Besides mechanistic explanations of phenomena, which have been seriously investigated in the last decade, biology and ecology also include explanations that pinpoint specific mathematical properties as explanatory of the explanandum under focus. Among these structural explanations, one finds topological explanations, and recent science pervasively relies on them. This reliance is especially due to the necessity to model large sets of data with no practical possibility to track the proper activities of all the numerous entities. The paper first defines (...) class='Hi'>topological explanations and then explains why topological explanations and mechanisms are different in principle. Then it shows that they are pervasive both in the study of networks—whose importance has been increasingly acknowledged at each level of the biological hierarchy—and in contexts where the notion of selective neutrality is crucial; this allows me to capture the difference between mechanisms and topological explanations in terms of practical modelling practices. The rest of the paper investigates how in practice mechanisms and topologies are combined. They may be articulated in theoretical structures and explanatory strategies, first through a relation of constraint, second in interlevel theories, or they may condition each other. Finally, I explore how a particular model can integrate mechanistic informations, by focusing on the recent practice of merging networks in ecology and its consequences upon multiscale modelling. (shrink)

The aim of this article is to investigate specific aspects connected with visualization in the practice of a mathematical subfield: low-dimensional topology. Through a case study, it will be established that visualization can play an epistemic role. The background assumption is that the consideration of the actual practice of mathematics is relevant to address epistemological issues. It will be shown that in low-dimensional topology, justifications can be based on sequences of pictures. Three theses will be defended. First, the representations (...) used in the practice are an integral part of the mathematical reasoning. As a matter of fact, they convey in a material form the relevant transitions and thus allow experts to draw inferential connections. Second, in low-dimensional topology experts exploit a particular type of manipulative imagination which is connected to intuition of two- and three-dimensional space and motor agency. This imagination allows recognizing the transformations which connect different pictures in an argument. Third, the epistemic—and inferential—actions performed are permissible only within a specific practice: this form of reasoning is subject-matter dependent. Local criteria of validity are established to assure the soundness of representationally heterogeneous arguments in low-dimensional topology. (shrink)

We study the Medvedev degrees of mass problems with distinguished topological properties, such as denseness, closedness, or discreteness. We investigate the sublattices generated by these degrees; the prime ideal generated by the dense degrees and its complement, a prime filter; the filter generated by the nonzero closed degrees and the filter generated by the nonzero discrete degrees. We give a complete picture of the relationships of inclusion holding between these sublattices, these filters, and this ideal. We show that the (...) sublattice of the closed Medvedev degrees is not a Brouwer algebra. We investigate the dense degrees of mass problems that are closed under Turing equivalence, and we prove that the dense degrees form an automorphism base for the Medvedev lattice. The results hold for both the Medvedev lattice on the Baire space and the Medvedev lattice on the Cantor space. (shrink)

Electron-phonon interactions appear to explain some of, but not all, the features of the dispersive kinks in recently observed by angle-resolved photoemission spectroscopy. The remaining anomalies have led to suggestions that magnon, and not phonon, interactions are involved. If one abandons the effective-medium approximation and replaces it with dopant-centred filamentary percolation, a picture emerges that is consistent not only with the photoemission experiments but also with tunnelling, infrared and neutron scattering data.

Reverse mathematics is a program in the foundations of mathematics founded by Friedman and developed extensively by Simpson and others. The aim of RM is to find the minimal axioms needed to prove a theorem of ordinary, that is, non-set-theoretic, mathematics. As suggested by the title, this paper deals with the study of the topological notions of dimension and paracompactness, inside Kohlenbach’s higher-order RM. As to splittings, there are some examples in RM of theorems A, B, C such that (...) A ↔, that is, A can be split into two independent parts B and C, and the aforementioned topological notions give rise to a number of splittings involving highly natural A, B, C. Nonetheless, the higher-order picture is markedly different from the second-one: in terms of comprehension axioms, the proof in higher-order RM of, for example, the paracompactness of the unit interval requires full second-order arithmetic, while the second-order/countable version of paracompactness of the unit interval is provable in the base theory RCA 0. We obtain similarly “exceptional” results for the Urysohn identity, the Lindelöf lemma, and partitions of unity. We show that our results exhibit a certain robustness, in that they do not depend on the exact definition of cover, even in the absence of the axiom of choice. (shrink)

Our reality constitutes itself as being one of pictures. Landscape is a product of aesthetic reflection as well as the perception of reality and virtual reality of the first order (VR 1). Pictorial representation of a landscape is virtual reality of the second order (VR 2). A picture is a structure of relations with a specific topology or an interrelationship. A picture is set in relation. Topology relates to relational similarities and differences as well as their transfer into other (...) interrelationships, other topologies. The differences between nature, landscape (VR 1), metaphor of landscape and pictorial representation of landscape, etc. (VR 2) describe a change of the interrelationship. These changes happen in a physical-psychic-mental production of reality. We are involved in the events of particular relationships of the picture. The Topology of Being and Appearance reflects the inconspicuous similarities of relations and proportions in the relationship of the world in association with our physical-psychic-mental existential orientation in the world. (shrink)

The search for a systems‐level picture of metabolism as a web of molecular interactions provides a paradigmatic example of how the methods used to characterize a system can bias the interpretation of its functional meaning. Metabolic maps have been analyzed using novel techniques from network theory, revealing some non‐trivial, functionally relevant properties. These include a small‐world structure and hierarchical modularity. However, as discussed here, some of these properties might actually result from an inappropriate way of defining network interactions. Starting from (...) the so‐called bipartite organization of metabolism, where the two meaningful subsets (reactions and metabolites) are considered, most current works use only one of the subsets by means of so‐called graph projections. Unfortunately, projected graphs often ignore relevant biological and chemical constraints, thus leading to statistical artifacts. Some of these drawbacks and alternative approaches need to be properly addressed. (shrink)

