Since the early days of physics, space has called for means to represent, experiment, and reason about it. Apart from physicists, the concept of space has intrigued also philosophers, mathematicians and, more recently, computer scientists. This longstanding interest has left us with a plethora of mathematical tools developed to represent and work with space. Here we take a special look at this evolution by considering the perspective of Logic. From the initial axiomatic efforts of Euclid, we revisit the major (...) milestones in the logical representation of space and investigate current trends. In doing so, we do not only consider classical logic, but we indulge ourselves with modal logics. These present themselves naturally by providing simple axiomatizations of different geometries, topologies, space-time causality, and vector spaces. (shrink)

Defending Robert Rosen’s claim that in every confrontation between physics and biology it is physics that has always had to give ground, it is shown that many of the most important advances in mathematics and physics over the last two centuries have followed from Schelling’s demand for a new physics that could make the emergence of life intelligible. Consequently, while reductionism prevails in biology, many biophysicists are resolutely anti-reductionist. This history is used to identify and defend a fragmented but progressive (...) tradition of anti-reductionist biomathematics. It is shown that the mathematicoephysico echemical morphology research program, the biosemiotics movement, and the relational biology of Rosen, although they have developed independently of each other, are built on and advance this antireductionist tradition of thought. It is suggested that understanding this history and its relationship to the broader history of post-Newtonian science could provide guidance for and justify both the integration of these strands and radically new work in post-reductionist biomathematics. (shrink)

There are many ways of understanding the nature of philosophical questions. One may consider their morphology, semantics, relevance, or scope. This article introduces a different approach, based on the kind of informational resources required to answer them. The result is a definition of philosophical questions as questions whose answers are in principle open to informed, rational, and honest disagreement, ultimate but not absolute, closed under further questioning, possibly constrained by empirical and logico-mathematical resources, but requiring noetic resources to (...) be answered. The article concludes with a discussion of some of the consequences of this definition for a conception of philosophy as the study (or “science”) of open questions, which uses conceptual design to analyse and answer them. (shrink)

According to the computational theory of mind , to think is to compute. But what is meant by the word 'compute'? The generally given answer is this: Every case of computing is a case of manipulating symbols, but not vice versa - a manipulation of symbols must be driven exclusively by the formal properties of those symbols if it is qualify as a computation. In this paper, I will present the following argument. Words like 'form' and 'formal' are ambiguous, as (...) they can refer to form in either the syntactic or the morphological sense. CTM fails on each disambiguation, and the arguments for CTM immediately cease to be compelling once we register that ambiguity. The terms 'mechanical' and 'automatic' are comparably ambiguous. Once these ambiguities are exposed, it turns out that there is no possibility of mechanizing thought, even if we confine ourselves to domains where all problems can be settled through decision-procedures. The impossibility of mechanizing thought thus has nothing to do with recherché mathematical theorems, such as those proven by Gödel and Rosser. A related point is that CTM involves, and is guilty of reinforcing, a misunderstanding of the concept of an algorithm. (shrink)

Arguments of stability, intended in a wide sense, including the discussion of the conditions of the onset of instability and of stability changes, play a central role in the main theorizations of morphogenesis in 20th century theoretical biology. The aim of this essay is to shed light on concepts and images mobilized in the construction of arguments of stability in theorizing morphogenesis, since they are pivotal in establishing meaningful relationships between mathematical models and empirical morphologies.

In this paper, a formal theory is presented that describes syntactic and semantic mechanisms of philosophical discourses. They are treated as peculiar language systems possessing deep derivational structures called architectonic forms of philosophical systems, encoded in philosophical mind. Architectonic forms are constituents of more complex structures called architectonic spaces of philosophy. They are understood as formal and algorithmic representations of various philosophical traditions. The formal derivational machinery of a given space determines its class of all possible architectonic forms. Some of (...) them stand under factual historical philosophical systems and they organize processes of doing philosophy within these systems. Many architectonic forms have never been realized in the history of philosophy. The presented theory may be interpreted as falling under Hegel’s paradigm of comprehending cultural texts. This paradigm is enriched and inspired with Propp’s formal, morphological view on texts. The peculiarity of this modification of the Hegel-Propp paradigm consists of the use of algebraic and algorithmic tools of modeling processes of cultural development. To speak metaphorically, the theory is an attempt at the mathematical and logical history of philosophy inspired by the Internet metaphor. And that is why it belongs to the tradition of doing metaphilosophy in The Lvov-Warsaw School, which is continued today mainly by Woleński, Pelc, Perzanowski, and Jadacki. (shrink)

