This review investigates the potential of non-orientable topology as a fundamental framework for understanding the Poincaré conjecture and its implications across various scientific disciplines. Integrating insights from Dokuchaev (2020), Rapoport, Christianto, Chandra, Smarandache (under review), and other pioneering works, this article explores the theoretical foundations linking non-orientable spaces to resolving the Poincaré conjecture and its broader implications in theoretical physics, geology, cosmology, and biology.

In the last decade, the Poincaré conjecture has probably been the most famous statement among all the contributions of Poincaré to the mathematics community. There have been many papers and books that describe various attempts and the final works of Perelman leading to a positive solution to the conjecture, but the evolution of Poincaré’s works leading to this conjecture has not been carefully discussed or described, and some other historical aspects about it have not (...) been addressed either. For example, one question is how it fits into the overall work of Poincaré in topology, and what are some other related questions that he had raised. Since Poincaré did not state the Poincaré conjecture as a conjecture but rather raised it as a question, one natural question is why he did this. In order to address these issues, in this paper, we examine Poincaré’s works in topology in the framework of classifying manifolds through numerical and algebraic invariants. Consequently, we also provide a full history of the formulation of the Poincaré conjecture which is richer than what is usually described and accepted and hence gain a better understanding of overall works of Poincaré in topology. In addition, this analysis clarifies a puzzling question on the relation between Poincaré’s stated motivations for topology and the Poincaré conjecture. (shrink)

Henri Poincaré was not just one of the most inventive, versatile, and productive mathematicians of all time--he was also a leading physicist who almost won a Nobel Prize for physics and a prominent philosopher of science whose fresh and surprising essays are still in print a century later. The first in-depth and comprehensive look at his many accomplishments, Henri Poincaré explores all the fields that Poincaré touched, the debates sparked by his original investigations, and how his discoveries (...) still contribute to society today. Math historian Jeremy Gray shows that Poincaré's influence was wide-ranging and permanent. His novel interpretation of non-Euclidean geometry challenged contemporary ideas about space, stirred heated discussion, and led to flourishing research. His work in topology began the modern study of the subject, recently highlighted by the successful resolution of the famous Poincaré conjecture. And Poincaré's reformulation of celestial mechanics and discovery of chaotic motion started the modern theory of dynamical systems. In physics, his insights on the Lorentz group preceded Einstein's, and he was the first to indicate that space and time might be fundamentally atomic. Poincaré the public intellectual did not shy away from scientific controversy, and he defended mathematics against the attacks of logicians such as Bertrand Russell, opposed the views of Catholic apologists, and served as an expert witness in probability for the notorious Dreyfus case that polarized France. Richly informed by letters and documents, Henri Poincaré demonstrates how one man's work revolutionized math, science, and the greater world. (shrink)

Testimony about episodes of artistic creativity often describes a puzzling combination of deliberate and involuntary elements. For example, Vincent Van Gogh wrote that it was possible for him to make an especially expressive picture, or as he put it, something with “feeling” in it, because the picture had already spontaneously taken form in his mind before he started drawing. He added, however, that if there was something worthwhile in the picture, this was “not by accident but because of real intention (...) and purpose.” Reflection on such testimony and on his own experience as a poet led Paul Valéry to conclude that artistic creation always involves a combination of “conscious acts” and “spontaneous formation”; only their relative proportion varies. If this point is granted, the outstanding and notoriously difficult problem is to understand how such different elements combine in the creative process. In other words, how is inspiration, or the work of the muse, related to the artist’s deliberations, plans, rational choices, and intentional actions? In what follows I shall develop a conjecture that falls within the conceptual space defined by two extreme theses--the popular, inspirationist idea that artistic creativity is a sudden, involuntary, and ultimately inexplicable event, and the dubious, rationalistic counterthesis, which characterizes artistic creation as a principled, deliberate selection from amongst an array of previously known options, each of which is associated with an expected quantity of artistic value. In my first section I discuss Henri Poincaré’s reflections on creativity in his 1908 essay, “L’Invention mathématique.” Although some of Poincaré’s ideas have been restated and elaborated upon in the literature on creativity, I contend that his most important claims still merit a closer look. Taking Poincairé’s general model as my point of departure, I sketch a new conjecture about artistic creativity in section 2; I also discuss a kindred proposal by Jon Elster. In section 3 I further illustrate and explain this conjecture with reference to Virginia Woolf’s artistic breakthrough in the writing of Jacob’s Room. In my final section I take up and respond to objections that may be raised against these claims about artistic creativity. (shrink)