Robustness is a pervasive property of living systems, instantiated at all levels of the biological hierarchies. As several other usual concepts in evolutionary biology, such as plasticity or dominance, it has been questioned from the viewpoint of its consequences upon evolution as well as from the side of its causes, on an ultimate or proximate viewpoint. It is therefore equally the explanandum for some enquiries in evolution in ecology, and the explanans for some interesting evolutionary phenomena such as evolvability. This (...) epistemological fact instantiates general property of biological evolution that I call “explanatory reversibility”. In this chapter, I attempt to systematize the explanatory projects regarding robustness by distinguishing a set of epistemological questions. Are they the various expressions of one general project with specific key concepts and methods, or very disparate epistemic projects, unified by the mere homonymy of the term “robustness”? More precisely, are there specific kinds of explanations suited to explain robustness? Finally, how does robustness as an explanandum connect with other explananda in which evolutionists have been massively interested recently such as complexity, modularity or evolvability? After having initially explored various meanings of the concept of robustness and surveyed its instances in biology, I will propose a distinction between mechanical and structural explanations of robustness in evolutionary and functional biology. Then, among the latter, I will highlight the class of “topological explanations,” and the subclass of explanations based on networks, as a major explanatory tool to address robustness. Focusing on evolutionary issues, I will eventually address the “explanatory reversibility” of robustness and consider its relation to key evolutionary concepts that are also explanatorily revertible such as modularity, evolvability and complexity. (shrink)

Computational modeling should play a central role in philosophy. In this introduction to our topical collection, we propose a small topology of computational modeling in philosophy in general, and show how the various contributions to our topical collection ft into this overall picture. On this basis, we describe some of the ways in which computational models from other disciplines have found their way into philosophy, and how the principles one found here still underlie current trends in the feld. Moreover, we (...) argue that philosophers contribute to computational modeling not only by building their own models, but also by thinking about the various applications of the method in philosophy and the sciences. In this context, we note that models in philosophy are usually simple, while models in the sciences are often more complex and empirically grounded. Bridging certain methodological gaps that arise from this discrepancy may prove to be challenging and fruitful for the further development of computational modeling in philosophy and beyond. (shrink)

Beyond Peaceful Coexistence: The Emergence of Space, Time and Quantum brings together leading academics in mathematics and physics to address going beyond the 'peaceful coexistence' of space-time descriptions (local and continuous ones) and quantum events (discrete and non-commutative ones). Formidable challenges waiting beyond the Standard Model require a new semantic consistency within the theories in order to build new ways of understanding, working and relating to them. The original A. Shimony meaning of the peaceful coexistence (the collapse postulate and non-locality) (...) appear to be just the tip of the iceberg in relation to more serious fundamental issues across physics as a whole. Chapters in this book present perspectives on emergent, discrete, geometrodynamic and topological approaches, as well as a new interpretative spectrum of quantum theories after Copenhagen, discrete time theories, time-less approaches and 'super-fluid' pictures of space-time. As well as stimulating further research among established theoretical physicists, the book can also be used in courses on the philosophy and mathematics of theoretical physics. (shrink)

What is the metaphysical relationship between the fetus/embryo and the pregnant organism? In this paper I apply a substance metaphysics view developed by Barry Smith and Berit Brogaard to argue, on the basis of topological connectedness, that fetuses/embryos are Lady-Parts: part of the maternal organism up until birth. This leaves two options. Either mammalian organisms begin at birth, or we revise our conception of organisms such that mammalian organisms can be part of other mammals. The first option has some (...) advantages: it is numerically neat; aligns with an intuitive picture of organisms as physically distinct individuals; and ties ‘coming into existence’ to a suitably recognisable and important event: birth. But it denies that the fetus survives birth, or that human organisms existed prior to their birth. The second option allows us to recognise that human organisms exist prior to and survive their birth, but at a cost: it leaves the question of when an organism comes into existence unanswered, and demands potentially far-reaching conceptual revision across a range of domains. (shrink)

In this paper we study a new approach to classify mathematical theorems according to their computational content. Basically, we are asking the question which theorems can be continuously or computably transferred into each other? For this purpose theorems are considered via their realizers which are operations with certain input and output data. The technical tool to express continuous or computable relations between such operations is Weihrauch reducibility and the partially ordered degree structure induced by it. We have identified certain choice (...) principles such as co-finite choice, discrete choice, interval choice, compact choice and closed choice, which are cornerstones among Weihrauch degrees and it turns out that certain core theorems in analysis can be classified naturally in this structure. In particular, we study theorems such as the Intermediate Value Theorem, the Baire Category Theorem, the Banach Inverse Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Theorem. We also explore how existing classifications of the Hahn—Banach Theorem and Weak Kőnig's Lemma fit into this picture. Well-known omniscience principles from constructive mathematics such as LPO and LLPO can also naturally be considered as Weihrauch degrees and they play an important role in our classification. Based on this we compare the results of our classification with existing classifications in constructive and reverse mathematics and we claim that in a certain sense our classification is finer and sheds some new light on the computational content of the respective theorems. Our classification scheme does not require any particular logical framework or axiomatic setting, but it can be carried out in the framework of classical mathematics using tools of topology, computability theory and computable analysis. We develop a number of separation techniques based on a new parallelization principle, on certain invariance properties of Weihrauch reducibility, on the Low Basis Theorem of Jockusch and Soare and based on the Baire Category Theorem. Finally, we present a number of metatheorems that allow to derive upper bounds for the classification of the Weihrauch degree of many theorems and we discuss the Brouwer Fixed Point Theorem as an example. (shrink)

According to the identity version of spacetime supersubstantivalism, material objects are numerically identical to spacetime regions. While the view has been commended for its parsimony and concordance with physics, it has not properly been assessed in light of the metaphysics of properties. The present essay fills this void by discerning several subspecies of identity supersubstantivalism, corresponding to various property ontologies. Finding them all wanting, the essay develops and defends a novel brand of supersubstantivalism based on the trope ontology. On this (...) view, here dubbed ‘supertropestantivalism’, regions are pluralities of geometrically and topologically alike tropes, some of which bundle together as material objects. The resulting picture is not itself an identity version of supersubstantivalism, but it retains the originally attractive features of the identity version while avoiding the shortcomings of competing ontologies. (shrink)

