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)

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)

The relationship between topological explanation and mechanistic explanation is unclear. Most philosophers agree that at least some topological explanations are mechanistic explanations. The crucial question is how to make sense of this claim. Zednik (Philos Psychol 32(1):23–51, 2019) argues that topological explanations are mechanistic if they (i) describe mechanism sketches that (ii) pick out organizational properties of mechanisms. While we agree with Zednik’s conclusion, we critically discuss Zednik’s account and show that it fails as a (...) general account of how and when topological explanations are mechanistic. First, if topological explanations were just mechanism sketches, this implies that they could be enriched by replacing topological terms with mechanistic detail. This, however, conflicts how topological explanations are used in scientific practice. Second, Zednik’s account fails to show how topological properties can be organizational properties of mechanisms that have a place in mechanistic explanation. The core issue is that Zednik’s account ignores that topological properties often are _global_ properties while mechanistic explanantia refer to _local_ properties. We demonstrate how these problems can be solved by a recent account of mechanistic completeness (Craver and Kaplan in Br J Philos Sci 71(1):287–319, 2020; Kohár and Krickel in Calzavarini and Viola (eds) Neural mechanisms—new challenges in the philosophy of neuroscience, Springer, New York, 2021) and use a multilayer network model of Alzheimer’s Disease to illustrate this. (shrink)

Proponents of ontic conceptions of explanation require all explanations to be backed by causal, constitutive, or similar relations. Among their justifications is that only ontic conceptions can do justice to the ‘directionality’ of explanation, i.e., the requirement that if X explains Y , then not-Y does not explain not-X . Using topological explanations as an illustration, we argue that non-ontic conceptions of explanation have ample resources for securing the directionality of explanations. The different ways in which (...) neuroscientists rely on multiplexes involving both functional and anatomical connectivity in their topological explanations vividly illustrate why ontic considerations are frequently (if not always) irrelevant to explanatory directionality. Therefore, directionality poses no problem to non-ontic conceptions of explanation. (shrink)

In this paper, I present a general theory of topological explanations, and illustrate its fruitfulness by showing how it accounts for explanatory asymmetry. My argument is developed in three steps. In the first step, I show what it is for some topological property A to explain some physical or dynamical property B. Based on that, I derive three key criteria of successful topological explanations: a criterion concerning the facticity of topological explanations, i.e. what makes it true (...) of a particular system; a criterion for describing counterfactual dependencies in two explanatory modes, i.e. the vertical and the horizontal; and, finally, a third perspectival one that tells us when to use the vertical and when to use the horizontal mode. In the second step, I show how this general theory of topological explanations accounts for explanatory asymmetry in both the vertical and horizontal explanatory modes. Finally, in the third step, I argue that this theory is universally applicable across biological sciences, which helps to unify essential concepts of biological networks. (shrink)

While mechanistic explanation and, to a lesser extent, nomological explanation are well-explored topics in the philosophy of biology, topological explanation is not. Nor is the role of diagrams in topological explanations. These explanations do not appeal to the operation of mechanisms or laws, and extant accounts of the role of diagrams in biological science explain neither why scientists might prefer diagrammatic representations of topological information to sentential equivalents nor how such representations might facilitate important (...) processes of explanatory reasoning unavailable to scientists who restrict themselves to sentential representations. Accordingly, relying upon a case study about immune system vulnerability to attacks on CD4+ T-cells, I argue that diagrams group together information in a way that avoids repetition in representing topological structure, facilitate identification of specific topological properties of those structures, and make available to controlled processing explanatorily salient counterfactual information about topological structures, all in ways that sentential counterparts of diagrams do not. (shrink)

So-called ‘distinctively mathematical explanations’ (DMEs) are said to explain physical phenomena, not in terms of contingent causal laws, but rather in terms of mathematical necessities that constrain the physical system in question. Lange argues that the existence of four or more equilibrium positions of any double pendulum has a DME. Here we refute both Lange’s claim itself and a strengthened and extended version of the claim that would pertain to any n-tuple pendulum system on the ground that such explanations are (...) actually causal explanations in disguise and their associated modal conditionals are not general enough to explain the said features of such dynamical systems. We argue and show that if circumscribing the antecedent for a necessarily true conditional in such explanations involves making a causal analysis of the problem, then the resulting explanation is not distinctively mathematical or non-causal. Our argument generalises to other dynamical systems that may have purported DMEs analogous to the one proposed by Lange, and even to some other counterfactual accounts of non-causal explanation given by Reutlinger and Rice. (shrink)

In this paper, I outline a heuristic for thinking about the relation between explanation and understanding that can be used to capture various levels of “intimacy”, between them. I argue that the level of complexity in the structure of explanation is inversely proportional to the level of intimacy between explanation and understanding, i.e. the more complexity the less intimacy. I further argue that the level of complexity in the structure of explanation also affects the explanatory depth (...) in a similar way to intimacy between explanation and understanding, i.e. the less complexity the greater explanatory depth and vice versa. (shrink)

