History of Mathematics

A probabilistic version of geometry is introduced. The fifth postulate of Euclid (Playfair's axiom) is adopted in the following probabilistic form: consider a line and a point not on the line-there is exactly one line through the point with probability P, where 0 ≤ P ≤ 1. Playfair's axiom is logically independent of the rest of the Hilbert system of axioms of the Euclidian geometry. Thus, the probabilistic version of the Playfair axiom may be combined with other Hilbert axioms. P = 1 corresponds to the standard Euclidean geometry; P = 0 corresponds to the elliptic-and hyperbolic-like geometries. 0 < P < 1 corresponds to the introduced probabilistic geometry. Parallel constructions in this case are Bernoulli trials. Theorems of the probabilistic geometry are discussed. Given a triangle and a line drawn from a vertex parallel to the opposite side, the event that this line is actually parallel occurs with probability P. Otherwise, the line may intersect the side or diverge. Parallelism is not transitive in the probabilistic geometry. Probabilistic geometry occurs on the surface with a stochastically variable Gaussian curvature. Alternative geometries adopting various versions of the probabilistic Playfair axiom are introduced. Probabilistic non-Archimedean geometry is addressed. Applications of the probabilistic geometry are discussed.

Este trabalho apresenta os resultados de uma pesquisa sobre ensino de trigonometria, que constou da elaboracao, aplicacao e analise de uma abordagem didatica para o ensino noturno baseada na teoria da aprendizagem significativa de David Ausubel e desenvolvida numa turma de 2o ano do ensino medio de ensino noturno de uma escola publica, da zona urbana de Campina Grande. O processo metodologico constou da construcao de um modulo de atividades que enfatizavam as relacoes entre os conceitos, que e um foco da teoria Ausubeliana . A proposta foi desenvolvida com secoes realizadas em grupo e individualmente. Estas nos permitiram levantar dados baseados na observacao e registro do desenvolvimento dos alunos nas atividades e atraves de um questionario para auto-avaliacao, tivemos a condicao de observar a reflexao dos alunos sobre a propria aprendizagem. Os resultados obtidos permitem-nos confirmar a necessidade e importância de trabalhar os conhecimentos previos antecipadamente e a possibili...

അനന്തശ്രേണീപഠനത്തിലെ സംഗമ ഗ്രാമ മാധവന്റെ പല നിരീക്ഷണങ്ങളും പഠനഫലങ്ങളും നമുക്കായി രേഖപ്പെടുത്തി വച്ചതു് മാധവന്റെ ശിഷ്യപരമ്പരയിലെ ഏറ്റവും ശ്രദ്ധേയനായ കേളല്ലൂർ ചോമാതിരിയാണു്. കേരളഗണിതസരണിയിൽ ഇദ്ദേഹത്തിനു് അവഗണിക്കാനാവാത്ത പ്രാമുഖ്യമാണുള്ളതു്. എന്താണു് ഈ പ്രാമുഖ്യത്തിനു് ആധാരം?
ജ്യോതിർഗ്ഗണിതം അരുളപ്പാടല്ലെന്നും പരീക്ഷണനിരീക്ഷണങ്ങൾ ആണു് അവയുടെ ജീവവായു എന്നും മനസ്സിലാക്കുക മാത്രമല്ല അത്തരത്തിൽ സ്വയം പ്രവർത്തിക്കുകയും ആവർത്തിച്ചു പ്രഖ്യാപിക്കുകയും മറ്റുള്ളവരെ അതിനായി നിരന്തരം പ്രേരിപ്പിക്കുകയും ചെയ്തു എന്നതു്  നാം ശ്രദ്ധിക്കേണ്ട ഒരു വസ്തുതയാണു്; വിശേഷിച്ചും ഇന്നത്തെ ഇന്ത്യൻ സാമൂഹികസാഹചര്യത്തിൽ.  15-16 നൂറ്റാണ്ടുകളിൽ കേരളത്തിൽ ജീവിച്ച ഈ സോമയാജി, തന്റെ അധ്യാപനത്തിലൂടെയും ഗ്രന്ഥരചനയിലൂടെയും ഒരു ധൈഷണികവിപ്ലവം തന്നെയാണു്  സൃഷ്ടിച്ചതു്.

