The aim of this article is to investigate the roles of commutative diagrams (CDs) in a specific mathematical domain, and to unveil the reasons underlying their effectiveness as a mathematical notation; this will be done through a case study. It will be shown that CDs do not depict spatial relations, but represent mathematical structures. CDs will be interpreted as a hybrid notation that goes beyond the traditional bipartition of mathematical representations into diagrammatic and linguistic. It will be argued that one (...) of the reasons why CDs form a good notation is that they are highly mathematically tractable: experts can obtain valid results by ‘calculating’ with CDs. These calculations, take the form of ‘diagram chases’. In order to draw inferences, experts move algebraic elements around the diagrams. It will be argued that these diagrams are dynamic. It is thanks to their dynamicity that CDs can externalize the relevant reasoning and allow experts to draw conclusions directly by manipulating them. Lastly, it will be shown that CDs play essential roles in the context of proof as well as in other phases of the mathematical enterprise, such as discovery and conjecture formation. (shrink)

We describe an organizing framework for the study of infinitary combinatorics. This framework is Čech cohomology. It describes ZFC principles distinguishing among the ordinals of the form $\omega _n$. More precisely, this framework correlates each $\omega _n$ with an $$ -dimensional generalization of Todorcevic’s walks technique, and begins to account for that technique’s “unreasonable effectiveness” on $\omega _1$.We show in contrast that on higher cardinals $\kappa $, the existence of these principles is frequently independent of the ZFC axioms. Finally, we (...) detail implications of these phenomena for the computation of strong homology groups and higher derived limits, deriving independence results in algebraic topology and homological algebra, respectively, in the process.Abstract prepared by Jeffrey Bergfalk.E-mail: [email protected]. (shrink)

We present some results related to Zilber’s Exponential-Algebraic Closedness Conjecture, showing that various systems of equations involving algebraic operations and certain analytic functions admit solutions in the complex numbers. These results are inspired by Zilber’s theorems on raising to powers.We show that algebraic varieties which split as a product of a linear subspace of an additive group and an algebraic subvariety of a multiplicative group intersect the graph of the exponential function, provided that they satisfy Zilber’s freeness and rotundity conditions, (...) using techniques from tropical geometry.We then move on to prove a similar theorem, establishing that varieties which split as a product of a linear subspace and a subvariety of an abelian variety A intersect the graph of the exponential map of A (again under the analogues of the freeness and rotundity conditions). The proof uses homology and cohomology of manifolds.Finally, we show that the graph of the modular j-function intersects varieties which satisfy freeness and broadness and split as a product of a Möbius subvariety of a power of the upper-half plane and a complex algebraic variety, using Ratner’s orbit closure theorem to study the images under j of Möbius varieties.Abstract prepared by Francesco Paolo GallinaroE-mail: [email protected]: https://etheses.whiterose.ac.uk/31077/. (shrink)

An MV-algebra A=(A,0,¬,⊕) is an abelian monoid (A,0,⊕) equipped with a unary operation ¬ such that ¬¬x=x,x⊕¬0=¬0, and y⊕¬(y⊕¬x)=x⊕¬(x⊕¬y). Chang proved that the equational class of MV-algebras is generated by the real unit interval [0,1] equipped with the operations ¬x=1−x and x⊕y=min(1,x+y). Therefore, the free n-generated MV-algebra Free n is the algebra of [0,1]-valued functions over the n-cube [0,1] n generated by the coordinate functions ξ i ,i=1, . . . ,n, with pointwise operations. Any such function (...) f is a McNaughton function, i.e., f is continuous, piecewise linear, and each piece has integer coefficients. Conversely, McNaughton proved that all McNaughton functions f: [0,1] n →[0,1] are in Free n . The elements of Free n are logical equivalence classes of n-variable formulas in the infinite-valued calculus of Łukasiewicz. The aim of this paper is to provide an alternative, representation-free, characterization of Free n. (shrink)

This unique book provides a self-contained conceptual and technical introduction to the theory of differential sheaves. This serves both the newcomer and the experienced researcher in undertaking a background-independent, natural and relational approach to "physical geometry". In this manner, this book is situated at the crossroads between the foundations of mathematical analysis with a view toward differential geometry and the foundations of theoretical physics with a view toward quantum mechanics and quantum gravity. The unifying thread is provided by the theory (...) of adjoint functors in category theory and the elucidation of the concepts of sheaf theory and homological algebra in relation to the description and analysis of dynamically constituted physical geometric spectrums. (shrink)