In the last couple of years, a few seemingly independent debates on scientific explanation have emerged, with several key questions that take different forms in different areas. For example, the questions what makes an explanation distinctly mathematical and are there any non-causal explanations in sciences (i.e., explanations that don’t cite causes in the explanans) sometimes take a form of the question of what makes mathematical models explanatory, especially whether highly idealized models in science can be explanatory and in (...) virtue of what they are explanatory. These questions raise further issues about counterfactuals, modality, and explanatory asymmetries: i.e., do mathematical and non-causal explanations support counterfactuals, and how ought we to understand explanatory asymmetries in non-causal explanations? Even though these are very common issues in the philosophy of physics and mathematics, they can be found in different guises in the philosophy of biology where there is the statistical interpretation of the Modern Synthesis theory of evolution, according to which the post-Darwinian theory of natural selection explains evolutionary change by citing statistical properties of populations and not the causes of changes. These questions also arise in philosophy of ecology or neuroscience in regard to the nature of topological explanations. The question here is can the mathematical (or more precisely topological) properties in network models in biology, ecology, neuroscience, and computer science be explanatory of physical phenomena, or are they just different ways to represent causal structures. The aim of this special issue is to unify all these debates around several overlapping questions. These questions are: are there genuinely or distinctively mathematical and non-causal explanations?; are all distinctively mathematical explanations also non-causal; in virtue of what they are explanatory; does the instantiation, implementation, or in general, applicability of mathematical structures to a variety of phenomena and systems play any explanatory role? This special issue provides a platform for unifying the debates around several key issues and thus opens up avenues for better understanding of mathematical and non-causal explanations in general, but also, it will enable even better understanding of key issues within each of the debates. (shrink)

In the last two decades, philosophy of neuroscience has predominantly focused on explanation. Indeed, it has been argued that mechanistic models are the standards of explanatory success in neuroscience over, among other things, topological models. However, explanatory power is only one virtue of a scientific model. Another is its predictive power. Unfortunately, the notion of prediction has received comparatively little attention in the philosophy of neuroscience, in part because predictions seem disconnected from interventions. In contrast, we argue that (...) topological predictions can and do guide interventions in science, both inside and outside of neuroscience. Topological models allow researchers to predict many phenomena, including diseases, treatment outcomes, aging, and cognition, among others. Moreover, we argue that these predictions also offer strategies for useful interventions. Topology-based predictions play this role regardless of whether they do or can receive a mechanistic interpretation. We conclude by making a case for philosophers to focus on prediction in neuroscience in addition to explanation alone. (shrink)

Philippe Huneman has recently questioned the widespread application of mechanistic models of scientific explanation based on the existence of structural explanations, i.e. explanations that account for the phenomenon to be explained in virtue of the mathematical properties of the system where the phenomenon obtains, rather than in terms of the mechanisms that causally produce the phenomenon. Structural explanations are very diverse, including cases like explanations in terms of bowtie structures, in terms of the topological properties of the system, (...) or in terms of equilibrium. The role of mathematics in bowtie structured systems and in topologically constrained systems has recently been examined in different papers. However, the specific role that mathematical properties play in equilibrium explanations requires further examination, as different authors defend different interpretations, some of them closer to the new-mechanistic approach than to the structural model advocated by Huneman. In this paper, we cover this gap by investigating the explanatory role that mathematics play in Blaser and Kirschner’s nested equilibrium model of the stability of persistent long-term human-microbe associations. We argue that their model is explanatory because: i) it provides a mathematical structure in the form of a set of differential equations that together satisfy an ESS; ii) that the nested nature of the ESSs makes the explanation of host-microbe persistent associations robust to any perturbation; iii) that this is so because the properties of the ESS directly mirror the properties of the biological system in a non-causal way. The combination of these three theses make equilibrium explanations look more similar to structural explanations than to causal-mechanistic explanation. (shrink)

In non-causal explanations, some non-causal facts (such as mathematical, modal or metaphysical) are used to explain some physical facts. However, precisely because these explanations abstract away from causal facts, they face two challenges: 1) it is not clear why would one rather than the other non-causal explanantia be relevant for the explanandum; and 2) why would standing in a particular explanatory relation (e.g., “counterfactual dependence”, “constraint”, “entailment”, “constitution”, “grounding”, and so on), and not in some other, be explanatory. I develop (...) an explanatory relevance account which is based on erotetic constraints and show how it addresses these two challenges. (shrink)

In this work we show that our self-awareness and perception may be successfully explained using two dimensional holistic structures with closed topology embedded into our brains - elastic membranes. These membranes are able to preserve their structure during conscious processes. Their elastic oscillations may be associated with our perceptions, where the frequency of the oscillations is responsible for the perception of different colors, sounds and other stimuli, while the amplitude of the oscillations is responsible for the feeling of a distance. (...) According to the model the squeezed regions of a membrane correspond to the brain zones involved into awareness and attention. The model may be useful for prediction, explanation and interpretation of various conscious phenomena. (shrink)