Most philosophical accounts on scientific theories are affected by three dogmas or ingrained attitudes. These dogmas have led philosophers to choose between analyzing the internal structure of theories or their historical evolution. In this paper, I turn these three dogmas upside down. I argue (i) that mathematical practices are not epistemically neutral, (ii) that the morphology of theories can be very complex, and (iii) that one should view theoretical knowledge as the combination of internal factors and their intrinsic (...) historicity. (shrink)

Few metaphors in biology are more enduring than the idea of Adaptive Landscapes, originally proposed by Sewall Wright (1932) as a way to visually present to an audience of typically non- mathematically savvy biologists his ideas about the relative role of natural selection and genetic drift in the course of evolution. The metaphor, how- ever, was born troubled, not the least reason for which is the fact that Wright presented different diagrams in his original paper that simply can- not refer (...) to the same concept and are therefore hard to reconcile with each other (Pigliucci 2008). For instance, in some usages, the landscape’s non- fitness axes represent combinations of individual genotypes (which cannot sensibly be aligned on a linear axis, and accordingly were drawn by Wright as polyhedrons of increasing dimensionality). In other usages, however, the points on the diagram represent allele or genotypic frequencies, and so are actually populations, not individuals (and these can indeed be coherently represented along continuous axes). (shrink)

Nền kinh tế nước ta đang chuyển biến mạnh mẽ từ nền kinh tế bao cấp sang kinh tế thị trường, nhất là từ khi nước ta gia nhập WTO. Đảng và chính phủ đã đề ra rất nhiều các chính sách để cải tiến các thể chế quản lý nền kinh tế và tài chính. Thị trường chứng khoán Việt Nam đã ra đời và đang đóng một vai trò quan trọng trong việc huy động vốn phục vụ cho (...) việc phát triển nền kinh tế quốc dân. Việc phân tích tác động của các chính sách đến nền kinh tế vĩ mô và nền tài chính, những đối sách của chính phủ ta và chính phủ các nước đối với cuộc khủng khoảng tài chính toàn cầu là các vấn đề rất thời sự đang thu hút sự chú ý của nhiều nhà kinh tế và toán học. (shrink)

The emerging contemporary natural philosophy provides a common ground for the integrative view of the natural, the artificial, and the human-social knowledge and practices. Learning process is central for acquiring, maintaining, and managing knowledge, both theoretical and practical. This paper explores the relationships between the present advances in understanding of learning in the sciences of the artificial, natural sciences, and philosophy. The question is, what at this stage of the development the inspiration from nature, specifically its computational models such as (...) info-computation through morphological computing, can contribute to machine learning and artificial intelligence, and how much on the other hand models and experiments in machine learning and robotics can motivate, justify, and inform research in computational cognitive science, neurosciences, and computing nature. We propose that one contribution can be understanding of the mechanisms of ‘learning to learn’, as a step towards deep learning with symbolic layer of computation/information processing in a framework linking connectionism with symbolism. As all natural systems possessing intelligence are cognitive systems, we describe the evolutionary arguments for the necessity of learning to learn for a system to reach humanlevel intelligence through evolution and development. The paper thus presents a contribution to the epistemology of the contemporary philosophy of nature. (shrink)

This article introduces morphological typology, exploring the patterns and structures underlying word formation and grammatical encoding across languages. A systematic literature review examines fundamental concepts, including distinctions between analytic and synthetic languages, features of agglutinating, fusional, and inflectional morphologies, and phenomena like suppletion and polysynthetic structures. Readers gain insights into language classification based on morphological characteristics, challenging strict categorical distinctions and emphasizing the continuum across types. The study highlights the diversity and complexity of morphological systems. Suppletion, where stems are irregularly (...) replaced in inflectional patterns, and polysynthetic languages, encoding entire sentences within single complex words, are explored in depth. This work offers a concise overview of morphological typology by synthesizing reputable sources, including books, journals, and online resources. It is an accessible resource for language learners, linguists, and anyone interested in understanding the intricate morphological underpinnings of human language. (shrink)

The purpose of this paper is to argue against the claim that morphological computation is substantially different from other kinds of physical computation. I show that some (but not all) purported cases of morphological computation do not count as specifically computational, and that those that do are solely physical computational systems. These latter cases are not, however, specific enough: all computational systems, not only morphological ones, may (and sometimes should) be studied in various ways, including their energy efficiency, cost, reliability, (...) and durability. Second, I critically analyze the notion of “offloading” computation to the morphology of an agent or robot, by showing that, literally, computation is sometimes not offloaded but simply avoided. Third, I point out that while the morphology of any agent is indicative of the environment that it is adapted to, or informative about that environment, it does not follow that every agent has access to its morphology as the model of its environment. (shrink)

The paper aims to understand whether morphology can be framed as a language for aesthetics. In particular, whether Olaf Breidbach’s contribution can determine its fundamental terms. These are related to the notion of forms and images. Hence, the paper is structured into three parts: i) framing of research on morphology in Ger-many; ii) analysis of Goethe’s method and vocabulary from an aesthetic standpoint; iii) presentation of Breidbach’s proposal in relation to Goethe.