This rare publication continues an exploratory journey in relational biology, a study of biology in terms of the organization of networked connections in living systems. It builds on the author's two earlier monographs which looked at the epistemology of life and the ontogeny of life. Here the emphasis is on the intangibility of life, that the real nature of living systems is conveyed not by their tangible material basis but by their intangible inherent processes. Relational biology is the approach that (...) hails 'function dictates structure'; it is mathematics decoded into biological realizations. Therefore, the work begins with a concise introduction to category theory, equiping the reader with the mathematical metalanguage of relation biology. The book is organized around three parts: Part I is a comprehensive study of the most important functor in relational biology, the power set functor. The author lays the set-theoretic foundations of the functorial connections in relational biology, exploring relations, mappings, and set-valued mappings. In Part II, Natural Law receives a new mathematical formulation founded on two axioms: 'Everything is a set.' and 'Every process is a set-valued mapping.' The reader sees how Metabolism-Repair networks, equipped with set-valued processors, expand their role from models of biological entities to generic models of all natural systems. Part III expounds the various shades of invertibility in general, and the inversion of encoding to decoding in particular. A plethora of mathematical and biological examples illustrate the category-theoretic concepts of equivalence and adjunction. This book's algebraic approach to biological models will appeal to researchers and graduate students in mathematics, biology, and the philosophy of science. (shrink)

We explore an application of homological algebra to set theoretic objects by developing a cohomology theory for Hausdorff gaps. This leads to a natural equivalence notion for gaps about which we answer questions by constructing many simultaneous gaps. The first result is proved in ZFC while new combinatorial hypotheses generalizing ♣ are introduced to prove the second result. The cohomology theory is introduced with enough generality to be applicable to other questions in set theory. Additionally, the notion of (...) an incollapsible gap is introduced and the existence of such a gap is shown to be independent of ZFC. (shrink)

We explore an application of homological algebra by developing a cohomology theory for a class of Aronszajn trees. Properties of this class, called Todorevi trees, are examined. The system is compared to that for Hausdorff gaps introduced in the author's previous work and general results about both tree and gap systems are also proven.

We exhibit a bridge between the theory of cellular categories, used in algebraic topology and homological algebra, and the model-theoretic notion of stable independence. Roughly speaking, we show that the combinatorial cellular categories (those where, in a precise sense, the cellular morphisms are generated by a set) are exactly those that give rise to stable independence notions. We give two applications: on the one hand, we show that the abstract elementary classes of roots of Ext studied by Baldwin–Eklof–Trlifaj (...) are stable and tame. On the other hand, we give a simpler proof (in a special case) that combinatorial categories are closed under 2-limits, a theorem of Makkai and Rosický. (shrink)

This volume explores the many different meanings of the notion of the axiomatic method, offering an insightful historical and philosophical discussion about how these notions changed over the millennia. The author, a well-known philosopher and historian of mathematics, first examines Euclid, who is considered the father of the axiomatic method, before moving onto Hilbert and Lawvere. He then presents a deep textual analysis of each writer and describes how their ideas are different and even how their ideas progressed over time. (...) Next, the book explores category theory and details how it has revolutionized the notion of the axiomatic method. It considers the question of identity/equality in mathematics as well as examines the received theories of mathematical structuralism. In the end, Rodin presents a hypothetical New Axiomatic Method, which establishes closer relationships between mathematics and physics. Lawvere's axiomatization of topos theory and Voevodsky's axiomatization of higher homotopy theory exemplify a new way of axiomatic theory building, which goes beyond the classical Hilbert-style Axiomatic Method. The new notion of Axiomatic Method that emerges in categorical logic opens new possibilities for using this method in physics and other natural sciences. This volume offers readers a coherent look at the past, present and anticipated future of the Axiomatic Method. (shrink)

Homologous operational localization processes are effectuated in terms of generalized topological covering systems on structures of physical events. We study localization systems of quantum events' structures by means of Gtothendieck topologies on the base category of Boolean events' algebras. We show that a quantum events algebra is represented by means of a Grothendieck sheaf-theoretic fibred structure, with respect to the global partial order of quantum events' fibres over the base category of local Boolean frames.

