Certain mathematical objects bear the name “pathological”. They either occur as unexpected and unwilling in mathematical research practice, or are constructed deliberately, for instance in order to delimit the scope of application of a theorem. I discuss examples of mathematical pathologies and the circumstances of their emergence. I focus my attention on the creative role of pathologies in the development of mathematics. Finally, I propose a few reflections concerning the degree of cognitive accessibility of mathematical objects. I believe that the (...) problems discussed in the paper may attract the attention of philosophers interested in concept formation and the development of mathematical ideas. (shrink)

Cantor’s theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one association between two sets. As is well known, all countable infinite sets have the same ‘size’ in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collectionAis properly included in a collectionBthen the ‘size’ ofAshould be less than the (...) ‘size’ ofB(part–whole principle). This second intuition was not developed mathematically in a satisfactory way until quite recently. In this article I begin by reviewing the contributions of some thinkers who argued in favor of the assignment of different sizes to infinite collections of natural numbers (Thabit ibn Qurra, Grosseteste, Maignan, Bolzano). Then, I review some recent mathematical developments that generalize the part–whole principle to infinite sets in a coherent fashion (Katz, Benci, Di Nasso, Forti). Finally, I show how these new developments are important for a proper evaluation of a number of positions in philosophy of mathematics which argue either for the inevitability of the Cantorian notion of infinite number (Gödel) or for the rational nature of the Cantorian generalization as opposed to that, based on the part–whole principle, envisaged by Bolzano (Kitcher). (shrink)

While the Enlightenment promoted thinking for oneself independent of religious authority, the ‘Endarkenment’ (Millgram 2015) concerns deference to a new authority: the specialist, a hyperspecializer. Non-specialists need to defer to such authorities as they are unable to understand their reasoning. Millgram describes how humans are capable of being serial hyperspecializers, able to move from one specialism to another. We support the basic thrust of Millgram’s position, and seek to articulate how the core idea is deployed in very different ways in (...) relation to extremely different philosophical areas. We attend to the issue of the degree of isolation of different specialists and we urge greater emphasis on parallel hyperspecialization, which describes how different specialisms can be embodied in one person at one time. (shrink)

A debated issue in the mathematical foundations in at least the last two decades is whether one can plausibly argue for the merits of treating undecidable questions of mathematics, e.g., the Continuum Hypothesis, by relying on the existence of a plurality of set-theoretical universes except for a single one, i.e., the well-known set-theoretical universe V associated with the cumulative hierarchy of sets. The multiverse approach has some varying versions of the general concept of multiverse yet my intention is to primarily (...) address ontological multiversism as advocated, for instance, by Hamkins or Väätänen, precisely for the reason that they proclaim, to the one or the other extent, ontological preoccupations for the introduction of respective multiverse theories. Taking also into account Woodin’s and Steel’s multiverse versions, I take up an argumentation against multiversism, and in a certain sense against platonism in mathematical foundations, mainly on subjectively founded grounds, while keeping an eye on Clarke-Doane’s concern with Benacerraf’s challenge. I note that even though the paper is rather technically constructed in arguing against multiversism, the non-negligible philosophical part is influenced to a certain extent by a phenomenologically motivated view of the matter. (shrink)

In a paper titled, “The Unreasonable Effectiveness of Mathematics”, published 20 years after Wigner’s seminal paper, the mathematician Richard W. Hamming discussed what he took to be Wigner’s problem of Unreasonable Effectiveness and offered some partial explanations for this phenomenon. Whether Hamming succeeds in his explanations as answers to Wigner’s puzzle is addressed by other scholars in recent years I, on the other hand, raise a more fundamental question: does Hamming succeed in raising the same question as Wigner? The answer (...) is no. My goal is to show that Hamming’s reading misses Wigner’s highly original formulation of the problem.Through a close and contextual reading of Wigner’s work, as I will show, we are led in new directions in addressing and solving the applicability problem. (shrink)

In his seminal 1960 paper, the physicist Eugene Wigner formulated the question of the applicability of mathematics in physics in a way nobody had before. This formulation has been entirely overlooked due to an exclusive concern with solving Wigner’s problem and explaining the effectiveness of mathematics in the natural sciences, in one way or another. Many have attempted to attribute Wigner’s unjustified conclusion—that mathematics is unreasonably effective in the natural sciences—to his formalist views on mathematics. My goal is to show (...) that this reading misses out on Wigner’s highly original formulation of the problem which is presented throughout his body of work in physics as well as in philosophy. This formulation, as I will show, leads us in a new direction in solving the applicability problem. (shrink)

