An influential position in the philosophy of biology claims that there are no biological laws, since any apparently biological generalization is either too accidental, fact-like or contingent to be named a law, or is simply reducible to physical laws that regulate electrical and chemical interactions taking place between merely physical systems. In the following I will stress a neglected aspect of the debate that emerges directly from the growing importance of mathematical models of biological phenomena. My main aim (...) is to defend, as well as reinforce, the view that there are indeed laws also in biology, and that their difference in stability, contingency or resilience with respect to physical laws is one of degrees, and not of kind . (shrink)

This essay proposes that mathematical biology can be used as a fruitful exemplar for the introduction of scientific principles to history. After reviewing the antecedents of the application of mathematics to biology, in particular evolutionary biology, I describe in detail a mathematical model of cultural diffusion based on an analogy with population genetics. Subsequently, as a case study, this model is used to investigate the dynamics of the early modern European witch-crazes in Bavaria, England, Hungary (...) and Finland. In the second part of the article, I sketch the methodological significance of this type of 'scientific history' and, in particular, I identify three lessons that mathematical biology can contribute to historiography. The first lesson is on the fundamental distinction between agent's purposes and structural social processes. I argue that mathematical modeling can be fruitfully applied to describe social processes, while agents' purposes ought to be addressed following an hermeneutic tradition. The second lesson is on the aim of mathematical modeling. Here I argue that the object of modeling, rather than being the prediction or retrodiction of events , is the understanding of the factors involved in the dynamics of social processes . Finally, the third lesson is on the new understanding of science after the collapse of the standard view. In summary, while mathematical modeling can provide an extremely powerful approach to clarify the dynamics of certain macro-historical processes, scientific methods ought to be regarded as a complement, not a substitute, to classical historiography. (shrink)

Category theory has been proposed as the ultimate algebraic model for biology. We review the Ehresmann–Vanbremeersch theory in the context of other mathematical approaches.

This seminal, multidisciplinary book shows how mathematics can be used to study the first principles of DNA. Most importantly, it enriches the so-called "Chargaff's grammar of biology" by providing the conceptual theoretical framework necessary to generalize Chargaff's rules. Starting with a simple example of DNA mathematical modeling where human nucleotide frequencies are associated to the Fibonacci sequence and the Golden Ratio through an optimization problem, its breakthrough is showing that the reverse, complement and reverse-complement operators defined over oligonucleotides (...) induce a natural set partition of DNA words of fixed-size. These equivalence classes, when organized into a matrix form, reveal hidden patterns within the DNA sequence of every living organism. Intended for undergraduate and graduate students both in mathematics and in life sciences, it is also a valuable resource for researchers interested in studying invariant genomic properties. (shrink)

Leonardo da Vinci’s extensive drawings and notes devoted to anatomy do not arise in a medical context. He does not engage with surgery or “physic.” Rather, his aim is to reveal what he understood to be the divine engineering of God’s greatest creation. His earliest anatomical drawings map the conduits for the “spirits” at a deep level not practiced by other artists interested in the human body. The first set of drawings he produced in 1489 describes skulls with brilliant draftsmanship. (...) His notes for these drawings are predominantly concerned with the location of the “common sense” (sensus communis) at the geometrical center of the cranium. The next great set of drawings completed in ca. 1510 expounds the form and functions of the bones and muscles in meticulous mechanical detail, probably in collaboration with the young Paduan doctor, Marcantonio della Torre. Leonardo’s greatest realization of mathematics in anatomy comes when he looks at the valves of the heart; in studying the heart he also conceived a casting technique that he adapted to determine the form of the ventricles in the brain, radically revising the traditional arrangement of these structures. In a range of portrayals from diagrammatic to pictorial and from static to dynamic, Leonardo’s anatomical research is unrivalled in revealing the mechanics of the human body. His work in this context is an underappreciated precursor of the modern science of biomechanics. (Larger-sized versions of the Leonardo drawings presented here are available as supplementary material in the online version of this article.). (shrink)