With contributions from a number of pioneering researchers in the field, this collection is aimed not only at researchers and scientists in nonlinear dynamics but also at a broader audience interested in understanding and exploring how modern chaos theory has developed since the days of Poincaré. This book was motivated by and is an outcome of the CHAOS 2015 meeting held at the Henri Poincaré Institute in Paris, which provided a perfect opportunity to gain inspiration and discuss new perspectives on (...) the history, development and modern aspects of chaos theory. Henri Poincaré is remembered as a great mind in mathematics, physics and astronomy. His works, well beyond their rigorous mathematical and analytical style, are known for their deep insights into science and research in general, and the philosophy of science in particular. The Poincaré conjecture (only proved in 2006) along with his work on the three-body problem are considered to be the foundation of modern chaos theory. (shrink)

This paper concerns the relations between the relative Manin–Mumford conjecture and Pink’s conjecture on unlikely intersections in mixed Shimura varieties. The variety under study is the 4-dimensional Poincaré biextension attached to a universal elliptic curve. A detailed list of its special subvarieties is drawn up, providing partial verifications of Pink’s conjecture in this case, and two open problems are stated in order to complete its proof.

A homeomorphism is built between the separable complex Hilbert space (quantum mechanics) and Minkowski space (special relativity) by meditation of quantum information (i.e. qubit by qubit). That homeomorphism can be interpreted physically as the invariance to a reference frame within a system and its unambiguous counterpart out of the system. The same idea can be applied to Poincaré’s conjecture (proved by G. Perelman) hinting at another way for proving it, more concise and meaningful physically. Furthermore, the conjecture (...) can be generalized and interpreted in relation to the pseudo-Riemannian space of general relativity therefore allowing for both mathematical and philosophical interpretations of the force of gravitation due to the mismatch of choice and ordering and resulting into the “curving of information” (e.g. entanglement). Mathematically, that homeomorphism means the invariance to choice, the axiom of choice, well-ordering, and well-ordering “theorem” (or “principle”) and can be defined generally as “information invariance”. Philosophically, the same homeomorphism implies transcendentalism once the philosophical category of the totality is defined formally. The fundamental concepts of “choice”, “ordering” and “information” unify physics, mathematics, and philosophy and should be related to their shared foundations. (shrink)

The paper continues the consideration of Hilbert mathematics to mathematics itself as an additional “dimension” allowing for the most difficult and fundamental problems to be attacked in a new general and universal way shareable between all of them. That dimension consists in the parameter of the “distance between finiteness and infinity”, particularly able to interpret standard mathematics as a particular case, the basis of which are arithmetic, set theory and propositional logic: that is as a special “flat” case of Hilbert (...) mathematics. The following four essential problems are considered for the idea to be elucidated: Fermat’s last theorem proved by Andrew Wiles; Poincaré’s conjecture proved by Grigori Perelman and the only resolved from the seven Millennium problems offered by the Clay Mathematics Institute (CMI); the four-color theorem proved “machine-likely” by enumerating all cases and the crucial software assistance; the Yang-Mills existence and mass gap problem also suggested by CMI and yet unresolved. They are intentionally chosen to belong to quite different mathematical areas (number theory, topology, mathematical physics) just to demonstrate the power of the approach able to unite and even unify them from the viewpoint of Hilbert mathematics. Also, specific ideas relevant to each of them are considered. Fermat’s last theorem is shown as a Gödel insoluble statement by means of Yablo’s paradox. Thus, Wiles’s proof as a corollary from the modularity theorem and thus needing both arithmetic and set theory involves necessarily an inverse Grothendieck universe. On the contrary, its proof in “Fermat arithmetic” introduced by “epoché to infinity” (following the pattern of Husserl’s original “epoché to reality”) can be suggested by Hilbert arithmetic relevant to Hilbert mathematics, the mediation of which can be removed in the final analysis as a “Wittgenstein ladder”. Poincaré’s conjecture can be reinterpreted physically by Minkowski space and thus reduced to the “nonstandard homeomorphism” of a bit of information mathematically. Perelman’s proof can be accordingly reinterpreted. However, it is valid in Gödel (or Gödelian) mathematics, but not in Hilbert mathematics in general, where the question of whether it holds remains open. The four-color theorem can be also deduced from the nonstandard homeomorphism at issue, but the available proof by enumerating a finite set of all possible cases is more general and relevant to Hilbert mathematics as well, therefore being an indirect argument in favor of the validity of Poincaré’s conjecture in Hilbert mathematics. The Yang-Mills existence and mass gap problem furthermore suggests the most general viewpoint to the relation of Hilbert and Gödel mathematics justifying the qubit Hilbert space as the dual counterpart of Hilbert arithmetic in a narrow sense, in turn being inferable from Hilbert arithmetic in a wide sense. The conjecture that many if not almost all great problems in contemporary mathematics rely on (or at least relate to) the Gödel incompleteness is suggested. It implies that Hilbert mathematics is the natural medium for their discussion or eventual solutions. (shrink)