The philosophical literature on scientific explanation in neuroscience has been dominated by the idea of mechanisms. The mechanist philosophers often claim that neuroscience is in the business of finding mechanisms. This view has been challenged in numerous ways by showing that there are other successful and widespread explanatory strategies in neuroscience. However, the empirical evidence for all these claims was hitherto lacking. Empirical evidence about the pervasiveness and uses of various explanatory strategies in neuroscience is particularly needed because examples (...) and case studies that are used to illustrate philosophical claims so far tend to be hand-picked. The risk of confirmation bias is therefore considerable: when looking for white swans, all one finds is that swans are white. The more systematic quantitative and qualitative bibliometric study of a large body of relevant literature that we present in this paper can put such claims into perspective. Using bibliometric tools, we identify the typical linguistic patterns used in the alleged mechanistic, dynamical, and topological explanations in the literature, their preponderance and how they change over time. Our findings show abundant use of mechanistic language, but also the presence of a significant neuroscience literature using topological and dynamical explanatory language, which grows over time and increasingly differentiates from each other and from mechanistic explanations. (shrink)

The mathematical structure of realist quantum theories has given rise to a debate about how our ordinary 3-dimensional space is related to the 3N-dimensional configuration space on which the wave function is defined. Which of the two spaces is our (more) fundamental physical space? I review the debate between 3N-Fundamentalists and 3D-Fundamentalists and evaluate it based on three criteria. I argue that when we consider which view leads to a deeper understanding of the physical world, especially given the deeper (...) class='Hi'>topological explanation from the unordered configurations to the Symmetrization Postulate, we have strong reasons in favor of 3D-Fundamentalism. I conclude that our evidence favors the view that our fundamental physical space in a quantum world is 3-dimensional rather than 3N-dimensional. I outline lines of future research where the evidential balance can be restored or reversed. Finally, I draw lessons from this case study to the debate about theoretical equivalence. (shrink)

Over the last decades, network-based approaches have become highly popular in diverse fields of biology, including neuroscience, ecology, molecular biology and genetics. While these approaches continue to grow very rapidly, some of their conceptual and methodological aspects still require a programmatic foundation. This challenge particularly concerns the question of whether a generalized account of explanatory, organisational and descriptive levels of networks can be applied universally across biological sciences. To this end, this highly interdisciplinary theme issue focuses on the definition, motivation (...) and application of key concepts in biological network science, such as explanatory power of distinctively network explanations, network levels, and network hierarchies. (shrink)

We provide two programmatic frameworks for integrating philosophical research on understanding with complementary work in computer science, psychology, and neuroscience. First, philosophical theories of understanding have consequences about how agents should reason if they are to understand that can then be evaluated empirically by their concordance with findings in scientific studies of reasoning. Second, these studies use a multitude of explanations, and a philosophical theory of understanding is well suited to integrating these explanations in illuminating ways.

Explanation is a foundational goal in the exact sciences. Besides the contemporary considerations on ‘description’, ‘classification’, and ‘prediction’, we often see these terms in thriving applications of artificial intelligence (AI) in chemistry hypothesis generation. Going beyond describing ‘things in the world’, these applications can make accurate numerical property calculations from theoretical or topological descriptors. This association makes an interesting case for a logic of discovery in chemistry: are these induction-led ventures showing a shift in how chemists can problematize (...) research questions? In this article, I present a fresh perspective on the current context of discovery in chemistry. I argue how data-driven statistical predictions in chemistry can be explained as a quasi-logical process for generating chemical theories, beyond the classic examples of organic and theoretical chemistry. Through my position on formal models of scientific explanation, I demonstrate how the dawn of AI can provide novel insights into the explanatory power of scientific endeavors. (shrink)

In this paper, we explore the conceptual problems arising when using network analysis in person- centered care (PCC) in psychiatry. Personalized network models are potentially helpful tools for PCC, but we argue that using them in psychiatric practice raises boundary problems, i.e., problems in demarcating what should and should not be included in the model, which may limit their ability to provide clinically-relevant knowledge. Models can have explanatory and representational boundaries, among others. We argue that we can make more explicit (...) what kind of questions personalized network models can address in PCC, given their representational and explanatory boundaries, using perspectival reasoning. (shrink)

