History and Philosophy of Geometry

A century and a half ago, a revolution in human thought began that has gone largely unrecognized by modern scholars: A system of non-Euclidean geometries was developed that literally changed the way that we view our world. At first, some thought that space itself was non-Euclidean and four-dimensional, but Einstein ended their 'speculations' when he declared that time was the fourth dimension. Yet our commonly perceived space is truly four-dimensional. Einstein unwittingly circumvented that particular revolution in thought and delayed its completion for a later day, although his work was also necessary for the completion of that revolution. That later day is now approaching. The natural progress of science has brought us back to the point where science again needs to consider the physical reality of a higher-dimensional space. Science must acknowledge the truth that space is four-dimensional and space-time is five-dimensional, as required by accepted physical theories and observations, before it can move forward with a new unified fundamental theory of physical reality.

A significant number of works have set forth, over the past decades, the emphasis laid by seventeenth-century mathematicians and philosophers on motion and kinematic notions in geometry. These works demonstrated the crucial role attributed in this context to genetic definitions, which state the mode of generation of geometrical objects instead of their essential properties. While the growing importance of genetic definitions in sixteenth-century commentaries on Euclid’s Elements has been underlined, the place, uses and status of motion in this geometrical tradition has however never been thoroughly and comprehensively studied. This book therefore undertakes to fill a gap in the history of early modern geometry and philosophy of mathematics by investigating the different treatments of motion and genetic definitions by seven major sixteenth-century commentators on Euclid’s Elements, from Oronce Fine (1494–1555) to Christoph Clavius (1538–1612), including Jacques Peletier (1517–1582), John Dee (1527–1608/1609) and Henry Billingsley (d. 1606), among others. By investigating the ontological and epistemological conceptions underlying the introduction and uses of kinematic notions in their interpretation of Euclidean geometry, this study displays the richness of the conceptual framework, philosophical and mathematical, inherent to the sixteenth-century Euclidean tradition and shows how it contributed to a more generalised acceptance and promotion of kinematic approaches to geometry in the early modern period.

This paper deals with the story of the shortening of circle and how and when artists and scientists became aware of the possibility to represent it as an ellipse.
Some cases are described just to give an idea of how we can fail in detecting the form a curve, the ellipse in particular, when we are facing with an elongated circular form.
Euclid, the shape of the amphitheaters, the rings in the Ptolemaic description of an armillary sphere, the section of the Dome of Santa Maria del Fiore in Florence by Brunelleschi, the aureolas in Piero della Francesca's frescos, the graphical (wrong) description of an ellipse by Dürer, and the ellipse in the cosmographical vision of Kepler are chosen as examples of my epistemological point of view.
The circle has been perceived and represented in linguistic form - as the shortening of the circle in function of the position of the eye - as well as in the graphic form as a mechanical drawing.

La axiomatización de la geometría euclídea llevada a cabo por David Hilbert (1862-1943) en su monografía Fundamentos de la geometría (1899) es considerada no sólo una de sus contribuciones más importantes a la matemática moderna, sino también un hito fundamental en la concepción abstracta del método axiomático. Hilbert presentó allí un sistema axiomático formal para la geometría euclídea elemental, a través del cual era posible construir esta teoría matemática de un modo riguroso y sistemático. Dicha presentación axiomática de la geometría trajo así aparejada una nueva manera de entender la naturaleza de las teoría matemáticas, que logró capturar magistralmente la marcada tendencia hacia la abstracción y la sistematización experimentada por la matemática en la segunda mitad del siglo XIX.
El objetivo del presente libro es ofrecer una exposición detallada de las investigaciones axiomáticas de Hilbert sobre los fundamentos de la geometría y de su concepción temprana del método axiomático. Un aspecto novedoso de este trabajo es que hace un uso extensivo de un conjunto de fuentes manuscritas, accesibles ahora por primera vez en lengua castellana. La importancia de estas fuentes reside en que permiten ofrecer una interpretación mejor contextualizada e históricamente más adecuada de la concepción del método axiomático defendida por Hilbert en este período inicial de sus trabajos sobre los fundamentos de la matemática, cuyo punto culminante fue la publicación de Fundamentos de la geometría.

This talk explores the roots of the power and fertility of mathematical imagination manifested in Riemann’s mathematics, one of the greatest and most imaginative mathematicians ever lived. So my main aim is to present a case study regarding how and to what extend philosophy can be relevant in developing a new understanding of geometry.