We continue the exploration of various aspects of divisibility of ultrafilters, adding one more relation to the picture: multiplicative finite embeddability. We show that it lies between divisibility relations \ and \. The set of its minimal elements proves to be very rich, and the \-hierarchy is used to get a better intuition of this richness. We find the place of the set of \-maximal ultrafilters among some known families of ultrafilters. Finally, we introduce new notions of largeness of subsets (...) of \, and compare it to other such notions, important for infinite combinatorics and topological dynamics. (shrink)

Given their commitment to practices, science studies have bestowed considerable attention upon objects. We have the boundary object, the standardized package, the network object, the immutable mobile, the fluid object, even a fire object has entered the scene. However, these objects do not provide us with a way of understanding their historicity. They are timeless, motionless pictures rather than things that change over time, and while enacting ‘historical moments’ they do not make visible the histories they contain within them. (...) What kind of object could embody history and make that history visible? Inspired by Michel Serres, I suggest the folded object is a way to attend to the temporality and spatiality of objects. In this article I explore this new object by unravelling the history of a DNA reference sequence. I show how, ever since it was produced in the early 1980s, attempts have been made to filter race out of the sequence. That effort has failed due to what one could call ‘political noise’. Making and remaking the sequence have left traces that cannot be erased. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Newton's Metaphysics: Essays by Eric SchliesserMarius StanEric Schliesser. Newton's Metaphysics: Essays. Oxford: Oxford University Press, 2021. Pp. 328. Hardback, $99.90.Newton owes his high regard to the quantitative science he left us, but his overall picture of the world had some robustly metaphysical threads woven in as well. Posthumous judgment about the value of these threads has varied wildly. Christian Wolff thought him a metaphysical rustic, as did Hans (...) Reichenbach some two centuries later ("Die Bewegungslehre bei Newton, Leibniz und Huygens," Kant-Studien 29 [1924]: 416–38). In the 1960s, the tide would turn, as Howard Stein and James Edward McGuire separately began to show that Newton's metaphysics was not just sophisticated, but often more compelling than its early modern alternatives (see respectively "Newtonian Spacetime," Texas Quarterly 10 [1967]: 174–200; and Tradition and Innovation: Newton's Metaphysics of Nature [Dordrecht: Kluwer Academic Publishers, 1995]). Eric Schliesser's book unfolds in that same register of appreciative, high scholarship, and it attests to the enduring attraction of its subject.The book grew out of previous discrete papers, to which he added a few more for this occasion. Accordingly, it does not come with a master argument. Rather, it is an in-depth exploration of some key metaphysical themes in Newton. As such, it befits its subject figure, who reflected on themes and concepts while stopping short of working out systems.The first major theme is Newton and Spinozism. The latter does not denote Spinoza's metaphysics. In fact, Schliesser explains, Newton shared with Spinoza some weighty commitments, for example, to space being actually infinite, and to substance monism: "for Newton, there is strictly speaking only one genuine substance," namely, God (33). Rather, "Spinozism" is an interpreter's category for a bundle of three theses: identifying God and nature; denying final causation in the physical world; and the assumption of "blind" metaphysical necessity. Some counted Hobbes and John Toland as Spinozist in this sense, and even Epicurus, avant la lettre. Newton and his followers argued vehemently against this package, as Schliesser shows in chapters 4, 5, and 8. Their chief complaint was that Spinozism is unable to account for the "origin of motion," for a certain type of order, and for the stability of cosmological structure—and to do so in a way that "meets the standards of Newtonian mechanics" (122). This interpretive lens allows Schliesser to draw Kant in, by reading his youthful Theory of Heaven as a possible Spinozist reply to their objections (chapter 3). [End Page 157]A second theme is the metaphysics of mechanics, where Schliesser weaves together several strands. One is the ontology of gravity, for which he offers a novel construal: for Newton, gravity counts as real, but in a qualified sense. Namely, it is not essential and not intrinsic: "even after creation, a lonely partless particle of matter in the universe would not be said to gravitate." And gravity is relational: a "shared quality" of two or more bits of matter, which obtains in virtue of their sharing a nature (19). Another strand is Newton's ontology of time—really, a subtle, difficult topic long neglected. Famously, Newton in Principia defends absolute, true, and mathematical time. The question is what these three qualifiers denote, and whether Newton thought they were synonymous. Schliesser in chapter 7 argues for the provocative view that "absolute" and "true" time are not identical concepts; rather, they pick out different entities. These entities share a common structure, namely, metric and topological. That makes them species of "mathematical" time (179). But, Schliesser contends, Newton has unequal warrant for his two concepts of time.Yet another theme is formal causation. It comes to the fore in chapter 5, which grapples with Newton's opaque idea that space is an "emanative effect." Schliesser explains that here, "emanation picks out a species of formal causation" (147), but in a novel, early modern sense that comes from Bacon, not Aristotle. If space has a formal cause, then so has its twin brother, time, the topic of chapter 7. God is their common formal cause. Then in chapter 6 (coauthored with Zvi Biener) Schliesser adds that, for Newton, laws... (shrink)

Recently, megastable systems have grabbed many researchers’ interests in the area of nonlinear dynamics and chaotic systems. In this paper, the oscillatory terms’ coefficients of the simplest megastable oscillator are forced to blink in time. The forced system can generate an infinitive number of hidden attractors without changing parameters. The behavior of these hidden attractors can be chaotic, tori, and limit cycle. The attractors’ topology of the system seems unique and looks like picture frames. Besides, the existence of different coexisting (...) attractors with different kinds of behaviors reflects the system's high sensitivity. Using the sample entropy algorithm, the system’s complexity for different initial values is assessed. In addition, the circuit of the introduced forced system is designed, and the possibility of implicating the system with analog elements is investigated. (shrink)

Computational modeling should play a central role in philosophy. In this introduction to our topical collection, we propose a small topology of computational modeling in philosophy in general, and show how the various contributions to our topical collection fit into this overall picture. On this basis, we describe some of the ways in which computational models from other disciplines have found their way into philosophy, and how the principles one found here still underlie current trends in the field. Moreover, we (...) argue that philosophers contribute to computational modeling not only by building their own models, but also by thinking about the various applications of the method in philosophy and the sciences. In this context, we note that models in philosophy are usually simple, while models in the sciences are often more complex and empirically grounded. Bridging certain methodological gaps that arise from this discrepancy may prove to be challenging and fruitful for the further development of computational modeling in philosophy and beyond. (shrink)

Introduces philosophers to John Willats' effective new drawing systems vocabulary for describing drawings and related images, also stresses topological-space values in pictures, vs psychology's projective tendencies (illus).