The paper discusses how systems biology is working toward complex accounts that integrate explanation in terms of mechanisms and explanation by mathematical models—which some philosophers have viewed as rival models of explanation. Systems biology is an integrative approach, and it strongly relies on mathematical modeling. Philosophical accounts of mechanisms capture integrative in the sense of multilevel and multifield explanations, yet accounts of mechanistic explanation have failed to address how a mathematical model could contribute to such (...) explanations. I discuss how mathematical equations can be explanatorily relevant. Several cases from systems biology are discussed to illustrate the interplay between mechanistic research and mathematical modeling, and I point to questions about qualitative phenomena, where quantitative models are still indispensable to the explanation. Systems biology shows that a broader philosophical conception of mechanisms is needed, which takes into account functional-dynamical aspects, interaction in complex networks with feedback loops, system-wide functional properties such as distributed functionality and robustness, and a mechanism’s ability to respond to perturbations. I offer general conclusions for philosophical accounts of explanation. (shrink)

The aim of this paper is to describe and analyze the epistemological justification of a proposal initially made by the biomathematician Robert Rosen in 1958. In this theoretical proposal, Rosen suggests using the mathematical concept of “category” and the correlative concept of “natural equivalence” in mathematical modeling applied to living beings. Our questions are the following: According to Rosen, to what extent does the mathematical notion of category give access to more “natural” formalisms in the modeling of (...) living beings? Is the so -called “naturalness” of some kinds of equivalences (which the mathematical notion of category makes it possible to generalize and to put at the forefront) analogous to the naturalness of living systems? Rosen appears to answer “yes” and to ground this transfer of the concept of “natural equivalence” in biology on such an analogy. But this hypothesis, although fertile, remains debatable. Finally, this paper makes a brief account of the later evolution of Rosen’s arguments about this topic. In particular, it sheds light on the new role played by the notion of “category” in his more recent objections to the computational models that have pervaded almost every domain of biology since the 1990s. (shrink)

Mathematical models of biological patterns are central to contemporary biology. This paper aims to consider what these models contribute to biology through the detailed consideration of an important case: Hamilton’s selfish herd. While highly abstract and idealized, Hamilton’s models have generated an extensive amount of research and have arguably led to an accurate understanding of an important factor in the evolution of gregarious behaviors like herding and flocking. I propose an account of what these models are able (...) to achieve and how they can support a successful scientific research program. I argue that the way these models are interpreted is central to the success of such programs. (shrink)

This article presents a challenge that those philosophers who deny the causal interpretation of explanations provided by population genetics might have to address. Indeed, some philosophers, known as statisticalists, claim that the concept of natural selection is statistical in character and cannot be construed in causal terms. On the contrary, other philosophers, known as causalists, argue against the statistical view and support the causal interpretation of natural selection. The problem I am concerned with here arises for the statisticalists because the (...) debate on the nature of natural selection intersects the debate on whether mathematical explanations of empirical facts are genuine scientific explanations. I argue that if the explanations provided by population genetics are regarded by the statisticalists as non-causal explanations of that kind, then statisticalism risks being incompatible with a naturalist stance. The statisticalist faces a dilemma: either she maintains statisticalism but has to renounce naturalism; or she maintains naturalism but has to content herself with an account of the explanations provided by population genetics that she deems unsatisfactory. This challenge is relevant to the statisticalists because many of them see themselves as naturalists. (shrink)

Philosophy can shed light on mathematical modeling and the juxtaposition of modeling and empirical data. This paper explores three philosophical traditions of the structure of scientific theory—Syntactic, Semantic, and Pragmatic—to show that each illuminates mathematical modeling. The Pragmatic View identifies four critical functions of mathematical modeling: (1) unification of both models and data, (2) model fitting to data, (3) mechanism identification accounting for observation, and (4) prediction of future observations. Such facets are explored using a recent exchange (...) between two groups of mathematical modelers in plant biology. Scientific debate can arise from different modeling philosophies. (shrink)

Pure mathematics can play an indispensable role explaining empirical phenomena if recent accounts of insect evolution are correct. In particular, the prime life cycles of cicadas and the geometric structure of honeycombs are taken to undergird an inference to the best explanation about mathematical entities. Neither example supports this inference or the mathematical realism it is intended to establish. Both incorrectly assume that facts about mathematical optimality drove selection for the respective traits and explain why they exist. (...) We show how this problem can be avoided, identify limitations of explanatory indispensability arguments, and attempt to clarify the nature of mathematical explanation. (shrink)

In his famous article “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” Eugen Wigner argues for a unique tie between mathematics and physics, invoking even religious language: “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve”. The possible existence of such a unique match between mathematics and physics has been extensively discussed by philosophers and historians of mathematics. Whatever the merits (...) of this claim are, a further question can be posed with regard to mathematization in science more generally: What happens when we leave the area of theories and laws of physics and move over to the realm of mathematical modeling in interdisciplinary contexts? Namely, in modeling the phenomena specific to biology or economics, for instance, scientists often use methods that have their origin in physics. How is this kind of mathematical modeling justified? (shrink)