The recognition at the beginning of the nineteenth century of the existence of a non-Euclidean geometry, thanks to Gauss, Lobachevskij and Bolyai, based on purely axiomatic (conceptual) considerations is considered as a symptom of faith in the truth content of pure thought, which was also expressed by German idealism though in a different way. A short time later this faith extended only to mathematical contents and consequently made the mathematisation of the natural sciences and the rejection of romantic natural philosophy necessary in accordance with currents of thought prevailing at that time. The integration of the non-Euclidean geometries into mathematics, which began in the sixties of the nineteenth century, played a part in the definitive separation of mathematical and especially geometrical abstraction from a way of looking at things determined by sensory perception. Once that mathematics had become conscious of its spiritual (abstract) strength, it began a decisive campaign of conquest throughout all branches of knowledge. On the other hand, because of this development, possibilities, which had not existed before to the same extent, were created for the training of pure thought and of supersensory imagination with the help of mathematics.
Rudolf Steiner was responsible for a new understanding of mathematics as preparatory training for spiritual comprehension as well as a methodological principle for the acquisition of knowledge in the natural sciences. Steiner threw clear light on the possibilities and limits of mathematics. Its strong point in developing consciousness lies in its methodological purity and universality, its weak point is its limitations in describing sensory phenomena because of the special form of their contents. Certain contents and methodological principles of mathematics, however, point towards areas beyond themselves, for example, synthetic projective geometry and the non-Euclidean geometries embedded in it, and, thus, these disciplines offer prospects of a mathematics of invigorative principles (mathematical physics and mathematics of living organisms).
Non-Euclidean as well as synthetic projective geometry plays an important role during a stage in the evolution of mathematics as it frees itself from the way of looking at things determined by sensory perception and becomes refined by complete abstraction, and also in the evolution of the natural sciences, which subsequently restrict themselves to the formal mathematical method. On the other hand, these geometries at the same time prove to be a powerful instrument for training pure thought and imagination free of sense perception. Thus, they are a means for leading man back to the invigorative reality of nature and spirit.

Popper, Platon'un "geometri bilmeyen girmesin" sözünün sanki Platon matematik çalışmalarının felsefe açısından faydalı olacağını söylemiş gibi düzanlamıyla ya da yalnızca Platoncu öğretiyi göz önünde bulundurarak geometrinin idealar kuramı için bir mihenk taşı vazifesi gördüğü biçiminde anlaşılmaması gerektiğini söyler. Bu yorumlar kısmen doğru olsalar da Popper bu sözün daha kapsamlı bir yorumu hak ettiğini iddia eder ve bir kestirimde bulunur. Ona göre irrasyonel sayıların keşfinin ardından doğal sayılara dayanan Pythagorasçı aritmetik artık yeterli değildi. İrrasyonel doğru ve sayılarla meşgul olan geometri bundan böyle en temel bilimdi. Platon bu olguyu görmüş ve Timaios'ta geometriye dayalı bir kozmoloji inşa etmeye uğraşmıştı. Bu kozmoloji aynı zamanda Platon'un Pythagorasçı düşünce mirasını irrasyonel sayı ve doğrular aracılığıyla geliştirerek aşması anlamına geliyordu. Bu yazıda amacım Popper'in Platon'un sözüne ilişkin görüşlerini detaylı biçimde ele almak olacak.

It is generally believed in the scientific community that no changes in the physical concept space occurred during the period of time just before the onset of the Second Scientific Revolution except for Mach’s successful logical argument against the possibility of Newton’s absolute space. This belief is untrue. The concept of space in general and absolute space in particular was undergoing a complete rehabilitation during the era. The proliferation of new wave of aether theories that were initiated by Maxwell’s electromagnetic theory were merely a side issue to the question of absolute space. This scenario fits well the foregone conclusions that Newtonian science had reached its zenith but had begun to stagnate. The stagnation was marked by wild speculations regarding the aether that were being thrown around the scientific community prior to the coming revolution. Therefore, when experiments to detect the aether failed and explanations of the emission spectra of atoms and molecules could not be explained within the Newtonian paradigm, the next logical step was the scientific revolution. At least this is the general story of late nineteenth century physics that is perpetuated by the early winners of the Second Scientific Revolution. This story is necessary to demonstrate the corruptness of the Newtonian paradigm with regard to the new physics and show how the new physics of the twentieth century is superior to the Newtonian paradigm. So this story is grossly incomplete on many issues if not outright untrue.