General Morphological Analysis (GMA) is a method for structuring a conceptual problem space – called a morphospace – and, through a process of existential combinatorics, synthesizing a solution space. As such, it is a basic modelling method, on a par with other scientific modelling methods including System Dynamics Modelling, Bayesian Networks and various types graph-based “influence diagrams”. The purpose of this article is 1) to present the theoretical and methodological basics of morphological modelling; 2) to situate GMA within a broader (...) modelling theoretical framework by developing a (morphological) model representing different modelling methods, and 3) to demonstrate some of the basic modelling techniques that can be carried out with GMA using dedicated computer support. (shrink)

General Morphological Analysis (GMA) is a computer-aided, non-quantified modelling method employing (discrete) category variables for identifying and investigating the total set of possible relationships contained in a given problem complex. The epistemological principle underlying discrete variable morphological modelling is that of decomposing a complex (multivariate) concept into a number of(“simple”) one dimensional concepts (i.e. category variables), the domains of which can then be recombined and recomposed in order to discover all of the other possible (multidimensional) concepts which can be generated (...) combinatorically. (shrink)

The contribution of the body to cognition and control in natural and artificial agents is increasingly described as “off-loading computation from the brain to the body”, where the body is said to perform “morphological computation”. Our investigation of four characteristic cases of morphological computation in animals and robots shows that the ‘off-loading’ perspective is misleading. Actually, the contribution of body morphology to cognition and control is rarely computational, in any useful sense of the word. We thus distinguish (1) (...) class='Hi'>morphology that facilitates control, (2) morphology that facilitates perception and the rare cases of (3) morphological computation proper, such as ‘reservoir computing.’ where the body is actually used for computation. This result contributes to the understanding of the relation between embodiment and computation: The question for robot design and cognitive science is not whether computation is offloaded to the body, but to what extent the body facilitates cognition and control – how it contributes to the overall ‘orchestration’ of intelligent behaviour. (shrink)

Call an explanation in which a non-mathematical fact is explained—in part or in whole—by mathematical facts: an extra-mathematical explanation. Such explanations have attracted a great deal of interest recently in arguments over mathematical realism. In this article, a theory of extra-mathematical explanation is developed. The theory is modelled on a deductive-nomological theory of scientific explanation. A basic DN account of extra-mathematical explanation is proposed and then redeveloped in the light of two difficulties that the (...) basic theory faces. The final view appeals to relevance logic and uses resources in information theory to understand the explanatory relationship between mathematical and physical facts. 1Introduction2Anchoring3The Basic Deductive-Mathematical Account4The Genuineness Problem5Irrelevance6Relevance and Information7Objections and Replies 7.1Against relevance logic7.2Too epistemic7.3Informational containment8Conclusion. (shrink)

out during January to December 1998 at Bangabandhu Sheikh Mujib Medical University, Dhaka to expand the knowledge of gross anatomy of the umbilical cord of Bangladesh. The length of the cords were irrespective of sex ranged from 28 to 93 cm with a mean (±SD) of 55.6 (±10.78). The length of the umbilical cords of males were significantly longer than female (P<0.001). The diameter of the cords irrespective of sex were varied from 1 to 1.9 cm with a mean (±SD) (...) of 1.45±0.31 cm. The mean circumference length percentage ratio index of umbilical cord was 8.31. Thirty-three (66%) cords were inserted eccentrically, all being paracentral in position. The rest were inserted centrally. False knots were more frequent (47; 94%). Only one (2%) showed a true knot in addition of false knot. In 2(4%) cases cord had not any true or false knot. It is concluded that the gross morphological and morphometrical features of the umbilical cord in Bangladesh appear to be similar to those described in western literature. (shrink)

We demonstrate how real progress can be made in the debate surrounding the enhanced indispensability argument. Drawing on a counterfactual theory of explanation, well-motivated independently of the debate, we provide a novel analysis of ‘explanatory generality’ and how mathematics is involved in its procurement. On our analysis, mathematics’ sole explanatory contribution to the procurement of explanatory generality is to make counterfactual information about physical dependencies easier to grasp and reason with for creatures like us. This gives precise content to key (...) intuitions traded in the debate, regarding mathematics’ procurement of explanatory generality, and adjudicates unambiguously in favour of the nominalist, at least as far as explanatory generality is concerned. (shrink)