In his seminal 1960 paper, the physicist Eugene Wigner formulated the question of the applicability of mathematics in physics in a way nobody had before. This formulation has been entirely overlooked due to an exclusive concern with solving Wigner’s problem and explaining the effectiveness of mathematics in the natural sciences, in one way or another. Many have attempted to attribute Wigner’s unjustified conclusion—that mathematics is unreasonably effective in the natural sciences—to his formalist views on mathematics. My goal is to show (...) that this reading misses out on Wigner’s highly original formulation of the problem which is presented throughout his body of work in physics as well as in philosophy. This formulation, as I will show, leads us in a new direction in solving the applicability problem. (shrink)

Recent experimental evidence from developmental psychology and cognitive neuroscience indicates that humans are equipped with unlearned elementary mathematical skills. However, formal mathematics has properties that cannot be reduced to these elementary cognitive capacities. The question then arises how human beings cognitively deal with more advanced mathematical ideas. This paper draws on the extended mind thesis to suggest that mathematical symbols enable us to delegate some mathematical operations to the external environment. In this view, mathematical symbols are not only used to (...) express mathematical concepts—they are constitutive of the mathematical concepts themselves. Mathematical symbols are epistemic actions, because they enable us to represent concepts that are literally unthinkable with our bare brains. Using case-studies from the history of mathematics and from educational psychology, we argue for an intimate relationship between mathematical symbols and mathematical cognition. (shrink)

The nonidentity problem is a deep puzzle challenging the moral intuition that what is bad must be bad for someone. The first part of the paper constructs a new theory of harming, whereas the second part builds on the conclusions of the first to offer a new solution to the NIP. The first part discusses the neglected question of when a burden inflicted in the context of overall benefitting can be discretized as a separate entity—only when it can, is it (...) possible to identify the burden as harm, and only then is it possible to harm in bringing into overall good existence. The second part explains how, in those cases where creating is indeed harming, we can use the logic of concept expansion to construct a concept of wronging that applies to creation cases. (shrink)

The 1981 Dahlem conference was a catalyst for contemporary evolutionary developmental biology (Evo-devo). This introductory chapter rehearses some of the details of the history surrounding the original conference and its associated edited volume, explicates the philosophical problem of conceptual change that provided the rationale for a workshop devoted to evaluating the epistemic revisions and transformations that occurred in the interim, explores conceptual change with respect to the concept of evolutionary novelty, and highlights some of the themes and patterns in the (...) different contributions to the present volume, Conceptual Change in Biology: Scientific and Philosophical Perspectives on Evolution and Development. (shrink)

Evo-Devo exhibits a plurality of scientific “cultures” of practice and theory. When are the cultures acting—individually or collectively—in ways that actually move research forward, empirically, theoretically, and ethically? When do they become imperialistic, in the sense of excluding and subordinating other cultures? This chapter identifies six cultures – three /styles/ (mathematical modeling, mechanism, and history) and three /paradigms/ (adaptationism, structuralism, and cladism). The key assumptions standing behind, under, or within each of these cultures are explored. Characterizing the internal structure of (...) the cultures is necessary for understanding how they collaborate or compete, and how they are fragmented or integrated, in the rich interdisciplinary /trading zone/ (Galison 1997) of Evo-Devo. Evo-Devo is an important example of how science can progress through a radical plurality of perspectives and cultures. (shrink)

The paper has three objectives: to expound a set-theoretical triplet model of concepts; to introduce some triplet relations (symbolic, logical, and mathematical formalization; equivalence, intersection, disjointness) between object concepts, and to instantiate them by relations between certain physical object concepts.