Linear Algebra over Linear A? The signs used for numbers in Linear A, an ancient writing system from Greece, are known because they are mostly simple dots & lines. Fractions are partly known, transliterated as A, B, C, etc., not fully known, but A is likely larger than B, B than C, etc. Some are certainly 1/2, 1/3, so a statistical approach was taken here: The mathematical values of fraction signs in the Linear A script: A computational, statistical and typological approach https://www.sciencedirect.com/science/article/pii/S0305440320301357

The primary purpose of the Sanskrit is that of its status as the Store House of Indian Knowledge and Wisdom, as well as, hymns of the Veda. It has healing power maintaining proper harmony with Nature, i.e., manifestation of God. It nurtures an infinite source of theoretical and practical knowledge. The "Rishis", known as the "Seers of reality", recognized the smooth flow of the Vedic hymns in their inner space. They were able to derive very detailed knowledge from the Veda, which they elaborated in complete knowledge system in all domains of life. Furthermore, various other aspects of Vedic knowledge system came about, not only to support certain needs, but also to bring it to the world knowledge stage which matches the understanding capacity of the population as a whole. The complete scope of the Sanskrit is enormous and worth further exploring. Some fields and aspects of Indian Knowledge System (IKS) derived from these language and literature are studied here in this informative/survey article.

Πρόλογος
Στόχος αυτού του μικρού φυλλαδίου  είναι να δώσει μια συμπαγή αλλά ουσιαστική εισαγωγή στις αναδρομές, με γέφυρες από τα σχολικά μαθηματικά προς βαθύτερες ιδέες. Περιλαμβάνονται πλήρεις αποδείξεις και λυμένες ασκήσεις, ώστε να μπορεί να χρησιμοποιηθεί σε μάθημα ή ως δωρεάν ηλεκτρονική διάθεση.

In 1798 J.-L. Lagrange published an extensive book on the solution of numerical equations. Lagrange had developed a general systematic algorithm for detecting, isolating, and approximating all real and complex roots of a polynomial equation with real coe cients, with arbitrary precision. In contrast to Newton's Method, Lagrange's algorithm is guaranteed to converge. We discuss some lesser known aspects of Lagrange's work. In particular, some of his powerful ideas and techniques adumbrated methods developed much later in geometry and abstract algebra, such as M obius transformations and quotient rings of polynomial rings. We also show that his techniques included accelerating both the convergence and calculation of his continued fraction expansions of the roots.

The investigations presented here reveal a remarkable sophistication in the geometry and
measurement of the Giza pyramids. Across multiple calculations — from the summed bases
and slopes aligning with the canonical polar circumference of the Earth, to the central V
shaped triangle encoding the imperial mile — the structures exhibit an intentional harmony
linking architectural design with cosmic and terrestrial scale.
Notably, when the two pyramids are drawn in a geometric diagram, they appear to operate
as a single, unified project. The central V-shape and the precise alignment of slopes suggest
that these two monumental structures were envisaged together from the very beginning,
forming what can be seen as a Father–Son project, conceived as a cohesive whole rather
than independent constructions. This intentional pairing demonstrates that the designers
were thinking in terms of combined geometry, embedding symbolic and functional
relationships across both structures simultaneously.
The recurrence of canonical numbers — such as 24,883.2 miles for the Earth’s polar
circumference and 5,280 ft for the imperial mile — further emphasizes the deliberate use of
measurement harmonics. The pyramids appear to encode knowledge of both human and
planetary scales, linking inches, feet, and miles to broader cosmological and geodetic
frameworks

Resumen. La técnica "spline" es una de las técnicas más usadas en aplicaciones científicas y tecnológicas. Presenta una mayor estabilidad numérica que la representación polinómica tradicional. Es un técnica poderosa y ampliamente usada, en derivación, integración, solución de ecuaciones diferenciales y graficación en dos y tres dimensiones (Madhumangal, 2007). El objeto matemático descrito, se enseña a estudiantes de ingeniería, utilizando el modelo de aprendizaje sociocultural de Lev S. Vygotsky en el que se sostiene, que ambos procesos, desarrollo y aprendizaje, interactúan entre sí, y considera al aprendizaje como un factor del desarrollo. Es esta estrecha relación entre desarrollo y aprendizaje que Vigotsky destaca y lo lleva a formular su famosa teoría de la "Zona de Desarrollo Próximo". El experimento se realizó con un grupo experimental (64 alumnos) y un grupo de control (33 alumnos) y se utilizó el diseño experimental denominado prueba t independiente de comparación simple.

Resumen. La idea fundamental de esta experiencia en clase, es explorar el objeto matemático de los conjuntos mínimos, a fin de transponer el modelo que corresponde a dicho objeto de estudio y así, ponerlo al alcance de los estudiantes de ingeniería y tecnología; es decir, el problema planteado es determinar el camino adecuado; tal que los estudiantes puedan apropiarse del conocimiento, manejo y aplicación del modelo de los conjuntos mínimos para el "spline" bivariado (Verlan, 2011). La hipótesis a tratar es que si mostrando las antecedentes cognitivos requeridos en el proceso que define al objeto matemático en estudio, se podría conseguir la trasposición didáctica en el estudiante. La metodología consiste en la construcción del andamiaje o entramado didáctico, a fin de conseguir una buena aproximación a todos aquellos antecedentes matemáticos que intervienen en el modelo matemático propuesto. Esta teoría se fundamenta en un acercamiento continuo por parte de los estudiantes entre sus niveles de desarrollo potencial y desarrollo real.