When this article was first planned, writing was going to be exclusively about two things - the origin of life and human evolution. But it turned out to be out of the question for the author to restrict himself to these biological and anthropological topics. A proper understanding of them required answering questions like “What is the nature of the universe – the home of life – and how did it originate?”, “How can time travel be removed from fantasy and (...) science fiction, to be made scientific and practical?”, and “How can the proposed young age of genus Homo be made to actually be reasonable – when simply stating it would be solid ground for instant rejection and dismissal?” The result is that the article also talks about subjects like Artificial Intelligence, General Relativity, and cosmology. -/- From where did life originate? God? Evolution? Panspermia? If the tendency of humans and scientists to regard undiscovered science as pseudoscience can be overcome, Einstein gave another alternative to consider when he introduced General Relativity. Time isn’t linear – progressing in a straight line from past to present to future. That assumption ignores Relativity which states that space AND TIME are curved. Where did life and the genetic code come from? Can the answer build AI? -/- The first question can be answered by the section of this article titled SETI, Evolution, and Time which says life (possibly multicellular and intelligent) and the genetic code came from humans acquiring knowledge of these things over the centuries, then applying that knowledge – via terraforming, accumulation of raw materials like amino acids and nucleic acids, genetic engineering - to a time in the past when life didn’t exist. From that origin, life evolved through innumerable mutations and adaptations, with humans once again acquiring knowledge of it in cyclic (nonlinear) time. -/- The second question is answered by saying artificial intelligence (AI) as the product of life is only half of the equation. The other half refers to Relativity’s curved space-time and violation of the notion that time always travels from past to future. We have always lived in an artificially intelligent, non-probabilistic universe where everything in time and space is connected into one thing by quantum entanglement – making the brain and genes products of binary-digit activity or artificial intelligence (life is not merely dependent on biology’s “lock and key” mechanisms but also possesses AI). -/- The earliest documented representative of the genus Homo is Homo habilis, which evolved around 2.8 million years ago. Scientists used to believe there was a straight line from H. habilis to us, Homo sapiens. This article will use the “advanced” waves loved by Physics Nobel laureate Richard Feynman, view the history of science through the lens of Conic Sections applied to Relativity’s curved space-time, and incorporate the necessity of so-called imaginary time * – popularized by Prof. Stephen Hawking. While the evolutionary proposals are more in agreement with this early straight line than with modern theories, Albert Einstein’s General Relativity is used to transform the straight line into a curved line, ultimately concluding that Homo habilis (H. habilis) originated only (and unbelievably, as far as today’s science and technology is concerned) ~250,000 years ago. Other branches and dead ends of Homo – e.g. Neanderthals – are the result of mutations and adaptations, with the resultant modifications to anatomy and physiology. The surprisingly young age of H. habilis allows nearly 200,000 years for habilis, or one of its descendants, to reach Australia … if this country’s indigenous Aboriginal population did, as claimed, reach this “island continent” 60,000 years ago. -/- * The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, is a failure of classical physics to predict observed phenomena: it can be shown that a blackbody - a hypothetical perfect absorber and radiator of energy - would release an infinite amount of energy, contradicting the principles of conservation of energy and indicating that a new model for the behaviour of blackbodies was needed. At the start of the 20th century, physicist Max Planck derived the correct solution by making some strange (for the time) assumptions. In particular, Planck assumed that electromagnetic radiation can only be emitted or absorbed in discrete packets, called quanta. Albert Einstein postulated that Planck's quanta were real physical particles (what we now call photons), not just a mathematical fiction. From there, Einstein developed his explanation of the photoelectric effect (when quanta or photons of light shine on certain metals, electrons are released and can form an electric current). So it appears entirely possible that another supposed mathematical trickery (imaginary time and the y-axis of Wick rotation) will find practical application in the future. -/- The article includes mathematical references to cosmology (spoiler alert – you’ll read about things like Vector-Tensor-Scalar Geometry, topology, the “eternal present”, Einstein’s Unified Field, the inverse-square law, and there being no Big Bang and no multiverse - but there will also be no equations). -/- The other subheadings in this essay are – -/- NONLINEAR TIME AND ELECTRICAL ENGINEERING (about a 2009 electrical engineering experiment at America’s Yale University, and cosmic wormholes) -/- BITS AND TOPOLOGY (base-2 maths aka Binary digiTS, Mobius strips, and figure-8 Klein bottles) -/- WICK ROTATION, CAUSALITY, AND UNITING TIME (do past, present and future co-exist in an “eternal present”?) -/- DIGITAL BRAIN, DIGITAL UNIVERSE (if the brain and the universe are ultimately composed of binary digits, we'll someday be able to do the same things with the brain and universe that we now do with computers) -/- PROPOSAL: HUMAN AND ANIMAL INSTINCTS ARE THE RESULT OF THE UNIVERSE BEING UNIFIED BY BINARY DIGITS (AND TOPOLOGY) (If everything in the universe is ultimately composed of electronic BITS, then the universe must possess Artificial Intelligence - some prefer the term Cosmic Consciousness) -/- INFORMATION THEORY CONQUERS A RED GIANT (preserving Earth by keeping the Sun near today’s level of activity forever). 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We aim to show that Kant’s theory of time is consistent by providing axioms whose models validate all synthetic a priori principles for time proposed in the Critique of Pure Reason. In this paper we focus on the distinction between time as form of intuition and time as formal intuition, for which Kant’s own explanations are all too brief. We provide axioms that allow us to construct ‘time as formal intuition’ as a pair of continua, corresponding to time as ‘inner (...) sense’ and the external representation of time as a line Both continua are replete with infinitesimals, which we use to elucidate an enigmatic discussion of ‘rest’ in the Metaphysical foundations of natural science. Our main formal tools are Alexandroff topologies, inverse systems and the ring of dual numbers. (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)