The current implementation of the Neo-Darwinian model of evolution typically assumes that the set of possible phenotypes is organized into a highly symmetric and regular space. Most conveniently, a Euclidean vector space is used, representing phenotypic properties by real-valued variables. Computational work on the biophysical genotype-phenotype model of RNA folding, however, suggests a rather different picture. If phenotypes are organized according to genetic accessibility, the resulting space lacks a metric and can be formalized only in terms of a relatively unfamiliar (...) structure. Patterns of phenotypic evolution—such as punctuation, irreversibility, and modularity—result naturally from the properties of the genotype-phenotype map, which, given the genetic accessibility structure, define accessibility in the phenotype space. The classical framework, however, addresses these patterns exclusively in terms of natural selection on suitably constructed fitness landscapes. Recent work has extended the explanatory level for phenotypic evolution from fitness considerations alone to include the topological structure of phenotype space as induced by the genotype-phenotype map. Lewontin’s notion of “quasi-independence” of characters can also be formalized in topological terms: it corresponds to the assumption that a region of the phenotype space is represented by a product space of orthogonal factors. In this picture, each character corresponds to a factor of a region of the phenotype space. We consider any region of the phenotype space that has a given factorization as a “type”, i.e., as a set of phenotypes that share the same set of phenotypic characters. Thus, a theory of character identity can be developed that is based on the correspondence of local factors in different regions of the phenotype space. (shrink)

In this fascinating but sometimes baffling book, the reader engages with a series of conditionals like the following: “If [the psychiatrist] Clérimbault manifests a delirium, it is because he discovers the tiny hallucinatory perceptions of ether addicts in the folds of clothing”. “If Leibniz’s principles [of identity and sufficient reason] appear to us as cries, it is because each one signals the presence of a class of beings that are themselves crying and draw attention to themselves by these cries...”. Deleuze’s (...) study is concerned with Leibniz and with leibnizianism; with the Baroque considered as a historical period and with the baroque considered as a persistent impulse in architecture, decoration, and in human thinking and system construction. Baroque and baroque are explorations, in his view, of curvilinearity, and Deleuze explores Leibniz’s use of mathematical analogies from topology, projective geometry and the calculus. But his attention in the book is mainly fixed on two Leibnizian images. The first image is that of the two-storied house, typically represented in baroque painting by an angelic realm in the upper half of the picture hovering above the human realm. In Leibniz’s writings, this translates into the world of matter and the world of intelligent, self-conscious spirits on the upper level. The second image is that of the fold, which appears in the pleats and draperies of Baroque costume, sculpture, and interior decoration, and which reappears in Leibniz’s mind-as-folded curtain, in his enveloped organisms, preformed under folds, and in the continuum, from whose recesses ever more numbers or particles can be pulled out. In clothing, waves, the brain, pleats of all sorts, surface disappears into interior. Deconstruction! But is it philosophy? Yes! Though Deleuze pulls his comparison-objects out of every corner of contemporary French culture, they are never arbitrary, and the book is focused clearly on its subject. (shrink)

On our ordinary representations of space, space is composed of indivisible, dimensionless points; extended regions are understood as infinite sets of points. Region-based theories of space reverse this atomistic picture, by taking as primitive several relations on extended regions, and recovering points as higher-order abstractions from regions. Over the years, such theories have focused almost exclusively on the topological and geometric structure of space. We introduce to region-based theories of space a new primitive binary relation (‘qualitative probability’) that is (...) tied to _measure_. It expresses that one region is _smaller than or equal in size_ to another. Algebraic models of our theory are _separation_ _σ_-_algebras with qualitative probability_: \((\mathbb {B}, \ll, \preceq )\), where \(\mathbb {B}\) is a Boolean _σ_-algebra, ≪ is a separation relation on \(\mathbb {B}\), and ≼ is a qualitative probability on \(\mathbb {B}\). We show that from algebraic models of this kind we can, in an interesting class of cases, recover a compact Hausdorff topology _X_, together with a countably additive measure _μ_ on a _σ_-field of Borel subsets of that topology, and that \((\mathbb {B}, \ll, \preceq )\) is isomorphic to a ‘standard model’ arising out of the pair (_X_, _μ_). It follows from one of our main results that any closed ball in Euclidean space, \(\mathbb {R}^{n}\), together with Lebesgue measure arises in this way from a separation _σ_-algebra with qualitative probability. (shrink)

A general duality connecting the level of a formal theory and of a metatheory is proposed. Because of the role of natural numbers in a metatheory the existence of a dual theory is conjectured, in which the natural numbers become formal in the theory but in formalizing non-formal natural numbers taken from the dual metatheory these numbers become nonstandard. For any formal theory there may be in principle a dual theory. The dual shape of the lattice of projections over separable (...) Hilbert space is exhibited. The crucial ingredient of this dual theory is set-theoretical (Cohen) forcing. Then, some attempts have been made to exhibit the dual shape of the exotic smooth (small) structures on topological ℝ4. There are striking similarities of both pictures. Nonstandard solutions of the equations of motion of 10-dimensional superstring of IIB type are described as a model theoretic extension of standard ones. Based on Brans conjecture one can predict the existence of sources of gravity in 4 dimensions. By their dual origins they carry quantum information as well. The generalized Maldacena conjecture appears naturally, which sheds light on a definition of background independent string theory. (shrink)