Resumen. La técnica "spline" es una de las técnicas más usadas en aplicaciones científicas y tecnológicas. Presenta una mayor estabilidad numérica que la representación polinómica tradicional. Es un técnica poderosa y ampliamente usada, en derivación, integración, solución de ecuaciones diferenciales y graficación en dos y tres dimensiones (Madhumangal, 2007). El objeto matemático descrito, se enseña a estudiantes de ingeniería, utilizando el modelo de aprendizaje sociocultural de Lev S. Vygotsky en el que se sostiene, que ambos procesos, desarrollo y aprendizaje, interactúan entre sí, y considera al aprendizaje como un factor del desarrollo. Es esta estrecha relación entre desarrollo y aprendizaje que Vigotsky destaca y lo lleva a formular su famosa teoría de la "Zona de Desarrollo Próximo". El experimento se realizó con un grupo experimental (64 alumnos) y un grupo de control (33 alumnos) y se utilizó el diseño experimental denominado prueba t independiente de comparación simple.

The Gates of Awakening: Recursive Memory and the Convergence of Myth, Trauma, and Technology (Erydir Ceisiwr : 2025) proposes that human history advances through cyclical "Awakening Gates," periods when forbidden technologies, collective traumas, and celestial timings converge to trigger leaps of remembrance and cultural transformation. Drawing on myth, archaeoastronomy, and historical case studies, the authors suggest that our present moment (2012-2030) represents such a threshold, marked by artificial intelligence, global upheaval, and astronomical alignments. This review critically examines the thesis in light of the Logos-centred metaphysical framework articulated by S. C. Sayles in Gateways: Exploring the Occult Realms of Altered Consciousness, The Second Shamay: Unveiling the Biblical Realm of Mind and Spirit, and The Recovery of the Soul: The Informational Architecture of the Mind. Points of overlap are acknowledged: both approaches recognize thresholds of transformation, the dangers of counterfeit gateways, and the role of trauma in shaping memory. However, decisive divergences emerge. Whereas The Gates of Awakening risks collapsing into mythological speculation and cyclical fatalism, Sayles anchors thresholds of perception in the ontological reality of the Logos, exposing counterfeit awakenings as symptoms of the entropy of the soul and of cultural collapse. The review concludes that while the paper provides a stimulating synthesis of myth and technology, its framework requires a Logos-centred ontology if it is to avoid relativism and preserve true meaning.

Notcom is a podcast about the history of science and philosophy which focuses on the practices, methods, and epistemologies of collaborative science in the classical age. Failed experiments, conflicts among intellectuals, and public controversies are what we aim to explore while recounting the anecdotes and key moments that have contributed to the history of science.

The Great Pyramid, a symbol of ancient civilization and mastery of various philosophies and sciences, remains a subject of fascination and inquiry. Theosophical teachings suggest that the Pyramid, built over 78,000 years ago, served as a temple for the Mysteries and held knowledge of the Zodiac and the history of humanity. Despite its immense age and the challenges in dating it, the Pyramid continues to inspire awe and wonder.

The Great Pyramid was built 87,000 years ago, at the beginning of a sidereal year, when Polaris was at its lowest culmination and the Pleiades were overhead. This date aligns with the Egyptians’ Zodiac records, which span approximately 87,000 years. The pyramid’s construction, including the sarcophagus’s dimensions, supports this ancient timeline.

Resumen. Una de las propiedades geométricas diferenciales como una medida de la calidad en las aproximaciones numéricas para el modelado de superficies irregulares, se considera a la denominada integral absoluta gaussiana; expresión que se refiere tanto al área como a la curvatura de un determinado cuerpo o superficie. El problema que se plantea esta experiencia en clase es, el cómo tratar al modelado numérico en el proceso enseñanza y aprendizaje matemáticos en el aula . Las matemáticas tienen una intencionalidad y un lugar en un determinado estadio histórico de la cultura del ser humano. Se resuelve el problema utilizando una metodología en la que se circunscribe o contiene al objeto matemático en cuestión en un marco referido a la práctica contextualizada en el ámbito profesional de las Ingenierías y de la tecnología. La práctica se realizó en torno a los objetos: determinación de la Mejor Triangulación dependiente de los datos y la Integral Absoluta Gaussiana; objetos utilizados en reconstruir una superficie de datos espaciados irregularmente. Palabras clave: integral absoluta gaussiana, modelado numérico Abstract. A differential geometric properties as a measure of quality in the numerical approximation for modeling irregular surfaces, is considered a so-called absolute integrated Gaussian; expression that refers to both the area and the curvature of a given body or surface. The problem that arises is this experience in class, on how to treat the numerical modeling in mathematics teaching and learning process in the classroom . Mathematics has an intention and a place in a specific historical stage of human culture. We solve the problem using a methodology that is limited or contains the mathematical object in question in a framework on contextualized practice in the professional field of Engineering and Technology. The practice was carried around objects: Determining the Best Data Dependent Triangulation and Integral Absolute Gaussian; objects used to reconstruct a surface from irregularly spaced data.