Brouwer's philosophy of mathematics is usually regarded as an intra-subjective, even solipsistic approach, an approach that also underlies his mathematical intuitionism, as he strived to create a mathematics that develops out of something inner and a-linguistic. Thus, points of connection between Brouwer's mathematical views and his views about and the social world seem improbable and are rarely mentioned in the literature. The current paper aims to challenge and change that. The paper employs a socially oriented prism to examine (...) Brouwer's views on the construction, use, and practice of mathematics. It focuses on Brouwer's views on language, his social interactions, and the importance of group context as they appear in the significs dialogues. It does so by exploring the establishment and dissolution of the significs movement, focusing on Gerrit Mannoury's influence and relationship with Brouwer and analyzing several fragments from the significs dialogues while emphasizing the role Brouwer ascribed to groups in forming and sharing new ideas. The paper concludes by raising two questions that challenge common historical and philosophical readings of intuitionism. (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)

Morphological and morphometric study of umbilical cord of 50 newborn babies were carried out during January to December 1998 at Bangabandhu Sheikh Mujib Medical University, Dhaka to expand the knowledge of gross anatomy of the umbilical cord of Bangladesh. The length of the cords irrespective of sex was ranged from 28 to 93 cm with a mean (±SD) of 55.6 (±10.78).The length of the umbilical cord of male was significantly longer than female (P<0.001). The diameter of the cord irrespective of (...) sex varied from 1 to 1.9 cm with a mean (±SD) of 1.45 +0.31 cm. The mean circumference length percentage ratio index of umbilical cord was 8.31. Thirty-three (66%) cords were inserted eccentrically, all being paracentral in position. The rest were inserted centrally. False knots were more frequent (47; 94%). Only one (2%) showed a true knot in addition of false knot. In 2(4%) cases cord had not any true or false knot. It is concluded that the gross morphological and morphometrical features of the umbilical cord in Bangladesh appear to be similar to those described in western literature. -/- . (shrink)

We know very little about mathematical skepticism in modem times. Imre Lakatos once remarked that “in discussing modem efforts to establish foundations for mathematical knowledge one tends to forget that these are but a chapter in the great effort to overcome skepticism by establishing foundations for knowledge in general." And in a sense he was clearly right: modem thought — with its new discoveries in mathematical sciences, the mathematization of physics, the spreading of Pyrrhonist doctrines, the centrality (...) of epistemological foundationalism and the diffusion of the geometrical method in philosophy — was the most natural arena in which skepticism and mathematics could confront each other. The problem remains, however, that no investigation of the whole topic has yet been attempted. Thus, as far as we know, mathematical certainties should have clashed with skeptical doubts, but whether and to what extent there was indeed a historical debate on mathematical skepticism in modern thought remains to be ascertained. (shrink)

Morphological Computation is based on the observation that biological systems seem to carry out relevant computations with their morphology (physical body) in order to successfully interact with their environments. This can be observed in a whole range of systems and at many different scales. It has been studied in animals – e.g., while running, the functionality of coping with impact and slight unevenness in the ground is "delivered" by the shape of the legs and the damped elasticity of the (...) muscle-tendon system – and plants, but it has also been observed at the cellular and even at the molecular level – as seen, for example, in spontaneous self-assembly. The concept of morphological computation has served as an inspirational resource to build bio-inspired robots, design novel approaches for support systems in health care, implement computation with natural systems, but also in art and architecture. As a consequence, the field is highly interdisciplinary, which is also nicely reflected in the wide range of authors that are featured in this e-book. We have contributions from robotics, mechanical engineering, health, architecture, biology, philosophy, and others. (shrink)

An analysis of the counter-intuitive properties of infinity as understood differently in mathematics, classical physics and quantum physics allows the consideration of various paradoxes under a new light (e.g. Zeno’s dichotomy, Torricelli’s trumpet, and the weirdness of quantum physics). It provides strong support for the reality of abstractness and mathematical Platonism, and a plausible reason why there is something rather than nothing in the concrete universe. The conclusions are far reaching for science and philosophy.

Pluralist mathematical realism, the view that there exists more than one mathematical universe, has become an influential position in the philosophy of mathematics. I argue that, if mathematical pluralism is true (and we have good reason to believe that it is), then mathematical realism cannot (easily) be justified by arguments from the indispensability of mathematics to science. This is because any justificatory chain of inferences from mathematical applications in science to the total body of (...) class='Hi'>mathematical theorems can cover at most one mathematical universe. Indispensability arguments may thus lose their central role in the debate about mathematical ontology. (shrink)