David W. Wood, "Mathesis of the Mind": A Study of Fichte's Wissenschaftslehre and Geometry (New York/Amsterdam: Rodopi (Brill), 2012), Fichte-Studien-Supplementa 29, 304 pages: https://brill.com/view/title/29604

This book examines J.G. Fichte’s philosophy of mathematics and theory of geometry. It investigates both the external formal and internal cognitive parallels between the axioms, intuitions and constructions of geometry and the scientific methodology of the Fichtean system of philosophy. In contrast to “ordinary” Euclidean geometry, in his Erlanger Logik of 1805 Fichte posits a model of an “ursprüngliche” or original geometry – that is to say, a synthetic and constructivistic conception grounded in ideal archetypal elements that are grasped through geometrical or intelligible intuition. Accordingly, this study classifies Fichte’s philosophy of mathematics as a whole as a species of mathematical Platonism or neo-Platonism, and concludes that the Wissenschaftslehre itself may be read as an attempt at a new philosophical mathesis, or “mathesis of the mind".

The book furthermore situates Fichte's archetypal geometry in relation to the points at infinity in the tradition of the projective geometry of Girard Desargues, Jacob Steiner and K.G.C. von Staudt. In terms of the philosophy of mathematics, geometry, and space, it especially compares Fichte's treatment with the Kantian conceptions. Above all based on a close textual reading of Fichte´s own texts, it also takes note of the interpretations of Fichte by Jules Vuillemin (La philosophie de l'algèbre), Joachim Widmann (Die Grundstruktur des Transzendentalen Wissens) and Werner Csech (Die Raumlehre Johann Gottlieb Fichtes), as well as engaging with the latest Fichte scholarship in the German, English and French languages.

This document contains the cover, contents, and acknowledgement pages.

In 1915, Einstein introduced a radical and revolutionary new idea into science and human thought: Gravity was not a force. The observed gravitational attraction between two or more material bodies was the product of the material bodies following the shortest geodesic paths across the curvature of a space-time continuum. The amount or degree of curvature at any location depended on the local presence of matter. As radical as this idea was, the theory became a known and accepted part of physics within just a few years, which was quite rapid relative to the overall scale of changes in science. It was accepted so rapidly because a few scientists had been attempting to find a connection between non-Euclidean geometry (especially Riemannian) and physics for several decades as well as how the geometry of space (or space-time) was related to ultimate nature and structure of matter. However, Einstein’s theory also came as a surprise because no one had detected any specific flaw in Newtonian gravity theory and other researchers working on the problem of matter had expected matter to ultimately be electric (electromagnetic) in nature. Within this historical context, several scientists and mathematicians immediately noticed a problem with the new theory and took advantage of that problem to extend physics further than Einstein originally anticipated. Although a tensor placed at a point in space was used to represent the curvature through that point, the mathematics and physics at the point itself was not completely specified or explained. This flaw allowed others to develop non-Riemannian and pseudo-Riemannian geometries (also called tangent spaces) to explain geometry specifically at or in the individual points of space-time. Since they were independent of the Riemannian metric geometry of extended (metric) space, these new geometries could be used to explain the electromagnetic nature of matter without disturbing the metric or gravitational explanation given by general relativity. This not only began the search for a unified field theory, but it also demonstrated quite clearly that the attempted rigorization of mathematics that had occurred over the past two centuries was inadequate and mathematics was still intimately linked to physics. The story of how these new geometries developed is both interesting and intriguing, yet so much more. The development of physics over the next few decades cannot be completely understood without taking these events into account. This lost history explains fully both Einstein’s search for a unified field theory (with an emphasis on a metric or extension geometry) as well as the later development of quantum mechanics based on the uncertainty principle and how that relates to quantum field theories (using a point non-geometrical space or space-time), thus changing the whole historical context of modern physics.

Hölder’s philosophy of geometry might appear to be the most problematic part of his epistemology. He maintains that geometry depends on experience also after Poincaré’s fundamental criticism of Helmholtz. Nevertheless, I think that Hölder’s view is worth discussing, for two reasons. Firstly, the related methodological considerations were crucial for the development of his epistemology. Secondly, Poincaré uses the opposition between Kantianism and empiricism to argue for his geometrical conventionalism. Nevertheless, Hölder shows that an alternative strategy is not excluded: he profits from Kantian objections in order to develop a consistent empiricism. At the same time, especially in Die mathematische Methode [Hölder 1924], he vindicates the Kantian view that mathematics is synthetic. In this paper, I will consider Hölder’s defence of the deductive method in geometry in Anschauung und Denken in der Geometrie [Hölder 1900] in connection with his approach to the theory of quantity [Hölder 1901]. Moreover, I will discuss his connection with Kant. My suggestion is that Hölder’s methodological considerations enable him to foreshadow a relativized conception of the a priori.

