le biais de confirmation, tendent à inverser le rapport entre les faits et les opinions. On cherche les faits confirmant les opinions entretenues, on met en doute les faits récalcitrants et l’on crée des faits corroborants. En bref, on passe d’une problématique de la coopération à une logique de l’affiliation, segmentant l’internet en groupes doxastiques homogènes. (shrink)

sBuilding models as a practical aspect of ecological theory has as a principal purpose the determination of relations in formal language. In this paper, the authors provide a formalization of ecological models based on impure systems theory. Impure systems contain objects and subjects: subjects are human beings. We can distinguish a person as an observer that by definition is the subject himself and part of the system. In this case he acquires the category of object. Objects are significances, which are (...) the consequence of perceptual beliefs on the part of the subject about material or energetic objects with certain characteristics. The impure system approach is as follows: objects are perceptual significances of material or energetic objects. The set of these objects will form an impure set of the first order. The existing relations between these relative objects will be of two classes: transactions of matter and/or energy and inferential relations. Transactions can have alethic modality: necessity, possibility, impossibility and contingency. In this work we define measures which let us choose the more suitable variables to relate both the model with the ecosystem and with different models. In this way we define different comparison indexes. (shrink)

This study examines the relativistic aspects of Arnold Schoenberg's harmonic and aesthetic theories in the light of a framework of ideas presented in the early writings of Ludwig Wittgenstein, the logician, philosopher of language, and Schoenberg's contemporary and Austrian compatriot. The author has identified correspondences between the writings of Schoenberg, the early Wittgenstein, and the Vienna Circle of philosophers, on a wide range of topics and themes. Issues discussed include the nature and limits of language, musical universals, theoretical conventionalism, word-to-world (...) correspondence in language, the need for a fact- and comparison-based approach to art criticism, and the nature of music-theoretical formalism and mathematical modeling. Schoenberg and Wittgenstein are shown to have shared a vision that is remarkable for its uniformity and balance, one that points toward the reconciliation of the positivist-relativist dualism that has dominated recent discourse in music theory. Contrary to earlier accounts of Schoenberg's harmonic and aesthetic relativism, this study identifies a solid epistemological core underlying his thought, a view that was very much in step with Wittgenstein and the Vienna Circle, and thereby with the most vigorous and forward-looking stream in early twentieth century intellectual history. (shrink)

In this paper, we describe "metaphysical reductions", in which the well-defined terms and predicates of arbitrary mathematical theories are uniquely interpreted within an axiomatic, metaphysical theory of abstract objects. Once certain (constitutive) facts about a mathematical theory T have been added to the metaphysical theory of objects, theorems of the metaphysical theory yield both an analysis of the reference of the terms and predicates of T and an analysis of the truth of the sentences of T. The well-defined terms and (...) predicates of T are analyzed as denoting abstract objects and abstract relations, respectively, in the background metaphysics, and the sentences of T have a reading on which they are true. After the technical details are sketched, the paper concludes with some observations about the approach. One important observation concerns the fact that the proper axioms of the background theory abstract objects can be reformulated in a way that makes them sound more like logical axioms. Some philosophers have argued that we should accept (something like) them as being logical. (shrink)

According to the species of neo-logicism advanced by Hale and Wright, mathematical knowledge is essentially logical knowledge. Their view is found to be best understood as a set of related though independent theses: (1) neo-fregeanism-a general conception of the relation between language and reality; (2) the method of abstraction-a particular method for introducing concepts into language; (3) the scope of logic-second-order logic is logic. The criticisms of Boolos, Dummett, Field and Quine (amongst others) of these theses are explicated and assessed. (...) The issues discussed include reductionism, rejectionism, the Julius Caesar problem, the Bad Company objections, and the charge that second-order logic is set theory in disguise. (shrink)

Jurado: Mario Gómez-Torrente (presidente), Miguel Ángel Fernández Vargas (vocal), Santiago Echeverri Saldarriaga (secretario). [Graduado con Mención Honorífica.].