Crystallography has proven a rich source of ideas over several centuries. Among the many ways of looking at space groups, N. David Mermin has pioneered the Fourier-space approach. Recently, we have supplemented this approach with methods borrowed from algebraic topology. We now show what topology, which studies global properties of manifolds, has to do with crystallography. No mathematics is assumed beyond what the typical physics or crystallography student will have seen of group theory; in particular, the reader need not have (...) any prior exposure to topology or to cohomology of groups. (shrink)

This edited volume of 13 new essays aims to turn past discussions of natural kinds on their head. Instead of presenting a metaphysical view of kinds based largely on an unempirical vantage point, it pursues questions of kindedness which take the use of kinds and activities of kinding in practice as significant in the articulation of them as kinds. The book brings philosophical study of current and historical episodes and case studies from various scientific disciplines to bear on natural kinds (...) as traditionally conceived of within metaphysics. Focusing on these practices reveals the different knowledge-producing activities of kinding and processes involved in natural kind use, generation, and discovery. -/- Specialists in their field, the esteemed group of contributors use diverse empirically responsive approaches to explore the nature of kindhood. This groundbreaking volume presents detailed case studies that exemplify kinding in use. Newly written for this volume, each chapter engages with the activities of kinding across a variety of disciplines. Chapter topics include the nature of kinds, kindhood, kinding, and kind-making in linguistics, chemical classification, neuroscience, gene and protein classification, colour theory in applied mathematics, homology in comparative biology, sex and gender identity theory, memory research, race, extended cognition, symbolic algebra, cartography, and geographic information science. -/- The volume seeks to open up an as-yet unexplored area within the emerging field of philosophy of science in practice, and constitutes a valuable addition to the disciplines of philosophy and history of science, technology, engineering, and mathematics. -/- Contributions from a diverse group of established and junior scholars in the fields of Philosophy and History and Philosophy of Science including Hasok Chang, Jordi Cat, Sally Haslanger, Joyce C. Havstad, Catherine Kendig, Bernhard Nickel, Josipa Petrunic, Samuli Pöyhönen, Thomas A. C. Reydon, Quayshawn Spencer, Jackie Sullivan, Michael Wheeler, and Rasmus Grønfeldt Winther. (shrink)

Despite its long history and stunning experimental successes, the mathematical foundation of perturbative quantum field theory is still a subject of ongoing research. This book aims at presenting some of the most recent advances in the field, and at reflecting the diversity of approaches and tools invented and currently employed. Both leading experts and comparative newcomers to the field present their latest findings, helping readers to gain a better understanding of not only quantum but also classical field theories. Though the (...) book offers a valuable resource for mathematicians and physicists alike, the focus is more on mathematical developments. This volume consists of four parts: The first Part covers local aspects of perturbative quantum field theory, with an emphasis on the axiomatization of the algebra behind the operator product expansion. The second Part highlights Chern-Simons gauge theories, while the third examines (semi-)classical field theories. In closing, Part 4 addresses factorization homology and factorization algebras. (shrink)

The historical evolution of the homotopy concept for paths illustrates how the introduction of a concept (be it implicit or explicit) depends upon the interests of the mathematicians concerned and how it gradually acquires a more satisfactory definition. In our case the equivalence of paths first meant for certain mathematicians that they led to the same value of the integral of a given function or that they led to the same value of a multiple-valued function. (See for instance [Cau], [Pui], (...) [Rie].) Later this dependency upon given functions is dropped (by Jordan for surfaces, by Riemann and most explictly by Poincaré) and this leads to a concept which depends only upon the manifold. It thus becomes a concept which belongs to topology. As a consequence of this hesitant evolution, there was at first a confusion between concepts and hence no attention to relations between them. At a later stage these relations were investigated, as for instance the fact that homotopy equivalence implies homology equivalence: in1882, Klein gave an example of a closed curve on a surface which is the boundary of a part of the surface but could not be shrunk to a point. In 1904, Poincaré explicitly said that this curve is “homologue à zéro” (null homologous), but not “équivalent à zéro” (null homotopic). Poincaré obtains the homologies from the fundamental group by allowing changes in the order of the terms in the “équivalences”. This means also that the equivalences imply homologies but not vice versa. From a methodological standpoint, this situation of using properties without asking about relations between them or without even properly defining them is reminiscent of mathematics of a century earlier. An example is the way differentiability was used for the differentiation of functions without consciously questioning the properties of this concept until in the nineteenth century examples were given by Bolzano (1834), Weierstrass (1872), Riemann (1854) and others of functions which are continuous but nowhere differentiable [Kli]. Another example is the use eighteenth-century mathematicians made of series without inquiring about the validity of the operations they used on them. Another aspect which had its influence was the success of algebraic methods in topology which explains the preference for theories with “base point” and constrained deformation even though free deformation is a more natural concept. As we can see the evolution of the homotopy concept for paths did not progress without impediments. It also had an influence on the evolution of the abstract group concept and the basic principle of equivalence. These aspects make it a history which is not only intrinsically interesting (how did homotopic paths come to be?), but also because it illustrates relations between different branches of mathematics. (shrink)