The way of thinking of mathematicians and chemists in their respective disciplines seems to have very different levels of abstractions. While the firsts are involved in the most abstract of all sciences, the seconds are engaged in a practical, mainly experimental discipline. Therefore, it is surprising that many luminaries of the mathematics universe have studied chemistry as their main subject. Others have started studying chemistry before swapping to mathematics or have declared some admiration and even love for this discipline. Here (...) I reveal some of these mathematicians who were involved in chemistry from a biographical perspective. Then, I analyze what these remarkable mathematicians and statisticians could have learned while studying chemical subjects. I found analogies between code-breaking and molecular structure elucidation, inspiration for statistics in quantitative analytical chemistry, and on the role of topology in the study of some organic molecules. I also analyze some parallelisms between the way of thinking of organic chemists and mathematicians in terms of the use of backward analysis, search for patterns, and use of pictures in their respective researches. (shrink)

The author attempts to show how mereology, taken together with certain topological notions, can yield the foundations for future investigations in formal ontology. He also attempts to show how the mereological framework allows for the direct and natural formulation of a series of theses - for example pertaining to the concept of a boundary - which can be only indirectly formulated (if at all) in set-theoretic terms. The far-reaching ain of the present framework is to serve as a basis (...) for a formal ontology of the common-sense world. The author is interested in producing formally precise theories of structures of certain sorts in such a way that it is the structures themselves that intrest him and not formal machinery that has been set up to describe them. Hence the axioms are chosen primarily for the sake of the light they throw on the intended subject-matters (and not for their logical independence). The world itself - or its picture given in ordinary experience - is the only model the paper concentrates upon. Thus for example the presented system allows to prove that every boundary is the boundary of something, and that in particular no point exists in isolation from a large extended whole that is its boundary. (shrink)

Undoubtedly the Penrose-Hameroff Orch OR model may be considered as a good theory for describing information processing mechanisms and holistic phenomena in the human brain, but it doesn’t give us satisfactory explanation of human perception. In this work a new approach explaining our perception is introduced, which is in good agreement with Orch OR model and other mainstream science theories such as string theory, loop quantum gravity and holographic principle. It is shown that human perception cannot be explained in the (...) terms of elementary particles and we should introduce new indivisible holistic objects with geometry based on smooth infinitesimal analysis - elastic membranes. The example of such a membrane is our Universe which is an indivisible whole. It is shown that our perception may be considered as the result of elastic oscillations of two dimensional (2D) elastic membranes with closed topology embedded in our bodies. Only one elastic membrane responsible for its perceptions will correspond to the selected organism, but there may be other membranes, even at the cell level. In other words, reality may be considered as the process of time evolution of holistic energetically very weak macro objects - elastic membranes with the geometry based on smooth infinitesimal analysis. An embedded membrane in this multidimensional world will look different for the external and internal observers: from the outside it will look like a material object with smooth infinitesimal geometry, while from the inside our Universe-like space-time fabric. When interacting with elementary particles and other membranes, a membrane will transform their energy into its elastic energy (a new form of energy) - the energy of stretching of the infinitesimal segments. The theory postulates that these elastic deformations will not be observable from the point of view of the internal observer. Heisenberg’s uncertainty principle will work in this physics only from the point of view of the internal observer. For the external observer each embedded elastic membrane may be stretched and even a very small region will become observable. For example, living organisms play the role of internal observers of the Universe, and at the same time they serve as external observers for 2D membranes embedded into our Universe. We can observe our 2D self-membranes through our perceptions, which are encoded in elastic oscillations of the elastic membrane. According to the theory, elastic membranes occupy energetically favorable positions around microtubules involved into Orch OR. The theory not only gives us a really multidimensional holistic picture of reality, but it also provides us with a new method for understanding such phenomena as perception, self-awareness and will. (shrink)

A complete Boolean algebra ${\mathbb{B}}$ satisfies property ${(\hbar)}$ iff each sequence x in ${\mathbb{B}}$ has a subsequence y such that the equality lim sup z n = lim sup y n holds for each subsequence z of y. This property, providing an explicit definition of the a posteriori convergence in complete Boolean algebras with the sequential topology and a characterization of sequential compactness of such spaces, is closely related to the cellularity of Boolean algebras. Here we determine the position of (...) property ${(\hbar)}$ with respect to the hierarchy of conditions of the form κ-cc. So, answering a question from Kurilić and Pavlović (Ann Pure Appl Logic 148(1–3):49–62, 2007), we show that ${``\mathfrak{h}{\rm -cc}\Rightarrow (\hbar)"}$ is not a theorem of ZFC and that there is no cardinal ${\mathfrak{k}}$ , definable in ZFC, such that ${``\mathfrak{k} {\rm -cc} \Leftrightarrow (\hbar)"}$ is a theorem of ZFC. Also, we show that the set ${\{ \kappa : {\rm each}\, \kappa{\rm -cc\, c.B.a.\, has}\, (\hbar ) \}}$ is equal to ${[0, \mathfrak{h})}$ or ${[0, {\mathfrak h}]}$ and that both values are consistent, which, with the known equality ${{\{\kappa : {\rm each\, c.B.a.\, having }\, (\hbar )\, {\rm has\, the}\, \kappa {\rm -cc } \} =[{\mathfrak s}, \infty )}}$ completes the picture. (shrink)

Here is told about necessity to create new anthropological paradigm based on the ecological approach. The matter is a man in ecological world, and subjects are the phenomena of consciousness, deciding direction to this world. Ecological world differs from physical world. Ecological world is the ontological unity of person and natural world, and it is characterized with combination of all items, events, occurrence in each other. Person’s attitude to the ecological world determines with notion “ecological consciousness”, it describes person’s structural (...) attitude to the world and it has morally value character. The main description of ecological consciousness is its orientation to the ecological world. The methods of new anthropological paradigm are phenomenal and topological approaches, which help to create the complete ecological – anthropological picture of the world. (shrink)