If you have enjoyed any of the 7 (seven) other books I have published over 20 years, including literally thousands of pages of mathematical and topological concepts, Python programs and conceptually expanding papers, please consider buying this book for $20.00 on google play books. -/- Introduction: -/- Though the following pages provide extensive exposition and dedicated descriptions of the phenomenological velocity formulas, theory and mystery, I thought it appropriate to write this introduction as a partial explanation for what (...) phenomenal velocity is, and describe, briefly its theory and applications. Phenomenological Velocity is a method for solving for something that ought cancel out with itself, but there are specific implicit forms for this thing that, “ought cancel out with itself,” namely the Lorentz coefficient ought cancel out with itself when applied to the height of a cone derived from the difference between the circumferences of two circles applied to the Pythagorean Theorem, or, more generally, the height implied by application of the Pythagorean theorem to the difference between two arc lengths’ equaling a third arc length. These difference equations are essential to conceptualizing differentiation, and in these further chapters, I demonstrate that the phenomenological velocity is, indeed the conditional derivative in the chapter, “Conditional Integral of Phenomenological Velocity.” The phenomenological velocity algebraic solution to the velocity within the Lorentz coefficient when applied to the height function in such a way that it ought cancel out with itself is both constructive mathematics and it employs the concept of, “bracketing,” - first introduced by Edmund Husserl in his writings on the phenomenological reduction. Phenomenological Velocity’s algebraic solution from the difference between two arc lengths applied to the Pythagorean Theorem to solve for a theoretical height (which is a projected distance in space), employs bracketing, because we, “set aside,” the existence of an undefined solution, namely due to the presence of necessitated complex analytical forms by the architecture of the equation, or the “mathetecture,” of the algebraic form. -/- With respect to theology, the phenomenological velocity is somehow symbolic of the creation itself; symbolic of creation due to the fact that we find the canceling out of the Lorentz coefficient as, “impotent,” non-existent or non-effecting to the mathetecture of the height function. However, via the modus-ponens work around to phenomenological velocity, which in itself does not require the complex field, but embeds implied complex field solutions to the equation while maintaining logical consistency, we find existence from non-existence. This is directly linguistically applicable to the concept of the big-bang, the resurrection of Yeshua the Messiah, and opens analogies for us to draw relationships between the, “fall,” of Adam and Eve as the generation of error, or the introduction of paradox, as we see the phallus representing paradox topologically. -/- The phenomenological velocity is a gestalt concept, relevant to cosmology, because we find that it is the perfect language-form for discussing dark matter. It does, however, require the reader to re-conceive or re-frame rather, some of the fundamental aspects of assumed physical reality like time, experience, solidity of dark matter, etc. We find the hidden dimension of phenomenological velocity to have been an overlooked aspect of mathematical physics by the researchers of Bell’s theorem and undoubtedly a host of other theorems. Thus, raising awareness about the real existence and necessitated reality of phenomenological velocity is in no way an endeavor deserving further procrastination by the scientific community, for doing so would be intellectually dishonest and further the propagation of incomplete or misleading theories on reality. -/- This work details how the Lorentz coefficient, when applied to the height of a cone in such a way as to cancel out with itself, permits the velocity variable to have a solution to it anyway, even though it ought cancel out with itself. This mathe-tecture, so to speak has consequences for complex analysis, and pave the way for, "transcendental relativity," building an adaptive framework for consciousness and physical reality. (shrink)

What is so special and mysterious about the Continuum, this ancient, always topical, and alongside the concept of integers, most intuitively transparent and omnipresent conceptual and formal medium for mathematical constructions and the battle field of mathematical inquiries ? And why it resists the century long siege by best mathematical minds of all times committed to penetrate once and for all its set-theoretical enigma ? -/- The double-edged purpose of the present study is to save from the transfinite deadlock of (...) higher set theory the jewel of mathematical Continuum -- this genuine, even if mostly forgotten today raison d'etre of all set-theoretical enterprises to Infinity and beyond, from Georg Cantor to W. Hugh Woodin to Buzz Lightyear, by simultaneously exhibiting the limits and pitfalls of all old and new reductionist foundational approaches to mathematical truth: be it Cantor's or post-Cantorian Idealism, Brouwer's or post-Brouwerian Constructivism, Hilbert's or post-Hilbertian Formalism, Goedel's or post-Goedelian Platonism. -/- In the spirit of Zeno's paradoxes, but with the enormous historical advantage of hindsight, we claim that Cantor's set-theoretical methodology, powerful and reach in proof-theoretic and similar applications as it might be, is inherently limited by its epistemological framework of transfinite local causality, and neither can be held accountable for the properties of the Continuum already acquired through geometrical, analytical, and arithmetical studies, nor can it be used for an adequate, conceptually sensible, operationally workable, and axiomatically sustainable re-creation of the Continuum. -/- From a strictly mathematical point of view, this intrinsic limitation of the constative and explicative power of higher set theory finds its explanation in the identified in this study ultimate phenomenological obstacle to Cantor's transfinite construction, similar to topological obstacles in homotopy theory and theoretical physics: the entanglement capacity of the mathematical Continuum. (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)