Homology is a natural kind concept, but one that has been notoriously elusive to pin down. There has been sustained debate over the nature of correspondence and the units of comparison. But this continued debate over its meaning has focused on defining homology rather than on its use in practice. The aim of this chapter is to concentrate on the practices of homologizing. I define “homologizing” to be a concept-in-use. Practices of homologizing are kinds of rule following, the satisfaction of (...) which demarcates a category—that of being a homologue. Identifying, explaining, discovering, and understanding are exchanges that connect practice to concept through the performance of a rule by practitioners. These practices are constitutive of natural kinding activities. If homologizing is a kind of kinding, consideration of these practices of discovery, tracking, and identification not only clarifies the meaning, use, and progression of the concept of homology, but provides further understanding of the processes and progression of natural kinds and kinding practices in general. (shrink)

This article provides an algebraic study of the propositional system $\mathtt {InqB}$ of inquisitive logic. We also investigate the wider class of $\mathtt {DNA}$ -logics, which are negative variants of intermediate logics, and the corresponding algebraic structures, $\mathtt {DNA}$ -varieties. We prove that the lattice of $\mathtt {DNA}$ -logics is dually isomorphic to the lattice of $\mathtt {DNA}$ -varieties. We characterise maximal and minimal intermediate logics with the same negative variant, and we prove a suitable version of Birkhoff’s classic variety (...) theorems. We also introduce locally finite $\mathtt {DNA}$ -varieties and show that these varieties are axiomatised by the analogues of Jankov formulas. Finally, we prove that the lattice of extensions of $\mathtt {InqB}$ is dually isomorphic to the ordinal $\omega +1$ and give an axiomatisation of these logics via Jankov $\mathtt {DNA}$ -formulas. This shows that these extensions coincide with the so-called inquisitive hierarchy of [9].1. (shrink)

Recent work on inheritance systems can be divided into inclusive conceptions, according to which genetic and non-genetic inheritance are both involved in the development and transmission of nearly all animal behavioral traits, and more demanding conceptions of what it takes for non-genetic resources involved in development to qualify as a distinct inheritance system. It might be thought that, if a more stringent conception is adopted, homologies could not subsist across two distinct inheritance systems. Indeed, it is commonly assumed that homology (...) relations cannot survive a shift between genetic and cultural inheritance systems, and substantial reliance has been placed on that assumption in debates over the phylogenetic origins of hominin behavioral traits, such as male-initiated intergroup aggression. However, in the homology literature it is widely accepted that a trait can be homologous—that is, inherited continuously in two different lineages from a single common ancestor—despite divergence in the mechanisms involved in the trait’s development in the two lineages. In this paper, we argue that even on an extremely stringent understanding of what it takes for developmental resources to form a separate inheritance system, homologies can nonetheless subsist across shifts between distinct inheritance systems. We argue that this result is a merit of this way of characterizing what it is to be an inheritance system, that it has implications for adjudicating between alternative accounts of homology, and that it offers an important cautionary lesson about how to reason with the homology concept, particularly in the context of cultural species. (shrink)

Homology is a natural kind term and a precise account of what homology is has to come out of theories about the role of homologues in evolution and development. Definitions of homology are discussed with respect to the question as to whether they are able to give a non-circular account of the correspondence or sameness referred to by homology. It is argued that standard accounts tie homology to operational criteria or specific research projects, but are not yet able to offer (...) a concept of homology that does not presuppose a version of homology or a comparable notion of sameness. This is the case for phylogenetic definitions that trace structures back to the common ancestor as well as for developmental approaches such as Wagner's biological homology concept. In contrast, molecular homology is able to offer a definition of homology in genes and proteins that explicates homology by reference to more basic notions. Molecular correspondence originates by means of specific features of causal processes. It is speculated that further understanding of morphogenesis might enable biologists to give a theoretically deeper definition of homology along similar lines: an account which makes reference to the concrete mechanisms that operate in organisms. (shrink)