Topic model is a well proven tool to investigate the semantic content of textual corpora. Yet corpora sometimes include texts in several languages, making it impossible to apply language-specific computational approaches over their entire content. This is the problem we encountered when setting to analyze a philosophy of science corpus spanning over eight decades and including original articles in Dutch, German and French, on top of a large majority of articles in English. To circumvent this multilingual problem, we use machine-translation (...) tools to bulk translate non-English documents into English. Though largely imperfect, especially syntactically, these translations nevertheless provide correctly translated terms and preserve the semantic proximity of documents with respect to one another. To assess the quality of this translation step, we develop a “semantic topology preservation test” that relies on estimating the extent to which document-to-document distances have been preserved during translation. We then conduct an LDA topic-model analysis over the entire corpus of translated and English original texts, and compare it to a topic-model done over the English original texts only. We thereby identify the specific contribution of the translated texts. These studies reveal a more complete picture of main topics that can found in the philosophy of science literature, especially during the early days of the discipline when numerous articles were published in languages other than English. (shrink)

There are at least two passages in the jamesian corpus where he seems to establish a topology of "regional ontologies", or to set up multiple "language games". the first of these is "the principles of psychology" when he talks about "the many worlds", or "...sub-universes commonly discriminated from each other...", the second is in "pragmatism", where he notes that there "are...at least three well-characterized levels, stages, or types of thought about the world we live in..." two questions immediately come to (...) mind about these levels. first, is each of these areas of equal importance, epistemologically and ontologically speaking? second, how, if at all, are these regional areas related to each other? each of these questions has a seemingly obvious answer. to wit, the world of perception is more important than any of the other areas; and second, the regional areas are related in terms of the disjunctive and conjunctive transitions which james so strenuously upholds, resulting in an overall "concatenated" picture. i argue that both of these responses are insufficient, and that the issue is more complicated than it first appears. (edited). (shrink)

The thesis focuses on a quite recent research field lying in between Descriptive Set Theory and Automata Theory (for infinite objects). In both areas, one is often concerned with subsets of the Cantor space or of its homeomorphic copies. In Descriptive Set Theory, such subsets are usually stratified in topological hierarchies, like the Borel hierarchy, the Wadge hierarchy and the difference hierarchy; in Automata Theory, such sets are studied in terms of regularity, that is, the property of being recognised (...) by an automaton or, equivalently, of being expressible in Monadic Second-Order Logic. This double point of view leads to many interesting questions about the interplay and relationship between topological complexity and regularity. While we have a complete picture of what happens in the case of automata on words, the case of automata on trees is still a terra incognita. Some results have already been obtained for particular classes of languages, like Buchi languages, deterministic languages, ¨ and unambiguous languages. In this thesis we instead drop any restriction and prove some new results concerning arbitrary regular tree languages which belong to low levels of the Borel hierarchy and of the Wadge hierarchy. (shrink)

In this fascinating but sometimes baffling book, the reader engages with a series of conditionals like the following: “If [the psychiatrist] Clérimbault manifests a delirium, it is because he discovers the tiny hallucinatory perceptions of ether addicts in the folds of clothing”. “If Leibniz’s principles [of identity and sufficient reason] appear to us as cries, it is because each one signals the presence of a class of beings that are themselves crying and draw attention to themselves by these cries...”. Deleuze’s (...) study is concerned with Leibniz and with leibnizianism; with the Baroque considered as a historical period and with the baroque considered as a persistent impulse in architecture, decoration, and in human thinking and system construction. Baroque and baroque are explorations, in his view, of curvilinearity, and Deleuze explores Leibniz’s use of mathematical analogies from topology, projective geometry and the calculus. But his attention in the book is mainly fixed on two Leibnizian images. The first image is that of the two-storied house, typically represented in baroque painting by an angelic realm in the upper half of the picture hovering above the human realm. In Leibniz’s writings, this translates into the world of matter and the world of intelligent, self-conscious spirits on the upper level. The second image is that of the fold, which appears in the pleats and draperies of Baroque costume, sculpture, and interior decoration, and which reappears in Leibniz’s mind-as-folded curtain, in his enveloped organisms, preformed under folds, and in the continuum, from whose recesses ever more numbers or particles can be pulled out. In clothing, waves, the brain, pleats of all sorts, surface disappears into interior. Deconstruction! But is it philosophy? Yes! Though Deleuze pulls his comparison-objects out of every corner of contemporary French culture, they are never arbitrary, and the book is focused clearly on its subject. (shrink)

Volume 1 of this biography of L. E. J. Brouwer was published in 1999.1 The volume under review here covers the period from the early nineteen twenties until Brouwer's death in 1966. It also includes a short epilogue that discusses the disposition of Brouwer's estate after his death, his influence on others, the paths of some of his students and colleagues, and other matters. Van Dalen notes in the Preface that in preparing this volume he consulted some historical studies that (...) appeared after the first volume was published. He also used new material from various archives. The biography contains interesting quotations from unpublished materials in the Brouwer Archive and from correspondence. The bibliographical references to Brouwer's publications, it should be noted, are somewhat different in this volume. This volume, like the first, contains some nice photographs and reproductions. I noted that there were many typographical errors in the earlier book but Volume 2 is relatively free of them.As is the case in Volume 1, the discussion of Brouwer's mathematical and philosophical work is woven into the narrative of Brouwer's life and times. The story in this volume starts with Brouwer's first contacts with Paul Alexandrov and Paul Urysohn in 1923. The interaction began when Urysohn announced that he had found a mistake in Brouwer's definition of dimension in Brouwer's 1913 paper on natural dimension. Was it just a slip of the pen, as Brouwer always maintained , or something more substantial? Urysohn and Alexandrov ultimately came to agree with Brouwer on the matter, and Urysohn was prepared to grant Brouwer priority for the definition of dimension. There were, however, ups and downs along the way. Karl Menger, through his own work in topology and dimension theory, soon got into the picture, and Brouwer and Menger were to …. (shrink)

Any hopes roused by an informed philosopher's undertaking a task of such importance as this title indicates are quickly reduced to perplexity. Professor Reiser's "Scientific Humanism" turns out to be a sort of "philosophy-fiction" picture of the universe, woven out of concepts from an astounding variety of fields--semantics, logic, philosophy of mathematics, topology, cybernetics, relativity theory, social engineering, quantum mechanics, parapsychology, etc.--L. K. B.