View more Abstract If logical truth is necessitated by sheer syntax, mathematics is categorially unlike logic even if all mathematics derives from definitions and logical principles. This contrast gets obscured by the plausibility of the Synonym Substitution Principle implicit in conceptions of analyticity: synonym substitution cannot alter sentence sense. The Principle obviously fails with intercepting: nonuniform term substitution in logical sentences. ‘Televisions are televisions’ and ‘TVs are televisions’ neither sound alike nor are used interchangeably. Interception synonymy gets assumed because logical (...) sentences and their synomic interceptions have identical factual content, which seems to exhaust semantic content. However, intercepting alters syntax by eliminating term recurrence, the sole strictly syntactic means of ensuring necessary term coextension, and thereby syntactically securing necessary truth. Interceptional necessity is lexical, a notational artifact. The denial of interception nonsynonymy and the disregard of term recurrence in logic link with many misconceptions about propositions, logical form, conventions, and metalanguages. Mathematics is distinct from logic: its truth is not syntactic; it is transmitted by synonym substitution; term recurrence has no essential role. The ‘=’ of mathematics is an objectual relation between numbers; the ‘=’ of logic marks a syntactic relation of coreferring terms. -/- . (shrink)

Objectives: The most principal nutrition source of a bone is nutrient arteries. They are important at every stage of bone development. A nutrient artery enters a bone through the nutrient foramen, the largest hole on the outer surface of the bone. The foramen is important both morphologically and clinically. -/- Methods: A total of 414 adult human dry bones were investigated in this study to identify topographic and morphological features of nutrient foramina in the scapula, clavicle, humerus, radius and ulna. (...) Nutrient foramina were examined with a hand lens. Their dimensions and directions were determined with a 21-gauge needle, and thus major foramina were detected. Positions of nutrient foramina were noted according to surfaces of the bones, and to segments separated as proximal, middle and distal by calculating foraminal index. -/- Results: A single nutrient foramen was found in 71% of our samples. We observed that 94.2% of foramina in the clavicle, 89.3% of foramina in the humerus, 51.3% of foramina in the radius, and 67.7% of foramina in the ulna were located in the middle 1/3 segment of the bones. -/- Conclusion: Our findings were in line with the data in the literature. On account of pathologies associated with the nutrient foramen, our findings may be helpful for surgeons to design applications performed in the region. In addition, we think that our data by contributing to the literature may be a resource for clinicians due to the importance of the nutrient foramen for surgical procedures. (shrink)

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

I argue that certain species of belief, such as mathematical, logical, and normative beliefs, are insulated from a form of Harman-style debunking argument whereas moral beliefs, the primary target of such arguments, are not. Harman-style arguments have been misunderstood as attempts to directly undermine our moral beliefs. They are rather best given as burden-shifting arguments, concluding that we need additional reasons to maintain our moral beliefs. If we understand them this way, then we can see why moral beliefs are (...) vulnerable to such arguments while mathematical, logical, and normative beliefs are not—the very construction of Harman-style skeptical arguments requires the truth of significant fragments of our mathematical, logical, and normative beliefs, but requires no such thing of our moral beliefs. Given this property, Harman-style skeptical arguments against logical, mathematical, and normative beliefs are self-effacing; doubting these beliefs on the basis of such arguments results in the loss of our reasons for doubt. But we can cleanly doubt the truth of morality. (shrink)

The primary justification for mathematical structuralism is its capacity to explain two observations about mathematical objects, typically natural numbers. Non-eliminative structuralism attributes these features to the particular ontology of mathematics. I argue that attributing the features to an ontology of structural objects conflicts with claims often made by structuralists to the effect that their structuralist theses are versions of Quine’s ontological relativity or Putnam’s internal realism. I describe and argue for an alternative explanation for these features which instead (...) explains the attributes them to the mathematical practice of representing numbers using more concrete tokens, such as sets, strokes and so on. (shrink)

This paper presents a view of nature as a network of infocomputational agents organized in a dynamical hierarchy of levels. It provides a framework for unification of currently disparate understandings of natural, formal, technical, behavioral and social phenomena based on information as a structure, differences in one system that cause the differences in another system, and computation as its dynamics, i.e. physical process of morphological change in the informational structure. We address some of the frequent misunderstandings regarding the natural/morphological computational (...) models and their relationships to physical systems, especially cognitive systems such as living beings. Natural morphological infocomputation as a conceptual framework necessitates generalization of models of computation beyond the traditional Turing machine model presenting symbol manipulation, and requires agent-based concurrent resource-sensitive models of computation in order to be able to cover the whole range of phenomena from physics to cognition. The central role of agency, particularly material vs. cognitive agency is highlighted. (shrink)

Let mathematical justification be the kind of justification obtained when a mathematician provides a proof of a theorem. Are Gettier cases possible for this kind of justification? At first sight we might think not: The standard for mathematical justification is proof and, since proof is bound at the hip with truth, there is no possibility of having an epistemically lucky justification of a true mathematical proposition. In this paper, I argue that Gettier cases are possible (and indeed (...) actual) in mathematical reasoning. By analysing these cases, I suggest that the Gettier phenomenon indicates some upshots for actual mathematical practice. (shrink)