In the first part of this article we survey general similarities and differences between biological and social macroevolution. In the second (and main) part, we consider a concrete mathematical model capable of describing important features of both biological and social macroevolution. In mathematical models of historical macrodynamics, a hyperbolic pattern of world population growth arises from non-linear, second-order positive feedback between demographic growth and technological development. Based on diverse paleontological data and an analogy with macrosociological models, we suggest (...) that the hyperbolic character of biodiversity growth can be similarly accounted for by non-linear, second-order positive feedback between diversity growth and the complexity of community structure. We discuss how such positive feedback mechanisms can be modelled mathematically. (shrink)

Although mathematical descriptions of the dynamics of system are widely employed in the physical sciences, they are employed infrequently in the biological sciences. The explanation for this usually appeals to the complexity of biological systems. I contend that quite the opposite is true and that such descriptions, in fact, enable complexity to be tamed. Moreover, in those areas in which mathematical descriptions have been used in the biological sciences, they provide a powerful vehicle for expanding our understanding of (...) the systems through an exploration of the mathematical models. Such investigations yield surprising information about the dynamics of the system some of which cannot be discovered through empirical investigation alone. (shrink)

Although mathematical descriptions of the dynamics of system are widely employed in the physical sciences, they are employed infrequently in the biological sciences. The explanation for this usually appeals to the complexity of biological systems. I contend that quite the opposite is true and that such descriptions, in fact, enable complexity to be tamed. Moreover, in those areas in which mathematical descriptions have been used in the biological sciences, they provide a powerful vehicle for expanding our understanding of (...) the systems through an exploration of the mathematical models. Such investigations yield surprising information about the dynamics of the system some of which cannot be discovered through empirical investigation alone. (shrink)

Bioscientists generate far more data than their minds can handle, and this trend is likely to continue. With the aid of a small set of versatile tools for mathematical modeling and statistical assessment, bioscientists can explore their real-world systems without experiencing data overflow. This article outlines an approach for combining modern high-throughput, low-cost, but non-selective biospectroscopy measurements with soft, multivariate biochemometrics data modeling to overview complex systems, test hypotheses, and making new discoveries. From preliminary, broad hypotheses and goals, many (...) relevant samples are selected and measured with respect to many informative variables. The resulting tables represent a “cacophony” of data. From these, the most relevant and reliable “underlying harmonies and rhythms” are extracted and tested statistically, displayed for interpretation, and used for prediction. Outliers are detected automatically. Interesting subsets of samples can then be chosen for in-depth analyses in subsequent research cycles. This pragmatic, top-down approach takes advantage of developments in both “soft,” data-driven modeling and “hard,” knowledge-driven cultures. Data analytical examples show how information-rich biospectroscopy can be used for characterizing and quantifying known and unknown chemical constituents and physical phenomena in intact biosamples. This is based on a combination of deductive “hard” and inductive “soft” modeling. The examples represent NIR spectra of biochemical mixtures, FTIR spectra of a microbiological fermentation process, and FTIR analysis of fatty acids in milk for functional genomics. (shrink)

The human attempts to access, measure and organize physical phenomena have led to a manifold construction of mathematical and physical spaces. We will survey the evolution of geometries from Euclid to the Algebraic Geometry of the 20th century. The role of Persian/Arabic Algebra in this transition and its Western symbolic development is emphasized. In this relation, we will also discuss changes in the ontological attitudes toward mathematics and its applications. Historically, the encounter of geometric and algebraic perspectives enriched the (...) mathematical practices and their foundations. Yet, the collapse of Euclidean certitudes, of over 2300 years, and the crisis in the mathematical analysis of the 19th century, led to the exclusion of “geometric judgments” from the foundations of Mathematics. After the success and the limits of the logico-formal analysis, it is necessary to broaden our foundational tools and re-examine the interactions with natural sciences. In particular, the way the geometric and algebraic approaches organize knowledge is analyzed as a cross-disciplinary and cross-cultural issue and will be examined in Mathematical Physics and Biology. We finally discuss how the current notions of mathematical (phase) “space” should be revisited for the purposes of life sciences. (shrink)