The paper surveys different notions of implicit definition. In particular , we offer an examination of a kind of definition commonly used in formal axiomatics, which in general terms is understood as providing a definition of the primitive terminology of an axiomatic theory. We argue that such "structural definitions" can be semantically understood in two different ways, namely (i) as specifications of the meaning of the primitive terms of a theory and (ii) as definitions of higher-order mathematical concepts or structures. We analyze these two conceptions of structural definition both in the history of modern axiomatics and in contemporary philosophical debates. Based on that, we give a systematic assessment of the underlying semantics of these two ways of understanding the definiens of such definitions, by considering alternative model-theoretic and inferential accounts of meaning.

Many physicists have thought that absolute time became otiose with the introduction of Special Relativity. William Lane Craig disagrees. Craig argues that although relativity is empirically adequate within a domain of application, relativity is literally false and should be supplanted by a Neo-Lorentzian alternative that allows for absolute time. Meanwhile, Craig and co-author James Sinclair have argued that physical cosmology supports the conclusion that physical reality began to exist at a finite time in the past. However, on their view, the beginning of physical reality requires the objective passage of absolute time, so that the beginning of physical reality stands or falls with Craig's Neo-Lorentzian metaphysics. Here, I raise doubts about whether, given Craig's NeoLorentzian metaphysics, physical cosmology could adequately support a beginning of physical reality within the finite past. Craig and Sinclair's conception of the beginning of the universe requires a past boundary to the universe. A past boundary to the universe cannot be directly observed and so must be inferred from the observed matter-energy distribution in conjunction with auxilary hypotheses drawn from a substantive physical theory. Craig's brand of Neo Lorentzianism has not been sufficiently well specified so as to infer either that there is a past boundary or that the boundary is located in the finite past. Consequently, Neo Lorentzianism implicitly introduces a form of skepticism that removes the ability that we might have otherwise had to infer a beginning of the universe. Furthermore, in analyzing traditional big bang models, I develop criteria that Neo-Lorentzians should deploy in thinking about the direction and duration of time in cosmological models generally. For my last task, I apply the same criteria to bounce cosmologies and show that Craig and Sinclair have been wrong to interpret bounce cosmologies as including a beginning of physical reality.

Es bien sabido que el surgimiento de las geometrías no-euclídeas provocó un controvertido debate filosófico acerca de la realidad o no de las mismas que involucró a pensadores tanto de formación más puramente filosófica como científica, principalmente matemáticos y físicos. Esta polémica dio lugar a las famosas propuestas de demostración empírica de tales geometrías por parte de Gauss, Lobachevsky o Riemann. Pero más allá de los fallidos intentos de experimentación, es importante recordar que el debate en filosofía era inevitable, puesto que ponía en duda el carácter sintético a priori que los juicios de la geometría habían tenido desde Kant. Esto generó numerosas posiciones filosóficas que trataron tanto de defender al pensador prusiano como de socavar todo su sistema metafísico. En esta disputa hay dos posiciones en especial que, tanto por su originalidad filosófica como por su fecundidad científica, merecen una consideración especial: el convencionalismo de Poincaré y el empirismo neokantiano de Helmholtz. La importancia de estas dos concepciones no radica sólo en su aportación al debate en cuestión, sino en que proporcionan dos nuevas maneras de pensar la ciencia que están muy próximas una de otra pero que, sin embargo, no son asimilables. Lo que trataremos de mostrar es cómo a partir de la reflexión sobre una misma disciplina científica y un problema concreto, se llegó a dos posiciones filosóficas que hoy en día todavía resultan fructíferas en las discusiones epistemológicas sobre la ciencia natural.

This paper addresses two primary questions about Aristotle’s account of time: the relationship between time and motion, and whether the now is extended or point-like. It claims that time is generated from motion by the use of point-like nows.  According to Aristotle motion is prior to time, but there is some ambiguity about the basis for this priority. A reconstruction of the role of the now can show the relationship between motion and time: by using the now to define a temporal unit, time as a measure is generated by abstraction from motion. Marking off a motion along an oriented continuity is to create a now. Once two have been marked, the soul draws the nows together into an act of defining a length between them. This generates a functionally indivisible unit: time. Thus by articulating and perceiving motion the soul can generate the temporal number as a new abstract being. Reconstructing this account of abstraction allows us to formulate more strongly Aristotle’s claim to the ontological dependence of time on motion. The paper gives a systematic overview of the relationships between the now and number in order to address the question of whether the now might be extended, whether motion depends on time, and how universal time is possible.