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)

The aim of this paper is to show that (elementary) topology may be useful for dealing with problems of epistemology and metaphysics. More precisely, I want to show that the introduction of topological structures may elucidate the role of the spatial structures (in a broad sense) that underly logic and cognition. In some detail I’ll deal with “Cassirer’s problem” that may be characterized as an early forrunner of Goodman’s “grue-bleen” problem. On a larger scale, topology turns out to be (...) useful in elaborating the approach of conceptual spaces that in the last twenty years or so has found quite a few applications in cognitive science, psychology, and linguistics. In particular, topology may help distinguish “natural” from “not-so-natural” concepts. This classical problem that up to now has withstood all efforts to solve (or dissolve) it by purely logical methods. Finally, in order to show that a topological perspective may also offer a fresh look on classical metaphysical problems, it is shown that Leibniz’s famous principle of the identity of indiscernibles is closely related to some well-known topological separation axioms. More precisely, the topological perspective gives rise in a natural way to some novel variations of Leibniz’s principle. (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)

Since antiquity well into the beginnings of the 20th century geometry was a central topic for philosophy. Since then, however, most philosophers of science, if they took notice of topology at all, considered it as an abstruse subdiscipline of mathematics lacking philosophical interest. Here it is argued that this neglect of topology by philosophy may be conceived of as the sign of a conceptual sea-change in philosophy of science that expelled geometry, and, more generally, mathematics, from the central position it (...) used to have in philosophy of science and placed logic at center stage in the 20th century philosophy of science. Only in recent decades logic has begun to loose its monopoly and geometry and topology received a new chance to find a place in philosophy of science. (shrink)

The general aim of this paper is to introduce some ideas of the theory of infinite topological games into the philosophical debate on supertasks. First, we discuss the elementary aspects of some infinite topological games, among them the Banach-Mazur game.Then it is shown that the Banach-Mazur game may be conceived as a Newtonian supertask.In section 4 we propose to conceive physical experiments as infinite games. This leads to the distinction between determined and undetermined experiments and the problem of (...) how it is related to that between determinism and indeter-minism. Finally the role of the Axiom of Choice as a source of indetermi-nacy of supertasks is discussed. (shrink)

Considering topology as an extension of mereology, this paper analyses topological variants of mereological essentialism (the thesis that an object could not have different parts than the ones it has). In particular, we examine de dicto and de re versions of two theses: (i) that an object cannot change its external connections (e.g., adjacent objects cannot be separated), and (ii) that an object cannot change its topological genus (e.g., a doughnut cannot turn into a sphere). Stronger forms of (...) structural essentialism, such as morphological essentialism (an object cannot change shape) and locative essentialism (an object cannot change position) are also examined. (shrink)

We now found nine new topologies, such as: NonStandard Topology, Largest Extended NonStandard Real Topology, Neutrosophic Triplet Weak/Strong Topologies, Neutrosophic Extended Triplet Weak/Strong Topologies, Neutrosophic Duplet Topology, Neutrosophic Extended Duplet Topology, Neutrosophic MultiSet Topology, and recall and improve the seven previously founded topologies in the years (2019-2023), namely: NonStandard Neutrosophic Topology, NeutroTopology, AntiTopology, Refined Neutrosophic Topology, Refined Neutrosophic Crisp Topology, SuperHyperTopology, and Neutrosophic SuperHyperTopology. They are called avantgarde topologies because of their innovative forms.

Scientific explanations must bear the proper relationship to the world: they must depict what, out in the world, is responsible for the explanandum. But explanations must also bear the proper relationship to their audience: they must be able to create human understanding. With few exceptions, philosophical accounts of explanation either ignore entirely the relationship between explanations and their audience or else demote this consideration to an ancillary role. In contrast, I argue that considering an explanation’s communicative role is (...) crucial to any satisfactory account of explanation. (shrink)

This paper offers a new account of metaphysical explanation. The account is modelled on Kitcher’s unificationist approach to scientific explanation. We begin, in Sect. 2, by briefly introducing the notion of metaphysical explanation and outlining the target of analysis. After that, we introduce a unificationist account of metaphysical explanation before arguing that such an account is capable of capturing four core features of metaphysical explanations: irreflexivity, non-monotonicity, asymmetry and relevance. Since the unificationist theory of metaphysical (...) class='Hi'>explanation inherits irreflexivity and non-monotonicity directly from the unificationist theory of scientific explanation that underwrites it, we focus on demonstrating how the account can secure asymmetry and relevance. (shrink)

In this paper I argue that the common practice of employing moral predicates as explaining phrases can be accommodated on an expressivist account of moral practice. This account does not treat moral explanations as in any way second-rate or derivative, since it subsumes moral explanations under the general theory of program explanations (as defended by Jackson and Pettit). It follows that the phenomenon of moral explanations cannot be used to adjudicate the debate between expressivism and its rivals.

