Geometrical Axioms Research Articles

Perceptual axioms are irreconcilable with Euclidean geometry.

There are different definitions of axioms, but the one that seems to have general approval is that axioms are statements whose truths are universally accepted but cannot be proven; they are the foundation from which further propositional truths are derived. Previous attempts, led by David Hilbert, to show that all of mathematics can be built into an axiomatic system that is complete and consistent failed when Kurt Gödel proved that there will always be statements which are known to be true but can never be proven within the same axiomatic system. But Gödel and his followers took no account of brain mechanisms that generate and mediate logic. In this largely theoretical paper, but backed by previous experiments and our new ones reported below, we show that in the case of so-called 'optical illusions', there exists a significant and irreconcilable difference between their visual perception and their description according to Euclidean geometry; when participants are asked to adjust, from an initial randomised state, the perceptual geometric axioms to conform to the Euclidean description, the two never match, although the degree of mismatch varies between individuals. These results provide evidence that perceptual axioms, or statements known to be perceptually true, cannot be described mathematically. Thus, the logic of the visual perceptual system is irreconcilable with the cognitive (mathematical) system and cannot be updated even when knowledge of the difference between the two is available. Hence, no one brain reality is more 'objective' than any other.

(114) Mathematical Model for Peyronie's Disease

Abstract Introduction Peyronie's disease may affect up to 20% of men worldwide. After the curvature, penile length loss is the most common complaint. The severity of the curve angle is associated to with penile length and volume loss. Objective Create a theoretical mathematical formula to help clinicians and patients to estimate the penile length and volume loss in patients with Peyronie's Disease. Methods Applying Euclidean Trigonometric principles an geometric axioms a mathematical formula was created. Inserting measurements that are commonly retrieved from patients with Peyronie's disease, i.e., Penile girth, length, and curvature angle in the formula, a 2-d dimensional triangle is created. Applying the torus equation the arc length and angular deviation is found and the the penis volume and lenght loss can be calculated. Results A mathematical formula to evaluate the penile volume and length loss was created accounting for the deviations in the penile geometry occuring in patients with monoplanar and multiplanar Peyronie’s disease deviations. Conclusions This formula provides a mathematically accurate method to predict potential loss of penile length and volume in patients with monoplane or multiplanar Peyronie's disease. The model can be an additional tool for advising patients between shortening or lengthening procedures and help physicians and patients set expectations for results after surgical procedures. Disclosure No.

Solitonic effect on relativistic string cloud spacetime attached with strange quark matter

<abstract><p>In this research paper, we discussed some geometric axioms of a relativistic string cloud spacetime attached with strange quark matter. We determined the conformal $ \eta $-Ricci soliton on a relativistic string cloud spacetime attached with strange quark matter with a $ \varphi(\mathcal{R}ic) $-vector field. In addition, we illustrated some physical significance of conformal pressure $ P $ in terms of conformal $ \eta $-Ricci soliton with the same vector field. Besides this, we deduced a generalized Liouville equation from the conformal $ \eta $-Ricci soliton. Furthermore, we examine the harmonic relevance of conformal $ \eta $-Ricci soliton on string cloud spacetime attached with strange quark matter with a potential function $ \psi $. Finally, we turned up a necessary and sufficient condition for the 1-form $ \eta $, which is the $ {g} $-dual of the vector field $ \gamma $ on a string cloud spacetime attached with strange quark matter to be a solution for the Schrödinger-Ricci equation.</p></abstract>

η-Ricci–Yamabe Solitons along Riemannian Submersions

In this paper, we investigate the geometrical axioms of Riemannian submersions in the context of the η-Ricci–Yamabe soliton (η-RY soliton) with a potential field. We give the categorization of each fiber of Riemannian submersion as an η-RY soliton, an η-Ricci soliton, and an η-Yamabe soliton. Additionally, we consider the many circumstances under which a target manifold of Riemannian submersion is an η-RY soliton, an η-Ricci soliton, an η-Yamabe soliton, or a quasi-Yamabe soliton. We deduce a Poisson equation on a Riemannian submersion in a specific scenario if the potential vector field ω of the soliton is of gradient type =:grad(γ) and provide some examples of an η-RY soliton, which illustrates our finding. Finally, we explore a number theoretic approach to Riemannian submersion with totally geodesic fibers.