“Functional homology” appears regularly in different areas of biological research and yet it is apparently a contradiction in terms—homology concerns identity of structure regardless of form and function. I argue that despite this conceptual tension there is a legitimate conception of ‘homology of function’, which can be recovered by utilizing a distinction from pre-Darwinian physiology (use versus activity) to identify an appropriate meaning of ‘function’. This account is directly applicable to molecular developmental biology and shares a connection to the theme (...) of hierarchy in homology. I situate ‘homology of function’ within existing definitions and criteria for structural assessments of homology, and introduce a criterion of ‘organization’ for judging function homologues, which focuses on hierarchically interconnected interdependencies (similar to relative position and connection for skeletal elements in structural homology). This analysis of biological concepts has at least three broad philosophical consequences: (1) it provides the grounds for the study of behavior and psychological categories as homologues; (2) it demonstrates that philosophers who take selected effect function as primary effectively ignore large portions of comparative, structural, and experimental research, thereby misconstruing biological reasoning and knowledge; and, (3) it underwrites causal generalizations, which illuminates inferences made from model organisms in experimental biology. (shrink)

In A. Kumar, & M. Banerjee [(2012). Definable and rough sets in covering-based approximation spaces. In T. Li. (eds.), Rough sets and knowledge technology (pp. 488–495). Springer-Verlag], A. Kumar, & M. Banerjee [(2015). Algebras of definable and rough sets in quasi order-based approximation spaces. Fundamenta Informaticae, 141(1), 37–55], authors proposed a pair of lower and upper approximation operators based on granules generated by quasiorders. This work is an extension of algebraic results presented therein. A characterisation has been presented for those (...) quasiorder-generated covering-based approximation spaces whose corresponding collections of definable and rough sets form Stone algebras. The notion of rough lattice was proposed in A. Kumar, & M. Banerjee [(2015). Algebras of definable and rough sets in quasi order-based approximation spaces. Fundamenta Informaticae, 141(1), 37–55], A. Kumar [(2020). A Study of Algebras and Logics of Rough Sets Based on Classical and Generalized Approximation Spaces. In Transactions on Rough Sets XXII, LNCS (Vol. 12485, pp. 123–251). Springer]. Some special rough lattices are introduced in this work, viz. rough Stone algebra, ∼1-complemented and ∼2-complemented rough lattices. Representations of these algebras in terms of rough sets are obtained. Moreover, logics for these algebras are shown to be sound and complete with respect to rough set semantics. (shrink)

In this paper we show that axiomatic extensions of H. Wansing’s connexive logic $$\textsf{C}$$ ( $$\textsf{C}^{\perp }$$ ) are algebraizable (in the sense of J.W. Blok and D. Pigozzi) with respect to sub-varieties of $$\textsf{C}$$ ( $$\textsf{C}^{\perp }$$ )-algebras. We develop the structure theory of $$\textsf{C}$$ ( $$\textsf{C}^{\perp }$$ )-algebras, and we prove their representability in terms of twist-like constructions over implicative lattices (Heyting algebras). As a consequence, we further clarify the relationship between the aforementioned classes. Finally, taking advantage of (...) the above machinery, we provide some preliminary remarks on the lattice of axiomatic extensions of $$\textsf{C}$$ ( $$\textsf{C}^{\perp }$$ ) as well as on some properties of their equivalent algebraic semantics. (shrink)

Over the past decade, it has been discovered that disparate aspects of morphology – often of distantly related groups of organisms – are regulated by the same genetic regulatory mechanisms. Those discoveries provide a new perspective on morphological evolutionary change. A conceptual framework for exploring these research findings is termed ‘deep homology’. A comparative framework for morphological relations of homology is provided that distinguishes analogy, homoplasy, plesiomorphy and synapomorphy. Four examples – three from plants and one from animals – demonstrate (...) that homologous developmental mechanisms can regulate a range of morphological relations including analogy, homoplasy and examples of uncertain homology. Deep homology is part of a much wider range of phenomena in which biological (genes, regulatory mechanisms, morphological traits) and phylogenetic levels of homology can both be disassociated. Therefore, to understand homology, precise, comparative, independent statements of both biological and phylogenetic levels of homology are necessary. (shrink)

I defend the view that many biological categories are defined by homology against a series of arguments designed to show that all biological categories are defined, at least in part, by selected function. I show that categories of homology are `abnormality inclusive'—something often alleged to be unique to selected function categories. I show that classifications by selected function are logically dependent on classifications by homology, but not vice-versa. Finally, I reject the view that biologists must use considerations of selected function (...) to abstract away from variation and pathology to form a canonical description of a class of biological systems. (shrink)

Serial homology, i.e., homology between repetitive structures in the same individual organism, is a debated concept in evolutionary developmental biology. The central question is the evolutionary interpretation of “sameness” in the context of the same body. This essay provides a synthetic analysis of the main issues involved in the debate, connecting conceptual problems with current experimental research. It is argued that a concept of serial homology that is not of the all-or-nothing kind can smooth several theoretical inconsistencies, while being more (...) in line with what we know about evolutionary change and the way we investigate it. (shrink)