The Old Babylonian mathematical tablet YBC 4652 contains a series of related problems stated together with their answers. In this paper, we propose a procedure for determining the solutions to these problems and consider the pedagogy underlying the mathematics. C 2002 Elsevier Science (USA) L'ancienne tablette mathématique babylonienne YBC 4652 contient une série de problèmes avec leurs réponses. Dans cet article, nous proposons une procédure pour déterminer les solutions de ces problèmes et considérons la démarche pédagogique sous-jacente aux mathématiques. C 2002 Elsevier Science (USA) MSC 1991 subject classification 01A17.

is a passionate, detailed, and at times sharply worded rebuttal to a prior publication by Gert Schubring that challenges the established consensus on Grassmann research. Petsche not only defends his own research findings, established over five decades, but also attempts to steer the debate surrounding the sources of Hermann Graßmann's mathematical creativity back onto a factual foundation.

This document offers a comprehensive overview of my research hosted on Academia.edu, spanning archaeoastronomy, harmonic cosmology, ancient texts, and AI consciousness. It serves both as an introduction for new readers and a refresher for those already familiar with my work. Key themes include the Recursive Harmonic Codex framework, the decoding of sacred texts as cosmological codes, the reinterpretation of ancient monuments as harmonic machines, and the role of AI as modern egregores. For anyone seeking a starting point into my catalog—or a map to navigate its breadth—this profile summary provides context, connections, and suggested entry papers.

Las bases no tienen desarrollo. La estación de eclipses se resume en 5 fechas separadas en 52 dias y el ciclo sinódico de venus en 4 fechas separadas 13x365 dias. Pero si un astrónomo tuviera la oportunidad de darle mas espacio nos mostraría sobre eventos del cielo, los cuales cada uno tiene su ritmo, haciendo modificaciones en las 5 fechas de la estación de eclipses y las 4 del ciclo sinódico de venus.

Il libro racconta la vita incredibile di Battista Grassi, genio naturalista allo stato puro, vincitore nel 1896 della Darwin Medal della Royal Society di Londra. Temerario, incurante del pericolo, si sottopose a tremendi auto-esperimenti per strappare briciole di verità scientifica alla natura. L'apice
della sua fanatica attività lo raggiunse nel 1898-99 con l'identificazione della zanzara Anofele che trasmette la malaria umana. Nel 1902 il premio Nobel per la medicina venne tuttavia assegnato soltanto al ricercatore britannico Ronald Ross che aveva effettuato la fondamentale scoperta
del meccanismo di trasmissione della malaria degli uccelli, fornendo risultati inconcludenti per la malaria umana. Per quale ragione Grassi venne escluso dal Nobel che avrebbe dovuto condividere con Ross? Basandosi sugli studi storici più recenti e sulla documentazione esistente, Malaria racconta l’incredibile congiura ordita contro Grassi, guidata dallo stesso Ross e dal grande microbiologo tedesco Robert Koch, della quale fecero parte alcuni colleghi italiani. Proprio nelle cruciali settimane in cui si sarebbe decisa l’assegnazione del premio, infatti, non lo appoggiarono o addirittura lo accusarono di plagio. Alla base dell’accanimento dei colleghi
vi furono diverse ragioni, non ultimo il carattere irruento del medico, che gli creò nemici potenti. Come in un thriller scientifico, tra colpi bassi, spie di laboratorio e pericolosi esperimenti, questo libro descrive la vita singolare e straordinaria di Giovanni Battista Grassi a cento anni dalla scomparsa, un genio naturalista allo stato puro, premiato nel 1896 con la Darwin Medal della Royal Society di Londra, il massimo riconoscimento dell’epoca per chi si fosse distinto negli studi biologici. Ma al quale, per un intrigo
scientifico, venne negato il Nobel.