The Frege–Hilbert controversy in context

This paper aims to show that Frege’s and Hilbert’s mutual disagreement results from different notions of Anschauung and their relation to axioms. In the first section of the paper, evidence is provided to support that Frege and Hilbert were influenced by the same developments of 19th-century geometry, in particular the work of Gauss, Plücker, and von Staudt. The second section of the paper shows that Frege and Hilbert take different approaches to deal with the problems that the developments in 19th-century geometry posed for the traditional Kantian philosophy of mathematics. Frege maintains that Anschauung is a source of knowledge by which we acknowledge geometrical axioms as true. For Hilbert, in contrast, axioms provide one of several correct “pictures” of reality. Hilbert’s position is thereby deeply influenced by epistemological ideas from Hertz and Helmholtz, and, in turn, influenced the neo-Kantian Cassirer.

Formalising Geometric Axioms for Minkowski Spacetime and Without-Loss-of-Generality Theorems

This contribution reports on the continued formalisation of an axiomatic system for Minkowski spacetime (as used in the study of Special Relativity) which is closer in spirit to Hilbert's axiomatic approach to Euclidean geometry than to the vector space approach employed by Minkowski. We present a brief overview of the axioms as well as of a formalisation of theorems relating to linear order. Proofs and excerpts of Isabelle/Isar scripts are discussed, with a focus on the use of symmetry and reasoning "without loss of generality".

Space Emerges from What We Know-Spatial Categorisations Induced by Information Constraints.

Seeking goals carried out by agents with a level of competency requires an “understanding” of the structure of their world. While abstract formal descriptions of a world structure in terms of geometric axioms can be formulated in principle, it is not likely that this is the representation that is actually employed by biological organisms or that should be used by biologically plausible models. Instead, we operate by the assumption that biological organisms are constrained in their information processing capacities, which in the past has led to a number of insightful hypotheses and models for biologically plausible behaviour generation. Here we use this approach to study various types of spatial categorizations that emerge through such informational constraints imposed on embodied agents. We will see that geometrically-rich spatial representations emerge when agents employ a trade-off between the minimisation of the Shannon information used to describe locations within the environment and the reduction of the location error generated by the resulting approximate spatial description. In addition, agents do not always need to construct these representations from the ground up, but they can obtain them by refining less precise spatial descriptions constructed previously. Importantly, we find that these can be optimal at both steps of refinement, as guaranteed by the successive refinement principle from information theory. Finally, clusters induced by these spatial representations via the information bottleneck method are able to reflect the environment’s topology without relying on an explicit geometric description of the environment’s structure. Our findings suggest that the fundamental geometric notions possessed by natural agents do not need to be part of their a priori knowledge but could emerge as a byproduct of the pressure to process information parsimoniously.

Group Geometrical Axioms for Magic States of Quantum Computing

Let H be a nontrivial subgroup of index d of a free group G and N be the normal closure of H in G. The coset organization in a subgroup H of G provides a group P of permutation gates whose common eigenstates are either stabilizer states of the Pauli group or magic states for universal quantum computing. A subset of magic states consists of states associated to minimal informationally complete measurements, called MIC states. It is shown that, in most cases, the existence of a MIC state entails the two conditions (i) N = G and (ii) no geometry (a triple of cosets cannot produce equal pairwise stabilizer subgroups) or that these conditions are both not satisfied. Our claim is verified by defining the low dimensional MIC states from subgroups of the fundamental group G = π 1 ( M ) of some manifolds encountered in our recent papers, e.g., the 3-manifolds attached to the trefoil knot and the figure-eight knot, and the 4-manifolds defined by 0-surgery of them. Exceptions to the aforementioned rule are classified in terms of geometric contextuality (which occurs when cosets on a line of the geometry do not all mutually commute).

Interpolation in Extensions of First-Order Logic

We prove a generalization of Maehara’s lemma to show that the extensions of classical and intuitionistic first-order logic with a special type of geometric axioms, called singular geometric axioms, have Craig’s interpolation property. As a corollary, we obtain a direct proof of interpolation for (classical and intuitionistic) first-order logic with identity, as well as interpolation for several mathematical theories, including the theory of equivalence relations, (strict) partial and linear orders, and various intuitionistic order theories such as apartness and positive partial and linear orders.