The first part of this paper highlights some key aspects of the differences in the use of mathematical tools in physics and in biology. Scientific knowledge is viewed as a network of interactions, some than as a hierarchically organized structure where mathematics would display the essence of phenomena. The concept of "unity" in the biological phenomenon is then discussed. In the second part, a foundational issue in mathematics is revisited, following recent perspective in the physiology of action. The (...) relevance of the historical formation of mathematical concepts is stressed in several parts. (shrink)

The Neo‐Darwinian concept of natural selection is plausible when one assumes a straightforward causation of phenotype by genotype. However, such simple 1:1 mapping must now give place to the modern concepts of gene regulatory networks and gene expression noise. Both can, in the absence of genetic mutations, jointly generate a diversity of inheritable randomly occupied phenotypic states that could also serve as a substrate for natural selection. This form of epigenetic dynamics challenges Neo‐Darwinism. It needs to incorporate the non‐linear, stochastic (...) dynamics of gene networks. A first step is to consider the mathematical correspondence between gene regulatory networks and Waddington's metaphoric ‘epigenetic landscape’, which actually represents the quasi‐potential function of global network dynamics. It explains the coexistence of multiple stable phenotypes within one genotype. The landscape's topography with its attractors is shaped by evolution through mutational re‐wiring of regulatory interactions – offering a link between genetic mutation and sudden, broad evolutionary changes. (shrink)

We argue that the mathematization of science should be understood as a normative activity of advocating for a particular methodology with its own criteria for evaluating good research. As a case study, we examine the mathematization of taxonomic classification in systematic biology. We show how mathematization is a normative activity by contrasting its distinctive features in numerical taxonomy in the 1960s with an earlier reform advocated by Ernst Mayr starting in the 1940s. Both Mayr and the numerical taxonomists sought (...) to formalize the work of classification, but Mayr introduced a qualitative formalism based on human judgment for determining the taxonomic rank of populations, while the numerical taxonomists introduced a quantitative formalism based on automated procedures for computing classifications. The key contrast between Mayr and the numerical taxonomists is how they conceptualized the temporal structure of the workflow of classification, specifically where they allowed meta-level discourse about difficulties in producing the classification. (shrink)

Evolution produced many species whose members are pre-programmed with almost all the competences and knowledge they will ever need. Others appear to start with very little and learn what they need, but appearances can deceive. I conjecture that evolution produced powerful innate meta-knowledge about a class of environments containing 3- D structures and processes involving materials of many kinds. In humans and several other species these innate learning mechanisms seem initially to use exploration techniques to capture a variety of useful (...) generalisations after which there is a "phase transition" in which learnt generalisations are displaced by a new generative architecture that allows novel situations and problems to be dealt with by reasoning -- a pre-cursor to explicit mathematical theorem proving in topology, geometry, arithmetic, and kinematics. This process seems to occur in some non-human animals and in preverbal human toddlers, but is clearest in the switch from pattern-based to syntax-based language use. The discovery of non-linguistic toddler theorems has largely gone unnoticed, though Piaget investigated some of the phenomena, and creative problem solving in some other animals also provides clues. A later evolutionary development seems to have enabled humans to cope with domains that involve both regularities and exceptions, explaining "U-shaped" language learning. Only humans appear to be able to develop meta-meta-competences needed for teaching learnt "theorems" and their proofs. I'll sketch a speculative theory, present examples, and propose a research programme, reducing the 'G' in AGI, while promising increased power in return. (shrink)

The theory suggests that quantum computations in brain neuronal dendritic-somatic microtubules regulate axonal firings to control conscious behavior. Within microtubule subunit proteins, collective dipoles in arrays of contiguous amino acid electron clouds enable suitable for topological dipole able to physically represent cognitive values, for example, those portrayed by Pothos & Busemeyer (P&B) as projections in abstract Hilbert space.

This is the story, told in the light of a new analysis of historical data, of a mathematical biology problem that was explored in the 1930s in Thomas Morgan’s laboratory at the California Institute of Technology. It is one of the early developments of evolutionary genetics and quantitative phylogeny, and deals with the identification and counting of chromosomal inversions in Drosophila species from comparisons of genetic maps. A re-analysis of the data produced in the 1930s using current mathematics (...) and computational technologies reveals how a team of biologists, with the help of a renowned mathematician and against their first intuition, came to an erroneous conclusion regarding the presence of phylogenetic signals in gene arrangements. This example illustrates two different aspects of a same piece: the appearance of a mathematical in biology problem solved with the development of a combinatorial algorithm, which was unusual at the time, and the role of errors in scientific activity. Also underlying is the possible influence of computational complexity in understanding the directions of research in biology. (shrink)