Although traditionally neglected, mathematical diagrams have recently begun to attract attention from philosophers of mathematics. By now, the literature includes several case studies investigating the role of diagrams both in discovery and justification. Certain preliminary questions have, however, been mostly bypassed. What are diagrams exactly? Are there different types of diagrams? In the scholarly literature, the term “mathematical diagram” is used in diverse ways. I propose a working definition that carves out the phenomena that are of most importance (...) for a taxonomy of diagrams in the context of a practice-based philosophy of mathematics, privileging examples from contemporary mathematics. In doing so, I move away from vague, ordinary notions. I define mathematical diagrams as forming notational systems and as being geometric/topological representations or two-dimensional representations. I also examine the relationship between mathematical diagrams and spatiotemporal intuition. By proposing an explication of diagrams, I explain certain controversies in the existing literature. Moreover, I shed light on why mathematical diagrams are so effective in certain instances, and, at other times, dangerously misleading. (shrink)

Indispensablists argue that when our belief system conflicts with our experiences, we can negate a mathematical belief but we do not because if we do, we would have to make an excessive revision of our belief system. Thus, we retain a mathematical belief not because we have good evidence for it but because it is convenient to do so. I call this view ‘ mathematical convenientism.’ I argue that mathematical convenientism commits the consequential fallacy and that (...) it demolishes the Quine-Putnam indispensability argument and Baker’s enhanced indispensability argument. (shrink)

“Morphological computation” is an increasingly important concept in robotics, artificial intelligence, and philosophy of the mind. It is used to understand how the body contributes to cognition and control of behavior. Its understanding in terms of "offloading" computation from the brain to the body has been criticized as misleading, and it has been suggested that the use of the concept conflates three classes of distinct processes. In fact, these criticisms implicitly hang on accepting a semantic definition of what constitutes computation. (...) Here, I argue that an alternative, mechanistic view on computation offers a significantly different understanding of what morphological computation is. These theoretical considerations are then used to analyze the existing research program in developmental biology, which understands morphogenesis, the process of development of shape in biological systems, as a computational process. This important line of research shows that cognition and intelligence can be found across all scales of life, as the proponents of the basal cognition research program propose. Hence, clarifying the connection between morphological computation and morphogenesis allows for strengthening the role of the former concept in this emerging research field. (shrink)

Analysing several characteristic mathematical models: natural and real numbers, Euclidean geometry, group theory, and set theory, I argue that a mathematical model in its final form is a junction of a set of axioms and an internal partial interpretation of the corresponding language. It follows from the analysis that (i) mathematical objects do not exist in the external world: they are our internally imagined objects, some of which, at least approximately, we can realize or represent; (ii) (...) class='Hi'>mathematical truths are not truths about the external world but specifications (formulations) of mathematical conceptions; (iii) mathematics is first and foremost our imagined tool by which, with certain assumptions about its applicability, we explore nature and synthesize our rational cognition of it. (shrink)

Mathematical diagrams are frequently used in contemporary mathematics. They are, however, widely seen as not contributing to the justificatory force of proofs: they are considered to be either mere illustrations or shorthand for non-diagrammatic expressions. Moreover, when they are used inferentially, they are seen as threatening the reliability of proofs. In this paper, I examine certain examples of diagrams that resist this type of dismissive characterization. By presenting two diagrammatic proofs, one from topology and one from algebra, I show (...) that diagrams form genuine notational systems, and I argue that this explains why they can play a role in the inferential structure of proofs without undermining their reliability. I then consider whether diagrams can be essential to the proofs in which they appear. (shrink)

This paper discusses the philosophical basis of mathematics by examining the perspectives of Kant and Hegel. It explores how Kant’s concept of the synthetic a priori, grounded in the intuitions of space and time, serves as a foundation for understanding mathematics. The paper then integrates Hegelian dialectics to propose a broader conception of mathematics, suggesting that the relationship between space and time is dialectically embedded in reality. By introducing the idea of a hypothetical transcendental subject, the paper attempts to overcome (...) a potential limitation of Kant’s framework, particularly regarding the application of mathematical truths to pre-human reality. This synthesis of Kantian and Hegelian thoughts offers a lens through which the connection between mathematics and reality can be understood, while also acknowledging limitations in both philosophical systems. (shrink)