Cette thèse vise à reconstituer la théorie de l'espace développée par Husserl dans les manuscrits des premières années quatre-vingt-dix qui composent le Livre de l'espace. On en exposera les thématiques principales et identifiera les influences exercées sur la pensée de Husserl durant sa formation philosophique, avec une attention particulière en ce qui concerne les aspects mathématiques et scientifiques du problème de l'espace.
Le Livre de l'espace représente encore un défi interprétatif, bien que Ingeborg Strohmeyer ait composé une édition critique de l'œuvre en suivant le sommaire esquissé par Husserl. Le corps textuel se fragmente dans une série des sections qui, à première vue, suggèrent que Husserl était "attiré" par un objet d'étude qui l'éloignait du cours principal de ses réflexions. Plusieurs chercheurs ont suivi cette lecture et ont interprété le Livre de l'espace comme une première exploration de la question des idéalités géométriques, en y repérant les concepts et les coordonnés théoriques des œuvres de la maturité. Selon une deuxième hypothèse complémentaire, bien que le Livre de l'espace ne réussisse pas à formuler une véritable théorie de l'espace, ses idées représentent néanmoins une ébauche de celles présentées dans le cours phénoménologique sur la chose de 1907. Ces interprétations abordent le problème en considérant seulement les textes publiés et, secondairement, elles tendent à lire les théories du jeune Husserl à travers les catégories de la phénoménologie tardive.
Même si certains aspects du Livre de l'espace anticipent les analyses des idéalités géométriques de la Crise des sciences européennes et de La chose et l'espace, on trouve des donnés textuels qui montrent l'impossibilité de réduire la théorie de l'espace préphénoménologique au contexte et au style des analyses phénoménologiques. Ces raisons émergent en concentrant l'analyse sur le chemin des recherches qui partent de la Philosophie de l'Arithmétique et s'ouvrent à des questions latérales, mais pas marginales. On aura ainsi l'occasion d'aborder des questions relatives à l'optique, la physiologie, les mathématiques et à la psychologie, qui sont largement traitées dans les inédits.
En abordant ces thématiques dans le cadre de la psychologie brentanienne, Husserl formule une théorie de l'espace qui, bien qu'irréductible aux catégories de la prochaine phénoménologie, n'est pas une théorie complète et cohérente. Le projet du Livre de l'espace est ainsi abandonné en faveur de certaines recherches plus urgentes qui conduiront Husserl vers le "tournant antipsychologiste" des Recherches Logiques. Cette thèse vise aussi à mettre en lumière les raisons qui ont poussé Husserl à abandonner sa première théorie de l'espace, en intégrant les réflexions de l'époque dans l'ensemble de la production husserlienne. À travers des références ponctuelles à la théorie phénoménologique de l'espace on montrera ses analogies et différences avec le Livre de l'espace et son contexte, en comprenant ainsi dans quel sens elle constitue l'héritage conceptuel des réflexions des années quatre-vingt-dix. Il s'agira donc de souligner le passage graduel de la théorie géométrique de l'espace à la théorie phénoménologique afin de tracer une histoire des différentes réponses que Husserl a données à la question de l'espace, entre psychologie et phénoménologie.