The purpose of this study is to compile a selection of the various formalisms found in conversation theory to introduce readers to Pask's discursive algebra. In this way, the text demonstrates how concept sharing and concept formation by means of the interaction of two participants may be formalized. The approach taken in this study is to examine the formal notation system used by Pask and demonstrate how such formalisms may be used to represent concept sharing and concept formation through (...) conversation. The compilation of the discursive algebra using the framework provided by conversation theory could potentially be used as an auxiliary framework to study conversational interactions in multi-agent systems theory. (shrink)

Famous for the number-theoretic first-order statement known as Goodstein's theorem, author R. L. Goodstein was also well known as a distinguished educator. With this text, he offers an elementary treatment that employs Boolean algebra as a simple medium for introducing important concepts of modern algebra. The text begins with an informal introduction to the algebra of classes, exploring union, intersection, and complementation; the commutative, associative, and distributive laws; difference and symmetric difference; and Venn diagrams. Professor Goodstein proceeds (...) to a detailed examination of three different axiomatizations, and an outline of a fourth system of axioms appears in the examples. The final chapter, on lattices, examines Boolean algebra in the setting of the theory of partial order. Numerous examples appear at the end of each chapter, with full solutions at the end. (shrink)

The meaning of the word homology has changed. From being a comparative concept in pre-Darwinian times, it became a historical concept, strictly signifying a common evolutionary origin for either anatomical structures or genes. This historical understanding of homology is not useful in classification; therefore I propose a return to its pre-Darwinian meaning.

Vorlesungen über die Algebra der Logik ist ein unveränderter, hochwertiger Nachdruck der Originalausgabe aus dem Jahr 1890. Hansebooks ist Herausgeber von Literatur zu unterschiedlichen Themengebieten wie Forschung und Wissenschaft, Reisen und Expeditionen, Kochen und Ernährung, Medizin und weiteren Genres. Der Schwerpunkt des Verlages liegt auf dem Erhalt historischer Literatur. Viele Werke historischer Schriftsteller und Wissenschaftler sind heute nur noch als Antiquitäten erhältlich. Hansebooks verlegt diese Bücher neu und trägt damit zum Erhalt selten gewordener Literatur und historischem Wissen auch für (...) die Zukunft bei. (shrink)

This article argues that at least very many judgments of homology rest on prior attributions of selected‐effect (SE) function, and that many of the “parts” of biological systems that are rightly classified as homologous are constituted by (are so classified in virtue of) their consequence etiologies. We claim that SE functions are often used in the prior identification of the parts deemed to be homologous and are often used to differentiate more restricted homologous kinds within less restricted ones. In doing (...) so, we discuss recent criticism of this view that has been offered (especially that offered by Paul Griffiths). *Received July 2007; revised November 2008. †To contact the authors, please write to: Department of Philosophy, 201 West Duke Building, Box 90743, Durham, NC 27708; e‐mail: [email protected] or [email protected]. (shrink)

This paper explores an important type of biological explanation called ‘homology thinking.’ Homology thinking explains the properties of a homologue by citing the history of a homologue. Homology thinking is significant in several ways. First, it offers more detailed explanations of biological phenomena than corresponding analogy explanations. Second, it provides an important explanation of character similarity and difference. Third, homology thinking offers a promising account of multiple realizability in biology.

Taxa and homologues can in our view be construed both as kinds and as individuals. However, the conceptualization of taxa as natural kinds in the sense of homeostatic property cluster kinds has been criticized by some systematists, as it seems that even such kinds cannot evolve due to their being homeostatic. We reply by arguing that the treatment of transformational and taxic homologies, respectively, as dynamic and static aspects of the same homeostatic property cluster kind represents a good perspective for (...) supporting the conceptualization of taxa as kinds. The focus on a phenomenon of homology based on causal processes (e.g., connectivity, activity-function, genetics, inheritance, and modularity) and implying relationship with modification yields a notion of natural kinds conforming to the phylogenetic-evolutionary framework. Nevertheless, homeostatic property cluster kinds in taxonomic and evolutionary practice must be rooted in the primacy of epistemological classification (homology as observational properties) over metaphysical generalization (series of transformation and common ancestry as unobservational processes). The perspective of individuating characters exclusively by historical-transformational independence instead of their developmental, structural, and functional independence fails to yield a sufficient practical interplay between theory and observation. Purely ontological and ostensional perspectives in evolution and phylogeny (e.g., an ideographic character concept and PhyloCode’s ‘individualism’ of clades) may be pragmatically contested in the case of urgent issues in biodiversity research, conservation, and systematics. (shrink)