In this review paper, we discuss how gravity and spin can be obtained as the realization of the local Conformal-Affine group of symmetry transformations. In particular, we show how gravitation is a gauge theory which can be obtained starting from some local invariance as the Poincaré local symmetry. We review previous results where the inhomogeneous connection coefficients, transforming under the Lorentz group, give rise to gravitational gauge potentials which can be used to define covariant derivatives accommodating minimal couplings of matter, (...) gauge fields (and then spin connections). After we show, in a self-contained approach, how the tetrads and the Lorentz group can be used to induce the spacetime metric and then the Invariance Induced Gravity can be directly obtained both in holonomic and anholonomic pictures. Besides, we show how tensor valued connection forms act as auxiliary dynamical fields associated with the dilation, special conformal and deformation (shear) degrees of freedom, inherent to the bundle manifold. As a result, this allows to determine the bundle curvature of the theory and then to construct boundary topological invariants which give rise to a prototype (source free) gravitational Lagrangian. Finally, the Bianchi identities, the covariant field equations and the gauge currents are obtained determining completely the dynamics. (shrink)

In “Topology of Balasaguni’s Kutadgu Bilig: Thinking the Between,” Onur Karamercan focuses on the philosophical dimension of Kutadgu Bilig, a poetic work of Yūsuf Balasaguni, an 11th century Central Asian thinker, poet, and statesman. Karamercan pays special attention to the meaning of betweenness and, in the first step of his argument, discusses the hermeneutic and topological implications of the between, distingushing the dynamic sense of betweenness from a static sense of in-betweenness. He then moves on to analyze Balasaguni’s notion (...) of language, which he interprets as an early critique of the instrumental account of language and, by examining several selected fragments from Kutadgu Bilig, illustrates Balasaguni’s designation of language as an inexhaustible phenomenon. In the process, he also points to the possible parallels between Balasaguni’s and Heidegger’s ideas of language. In the final section of the article, building on his argument, Karamercan thematizes the margins of Turkic languages and of Islamic philosophy, suggesting that they need to be reexamined. He problematizes the very meaning of Asia itself by decentering what he calls “its internal East–West antagonism” and puts forth instead a framework based on in-betweenness reinterpreted from a topological perspective, proposing it as an alternative view which might help us make sense of the hermeneutic neighborhoods of Asian philosophies. (shrink)

Volume 1 of this biography of L. E. J. Brouwer was published in 1999.1 The volume under review here covers the period from the early nineteen twenties until Brouwer's death in 1966. It also includes a short epilogue that discusses the disposition of Brouwer's estate after his death, his influence on others, the paths of some of his students and colleagues, and other matters. Van Dalen notes in the Preface that in preparing this volume he consulted some historical studies that (...) appeared after the first volume was published. He also used new material from various archives. The biography contains interesting quotations from unpublished materials in the Brouwer Archive and from correspondence. The bibliographical references to Brouwer's publications, it should be noted, are somewhat different in this volume. This volume, like the first, contains some nice photographs and reproductions. I noted that there were many typographical errors in the earlier book but Volume 2 is relatively free of them.As is the case in Volume 1, the discussion of Brouwer's mathematical and philosophical work is woven into the narrative of Brouwer's life and times. The story in this volume starts with Brouwer's first contacts with Paul Alexandrov and Paul Urysohn in 1923. The interaction began when Urysohn announced that he had found a mistake in Brouwer's definition of dimension in Brouwer's 1913 paper on natural dimension. Was it just a slip of the pen, as Brouwer always maintained, or something more substantial? Urysohn and Alexandrov ultimately came to agree with Brouwer on the matter, and Urysohn was prepared to grant Brouwer priority for the definition of dimension. There were, however, ups and downs along the way. Karl Menger, through his own work in topology and dimension theory, soon got into the picture, and Brouwer and Menger were to …. (shrink)

갈등은 일차적으로 개인(주관)의 심리문제일 것이지만, 결국은 사회(주체)의 정치적 문제가 된다. 그리고 이 정치적 문제는 갈등의 제거가 아니라 갈등의 승인에서 가능하다. 이는 갈등에 대한 정치적 논의가 사회적 현실에 대한 “탈구와 재구성”의 역학, 즉 ‘해방의 변증법’을 수반하는 ‘갈등의 위상학’을 통해 이루어져야 한다는 것을 의미한다. 주체에 억압된 욕망, 부연하자면 ‘존재론적 욕망’으로 회귀하며 “내재성”의 논리에 입각해 갈등의 ‘위상학’(topology)을 선명히 가리킨 것은 현대의 철학, 특히 후설 이후 메를로퐁티와 하이데거, 그리고 라캉과 들뢰즈의 철학이었다. 이 글은, 이들에 함축된 ‘시(詩)적 인식’의 위상학적 양상을 메를로퐁티의 관점으로 수렴하는 자리에서, 갈등이 (...) 갖는 해방의 변증법적 의의를 짚은 후, 우리사회에서 폭발하는 갈등이 위기가 아니라 사실은 〈외재적 초월의 유토피아적 허구를 넘는 ‘내재적 초월’의 현실에 입각해 정치적 상상력을 재정립할 기회〉의 한 계기임을 제시할 것이다. (shrink)

The spatial organization of the genome is intimately linked to its biological function, yet our understanding of higher order genomic structure is coarse, fragmented and incomplete. In the nucleus of eukaryotic cells, interphase chromosomes occupy distinct.