A general equation is derived describing the concentration of all possible complexes of a central molecule with a set of ligands bound to the central molecule. This deduction allows the reaction rate constants for the binding of a given molecule to the central molecule to depend on the species of molecules already bound and the location of the molecules already bound. The model thus allows for structural alteration of the central molecule by binding. Functions describing the concentration dependence of any (...) effect whatever depending on the distribution of complexes are deduced. Possible applications and methods of application are indicated.II est dérivé une équation qui décrit la concentration des toutes les complexes d'une molécule centrale avec une collection de ligands connectés à la molécule centrale. La déduction montre comme les coefficients de la rapidité de reaction entre la molécule centrale et les molécules de la collection dépend de l'espèce et de la location des molécules déjà connectées.La modèle dérivée de l'équation montre les changements structurelles par la connection. Les functions qui décrivent la dependence de chaque effet à la concentration, sont dérivés. Ces effects dépendent aussi à la distribution des complexes. Des applications et des méthodes sont indiqués.Eine allgemeine Vergleichung hat man bekommen welche die Konzentration aller möglichen Komplexen eines zentralen Moleküles mit einer Reihe von „Ligands” verbunden mit dem zentralen Molekül beschreibt. Diese Folgerung gestattet dass die Reaktiongeschwindigkeit Konstanten für die Bindung eines gegebenen Moleküles mit dem zentralen Molekül abhÄngt von den Arten schon gegebenen Molekülen und die Ortsbestimmung der schon gebundenen Molekülen. Das Model zeigt die strukturelle Änderungen des zentralen Moleküls durch Bindung. Funktionen, die die Konzentrationband jedes Effektes beschreiben dass abhÄngt von der Distribution der Komplexen, sind abgeleitet.Mögliche Anwendungen und Methoden von Anwendung sind dargelegt. (shrink)

Ever since the early decades of this century, there have emerged a number of competing schools of ecology that have attempted to weave the concepts underlying natural resource management and natural-historical traditions into a formal theoretical framework. It was widely believed that the discovery of the fundamental mechanisms underlying ecological phenomena would allow ecologists to articulate mathematically rigorous statements whose validity was not predicated on contingent factors. The formulation of such statements would elevate ecology to the standing of a rigorous (...) scientific discipline on a par with physics. However, there was no agreement as to the fundamental units of ecology. Systems ecologists sought to identify the fundamental organization that tied the physical and biological components of ecosystems into an irreducible unit: the ecosystem was their fundamental unit. Population ecologists sought, instead, to identify the biological mechanisms regulating the abundance and distribution of plant and animal species: to these ecologists, the individual organism was the fundamental unit of ecology, and the physical environment was nothing more than a stage upon which the play of individuals in perennial competition took place. As Joel Hagen has pointed out, the two schools were thus dividied by fundamentally different and irreconcilable assumptions about the nature of ecosystems.Notwithstanding these divisive efforts to elevate the image of ecology, the discipline remained in the shadows of American academia until the mid-1960s, when systems ecologists succeeded in projecting ecology onto the national scene. They did so by seeking closer involvement with practical problems: they argued before Congress that their approach to the theoretical problems of ecology was uniquely suited to the solution of the impending “environmental crisis.” With the establishment of the International Biological Program, they succeeded in attracting unprecedented levels of funding for systems ecology research. Theoretical population ecologists, on the other hand, found themselves consigned to the outer regions of this new institutional landscape. The systems ecologists' successful capture of the limelight and the purse brought the divisions between them and population ecologists into sharper relief — hence the hardening of the division of ecology observed by Hagen.45I have argued that the population biologist Richard Levins, prompted by these institutional developments, sought to challenge the social position of systems ecology, and to assert the intellectual priority of theoretical population ecology. He attempted to do so by articulating a nontrivial and rather carefully thought out classification of ecological models that led to the disqualification of systems analysis as a legitimate approach to the study of ecological phenomena. I have suggested that — ultimately —Levins's case against systems analysis in ecology rested on the view that an aspiration to realism and prediction was incompatible with an interest in theoretical issues, a concern that he equated with the search for generality. He sought to reinforce this argument by exploiting the fact that systems ecologists had staked their future on the provision of technical solutions to the problems of the “environmental crisis”: he associated systems ecologists' aspiration to realism and precision with a concern for practical issues, trading on the widely accepted view that practical imperatives are incompatible with the aims of scientific inquiry.46 These are plausible, but nonetheless questionable, claims which have now become an integral part of ecological knowledge. And finally, I hope to have shown how even the most abstract levels of scientific argument are shaped by political considerations, and how discussions of the conceptual development of modern ecology might benefit from a greater consideration of its historical and social dimensions.47. (shrink)