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Il presente lavoro mira a ricostruire la teoria dello spazio sviluppata da Husserl nei manoscritti dei primi anni Novanta che vanno a formare il cosiddetto Raumbuch, esponendone le tematiche principali ed individuando le influenze recepite da Husserl durante la sua formazione giovanile, con particolare attenzione ai risvolti matematico-scientifici del Raumproblem.
Sebbene Ingeborg Strohmeyer, nel comporre l'edizione critica dell'opera, abbia seguito l'indice abbozzato dallo stesso Husserl, comprendere il Raumbuch rappresenta ancora una sfida per l'interprete. Il corpo della trattazione è infatti frammentato in molteplici sezioni, più o meno sostanziose, che a un primo sguardo potrebbero dare l'idea di un Husserl attratto/distratto da una curiosità passeggera. Numerosi studiosi hanno assecondato questa lettura, scorgendo nel Raumbuch un primo confronto di Husserl col problema delle idealità geometriche e riconoscendovi i concetti e le coordinate teoriche che torneranno nelle opere mature. A questa prima ipotesi interpretativa se n'è associata un'altra, complementare, secondo cui il Raumbuch, pur non riuscendo a formulare una teoria dello spazio che possa definirsi tale, rappresenti un abbozzo della teoria dello spazio fenomenologica presentata nelle lezioni sulla cosa del '07. Entrambe le interpretazioni sono in realtà frutto di un approccio alla questione che tiene conto dei soli testi editi da un lato, e dall'altro cade tende a leggere le teorie giovanili attraverso le categorie della fenomenologia matura.
Anche se che alcuni aspetti del Raumbuch anticipano la trattazione delle idealità geometriche della Krisis der europaischen Wissenschaften und die transzendentale Phänomenologie e la teoria della Dingvorlesung, non si possono certo ignorare quelle ragioni che fanno della teoria dello spazio pre-fenomenologica un tentativo di affrontare il Raumproblem autonomo dal contesto delle riflessioni fenomenologiche, seppur ad esse evidentemente legato. Tali ragioni testuali emergono se si volge l'attenzione al panorama di problemi che occupano Husserl in quel periodo: solo seguendo il filo dell'indagine che dalla prima opera di Husserl si apre a considerazioni laterali ma non per questo marginali, si accede ad un orizzonte di problemi (ottici, fisiologici, matematici, psicologici) ampiamente dibattuti negli inediti.
Dall'incontro tra la discussione del problema della spazio e gli strumenti ereditati dalla psicologia di matrice brentaniana, nasce una teoria dello spazio indipendente dalle categorie della fenomenologia, ma non per questo completa e consistente con il corpus delle riflessioni husserliane che si andavano codificando durante quell'ultimo decennio del XIX secolo. Il progetto del Raumbuch viene infatti accantonato nell'ottobre del '93, a favore di indagini più pressanti che avrebbero condotto Husserl alla "svolta antipsicologistica" delle Logische Untersuchungen. Il presente lavoro mira a decifrare i motivi e le tempistiche che hanno condotto Husserl all'abbandono della prima teoria dello spazio, integrandone le riflessioni finora poco frequentate dalla letteratura critica nel complesso della produzione husserliana. A tal scopo, si troveranno alcuni rimandi alla teoria dello spazio fenomenologica che, nell'illustrare differenze e analogie con la temperie del Raumbuch, aiuteranno a comprendere in che senso la prima possa essere considerata erede della seconda. Mettendo in luce il graduale passaggio dalla teoria geometrica dello spazio alla teoria fenomenologica dello spazio, verrà tracciata una storia delle differenti risposte fornite da Husserl alla questione dello spazio, tra psicologia e fenomenologia.

The power of reflection is what drives traditional claims that we are disembodied minds or transcendental egos. This chapter argues that reflection is a bodily capacity, a power given to bodies with *logos* – the distinctive capacity for what Merleau-Ponty calls "symbolic conduct". By placing reflection in its bodily context and returning it to its roots in pre-reflective meaning, we also arrive at a new understanding of space as a tradition or *institution* — a way of making sense that our bodies have inherited from the living bodies that came before us.

The argument of this chapter is drawn principally from a dense and under-studied passage in Merleau-Ponty’s first book, *The Structure of Behaviour*. I follow the Structure’s strategy of illuminating human experience via the study of animal behaviour – in this case, that of the chimpanzee. By contrasting the chimp’s experience of space with our own, I argue that our experience of space as uniform and universal is a development of the animal’s sense of space as oriented around its own body.

This book offers a reconstruction of the debate on non-Euclidean geometry in neo-Kantianism between the second half of the nineteenth century and the first decades of the twentieth century. Kant famously characterized space and time as a priori forms of intuitions, which lie at the foundation of mathematical knowledge. The success of his philosophical account of space was due not least to the fact that Euclidean geometry was widely considered to be a model of certainty at his time. However, such later scientific developments as non-Euclidean geometries and Einstein’s general theory of relativity called into question the certainty of Euclidean geometry and posed the problem of reconsidering space as an open question for empirical research. The transformation of the concept of space from a source of knowledge to an object of research can be traced back to a tradition, which includes such mathematicians as Carl Friedrich Gauss, Bernhard Riemann, Richard Dedekind, Felix Klein, and Henri Poincaré, and which finds one of its clearest expressions in Hermann von Helmholtz’s epistemological works. Although Helmholtz formulated compelling objections to Kant, the author reconsiders different strategies for a philosophical account of the same transformation from a neo-Kantian perspective, and especially Hermann Cohen’s account of the aprioricity of mathematics in terms of applicability and Ernst Cassirer’s reformulation of the a priori of space in terms of a system of hypotheses. This book is ideal for students, scholars and researchers who wish to broaden their knowledge of non-Euclidean geometry or neo-Kantianism.