We consider interpretable topological spaces and topological groups in a p-adically closed field K. We identify a special class of “admissible topologies” with topological tameness properties like generic continuity, similar to the topology on definable subsets of Kn. We show that every interpretable set has at least one admissible topology, and that every interpretable group has a unique admissible group topology. We then consider definable compactness (in the sense of Fornasiero) on interpretable groups. We show that an (...) interpretable group is definably compact if and only if it has finitely satisfiable generics (fsg), generalizing an earlier result on definable groups. As a consequence, we see that fsg is a definable property in definable families of interpretable groups, and that any fsg interpretable group defined over Qp is definably isomorphic to a definable group. (shrink)

Hobbes emphasized that the state of nature is a state of war because it is characterized by fundamental and generalized distrust. Exiting the state of nature and the conflicts it inevitably fosters is therefore a matter of establishing trust. Extant discussions of trust in the philosophical literature, however, focus either on isolated dyads of trusting individuals or trust in large, faceless institutions. In this paper, I begin to fill the gap between these extremes by analyzing what I call the topology (...) of communities of trust. Such communities are best understood in terms of interlocking dyadic relationships that approximate the ideal of being symmetric, Euclidean, reflexive, and transitive. Few communities of trust live up to this demanding ideal, and those that do tend to be small (between three and fifteen individuals). Nevertheless, such communities of trust serve as the conditions for the possibility of various important prudential epistemic, cultural, and mental health goods. However, communities of trust also make possible various problematic phenomena. They can become insular and walled-off from the surrounding community, leading to distrust of out-groups. And they can lead their members to abandon public goods for tribal or parochial goods. These drawbacks of communities of trust arise from some of the same mecha-nisms that give them positive prudential, epistemic, cultural, and mental health value – and so can at most be mitigated, not eliminated. (shrink)

Dorian Gray pays a hefty price for years of sin and vice in this completely unabridged edition of The Picture of Dorian Gray.

Drawing from Andean cosmological, mythological and aesthetic lineages, this paper is about the possibility of a phenomenology of things at catastrophic ends. In this regard, I approach things under the sway of a (de)formative emptiness. In the first part, I develop a relational ontology on the basis of the Andean notion of pacha or cosmos, which provides a phenomenological frame for a determination of “place,” “world” and “topology.” I also contrast an elemental topology of the cosmos configured by ouranic sunlight (...) with an elemental catastrophism that overturns it. In the second part, I study a chronological series of artifacts in an Andean aesthetic lineage in order to attempt a phenomenology of “things” (utensils, buildings, mountains) that “spectrally” hover in overturns of the cosmos (pachakuti). (shrink)

This paper argues that besides mechanistic explanations, there is a kind of explanation that relies upon “topological” properties of systems in order to derive the explanandum as a consequence, and which does not consider mechanisms or causal processes. I first investigate topological explanations in the case of ecological research on the stability of ecosystems. Then I contrast them with mechanistic explanations, thereby distinguishing the kind of realization they involve from the realization relations entailed by mechanistic explanations, and explain (...) how both kinds of explanations may be articulated in practice. The second section, expanding on the case of ecological stability, considers the phenomenon of robustness at all levels of the biological hierarchy in order to show that topological explanations are indeed pervasive there. Reasons are suggested for this, in which “neutral network” explanations are singled out as a form of topological explanation that spans across many levels. Finally, I appeal to the distinction of explanatory regimes to cast light on a controversy in philosophy of biology, the issue of contingence in evolution, which is shown to essentially involve issues about realization. (shrink)

We provide three innovations to recent debates about whether topological or “network” explanations are a species of mechanistic explanation. First, we more precisely characterize the requirement that all topological explanations are mechanistic explanations and show scientific practice to belie such a requirement. Second, we provide an account that unifies mechanistic and non-mechanistic topological explanations, thereby enriching both the mechanist and autonomist programs by highlighting when and where topological explanations are mechanistic. Third, we defend this view against (...) some powerful mechanist objections. We conclude from this that topological explanations are autonomous from their mechanistic counterparts. (shrink)

This paper intends to further the understanding of the formal properties of (higher-order) vagueness by connecting theories of (higher-order) vagueness with more recent work in topology. First, we provide a “translation” of Bobzien's account of columnar higher-order vagueness into the logic of topological spaces. Since columnar vagueness is an essential ingredient of her solution to the Sorites paradox, a central problem of any theory of vagueness comes into contact with the modern mathematical theory of topology. Second, Rumfitt’s recent (...) class='Hi'>topological reconstruction of Sainsbury’s theory of prototypically defined concepts is shown to lead to the same class of spaces that characterize Bobzien’s account of columnar vagueness, namely, weakly scattered spaces. Rumfitt calls these spaces polar spaces. They turn out to be closely related to Gärdenfors’ conceptual spaces, which have come to play an ever more important role in cognitive science and related disciplines. Finally, Williamson’s “logic of clarity” is explicated in terms of a generalized topology (“locology”) that can be considered an alternative to standard topology. Arguably, locology has some conceptual advantages over topology with respect to the conceptualization of a boundary and a borderline. Moreover, in Williamson’s logic of clarity, vague concepts with respect to a notion of a locologically inspired notion of a “slim boundary” are (stably) columnar. Thus, Williamson’s logic of clarity also exhibits a certain affinity for columnar vagueness. In sum, a topological perspective is useful for a conceptual elucidation and unification of central aspects of a variety of contemporary accounts of vagueness. (shrink)

This chapter provides a systematic overview of topological explanations in the philosophy of science literature. It does so by presenting an account of topological explanation that I (Kostić and Khalifa 2021; Kostić 2020a; 2020b; 2018) have developed in other publications and then comparing this account to other accounts of topological explanation. Finally, this appraisal is opinionated because it highlights some problems in alternative accounts of topological explanations, and also it outlines responses to some of the main (...) criticisms raised by the so-called new mechanists. (shrink)