The distinction between the a priori and the a posteriori is an old and influential one. But both the distinction itself and the crucial notion of a priori knowledge face powerful philosophical challenges. Many philosophers worry that accepting the a priori is tantamount to accepting epistemic magic. In contrast, this Element argues that the a priori can be formulated clearly, made respectable, and used to do important epistemological work. The author's conception of the a priori and its role falls short (...) of what some historical proponents of the notion may have hoped for, but it allows us to accept and use the notion without abandoning either naturalism or empiricism, broadly understood. This Element argues that we can accept and use the a priori without magic. (shrink)

Neo-Fregeanism contends that knowledge of arithmetic may be acquired by second-order logical reflection upon Hume's principle. Heck argues that Hume's principle doesn't inform ordinary arithmetical reasoning and so knowledge derived from it cannot be genuinely arithmetical. To suppose otherwise, Heck claims, is to fail to comprehend the magnitude of Cantor's conceptual contribution to mathematics. Heck recommends that finite Hume's principle be employed instead to generate arithmetical knowledge. But a better understanding of Cantor's contribution is achieved if it is supposed that (...) Hume's principle really does inform arithmetical practice. More generally, Heck's arguments misconceive the epistemological character of neo-Fregeanism. (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)

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

Wittgenstein's conception of logical and conceptual truth is often thought to rival that of the logical positivists. This paper argues that there are important respects in which these conceptions complement each other. Analyticity, in the positivists' sense, coincides, not with Wittgenstein's notion of a grammatical proposition, but rather with his notion of a tautology. Grammatical propositions can usually be construed as analyticity postulates in Carnap's sense of the term. This account of grammatical and analytic propositions will be illustrated by appeal (...) to logical, conceptual and arithmetic truths. Its consequences for an analysis of corresponding modal notions will be indicated. (shrink)

Empiricist theories of knowledge are attractive for they offer the prospect of a unitary theory of knowledge based on relatively well understood physiological and cognitive processes. Mathematical knowledge, however, has been a traditional stumbling block for such theories. There are three primary features of mathematical knowledge which have led epistemologists to the conclusion that it cannot be accommodated within an empiricist framework: 1) mathematical propositions appear to be immune from empirical disconfirmation; 2) mathematical propositions appear to be known with certainty; (...) and 3) mathematical propositions are necessary. Epistemologists who believe that some nonmathematical propositions, such as logical or ethical propositions, cannot be known a posteriori also typically appeal to the three factors cited above in defending their position. The primary purpose of this paper is to examine whether any of these alleged features of mathematical propositions establishes that knowledge of such propositions cannot be a posteriori. (shrink)

In this paper, we investigate (1) what can be salvaged from the original project of "logicism" and (2) what is the best that can be done if we lower our sights a bit. Logicism is the view that "mathematics is reducible to logic alone", and there are a variety of reasons why it was a non-starter. We consider the various ways of weakening this claim so as to produce a "neologicism". Three ways are discussed: (1) expand the conception of logic (...) used in the reduction, (2) allow the addition of analytic-sounding principles to logic so that the reduction is not to "logic alone" but to logic and truths knowable a priori, and (3) revise the conception of "reducible". We show how the current versions of neologicism fit into this classification scheme, and then focus on a kind of neologicism which we take to have the most potential for achieving the epistemological goals of the original logicist project. We argue that that the "weaker" the form of neologicism, the more likely it is to be a new form of logicism, and show how our preferred system, though mathematically weak, is metaphysically and epistemogically strong, and can "reduce" arbitrary mathematical theories to logic and analytic truths, if given a legitimate new sense of "reduction". (shrink)

The goal of this paper is to explain the nature of philosophy as a distinct science with its own subject-matter. This is achieved through a comparative analysis of mathematical and philosophical knowledge that reveals a profound similarity between mathematics and philosophy as mutually complementary sciences exploring the field of abstract entities that can be comprehended only by purely a priori theoretical inquiry. By considering this complementarity, a general definition of philosophy can be obtained by dualizing the traditional Aristotelian definition of (...) mathematics as the “science of quantity”. Philosophy should thus be interpreted as an a priori science of the pure qualitative attributes of being. (shrink)

The problem of semantic content is the problem of explicating those features of brain processes by virtue of which they may properly be thought to possess meaning or reference. This paper criticizes the account of semantic content associated with fodor's version of cognitive science, And offers an alternative account based on mathematical communication theory. Its key concept is that of a neuronal representation maintaining a high-Level of mutual information with a designated external state of affairs under changing conditions of perceptual (...) presentation. (shrink)