A theory of magnitudes involves criteria for their equivalence, comparison and addition. In this paper we examine these aspects from an abstract viewpoint, by focusing on the so-called De Zolt's postulate in the theory of equivalence of plane polygons ("If a polygon is divided into polygonal parts in any given way, then the union of all but one of these parts is not equivalent to the given polygon"). We formulate an abstract version of this postulate and derive it from some selected principles for magnitudes. We also formulate and derive an abstract version of Euclid's Common Notion 5 ("The whole is greater than the part"), and analyze its logical relation to the former proposition. These results prove to be relevant for the clarification of some key conceptual aspects of Hilbert's proof of De Zolt's postulate, in his classical Foundations of Geometry (1899). Furthermore, our abstract treatment of this central proposition provides interesting insights for the development of a well-behaved theory of compatible magnitudes.

The famous Fermat's Last Theorem, formulated in 1637, states that no three positive integers a, b, and c can satisfy the equation a n + b n = c n if n is an integer greater than two. " I will not address the centuries long story of the proof attempts from Fermat to Wiles and would instead discuss the semantic space of the sought for proofs. 39 First, I would like to recall that the history of Mathematics is often quite instructive about the actual heuristic processes taking part even long time ago. Thus, in the case with Fermat's theorem, or, at the time and technically, the Fermat's Conjecture, on pains that a rigorous proof was not available in written form accessible to the mathematical community for evaluation of truth and rugour, he famously conjected that «he had a proof that was too large to fit in the margin» of his copy of the edition of Diophantus' Arithmetica, (Heath, 1910). As I have argued elsewhere (Grozdanoff, 2015), we often tend to limit the space of our semantic interpretation of mathematical formalism and particularly so when we work with numbers in arithmetic and algebras. In this case, clearly, Fer-mat's wrote the bases a, b and c as designating numbers (positive integers , explicit in the theorem) as well as the exponent n, which stood again for nothing but numbers, and, again, positive integers. Thus the semantic space of the theorem is within the theorem explicitly defined as the set of the positive integers or naturals, Z +. The minimal semantic sense which I would employ here, is the interpretation of the symbols and the alleged objects of their reference. The Z + space is a sub-domain of the realm of all numbers and does not include non-numerical inhabitants like geometric objects, functions or knots, for example. The operations on the space, both explicit and implicit in the theorem, are standard in number theory: the first order opera-39 The text is a part of an ongoing project and aims at no mathematical rigour. It hopes to shed some light on a different approach to Fermat's last theorem, which is based on a geometrical interpretation, just like in the standard interpretations of the Pythagorean Theorem, and not on purely arithmetical interpretations, which take the theorem as exclusively formulated in number theory.

En su opera prima. Kant propuso una geometria de todos los tipos posibles de espacio, y no sólo del espacio conocido. Como esos tipos de espacio Kant los referia cada uno a un mundo posible distinto, es razonable pensar que la "geometria suprema", como Kant la denominó, en realidad era el nombre genérico para un conjunto de geometrias diversas que describen espacios igualmente diversos. ¿Acaso entre ellas podrian encontrarse las que hoy conocemos como no euclídeas?

This paper aims to achieve two things. One is to give an account of how Riemann explains his notion of manifold. Riemann was influenced by Herbart's philosophy on the one hand and Gauss's mathematics on the other. For this reason, Erhard Scholz (1982) claims that the notion of manifold is a quasi-philosophical concept. It is therefore important to explicate the role philosophy plays in Riemann's notion of manifold as this will show how and to what extend philosophy can be relevant to mathematics and perhaps other fields of study as well. I will then discuss possible consequences of this notion for geometry (especially issues regarding the evolution of ideas of space) from an epistemological point of view.