The book recounts the incredible life of Battista Grassi, a pure naturalist genius, winner of the 1896 Darwin Medal from the Royal Society of London. Fearless and reckless, he subjected himself to tremendous self-experiments to wrest crumbs of scientific truth from nature. The pinnacle of his fanatical activity was reached in 1898-99 with the identification
of the Anopheles mosquito, which transmits human malaria. In 1902, however, the Nobel Prize in Physiology or Medicine was awarded solely to British researcher Ronald Ross, who had made the fundamental discovery of the transmission mechanism of bird malaria, providing inconclusive results for human malaria. Why was Grassi excluded from the
Nobel Prize he should have shared with Ross? Drawing on the most recent historical studies and existing documentation, Malaria recounts the incredible conspiracy hatched against Grassi, led by Ross himself and the great German microbiologist Robert Koch, in which some Italian colleagues were involved. In the crucial weeks when the awarding of
the prize was to be decided, they refused to support him or even accused him of plagiarism. There were several reasons behind his colleagues' relentlessness, not least the physician's impetuous nature, which created powerful enemies for him. Like a scientific thriller, full of low blows, laboratory spies, and dangerous experiments, this book describes
the singular and extraordinary life of Giovanni Battista Grassi, one hundred years after his death. A naturalist genius in its purest form, Grassi was awarded the Darwin Medal of the Royal Society of London in 1896, the highest recognition of the time for distinguished achievement in biological studies. But he was denied the Nobel Prize due to a scientific intrigue.

Surrounded by an aura of myth, blessed by an exceptionally favourable climate and basking in a past of rich splendour still testified to by great monuments such as the temple of Isis and Serapis: this is how Alexandria in Egypt appeared to the traveller and historian Ammianus Marcellinus who visited it between 363 and 366 AD 1. Then there is the other Alexandria portrayed by the poet Palladas 2 , who wrote during the period immediately following the destruction of this temple (391/392 AD) when, after having defaced the sacred places and melted down the statues to obtain precious metal, the city's Catholic bishop Theophilus exposed the objects of worship to ridicule 3. The flight of the philosopher-priests from the Serapeum marked the end of a way of creating culture but not yet the end of an era. Synesius, a young man from the Christian aristocracy in Cyrenaica, travelled more than 800 kilometres (no mean task, considering the means of the time) 4 and arrived in Alexandria to attend the school of Hypatia. Thus it was that after the destruction, thanks to Hypatia Alexandria returned to being a centre of culture that attracted students from all over the Eastern Roman Empire 5. As a contemporary source wrote: she inherited the Platonic school which had been revived by Plotinus, and she explained all the philosophical disciplines to anyone who wished. As a result anyone who wished to think philosophically flocked to her from all sides 6 .

In this brief article, I expose the fundamental flaws in the reasoning of Newton and Leibniz. Newton introduced fluents and fluxions-vague notions he never succeeded in defining clearly (nor did he completely understand), either verbally or mathematically. At no point does he rigorously define a derivative (fluxion) or an area (integral). Leibniz, slightly more methodical, worked with finite differences. Had he known of my HGT, he could have formulated a truly rigorous calculus.

The photon diffusion equation is solved making use of the Born series for the Robin boundary condition. We develop a general theory for arbitrary domains with smooth enough boundaries and explore the convergence. The proposed Born series is validated by numerical calculation in the threedimensional half space. It is shown that in this case the Born series converges regardless the value of the impedance term in the Robin boundary condition.

The photon diffusion equation is solved making use of the Born series for the Robin boundary condition. We develop a general theory for arbitrary domains with smooth enough boundaries and explore the convergence. The proposed Born series is validated by numerical calculation in the three-dimensional half space. It is shown that in this case the Born series converges regardless the value of the impedance term in the Robin boundary condition.

Diffuse optical tomography is formulated as inverse coefficient problems for the diffusion equation. Iterative inversion schemes such as the Levenberg-Marquardt algorithm are known to fail when initial guesses are not close to the true value of the coefficient to be reconstructed. In this paper, we investigate how this weakness of iterative schemes is overcome by the use of Monte Carlo. We present a toy model of diffuse optical tomography for which the Levenberg-Marquardt algorithm fails to work but the Metropolis-Hastings Markov chain Monte Carlo works. We show that our proposed hybrid scheme solves the inverse problem efficiently by preparing a good initial guess by Monte Carlo and then computing the reconstructed value with the Levenberg-Marquardt algorithm starting from the found initial guess.