This paper responds to the essay reviews by David Haig, Alan Love and Rachel Brown of my recently published book “Homology, Genes and Evolutionary Innovation”. The issues addressed here relate to: the notion of classes and individuals, issues of explanatory value of adaptive and structuralist explanations in evolutionary biology, the role of homology in evolutionary theory, the limits of a pluralist stance vis a vis alternative explanations of homology, as well as the question whether and to what extend the perspective (...) laid out in HGEI can be or should be transferred to other branches of study, like comparative behavioral biology. (shrink)

In this article, I examine the issue of the alleged circularity in the determination of homologies within cladistic analysis. More specifically, I focus on the claims made by the proponents of the dynamic homology approach, regarding the distinction (sometimes made in the literature) between primary and secondary homology. This distinction is sometimes invoked to dissolve the circularity issue, by upholding that characters in a cladistic data matrix have to be only primarily homologous, and thus can be determined independently of phylogenetic (...) hypotheses, by using the classical Owenian criteria (for morphological characters) or via multiple sequence alignment (for sequence data). However, since in the dynamic approach, sequence data can be analyzed without being pre-aligned, proponents have claimed that the distinction between primary and secondary homology has no place within cladistics. I will argue that this is not the case, since cladistic practice within the dynamic framework does presuppose primary homology statements at a higher level. (shrink)

Homology is the similarity between organisms due to common ancestry. Introduced by Richard Owen in 1843 in a paper entitled “Lectures on comparative anatomy and physiology of the invertebrate animals”, the concept of homology predates Darwin's “Origin of Species” and has been very influential throughout the history of evolutionary biology. Although homology is the central concept of all comparative biology and provides a logical basis for it, the definition of the term and the criteria of its application remain controversial. Here, (...) I will discuss homology in the context of the hierarchy of biological organization. I will provide insights gained from an exemplary case study in evolutionary developmental biology that indicates the uncoupling of homology at different levels of biological organization. I argue that continuity and hierarchy are separate but equally important issues of homology. BioEssays 30:653–658, 2008. © 2008 Wiley Periodicals, Inc. (shrink)

The present paper analyzes the use and understanding of the homology concept across different biological disciplines. It is argued that in its history, the homology concept underwent a sort of adaptive radiation. Once it migrated from comparative anatomy into new biological fields, the homology concept changed in accordance with the theoretical aims and interests of these disciplines. The paper gives a case study of the theoretical role that homology plays in comparative and evolutionary biology, in molecular biology, and in evolutionary (...) developmental biology. It is shown that the concept or variant of homology preferred by a particular biological field is used to bring about items of biological knowledge that are characteristic for this field. A particular branch of biology uses its homology concept to pursue its specific theoretical goals. (shrink)

The ‘byproduct account’ of female orgasm, a subject of renewed debate since Lloyd (The case of the female orgasm, Harvard University Press, Cambridge, 2005), is universally attributed to Symons (The evolution of human sexuality, Oxford University Press, Oxford, 1979). While this is correct to the extent that he linked it to the adaptive value of male orgasm, I argue that the attribution of the theory as we understand it to Symons is based on a serious and hitherto unrecognised misinterpretation. Symons (...) had a different explanation of why women can orgasm, and beneath this explanation lies an obscure line of argument, including a particularly obscure use of the word ‘homologous’. (shrink)

The concept of homology is the most solid theoretical basis elaborated by the morphological thinking during its history. The enucleation of some general criteria for the interpretation of homology is today a fundamental tool for life sciences, and for restoring their own opening to the question of qualitative innovation that arose so powerfully in the original Darwinian project. The aim of this paper is to verify the possible uses of the concept of compositional homology in order to provide of an (...) adequate understanding of the dynamics of creative thinking. (shrink)

This paper puts forth a class of algebraic structures, relativized Boolean algebras (RBAs), that provide semantics for propositional logic in which truth/validity is only defined relative to a local domain. In particular, the join of an event and its complement need not be the top element. Nonetheless, behavior is locally governed by the laws of propositional logic. By further endowing these structures with operators—akin to the theory of modal Algebras—RBAs serve as models of modal logics in which truth is relative. (...) In particular, modal RBAs provide semantics for various well-known awareness logics and an alternative view of possibility semantics. (shrink)