Wittgenstein has shown that that life, in the sense that applies in the first place to human beings, is inherently linguistic. In this paper, I ask what is involved in language, given that it is thus essential to life, answering that language – or concepts – must be both alive and the ground for life. This is explicated by a Wittgensteinian series of entailments of features. According to the first feature, concepts are not intentional engagements. The second feature brings life (...) back to concepts by describing them as inflectible: Attitudes, actions, conversations and other engagements inflect concepts, i.e., concepts take their particular characters in our actual engagements. However, inflections themselves would be reified together with the life they ground unless they could preserve the openness of concepts: hence the third feature of re-inflectibility. Finally, the openness of language must be revealed in actual life. This entails the possibility of conceptual ambivalence. (shrink)

Traditional attempts to delineate the distinctive rationality of modern science have taken it for granted that the purpose of empirical research is to test judgments. The choice of concepts to use in those judgments is therefore seen either a matter of indifference (Popper) or as important choice which must be made, so to speak, in advance of all empirical research (Carnap). I argue that scientific method aims precisely at empirical testing of concepts, and that even the simplest scientific ex- periment (...) or observation results in conceptual change. (shrink)

We consider an approach to some philosophical problems that I call the Method of Conceptual Articulation: to recognize that a question may lack any determinate answer, and to re-engineer concepts so that the question acquires a definite answer in such a way as to serve the epistemic motivations behind the question. As a case study we examine “Galileo’s Paradox”, that the perfect square numbers seem to be at once as numerous as the whole numbers, by one-to-one correspondence, and yet less (...) numerous, being a proper subset. I argue that Cantor resolved this paradox by a method at least close to that proposed—not by discovering the true nature of cardinal number, but by articulating several useful and appealing extensions of number to the infinite. Galileo was right to suggest that the concept of relative size did not apply to the infinite, for the concept he possessed did not. Nor was Bolzano simply wrong to reject Hume’s Principle (that one-to-one correspondence implies equal number) in the infinitary case, in favor of Euclid’s Common Notion 5 (that the whole is greater than the part), for the concept of cardinal number (in the sense of “number of elements”) was not clearly defined for infinite collections. Order extension theorems now suggest that a theory of cardinality upholding Euclid’s principle instead of Hume’s is possible. Cantor’s refinements of number are not the only ones possible, and they appear to have been shaped by motivations and fruitfulness, for they evolved in discernible stages correlated with emerging applications and results. Galileo, Bolzano, and Cantor shared interests in the particulate analysis of the continuum and in physical applications. Cantor’s concepts proved fruitful for those pursuits. Finally, Gödel was mistaken to claim that Cantor’s concept of cardinality is forced on us; though Gödel gives an intuitively compelling argument, he ignores the fact that Euclid’s Common Notion is also intuitively compelling, and we are therefore forced to make a choice. The success of Cantor’s concept of cardinality lies not in its truth (for concepts are not true or false), nor its uniqueness (for it is not the only extension of number possible), but in its intuitive appeal, and most of all, its usefulness to the understanding. (shrink)

I develop Special Relativity with backward-light-cone simultaneity, which I call, for reasons made clear in the paper, ‘Apparent Simultaneity’. In the first section I show some advantages of this approach. I then develop the kinematics in the second section. In the third section I apply the approach to the Twins Paradox: I show how it removes the paradox, and I explain why the paradox was a result of an artificial symmetry introduced to the description of the process by Einstein’s simultaneity (...) definition. In the fourth section I discuss some aspects of dynamics. I conclude, in a fifth section, with a discussion of the nature of light, according to which transmission of light energy is a form of action at a distance. -/- deposited in Pittsburgh PhilSci-Archive, March 2007. (shrink)

This thesis comprises three main chapters—each comprising one relatively standalone paper. The unifying theme is fragmentalism about truth, which is the view that the predicate “true” either expresses distinct concepts or expresses distinct properties. -/- In Chapter 1, I provide a formal development of alethic pluralism. Pluralism is the view that there are distinct truth properties associated with distinct domains of subject matter, where a truth property satisfies certain truth-characterizing principles. On behalf of pluralists, I propose an account of logic (...) and semantics that shows how they can answer central conceptual and logical challenges for their view. -/- In Chapter 2, I motivate and develop a modal account of propositions on the basis of an iterative conception of propositions, where the modality is logico-mathematical. The modal account of propositions takes the conception to motivate an inherently potential hierarchy of propositions. I show that the account helps provide satisfying solutions to the intensional paradoxes of Russell-Myhill, Kaplan, and Prior. -/- In Chapter 3, I propose that “true” is polysemous. I suggest that “true” is initially polysemous between correspondence truth and disquotational truth, and further polysemous between the meanings corresponding to the subconcepts of the concept truth generated by the indefinite extensibility of that concept. I show that the proposal provides satisfying solutions to the semantic paradoxes. (shrink)