Explanation is asymmetric: if A explains B, then B does not explain A. Tradition- ally, the asymmetry of explanation was thought to favor causal accounts of explanation over their rivals, such as those that take explanations to be inferences. In this paper, we develop a new inferential approach to explanation that outperforms causal approaches in accounting for the asymmetry of explanation.

I discuss the methodological passage in the begin- ning of Ethica Eudemia I.6 (1216b26-35), which has received attention in connection with Aristotle’s notion of dialectic and his methodology in Ethics. My central focus is not to discuss whether Aristotle is prescribing and using what has been called the method of endoxa. I will focus on how this passage coheres with the remaining parts of the same chapter, which also are advancing methodological remarks. My claim is that the meth- od of (...) Ethica Eudemia I.6 is in agreement with many features of Aristotle’s theory of explanation as presented in the Posterior Analytics: Aristotle’s main concern is a warning against misuses of explanatory arguments. (shrink)

The aim of this paper is to show that topology has a bearing on<br><br>combinatorial theories of possibility. The approach developed in this article is “mapping account” considering combinatorial worlds as mappings from individuals to properties. Topological structures are used to define constraints on the mappings thereby characterizing the “really possible” combinations. The mapping approach avoids the well-known incompatibility problems. Moreover, it is compatible with atomistic as well as with non-atomistic ontologies.It helps to elucidate the positions of logical atomism and (...) monism with theaid of topological separation axioms. (shrink)

Genetic explanations of religious belief, such as Freud’s analysis of theism as ‘a neurotic relic’, pose a problem for theists: how far do such explanations establish the irrationality of religious belief? I argue that genetic analyses of belief suffer from a number of limitations. Showing that some reason-irrelevant factor or factors were sufficient to produce conviction on some occasion would not establish that they were necessary in every case of religious conviction. Showing that reason-irrelevant factors were both necessary and sufficient (...) to produce conviction would not establish that reason-relevant factors were entirely absent. And showing that reason-relevant factors played no role in the adoption of the belief would not establish that they were absent in its continued retention. I conclude that we will profitably investigate the plausibility of religious beliefs by attending to the reasons that can be given for or against them rather than by speculating about their causes. (shrink)

Humeanism about laws of nature — the view that the laws reduce to the Humean mosaic — is a popular view, but currently existing versions face powerful objections. The non-supervenience objection, the non-fundamentality objection and the explanatory circularity objection have all been thought to cause problems for the Humean. However, these objections share a guiding thought — they are all based on the idea that there is a certain kind of divergence between the practice of science and the metaphysical picture (...) suggested by Humeanism. -/- I suggest that the Humean can respond to these objections not by rejecting this divergence, but by arguing that is appropriate. In particular the Humean can, in the spirit of Loewer (2012), distinguish between scientific and metaphysical explanation — this is motivated by differing aims of explanation in science and metaphysics. And they can further leverage this into distinctions between scientific and metaphysical fundamentality and scientific and metaphysical possibility. We can use these distinctions to respond to the objections that the Humean faces. (shrink)

There has been a growing trend to include non-causal models in accounts of scientific explanation. A worry addressed in this paper is that without a higher threshold for explanation there are no tools for distinguishing between models that provide genuine explanations and those that provide merely potential explanations. To remedy this, a condition is introduced that extends a veridicality requirement to models that are empirically underdetermined, highly-idealised, or otherwise non-causal. This condition is applied to models of electroweak symmetry (...) breaking beyond the Standard Model. (shrink)

This is a revised version of the introductory essay in C. Eschenbach, C. Habel and B. Smith (eds.), Topological Foundations of Cognitive Science, Hamburg: Graduiertenkolleg Kognitionswissenschaft, 1994, the text of a talk delivered at the First International Summer Institute in Cognitive Science in Buffalo in July 1994.

It is often thought that metaphysical grounding underwrites a distinctive sort of metaphysical explanation. However, it would be a mistake to think that all metaphysical explanations are underwritten by metaphysical grounding. In service of this claim, I offer a novel kind of metaphysical explanation called metaphysical explanation by constraint, examples of which have been neglected in the literature. I argue that metaphysical explanations by constraint are not well understood as grounding explanations.