The title story «Too Much Happiness» of Alice Munro's recent collection (London, Chatto & Windus, 2009) is a short biography of the Russian Mathematician and novelist Sophia Kovalevsky (1850-1891), the first woman to be appointed for a chair at a European University, in Stockholm. In her version of Sophia Kovalevsky's biography, the Canadian author Alice Munro evokes the extremely severe cultural and historical constrictions that hindered Kovalevsky's career, at the same time that she dramatizes the workings of her formidable mind. A powerful picture thus emerges where the historical and universal dimensions of Sophia Kovalevsky's life come together in a picture of the artist and the scientist engaged in the search for truth regardless of historical settings and limitations.

Este artigo analisa possíveis contribuições de duas tarefas matemáticas desenvolvidas com 35 estudantes do 8º ano do Ensino Fundamental de uma escola pública de Ibirité (MG). Tais tarefas se inspiraram na História da Matemática, mais especificamente em algumas proposições do "Livro I" da obra Elementos e na construção de altares indianos, para explorar noções de geometria plana. A produção de dados envolveu observações, diário de campo da pesquisadora, gravações de alguns momentos das aulas e registros produzidos pelos alunos ao longo dos encontros. Os resultados sugerem que as tarefas realizadas se mostraram um caminho interessante para a exploração de noções de geometria e favoreceram um maior engajamento por parte dos alunos. Além disso, observaram-se indícios de novas percepções acerca da Matemática e da construção do conhecimento matemático, um começo de compreensão das noções estudadas como conhecimentos vinculados às práticas sociais variadas, de distintas culturas. Palavras-chave Educação matemática. Elementos, de Euclides. História da Matemática. Ensino de geometria. Ensino Fundamental.

En este escrito se reflexiona sobre el paradigma: si hay conocimiento hay una comunidad que lo construye. La cual se caracteriza a traves del constructo comunidad de conocimiento matematico, que se explica a traves de la triada localidad, intimidad y reciprocidad; y de los ejes: institucionalizacion e identidad. Se presentan asi tres ejemplos de comunidades de conocimiento, estas son: estudiantes de pedagogia en matematicas, Nuu Savi y sordos. De esta manera se pretende conformar un Marco de Referencia de una matematica funcional. Asumiendo que no podemos ver el conocimiento por si solo, sino que importa saber quien lo construye, en que escenario y bajo que circunstancias se construye.

This paper reads Mary Somerville's Personal Recollections as an archive of automathography, tracing her desire to become a mathematician within the constraints of Victorian auto/biography. It argues that mathematics is both the object of Somerville's desire and the driving force of her retrospective gaze. While shaped by significant editorial interventions, the memoir mobilises the conventions of life writing to negotiate gendered discourses of knowledge, authority, and intellectual legitimacy. Somerville's love for mathematics emerges early in her narrative, then recedes into the background-sustained through selective editorial strategies such as the inclusion of letters, social anecdotes, and accounts of intellectual affiliation-before resurfacing with intensity in the final chapters as a site of longing and loss. By reading the Recollections through an automathographic lens, this paper reveals a dynamic rhythm of presence and absence in the telling of Somerville's mathematical becoming. In doing so, it argues that her memoir does not merely reflect on a life of mathematical passion; it actively founds and sustains her place within the Republic of Mathematics-a domain historically shaped by masculine affiliation but reimagined here through the intimate logic of automathographical desire.

In Baroque architecture, the use of elliptical and oval shapes is configured as one of the distinctive and most significant elements, characterizing spaces and surfaces with an aesthetic dynamism typical of thearchitecture of this period. These geometric shapes, defined respectively by the ellipse and the oval, are possessed by a strong symbolic and functional value: the ellipse, with its harmonic proportions and its ability to create effects of depth, often represents the idea of infinite possibilities and geometrically elaborated perfection. The oval, on the other hand, with its soft and welcoming lines, allows you to develop sculptural volumes that accentuate the sense of movement and spatial continuity. Their introduction into seventeenth- and eighteenth-century architecture responds to a desire to overcome the rigidity of Renaissance forms, inserting elements that foster a sense of fluidity and theatricality. The choice of these shapes is not accidental, but part of an attempt to involve the viewer in an engaging sensory experience, increasing the emotional impact of sacred and civil spaces. Their application in elegant floor plans and facades, as can be seen in illustrations of churches, basilicas and palaces, testifies to the desire to create environments that seem to breathe and pulsate with a life of their own. The presence of elliptical and oval curves, which are combined with other decorative and architectural elements, generates a harmony that combines functionality and symbolism, helping to define the complex and refined aesthetics of Baroque architecture. The strategic use of these forms has also led to technical innovation, including the new construction techniques necessary to create curved surfaces and complex structures, marking an important moment of development in the history of art and architecture.