An isomorphism is built between the separable complex Hilbert space (quantum mechanics) and Minkowski space (special relativity) by meditation of quantum information (i.e. qubit by qubit). That isomorphism can be interpreted physically as the invariance between a reference frame within a system and its unambiguous counterpart out of the system. The same idea can be applied to Poincaré’s conjecture (proved by G. Perelman) hinting another way for proving it, more concise and meaningful physically. Mathematically, the isomorphism means the (...) invariance to choice, the axiom of choice, well-ordering, and well-ordering “theorem” (or “principle”) and can be defined generally as “information invariance”. (shrink)

The topic of the paper is the "realism-Instrumentalism" debate concerning the status of scientific theories. Popper's contributions to this debate are critically examined. In the first part his arguments against instrumentalism are considered; it is claimed that none strikes home against better versions of the doctrine (specifically those developed by duhem and poincare). In the second part, Various arguments against realism propounded by duhem and/or poincare (and much discussed by more recent philosophers) are evaluated. These are the arguments from the (...) use of idealisations in science, From the "underdetermination" of scientific theories, And (especially) from the existence of radical scientific revolutions. A maximally strong version of realism-After due allowance has been made to these arguments-Is stated and defended. This position is close to popper's own "conjectural realism" but involves dropping entirely the idea that science has developed "via" theories possessing increasing verisimilitude. (shrink)

The paper introduces and utilizes a few new concepts: “nonstandard Peano arithmetic”, “complementary Peano arithmetic”, “Hilbert arithmetic”. They identify the foundations of both mathematics and physics demonstrating the equivalence of the newly introduced Hilbert arithmetic and the separable complex Hilbert space of quantum mechanics in turn underlying physics and all the world. That new both mathematical and physical ground can be recognized as information complemented and generalized by quantum information. A few fundamental mathematical problems of the present such as Fermat’s (...) last theorem, four-color theorem as well as its new-formulated generalization as “four-letter theorem”, Poincaré’s conjecture, “P vs NP” are considered over again, from and within the new-founding conceptual reference frame of information, as illustrations. Simple or crucially simplifying solutions and proofs are demonstrated. The link between the consistent completeness of the system mathematics-physics on the ground of information and all the great mathematical problems of the present (rather than the enumerated ones) is suggested. (shrink)

Monsters lurk within mathematical as well as literary haunts. I propose to trace some pathways between these two monstrous habitats. I start from Jeffrey Jerome Cohen’s influential account of monster culture and explore how well mathematical monsters fit each of his seven theses. The mathematical monsters I discuss are drawn primarily from three distinct but overlapping domains. Firstly, late nineteenth-century mathematicians made numerous unsettling discoveries that threatened their understanding of their own discipline and challenged their intuitions. The great French mathematician (...) Henri Poincaré characterised these anomalies as ‘monsters’, a name that stuck. Secondly, the twentieth-century philosopher Imre Lakatos composed a seminal work on the nature of mathematical proof, in which monsters play a conspicuous role. Lakatos coined such terms as ‘monster-barring’ and ‘monster-adjusting’ to describe strategies for dealing with entities whose properties seem to falsify a conjecture. Thirdly, and most recently, mathematicians dubbed the largest of the sporadic groups ‘the Monster’, because of its vast size and uncanny properties, and because its existence was suspected long before it could be confirmed. (shrink)

I argue for the possibility of knowledge by invention whch is neither á priori nor á posteriori. My conception of knowledge by invention evolves from Poincaré’s conventionalism, but unlike Poincaré’s conventions, propositions known by invention have a truth value. An individuating criteria for this type of knowledge is conjectured. The proposition known through invention is: gounded historically in the discipline to which it belongs; a result of the careful, sincere and objective quest and effort of the knower; chosen (...) freely by the inventer or knower; and, private in its invention but public once invented. I extend knowledge by invention to include the knowledge of the invented proposition by those who do not invent it but accept it as a convention for good reasons. Finally, knowledge by invention combined with a revisionist, Platonist definition of knowledge as actively justified true belief provides a pedagogical model reviving the proactive spirit of the Socratic method with an emphasis on invention and activity and a de-emphasis on information gathering and passivity. (shrink)