The problem of semantic content is the problem of explicating those features of brain processes by virtue of which they may properly be thought to possess meaning or reference. This paper criticizes the account of semantic content associated with fodor's version of cognitive science, And offers an alternative account based on mathematical communication theory. Its key concept is that of a neuronal representation maintaining a high-Level of mutual information with a designated external state of affairs under changing conditions of perceptual (...) presentation. (shrink)

W ciągu 30 lat dokonała się głęboka przemiana w relacji między Internetem a aktywnością naukową. Nastąpiło przesunięcie funkcji Internetu, który przestał być jedynie instrumentem współpracy akademickiej, a stał się również narzędziem używanym przez media społecznościowe, zezwalające na maksymalną dyfuzję irracjonalizmu i starego faktualizmu. Aby zrozumieć tę przemianę, należy ponownie zbadać mechanizmy zbieżności opinii. Wynika ona z istnienia wspólnego świata, który ukazuje nam te same fakty i determinuje nieustanną rewizję przekonań każdej osoby. W tym procesie asymptotycznej zbieżności opinii nie wymaga się (...) żadnej komunikacji między jednostkami. Oczywiście, jeśli fakty znane przez jednych są komunikowane innym, konwergencja znacznie przyspiesza, i to tak bardzo, że wiedzę musimy ująć jako rezultat zbiorowej aktywności, w której wymiana informacji jest jednym z jej istotnych źródeł. Z jednej strony takie poznawcze współdziałanie wydaje się naturalne i łatwe do implementacji. Biorąc pod uwagę właściwości informacji, które przechowujemy, nawet jeśli je udostępniamy, to dzielenie się informacją przestaje być przedmiotem zwykłych trudności związanych ze współpracą. Z drugiej strony takie współdziałanie jest produktywne z samej natury: współpraca między kimś, kto zna φ, i kimś, kto wie, że φ implikuje ψ u obydwu jako mających wiedzę ψ, której żaden z nich nie posiadał przed wymianą informacji. Wczesny Internet pozwolił na ekstremalną intensywność tej informacyjnej współpracy. Obecnie mamy do czynienia z odmienną sytuacją. Poszczególne wyznaczniki rosnącej porowatości między naukowcami a opinią publiczną wzmocniły rolę wymiany. Nie jest to już wymiana informacji, lecz wymiana opinii. Stronniczość tkwiąca w naturze ludzkiej, a szczególnie potwierdzenie tej stronniczości, prowadzi do odwrócenia relacji między faktami a opiniami. Szukamy faktów potwierdzających podzielane przez nas opinie, wątpiąc w te, które ich nie określają, i tworzymy fakty do potwierdzenia tego, o czym jesteśmy przekonani. Dlatego przechodzimy od współdziałania do przynależności, dzieląc Internet na homogeniczne grupy wyznawców. (shrink)

The purpose of this dissertation is to present historical evidence in favor of the thesis that many forms of dichotomy appearing in the history of epistemology are related to the duality represented by the mathematical concepts of continuity and discreteness. Parts 1 and 2 give a descriptive and historical account of epistemological dichotomies appearing during the development of mathematics and logic. In part 3, the implications of these dichotomies for general philosophy are explored by means of a collage of analytic, (...) historical, and factual materials, accompanied by brief commentaries, and aimed at establishing the following points: Acceptance of the primordial intuition of continuity distinguishes the constructive model from the structural model in epistemology; The constructive model emphasizes meaningful thought processes of the reflecting and acting subject, while the structural model emphasizes publicly recognizable and communicable thought processes based upon logic, linguistics, or natural law; An activity which translates between the constructive model and the structural model by means of symbols which include iconic components is to be called "aesthetic"; The function of aesthetics is to bring about harmonious and productive relations between individuals and the larger wholes to which they belong. ;The conclusion is reached that, from the point of view of the individual, aesthetics is important for motivation and for the development of cognitive powers and moral sensibilities; from the point of view of the species, aesthetics is important for long-term survival. Therefore, logic and aesthetics are of equal and complementary importance and deserve equal consideration in the study of epistemology. (shrink)