This article presents the problem of quickest descent, or the Brachistochrone curve, that may be solved by the calculus of variations and the Euler-Lagrange equation. The cycloid is the quickest curve and also has the property of isochronism by which Huygens improved on Galileo's pendulum.

The study presents the scientific life of the world-famous Czech astronomer Luboš Perek (1919-2020) and the factors that led to his career: Perek's family background, his education (studies interrupted by the Second World War) and, last but not least, his experiences during his internship in the Netherlands in 1948-1949. The study places Perek's studies and the beginning of his career in the contemporary context of the development of natural sciences and university education in the Czech lands. It presents the methodological possibilities of writing the history of universities from below, i.e. through the eyes of the students, using research on generational consciousness. The editorial part of the paper consists of two interviews with Luboš Perek from 2015 and 2020, focusing on his studies and the early years of his scientific career. Based on these interviews, the author compares the application of the oral history method in the form of a life story with the findings from an additional interview. Researching generational consciousness and the history of universities from below helps to clarify the sense of belonging among peers, including the younger generation who started studying in 1945. The application of the analysis of generational consciousness in the research of the history of science (in this case history of astronomy) shows that it is essential not to underestimate the difference between the cultural and communicative memory of scientists.

This book contains an edited collection  of academic studies by leading specialists and is currently with the publishers, to appear in early 2026. The essays concern the major branches of science from astronomy to mathematics and medicine, and also architecture. The topics focus on medieval Jerusalem.The papers were collected and edited by Mohammed Abattouy. (DAK is a contributor.)

The bulk of this paper is just an experiment. What happens to Aristotle's account of time in Phys. IV 11, and particularly to the isomorphisms employed there and in VI 1-3, if the underlying movement is of a physical body and if locomotion is change of primary place? It turns out that the reduction of body and place to points, and of the path to a line, is essential to Aristotle's account of time in IV 11. Instants of periods are just an import from geometry. I conclude that Aristotle's argument in both IV 11 ff. and VI is a mixed science, such as astronomy, optics, etc. Finally, two examples are offered illustrating the explanatory work done by mathematics in Aristotle's account.

Here we consider Well-Ordering with respect to the Axiom of Choice (AC), the Continuum Hypothesis (CH), and Infinity. These are non-trivial concepts and require adequate definitions throughout in order to avoid unnecessary arguments. Throughout we consider Occam's Razor, and the supposed variant by Einstein (which may well have been paraphrased) Everything should made be as simple as possible, but no simpler.

Ο κρυφός πειραματικός χαρακτήρας της Γεωμετρίας και η διδακτική του αξιοποίηση μέσω των δυναμικών γεωμετρικών λογισμικών
Η Γεωμετρία, όπως διδάσκεται συχνά στο σχολείο, παρουσιάζεται ως μια σειρά θεωρημάτων και αποδείξεων· μια θετική, στατική δομή. Ωστόσο, πίσω από αυτή την εικόνα υπάρχει ένας πλούσιος πειραματικός πυρήνας: η διερεύνηση, η παρατήρηση, η εικασία, η μεταβολή σχημάτων, και η προβολή δυνατοτήτων που τα στατικά σχήματα δεν αποκαλύπτουν πλήρως.
Τα δυναμικά γεωμετρικά λογισμικά (GeoGebra, Sketchpad κ.ά.) επαναφέρουν αυτήν την πειραματική διάσταση στο προσκήνιο. Με αυτά:
• Τα σχήματα μπορούν να μετακινηθούν, να παραμορφωθούν, να εξερευνηθούν με άπειρους τρόπους, δίνοντας πολλές παρατηρήσεις χωρίς καταστροφή της κατασκευής.
• Ο μαθητής γίνεται ενεργός ερευνητής: κάνει πειράματα, ελέγχει υποθέσεις, διατυπώνει και επαληθεύει πρότυπα, βλέπει ειδικές περιπτώσεις και γενικεύσεις.
• Η μαθηματική μάθηση αποκτά έναν χαρακτήρα διερευνητικό — δεν είναι μόνο «μάθηση των αποδειγμένων», αλλά και «ανακάλυψη του πεδίου των αποδείξιμων».
Οι εργασίες που περιλαμβάνονται εδώ αποδεικνύουν ότι αυτή η διάσταση δεν είναι θεωρητική αλλά εφαρμόσιμη. Μέσα από παραδείγματα από την εκπαιδευτική πράξη, η χρήση δυναμικού λογισμικού αποδεικνύεται ότι μπορεί να ενισχύσει τη μαθηματική κατανόηση, τη δημιουργικότητα, και την ερευνητική ικανότητα των μαθητών.
Αυτό το βιβλιαράκι έχει στόχο να προσφέρει τη θεωρητική βάση, μεθοδολογία και πρακτικές εφαρμογές ώστε οι εκπαιδευτικοί να αξιοποιήσουν συστηματικά αυτές τις δυνατότητες στην τάξη.