Objective: Atlas is located at a critical point close to the vital centers of the medulla oblongata, which can be compressed by the dislocation of the atlantoaxial complex or instability of the atlantooccipital joint. This study aimed to determine in detail the morphometric and morphological characteristics of the atlas to guide the reduction of the risk of complications and increase the success rate in various surgical approaches for the craniovertebral junction. -/- Methods: In this study, 17 atlas vertebrae whose measurement (...) parameters were pronounced and unknown gender, age, and ethnic characteristics were examined. -/- Results: Totally 16 parameters, 11 of which were bilateral and 5 were unilateral, were examined on the atlas. Also, no accessory foramen transversarium was found in these atlas vertebrae. Of the 23 foramina transversaria that were prominent and not broken, 7 were found to be round-shaped (30.43%), and 16 were oval-shaped (69.57%). -/- Conclusion: It is deducted that the results obtained in this study will help to have information about the morphometry and morphology of atlas vertebrae. Although information such as age, gender, and ethnic origin is not known about the bones evaluated, it is the advantage of this study that a large number of parameters are evaluated and compared with previous publications. Nevertheless, it seems that there is a need for studies in which much more cases are assessed, and information such as age, gender, and ethnic origin is known. (shrink)

Socialist buildings in the city of Kruja (Albania) date back after the Second World War between the years 1945-1990. These buildings were built during the time of the socialist Albanian dictatorship and the totalitarian communist regime. A questionnaire with 30 questions was conducted and 14 people were interviewed. The interviewed residents belong to a certain area of the city of Kruja. Based on the results obtained, diagrams have been conceived and mathematical regression models have been developed which will serve (...) as a base for drawing conclusions. The purpose of this study is to find a relationship that specifies the role of the inhabitants, the role of the dwellings, the physical-mechanical characteristics, and their energy efficiency, in order to improve the living conditions of the inhabitants. The independent variables interact with each other assuming positive or negative values depending on the probabilistic mathematical regression, giving different results based on the calculated data. The variables that co-exist among them are living space, quality of life, time spent at home, social interaction of residents, dwelling humidity level, dwelling orientation, satisfaction level of the inhabitants, dwelling improvements demand, and monthly electricity bills. (shrink)

There is a wide range of realist but non-Platonist philosophies of mathematics—naturalist or Aristotelian realisms. Held by Aristotle and Mill, they played little part in twentieth century philosophy of mathematics but have been revived recently. They assimilate mathematics to the rest of science. They hold that mathematics is the science of X, where X is some observable feature of the (physical or other non-abstract) world. Choices for X include quantity, structure, pattern, complexity, relations. The article lays out and compares these (...) options, including their accounts of what X is, the examples supporting each theory, and the reasons for identifying the science of X with (most or all of) mathematics. Some comparison of the options is undertaken, but the main aim is to display the spectrum of viable alternatives to Platonism and nominalism. It is explained how these views answer Frege’s widely accepted argument that arithmetic cannot be about real features of the physical world, and arguments that such mathematical objects as large infinities and perfect geometrical figures cannot be physically realized. (shrink)

The existence of fundamental moral disagreements is a central problem for moral realism and has often been contrasted with an alleged absence of disagreement in mathematics. However, mathematicians do in fact disagree on fundamental questions, for example on which set-theoretic axioms are true, and some philosophers have argued that this increases the plausibility of moral vis-à-vis mathematical realism. I argue that the analogy between mathematical and moral disagreement is not as straightforward as those arguments present it. In particular, (...) I argue that pluralist accounts of mathematics render fundamental mathematical disagreements compatible with mathematical realism in a way in which moral disagreements and moral realism are not. 11. (shrink)

Statistics and probability enabled students to better understand, process, and evaluate massive amounts of quantitative data that existed and had a probabilistic sense in uncertain situations. The research article aimed to elucidate the performance and self-efficacy as predictors of students' achievement in the statistics and probability courses. The study utilized a descriptive-predictive research method and was conducted at Sto. Tomas National High School, involving a sample of 263 grade 11 senior high school students. The gathered data were analyzed using descriptive (...) measures and multiple regression analysis. The study's results revealed that the performance in General Mathematics was very satisfactory, while self-efficacy was high. Moreover, the level of achievement in Statistics and Probability was very satisfactory. It was also revealed that both General Mathematics performance and self-efficacy had a positive and significant relationship with Statistics and Probability achievement. Through regression analysis, it was discovered that General Mathematics performance was the strongest predictor that influenced achievement in Statistics and Probability. The study identified a significant predictive model for Statistics and Probability achievement. These findings could provide valuable guidance to teachers in enhancing the achievement of senior high school students in Statistics and Probability. (shrink)

I discuss ways in which the linguistic form of mathimatics helps us think mathematically.