A un secolo e mezzo dalla pubblicazione dell'Origine delle specie, la nozione di omologia di Owen, che implica che due organi o due funzioni a confronto possono essere riconosciuti come “la stessa cosa” continua a dominare gli attuali approcci al problema dell’omologia. L'idea che i caratteri possano “rimanere se stessi” attraverso un numero indefinito (anche se finito) di stati alternativi che si susseguono nel corso dell'evoluzione è probabilmente basato su un'interpretazione idealistica di come gli organismi si evolvono. Se si dà (...) per scontata l'esistenza di caratteristiche del corpo che conservano la loro identità nel cambiamento, l'esistenza di omologhi può essere spiegata (i) in termini di leggi universali della forma, magari nella forma moderna dello strutturalismo di processo, o (ii) come il prodotto di discendenza comune, o (iii) in termini di cause prossime responsabili della comparsa di moduli di sviluppo conservati. Tuttavia, leggendo la storia della vita attraverso la lente dell’evoluzione, appare molto problematico descrivere un organismo come una somma di caratteri che, una volta apparsi come novità evolutive, rimangono poi come unità costruttive essenzialmente immutabili tra le quali possiamo predicare (o negare) omologia. Tutti i tratti sono meglio interpretati come incroci complessi e mutevoli di un numero indeterminato di elementi e dei processi che li modellano nel corso dell’ontogenesi e della filogenesi. Di conseguenza, appare raccomandabile un approccio fattoriale, o combinatorio, all’omologia. (shrink)

The Mind-Brain Identity Theory lived a short life as a respectable philosophical position in the late 1950s, until Hilary Putnam developed his famous argument on the multiple realizability of mental states. The argument was, and still is, taken as the definitive demonstration of the falsity of Identity Theory and the foundation on which contemporary functionalist computational cognitive science was to be grounded. In this paper, in the wake of some contemporary philosophers, we reopen the case for Identity Theory and offer (...) a solution to the problem of multiple realizabilty. The solution is based on the necessity, at the time of establishing identity relations, of appealing to the notions of “homology” and “analogy” developed in the nineteenth century by Richard Owen. We also suggest that these notions are useful in order to correct certain shortcomings of some recent attempts at rebutting the Multiple Realizability argument. (shrink)

This paper attempts to reconcile the conceptual conflict between typological and population thinking to provide a philosophical foundation for extended evolutionary synthesis. Typological thinking has been considered a pre-Darwinian, essentialist dogma incompatible with population thinking, which is the core notion of Darwinism. More recent philosophical and historical studies suggest that a non-essentialist form of typology has some advantages in the study of evolutionary biology. However, even if we adopt such an epistemological interpretation of typological thinking, there still remains an epistemological (...) and methodological conflict between these two styles of thinking. How can we relate typological thinking with population thinking in pursuit of more integrated or interconnected research into evolutionary biology? I propose that homology thinking, which is another style of thinking that recognizes homologous characters, provides a common basis for typological representations of character states and for character dynamics in an evolving population. Good examples of this bridging role are found in teratology and breeding, where variation and novelty are recognized in developmental and morphological traits, gene expression patterns, and so on. Essentialism-free, dynamic views of homology have great potential to reconcile typological and population thinking and to set the stage for the EES. (shrink)

An algebraic characterisation is given of the Mares-Goldblatt semantics for quantified extensions of relevant and modal logics. Some features of this more general semantic framework are investigated, and the relations to some recent work in algebraic semantics for quantified extensions of non-classical logics are considered.

Varios filósofos han propuesto un enfoque “algebraico” de los universales de acuerdo con el cual hay operaciones de universales a universales. No es obvio, sin embargo, cómo estas operaciones deban ser interpretadas y qué impacto tengan para las condiciones de identidad de los universales. Hay dos interpretaciones principales del álgebra de universales. Por una parte, se ha interpretado como formas de “construir” universales complejos. Por otra, ha sido interpretada como “morfismos” o “mapeos” entre universales, pero no como algo que “construye” (...) universales complejos desde otros más básicos. En este trabajo se evalúan las ventajas comparativas de las dos concepciones, y se ofrecen razones para preferir la segunda. (shrink)

The evolutionary embryologist Gavin Rylands de Beer can be viewed as one of the forerunners of modern evolutionary developmental biology in that he posed crucial questions and proposed relevant answers about the causal relationship between ontogeny and phylogeny. In his developmental approach to the phylogenetic phenomenon of homology, he emphasized that homology of morphological structures is to be identified neither with the sameness of the underlying developmental processes nor with the homology of the genes that are in involved in the (...) development of the structures. De Beer’s work on developmental evolution focused on the notion of heterochrony, arguing that paedomorphosis increases morphological evolvability and is thereby an important mode of evolution that accounts for the origin of many taxa, including higher taxa. (shrink)