The recognition at the beginning of the nineteenth century of the existence of a non-Euclidean geometry, thanks to Gauss, Lobachevskij and Bolyai, based on purely axiomatic (conceptual) considerations is considered as a symptom of faith in the truth content of pure thought, which was also expressed by German idealism though in a different way. A short time later this faith extended only to mathematical contents and consequently made the mathematisation of the natural sciences and the rejection of romantic natural philosophy necessary in accordance with currents of thought prevailing at that time. The integration of the non-Euclidean geometries into mathematics, which began in the sixties of the nineteenth century, played a part in the definitive separation of mathematical and especially geometrical abstraction from a way of looking at things determined by sensory perception. Once that mathematics had become conscious of its spiritual (abstract) strength, it began a decisive campaign of conquest throughout all branches of knowledge. On the other hand, because of this development, possibilities, which had not existed before to the same extent, were created for the training of pure thought and of supersensory imagination with the help of mathematics.
Rudolf Steiner was responsible for a new understanding of mathematics as preparatory training for spiritual comprehension as well as a methodological principle for the acquisition of knowledge in the natural sciences. Steiner threw clear light on the possibilities and limits of mathematics. Its strong point in developing consciousness lies in its methodological purity and universality, its weak point is its limitations in describing sensory phenomena because of the special form of their contents. Certain contents and methodological principles of mathematics, however, point towards areas beyond themselves, for example, synthetic projective geometry and the non-Euclidean geometries embedded in it, and, thus, these disciplines offer prospects of a mathematics of invigorative principles (mathematical physics and mathematics of living organisms).
Non-Euclidean as well as synthetic projective geometry plays an important role during a stage in the evolution of mathematics as it frees itself from the way of looking at things determined by sensory perception and becomes refined by complete abstraction, and also in the evolution of the natural sciences, which subsequently restrict themselves to the formal mathematical method. On the other hand, these geometries at the same time prove to be a powerful instrument for training pure thought and imagination free of sense perception. Thus, they are a means for leading man back to the invigorative reality of nature and spirit.

It is commonly believed that the first hyperspace theories in physics were developed in the early twentieth century - Kaluza’s five-dimensional extension of relativity is the best known, but this is untrue. It is also commonly believed that W.K. Clifford ‘speculated' on a higher space in 1870, had no followers and never published his theory (if he even had one). Nothing could be further from the historical truth. As early as 1869, Clifford, his followers and students began to develop a physical theory of matter based on a three-diemnsional space curved in four dimensions. Clifford began to publish his theory, but modern researchers cannot recognize his theoretical work because they look for something like Einstein's theory even though Clifford developed an electromagnetic theory. Clifford may not have directly influenced Einstein's relativity, but he made plausible arguments for the reality of space curvature, rendering the rapid acceptance of Einstein’s concept of curved space-time more plausible. Clifford’s work is either largely ignored by historians, scientists and other scholars or considered irrelevant because the early work on hyperspaces has been associated with ether theories, which were abandoned, and utilized quaternion algebras, which were replaced by vectors and tensors, and became associated with spiritualism.

In studies of history of non-Euclidean geometry and its relationship with Kant's critical philosophy we see Riemann stands in a special position with his conceptual mathematics and the concept of manifold which transformed our mathematical, physical, and philosophical understanding of spatiality. Although most of the philosophers accept the Riemann's special position I think that Riemann's philosophy of geometry has not been paid enough attention it deserves. Probably this is because of the fact that although the impact of mathematical greatness of Riemann's Habilitationsvortrag is unquestionable the same cannot be said of his epistemological conclusions, on whose very meaning all philosophers are not agreed. In this paper I aim to achieve two things. One is to examine Habilitationsvortrag in which Riemann explains his notion of manifold. My second aim is to show that Riemann's notion of manifold does not dismiss Kant's spatial intuition.

A theory of magnitudes involves criteria for their equivalence, comparison and addition. In this article we examine these aspects from an abstract viewpoint, by focusing on the so-called De Zolt’s postulate in the theory of equivalence of plane polygons (“If a polygon is divided into polygonal parts in any given way, then the union of all but one of these parts is not equivalent to the given polygon”). We formulate an abstract version of this postulate and derive it from some selected principles for magnitudes. We also formulate and derive an abstract version of Euclid’s Common Notion 5 (“The whole is greater than the part”), and analyze its logical relation to the former proposition. These results prove to be relevant for the clarification of some key conceptual aspects of Hilbert’s proof of De Zolt’s postulate, in his classical Foundations of Geometry (1899). Furthermore, our abstract treatment of this central proposition provides interesting insights for the development of a well...

The paper presents and analyzes a set of unpublished notes for lecture courses on geometry, which David Hilbert gave between 1891 and 1905. it will be argued that in these lecture courses the author elaborates the conception of geometry which underlies his axiomatic investigations in Foundations of Geometry (1899). On the one hand, it is claimed that this conception of geometry can be characterized by: i)
an abstract or formal axiomatic view; ii) an empiricist stance regarding the origins of geometry and its place among the different mathematical theories. On the other hand, it is asserted that the role that Hilbert assigns to geometrical intuition in the process of the (formal) axiomatization of this theory, can be used to perceive his clear opposition with regard to (radical) formalist views with which he is often identified.