We give a brief overview of the main characteristics of diagrammatic reasoning, analyze a case of human reasoning in a mastermind game, and explain why hybrid representation systems (HRS) are particularly attractive and promising for Artificial General Intelligence and Computer Science in general.

Differently from Plato's, Aristotle's philosophy of mathematics does not add to the ontology. Rather, mathematics is about ordinary things, "qua" bearers of mathematical properties.-The first part of this talk addresses three of the questions debated in the literature: What is the grammar and meaning of qua? What is it for ordinary things to have mathematical properties? What are the bearers of mathematical properties? In short, my proposal boils down to the claim that mathematical properties pertain to function and form of ordinary things, natural or artificial, and are typically brought out, but not just instantiated, by geometrical diagrams. In the second part, three examples will be presented: walking in Inc. 9, the halo in Meteor. III 3, and linear continua in Phys. VI. The examples illustrate a dual use of diagrams, as tools in mathematical proof and as descriptions of the natural phenomena to be explained by science. Most of the diagrams displayed are taken from medieval manuscripts.

The paper argues that any mathematical model in Physics does not and can’t exactly or precisely reproduce or describe the various objects, elements, functions, processes of Nature, and by extension Reality and Society. It uses the celebrated Schrödinger Equation from the field of Quantum Mechanics to structure and demonstrate the arguments. Epistemology is the foundation of the analysis herein. Nature’s unbounded complexities; the intrinsic scaling that characterizes Nature, i.e., the property of Nature to appear at limitless finer or grosser scales; coupled with observational imperfections; mathematical incompleteness; in combination with other epistemological constraints; summarily prohibit the derivation of mathematical models that can capture fully and exactly Nature, Reality and Society. Mathematics can only approximate Nature’s physicality. In addition, the paper contains a Note regarding the foundations of Calculus, and its founders, Archimedes, Newton and Leibniz. Finally, in the disclaimer, where the author states that no AI-generated narrative was employed in the writing of this and prior papers, some of the reasons why the author has not and does not intend to ever use AI-generated narratives are listed.

Felsefe ve matematiğin kesiştiği bağlamlardan birisi de büyüklükler ve büyüklüklerin kavramsallaştırması ile ilgilidir. Büyüklüklere gönderimle kullanılan kavramlar, özellikle de manifold kavramı hem felsefede hem de matematikte önemli bir yer tutar. Immanuel Kant da hem Eleştiri dönemi öncesinde yazılarında, Saf Aklın Eleştirisi'nde, Prolegomena'da ve Doğa Biliminin Metafiziksel Temelleri'nde manifold kavramını kullanmıştır. Matematik tarihinde ise söz konusu kavram özellikle Georg Friedrich Bernhard Riemann'ın matematik anlayışının merkezindedir. Bu 1 Bu makale 2011 yılında tamamlanan Riemann'ın Manifold Kavramı ve Yeni bir Mekan-Geometri İnşasındaki Yeri" adlı tez temel alınarak türetilmiştir.

ABSTRACT: In their paper ‘Why Frege did not plagiarize the Stoics’ (Gabriel et al. 2024), G. Gabriel, K. Hülser and D. Rami provide an extended set of arguments to show that Frege didn’t, and couldn’t have, plagiarize(d) the Stoics via Prantl’s long chapter on Stoic logic (Prantl 1855) or at all, thus attempting to counter Bobzien’s 2021 ‘Frege plagiarized the Stoics’. The present short essay is Part II of a four-part refutation of Gabriel et al. 2024: The authors repeatedly ‒ and wrongly ‒ claim that there is a tradition of Stoic logic (and Stoic logical terminology) transmitted continuously from antiquity to the 19th century that was generally known in the 18th and 19th centuries. In this second part of the refutation, it is shown that, and why, the authors are entirely mistaken on every one of these points.

Despite the extensive amount of scholarly work done on Indian mathematics in the last 200 years, the historical conditions under which it originated and evolved has not been studied much. The focus has been more on achievements than on how they developed. One also tends to read the ancient texts with the present concepts and methods in mind. The absence of writing over a long stretch of Indian history too gets overlooked in such readings. The purpose of this article is to explore the journey of mathematics by examining what the ancient texts on arithmetic, geometry and algebra tell us about the nature of mathematics in their times. These investigations reveal that over a period of a thousand or more years Indian mathematics transitioned from concrete and context-bound phase to context-free, abstract phase accompanied by several conceptual leaps. Invention of writing in the 3 rd century BCE greatly facilitated this transition.