While many different mechanisms contribute to the generation of spatial order in biological development, the formation of morphogenetic fields which in turn direct cell responses giving rise to pattern and form are of major importance and essential for embryogenesis and regeneration. Most likely the fields represent concentration patterns of substances produced by molecular kinetics. Short range autocatalytic activation in conjunction with longer range “lateral” inhibition or depletion effects is capable of generating such patterns (Gierer and Meinhardt, 1972). Non-linear reactions are (...) required, and mathematical criteria were derived to design molecular models capable of pattern generation. The classical embryological feature of proportion regulation can be incorporated into the models. The conditions are mathematically necessary for the simplest two-factor case, and are likely to be a fair approximation in multi-component systems in which activation and inhibition are systems parameters subsuming the action of several agents. Gradients, symmetric and periodic patterns, in one or two dimensions, stable or pulsing in time, can be generated on this basis. Our basic concept of autocatalysis in conjunction with lateral inhibition accounts for self-regulatory biological features, including the reproducible formation of structures from near-uniform initial conditions as required by the logic of the generation cycle. Real tissue form, for instance that of budding Hydra, may often be traced back to local curvature arising within an initially relatively flat cell sheet, the position of evagination being determined by morphogenetic fields. Shell theory developed for architecture may also be applied to such biological processes. (shrink)

How fast? How strong? How many? So what? Why do numbers matter in biology? Chromatin binding proteins are forever in motion, exchanging rapidly between bound and free pools. How do regulatory systems whose components are in constant flux ensure stability and flexibility? This review explores the application of quantitative and mathematical approaches to mechanisms of epigenetic regulation. We discuss methods for measuring kinetic parameters and protein quantities in living cells, and explore the insights that have been gained by (...) quantifying and modelling dynamics of chromatin binding proteins. (shrink)

The book starts with the assumption that vagueness is a fundamental property of this world. From a philosophical account of vagueness via the presentation of alternative mathematics of vagueness, the subsequent chapters explore how vagueness manifests itself in the various exact sciences: physics, chemistry, biology, medicine, computer science, and engineering.

In 1966 Richard Levins argued that applications of mathematics to population biology faced various constraints which forced mathematical modelers to trade-off at least one of realism, precision, or generality in their approach. Much traditional mathematical modeling in biology has prioritized generality and precision in the place of realism through strategies of idealization and simplification. This has at times created tensions with experimental biologists. The past 20 years however has seen an explosion in mathematical modeling of (...) biological systems with the rise of modern computational systems biology and many new collaborations between modelers and experimenters. In this paper I argue that many of these collaborations revolve around detail-driven modeling practices which in Levins’ terms trade-off generality for realism and precision. These practices apply mathematics by working from detailed accounts of biological systems, rather than from initially idealized or simplified representations. This is possible by virtue of modern computation. The form these practices take today suggest however Levins’ constraints on mathematical application no longer apply, transforming our understanding of what is possible with mathematics in biology. Further the engagement with realism and the ability to push realistic models in new directions aligns well with the epistemological and methodological views of experimenters, which helps explain their increased enthusiasm for biological modeling. (shrink)

In the first part of this article we survey general similarities and differences between biological and social macroevolution. In the second (and main) part, we consider a concrete mathematical model capable of describing important features of both biological and social macroevolution. In mathematical models of historical macrodynamics, a hyperbolic pattern of world population growth arises from non-linear, second-order positive feedback between demographic growth and technological development. This is more or less identical with the working of the collective learning (...) mechanism. Based on diverse paleontological data and an analogy with macrosociological models, we suggest that the hyperbolic character of biodiversity growth can be similarly accounted for by non-linear, second-order positive feedback between diversity growth and the complexity of community structure, suggesting the presence within the biosphere of a certain analogue of the collective learning mechanism. We discuss how such positive feedback mechanisms can be modelled mathematically. (shrink)

This paper presents two interpretations of the fiber bundle fonnalism that is applicable to all gauge field theories. The constructionist interpretation yields a substantival spacetime. The analytic interpretation yields a structural spacetime, a third option besides the familiar substantivalism and relationalism. That the same mathematical fonnalism can be derived in two different ways leading to two different ontological interpretations reveals the inadequacy of pure fonnal arguments.