There is a major debate as to whether there are non-causal mathematical explanations of physical facts that show how the facts under question arise from a degree of mathematical necessity considered stronger than that of contingent causal laws. We focus on Marc Lange’s account of distinctively mathematical explanations to argue that purported mathematical explanations are essentially causal explanations in disguise and are no different from ordinary applications of mathematics. This is because these explanations work not by (...) appealing to what the world must be like as a matter of mathematical necessity but by appealing to various contingent causal facts. (shrink)

Mathematics appears to play an explanatory role in science. This, in turn, is thought to pave a way toward mathematical Platonism. A central challenge for mathematical Platonists, however, is to provide an account of how mathematical explanations work. I propose a property-based account: physical systems possess mathematical properties, which either guarantee the presence of other mathematical properties and, by extension, the physical states that possess them; or rule out other mathematical properties, and their associated (...) physical states. I explain why Platonists should accept that physical systems have mathematical properties, and why a property based account is better than existing accounts of mathematical explanation. I close by considering whether nominalists can accept the view I propose here. I argue that they cannot. (shrink)

Mathematical thinking skills are very important in mathematics, both to learn math or as learning goals. Thinking skills can be seen from the description given answers in solving mathematical problems faced. Mathematical thinking skills can be seen from the types, levels, and process. Proportionally questions given to students at universities in Indonesia (semester I, III, V, and VII). These questions are a matter of description that belong to the higher-level thinking. Students choose 5 of 8 given problem. (...) Qualitatively, the answers were analyzed by descriptive to see the tendency to think mathematically used in completing the test. The results show that students tend to choose the issues relating to the calculation. They are more use cases, examples and not an example, to evaluate the conjecture and prove to belong to the numeric argumentation. Used mathematical thinking students are very personal (intelligence, interest, and experience), and the situation (problems encountered). Thus, the level of half of the students are not guaranteed and shows the level of mathematical thinking. (shrink)

Is shadowboxing an effective form of functional exercise? What physiological and morphological changes result from an exercise program based exclusively on shadowboxing for 3 weeks? To date, no empirical research has focused specifically on addressing these questions. Since mixed martial arts (MMA) is the fastest growing sport in the world, and since boxing and kickboxing fitness classes are among the most popular in gyms and fitness clubs worldwide, the lack of research on shadowboxing and martial arts-based fitness programs in the (...) extant literature is a shortcoming that the present article aims to address. This case study involved a previously sedentary individual engaging in an exercise program based exclusively on shadowboxing for 3 weeks. Body composition and heart rate data were collected before, throughout, and upon completion of the 3-week exercise program to determine the effectiveness of shadowboxing for functional fitness purposes. An original shadowboxing program prepared by an Everlast Master Instructor and NASM Certified Personal Trainer (NASM-CPT) and Performance Enhancement Specialist (NASM-PES) was used for this 3-week period. The original shadowboxing program with goals, techniques, and combinations to work on throughout the 3-week program is included in this article. This case study demonstrates that a 3-week exercise program based exclusively on shadowboxing can increase aerobic capacity, muscle mass, bone mass, basal metabolic rate, and daily calorie intake, and decrease resting heart rate, fat mass, body fat percentage, and visceral fat rating in a previously sedentary individual. The results of this research demonstrate that shadowboxing can be a safe and effective form of exercise leading to morphological and physiological improvements including fat loss and increased aerobic capacity. The results of this research also demonstrate that the Tanita BC-1500 is a reliable tool for individuals to evaluate their own fitness progress over time. (shrink)

The triumph of scientific materialism in the Seventeenth Century not only bifurcated nature into matter and mind and primary and secondary qualities, as Alfred North Whitehead pointed out in Science and the Modern World. It divided science and the humanities. The core of science is the effort to comprehend the cosmos through mathematics. The core of the humanities is the effort to comprehend history and human nature through narratives. The life sciences can be seen as the zone in which the (...) conflict between these two very different ways of comprehending the world collide. Evolutionary theory as defended by Schelling developed out of natural history, but efforts have been made to formulate neo-Darwinism through mathematical models. However, it is impossible to eliminate stories from biology. As Stuart Kauffman argued, mathematical models attempt to pre-state all possibilities, but in evolution there can be adjacent possibles that can be embraced by organisms but cannot be pre-stated. To account for such actions it is necessary to tell stories. Mathematics provides analytic precision allowing long chains of deductions, but tends to deny temporal becoming and cannot do justice to the openness of the future, while narratives focus on processes and events, but lack exactitude that would provide precise deductions and predictions. In advancing mathematics adequate to life, Robert Rosen argued that living beings as anticipatory systems must have models of themselves, and strove to develop a form of mathematics able to model life itself. It has been convincingly argued that narratives are central to human self-creation and they are lived out before being explicitly told. Their models of themselves are first and foremost, stories or narratives. If this is the case, might not living beings as biological entities be characterized by proto-stories or narratives in their models of themselves? Biosemiotics, largely inspired by C.S. Peirce, provides a bridge between mathematical and narrative comprehension, conceiving them as different forms of semiosis. The study of life through biosemiotics could reveal how mathematics and narratives can be understood in relation to each other. This could have implications for how we understand science and the humanities and their relationship to each other. In this paper I will examine work in theoretical biology that might advance these efforts. (shrink)