The Geometry of Otto Selz’s Natural Space

Following ideas elaborated by Hering (Grundzuge der Lehre vom Lichtsinn, Springer, Berlin, 1920) in his celebrated analysis of color, the psychologist and gestalt theorist Otto Selz developed in the 1930s a theory of “natural space”, i.e., space as it is conceived by us. Selz’s thesis is that the geometric laws of natural space describe how the points of this space are related to each other by directions which are ordered in the same way as the points on a sphere. At the end of one of his articles, Selz (Zeitschrift fur Psychologie 114:351–362, 1930a, p. 358ff) tries to derive within his framework two of Hilbert’s axioms for Euclidean geometry. Such derivations (if successful) would, according to Selz, disclose the psychological origin and meaning of the geometric axioms and would thus contribute to a clarification of their epistemological status. In the present article Selz’s theory is explained and analyzed, his basic principles are amended, and implicit assumptions are made explicit. It is shown that the resulting system is one of ordered affine geometry in which all of Hilbert’s axioms except the axioms of congruence and those of continuity are derivable. The primary aim of the present paper is to make explicit the basic principles behind Selz’s geometry of natural space. The question of the logical independence of these principle is not investigated.

„Cała matematyka to właściwie geometria”. Poglądy Gottloba Fregego na podstawy matematyki po upadku logicyzmu

Gottlob Frege abandoned his logicist program after Bertrand Russell had discovered that some assumptions of Frege’s system lead to contradiction (so called Russell’s paradox). Nevertheless, he proposed a new attempt for the foundations of mathematics in two last years of his life. According to this new program, the whole of mathematics is based on the geometrical source of knowledge. By the geometrical source of cognition Frege meant intuition which is the source of an infinite number of objects in arithmetic. In this article, I describe this final attempt of Frege to provide the foundations of mathematics. Furthermore, I compare Frege’s views of intuition from The Foundations of Arithmetic (and his later views) with the Kantian conception of pure intuition as the source of geometrical axioms. In the conclusion of the essay, I examine some implications for the debate between Hans Sluga and Michael Dummett concerning the realistic and idealistic interpretations of Frege’s philosophy.

Die Dichotomie analytisch-synthetisch bei Frege unter Berücksichtigung von Kant

Abstract I begin by commenting on Kant’s conception of analytic judgements. I then turn to Frege’s notion of analyticity. I argue that his definition of analytic truth in terms of provability from logical axioms and definitions is incomplete. The requisite analyticity of the logical axioms and the definitions, and accordingly the required justification of acknowledging them as true, must be explained in a non-deductive way. I further argue that analyticity in terms of deductive proof deviates significantly from Kant’s conception of analytic judgements. I conclude with two case studies. The first concerns Frege’s attempted justification of the synthetic nature of the geometrical axioms. The second deals with Hume’s Principle, which in his logicist project Frege must establish as an analytic truth.

SYMMETRY OF CHRACTERS IN INGA ĀBELE’S NOVEL “KLŪGU MŪKS”

The prototype of the central character in Inga Ābele’s (1972) award winning novel “Klūgu mūks” (Wicker Monk, 2014) is Francis Trasuns (1864–1926), a well-known Latgalian patriot, catholic priest, politician, cultural and social figure and writer. The novel is set at the end of 19th century and early 20th century, during the time of the National Awakening. F. Trasuns was one of the most significant personalities in the history of the first Latgalian awakening in St Petersburg. He contributed greatly to the creation and promotion of Latgalian self-confidence, spiritual and cultural development and political growth. He was the first Latgalian ever to be elected to the national parliament – State Duma of the Russian Empire. The private life of F. Trasuns was dramatic, difficult and marked by conflicts. On September 20, 1925, he was officially excommunicated – banished from the Catholic Church. He was accused of arrogance, defiance of his religious authorities, wearing civilian clothes and concubinage. The myth about F. Trasuns enjoying undivided social support in the last years of his life emerged soon after his death, when he became a symbol of struggle against ignorance and blind devotion to clerical dogmas. The educational and literary heritage of F. Trasuns (sketches, literary portraits, poems, a play, translations and feuilletons) should be understood as commentaries to his views, convictions and activities. The cultural philosophy of Richard Rorty and Euclid’s five geometrical axioms were used in the analysis of the novel to prove the symmetry of relationship between men and women characters. The symmetry of characters is revealed through the metaphor of wicker weaving (for example, a basket) that expresses many meanings including a parallel, cross, circle etc. Wicker weaving in the novel symbolises the product of a natural material and human work – interaction of nature and culture – the main condition for the existence of a small nation.