The book sets an ambitious goal. It devises a new account of scientific methodology that makes it possible to explain how scientists manage, at least occasionally, to find true models of reality. The new methods may be contrasted with all those currently available that employ “coherence theories” of knowledge. Under this designation are grouped positions that can seem very different (such as those of Poincaré, Duhem, Popper, Hempel, Quine, Kuhn, and Feyerabend) but are united by the idea that the (...) most general statements of science are merely hypotheses. They may be conjectures, opinions, conventions, posits, paradigms, or even myths. The most we can claim to know from such generalities is that they are internally consistent and coherent with empirical data. Consistency is insufficient to establish the truth of a conceptual system because many different systems, perhaps an infinite number, can be logically consistent and cohere with recorded data. Such is the well-known problem of the empirical under-determination of theories. Francis Bacon’s Skeptical Recipes for New Knowledge suggests a new methodology that solves this fundamental problem of knowledge. (shrink)

Here is an accurate and readable translation of a seminal article by Henri Poincaré that is a classic in the study of dynamical systems popularly called chaos theory. In an effort to understand the stability of orbits in the solar system, Poincaré applied a Hamiltonian formulation to the equations of planetary motion and studied these differential equations in the limited case of three bodies to arrive at properties of the equations' solutions, such as orbital resonances and horseshoe orbits. Poincaré wrote (...) for professional mathematicians and astronomers interested in celestial mechanics and differential equations. Contemporary historians of math or science and researchers in dynamical systems and planetary motion with an interest in the origin or history of their field will find his work fascinating. (shrink)

" Vivid . . . immense clarity . . . the product of a brilliant and extremely forceful intellect." — Journal of the Royal Naval Scientific Service "Still a sheer joy to read." — Mathematical Gazette "Should be read by any student, teacher or researcher in mathematics." — Mathematics Teacher The originator of algebraic topology and of the theory of analytic functions of several complex variables, Henri Poincare (1854–1912) excelled at explaining the complexities of scientific and mathematical ideas to lay (...) readers. Science and Method, written in 1908, has been appreciated by a wide audience of nonprofessionals and translated into many languages. It defines the basic methodology and psychology of scientific discovery, particularly in regard to mathematics and mathematical physics. Drawing on examples from many fields, it explains how scientists analyze and choose their working facts, and it explores the nature of experimentation, theory, and the mind. 1914 edition. Translated by Francis Maitland. (shrink)

Le savant et la science -- Le raisonnement mathématique -- La mécanique nouvelle -- La science astronomique.

THE VALUE OF SCIENCE INTRODUCTION The search for truth should be the goal of our activities; it is the sole end worthy of them. Doubtless we should first bend our efforts to assuage human suffering, but why ? Not to suffer is a negative ...

1. Number and magnitude.--2. Space.--3. Force.--4. Nature.

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work.This work is in the public domain in (...) the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

Poincaré tries to tackle the question "Can science sway us into materialism?". In other words, he analyzes if science - in particular Physics - and its theories are dependent on a materialistic ontological view of the world. His final answer appears to be "no". However, in order to reach this conclusion he resorts to a brief history of Physics, providing an insightful account on the evolution of the concept of atom from Democritus to the, then, recent discoveries on the (...) composition of the atom. In his opinion, science swung and will keep swinging between two opposing approaches over the constitution of nature: the discrete and the continuous approach. (shrink)

Nous reproduisons ici, avec l’aimable autorisation de la Hofbibliothek Aschaffenburg (Allemagne), la courte correspondance inédite entre Dingler et Henri Poincaré. Cette correspondance, conservée aux Archives Dingler dans la Hofbibliothek Aschaffenburg (et mentionnée dans le catalogue du Nachlass de Dingler [Wolters & Schroeder-Heister 1979]), consiste en trois documents : un brouillon en allemand de la main de Dingler, sa traduction française par une autre plume, mais avec des corrections ef...

"La Valeur de la Science" de Henri Poincaré. Mathématicien, physicien et philosophe français (1854-1912).

On the nature of mathematical reasoning -- Mathematical magnitude and experience -- Non-Euclidian geometries -- Space and geometry -- Experience and geometry -- Classical mechanics -- Relative and absolute motion -- Energy and thermodynamics -- Hypotheses in physics -- Theories of modern physics -- Probability calculus -- Optics and electricity -- Electrodynamics -- The end of matter.