Non-causal accounts of action explanation have long been criticized for lacking a positive thesis, relying primarily on negative arguments to undercut the standard Causal Theory of Action The Stanford Encyclopedia of Philosophy, 2016). Additionally, it is commonly thought that non-causal accounts fail to provide an answer to Donald Davidson’s challenge for theories of reasons explanations of actions. According to Davidson’s challenge, a plausible non-causal account of reasons explanations must provide a way of connecting an agent’s reasons, not only to (...) what she ought to do, but to what she actually does. That is, such explanations must be truth-apt, not mere rationalizations. My aim in this paper is to show how a non-causal account of action can provide explanations that are truth-apt and genuinely explanatory. To make this argument, I take as a given an account of the practical syllogism discussed by Michael Thompson and Eric Wiland, according to which the practical syllogism is truly practical rather than propositional in nature. Next, I present my primary positive thesis: reasons for actions have explanatory power in virtue of being parts of a structure—the practical syllogism—that contains the action being explained. I then argue that structural action explanations can meet Davidson’s challenge and that they genuinely explain actions. Finally, I conclude by addressing some objections to my argument. (shrink)

Recent work on interpretability in machine learning and AI has focused on the building of simplified models that approximate the true criteria used to make decisions. These models are a useful pedagogical device for teaching trained professionals how to predict what decisions will be made by the complex system, and most importantly how the system might break. However, when considering any such model it’s important to remember Box’s maxim that "All models are wrong but some are useful." We focus on (...) the distinction between these models and explanations in philosophy and sociology. These models can be understood as a "do it yourself kit" for explanations, allowing a practitioner to directly answer "what if questions" or generate contrastive explanations without external assistance. Although a valuable ability, giving these models as explanations appears more difficult than necessary, and other forms of explanation may not have the same trade-offs. We contrast the different schools of thought on what makes an explanation, and suggest that machine learning might benefit from viewing the problem more broadly. (shrink)

This paper presents and argues for an account of objectual understanding that aims to do justice to the full range of cases of scientific understanding, including cases in which one does not have an explanation of the understood phenomenon. According to the proposed account, one understands a phenomenon just in case one grasps a sufficiently accurate and comprehensive model of the ways in which it or its features are situated within a network of dependence relations; one’s degree of understanding (...) is proportional to the comprehensiveness and accuracy of such a model. I compare this account with accounts of scientific understanding that explicate understanding in terms of having an explanation of the understood phenomenon. I discuss three distinct types of cases in which scientific understanding does not amount to possessing an explanation of any kind, and argue that the proposed model-based account can accommodate these cases while still retaining a strong link between understanding and explanation. (shrink)

There is a widely shared belief that the higher-level sciences can provide better explanations than lower-level sciences. But there is little agreement about exactly why this is so. It is often suggested that higher-level explanations are better because they omit details. I will argue instead that the preference for higher-level explanations is just a special case of our general preference for informative, logically strong, beliefs. I argue that our preference for informative beliefs entirely accounts for why higher-level explanations are sometimes (...) better—and sometimes worse—than lower-level explanations. The result is a step in the direction of the unity of science hypothesis. 1Introduction2Background: Is Omitting Details an Explanatory Virtue? 2.1Anti-reductionist arguments2.2Reductionist argument2.3Logical strength3Bases, Links and Logical Strength4Functionalism and Fodor’s Argument5Two Generalizations6Should the Base Really Be Maximally Strong?7Anti-reductionist Arguments Regarding the Base8Should the Antecedent of the Link Really Be Maximally Weak? (shrink)

Normative explanations of why things are wrong, good, or unfair are ubiquitous in ordinary practice and normative theory. This paper argues that normative explanation is subject to a justification condition: a correct complete explanation of why a normative fact holds must identify features that would go at least some way towards justifying certain actions or attitudes. I first explain and motivate the condition I propose. I then support it by arguing that it fits well with various theories of (...) normative reasons, makes good sense of certain legitimate moves in ordinary normative explanatory discourse, and helps to make sense of our judgments about explanatory priority in certain cases of normative explanation. This last argument also helps to highlight respects in which normative explanation won’t be worryingly discontinuous with explanations in other domains even though these other explanations aren’t subject to the justification condition. Thus the paper aims not only to do some constructive theorizing about the relatively neglected topic of normative explanation but also to cast light on the broader question of how normative explanation may be similar to and different from explanations in other domains. (shrink)

A possible world is a junky world if and only if each thing in it is a proper part. The possibility of junky worlds contradicts the principle of general fusion. Bohn (2009) argues for the possibility of junky worlds, Watson (2010) suggests that Bohn‘s arguments are flawed. This paper shows that the arguments of both authors leave much to be desired. First, relying on the classical results of Cantor, Zermelo, Fraenkel, and von Neumann, this paper proves the possibility of junky (...) worlds for certain weak set theories. Second, the paradox of Burali-Forti shows that according to the Zermelo-Fraenkel set theory ZF, junky worlds are possible. Finally, it is shown that set theories are not the only sources for designing plausible models of junky worlds: Topology (and possibly other "algebraic" mathematical theories) may be used to construct models of junky worlds. In sum, junkyness is a relatively widespread feature among possible worlds. (shrink)