Algebro-geometric axioms for $\operatorname {DCF}_{0,m}$

We give an algebro-geometric first-order axiomatization of $\operatorname {DCF}_{0,m}$, the theory of differentially closed fields of characteristic zero with $m$ commuting derivations, in the spirit of the classical geometric axioms of $\operatorname {DC

Poincaré on the Foundations of Arithmetic and Geometry. Part 1: Against “Dependence-Hierarchy” Interpretations

The main goal of part 1 is to challenge the widely held view that Poincare orders the sciences in a hierarchy of dependence, such that all others presuppose arithmetic. Commentators have suggested that the intuition that grounds the use of induction in arithmetic also underlies the conception of a continuum, that the consistency of geometrical axioms must be proved through arithmetical induction, and that arithmetical induction licenses the supposition that certain operations form a group. I criticize each of these readings. More fully, I argue that the justification Poincare offers for the use of the group notion in geometry appears to extend to set-theoretic notions that would suffice to put arithmetic on a logical foundation, thus undermining his own case for the necessity of intuition in arithmetic. In part 2, I offer an interpretation of intuition’s role on which it justifies the use of group-theoretic, but not set-theoretic, notions.

CONSTRUCTIVE GEOMETRY AND THE PARALLEL POSTULATE

AbstractEuclidean geometry, as presented by Euclid, consists of straightedge-and-compass constructions and rigorous reasoning about the results of those constructions. We show that Euclidean geometry can be developed using only intuitionistic logic. This involves finding “uniform” constructions where normally a case distinction is used. For example, in finding a perpendicular to line L through point p, one usually uses two different constructions, “erecting” a perpendicular when p is on L, and “dropping” a perpendicular when p is not on L, but in constructive geometry, it must be done without a case distinction. Classically, the models of Euclidean (straightedge-and-compass) geometry are planes over Euclidean fields. We prove a similar theorem for constructive Euclidean geometry, by showing how to define addition and multiplication without a case distinction about the sign of the arguments. With intuitionistic logic, there are two possible definitions of Euclidean fields, which turn out to correspond to different versions of the parallel postulate.We consider three versions of Euclid’s parallel postulate. The two most important are Euclid’s own formulation in his Postulate 5, which says that under certain conditions two lines meet, and Playfair’s axiom (dating from 1795), which says there cannot be two distinct parallels to line L through the same point p. These differ in that Euclid 5 makes an existence assertion, while Playfair’s axiom does not. The third variant, which we call the strong parallel postulate, isolates the existence assertion from the geometry: it amounts to Playfair’s axiom plus the principle that two distinct lines that are not parallel do intersect. The first main result of this paper is that Euclid 5 suffices to define coordinates, addition, multiplication, and square roots geometrically.We completely settle the questions about implications between the three versions of the parallel postulate. The strong parallel postulate easily implies Euclid 5, and Euclid 5 also implies the strong parallel postulate, as a corollary of coordinatization and definability of arithmetic. We show that Playfair does not imply Euclid 5, and we also give some other independence results. Our independence proofs are given without discussing the exact choice of the other axioms of geometry; all we need is that one can interpret the geometric axioms in Euclidean field theory. The independence proofs use Kripke models of Euclidean field theories based on carefully constructed rings of real-valued functions. “Field elements” in these models are real-valued functions.

Hermann Cohen and Alois Riehl on Geometrical Empiricism

When non-Euclidean geometry was developed in the nineteenth century, both scientists and philosophers addressed the question as to whether the Kantian theory of space ought to be refurbished or even rejected. The possibility of considering a variety of hypotheses regarding physical space appeared to contradict Kant’s supposition of Euclid’s geometry as a priori knowledge and suggested the view that the geometry of space is a matter for empirical investigation. In this article, I discuss two different attempts to defend the Kantian theory of space against geometrical empiricism. Both Cohen and Riehl defended the apriority of geometrical axioms. Cohen pointed out that knowledge necessarily requires both sensibility and understanding. Therefore, he maintained that space can be proved to be a condition of experience without admitting that some statements about it are intrinsically necessary. By contrast, Riehl emphasized universality and necessity as intrinsic properties of a priori knowledge. He was one of t...

What Does It Mean That “Space Can Be Transcendental Without the Axioms Being So”?

In 1870, Hermann von Helmholtz criticized the Kantian conception of geometrical axioms as a priori synthetic judgments grounded in spatial intuition. However, during his dispute with Albrecht Krause (Kant und Helmholtz uber den Ursprung und die Bedeutung der Raumanschauung und der geometrischen Axiome. Lahr, Schauenburg, 1878), Helmholtz maintained that space can be transcendental without the axioms being so. In this paper, I will analyze Helmholtz’s claim in connection with his theory of measurement. Helmholtz uses a Kantian argument that can be summarized as follows: mathematical structures that can be defined independently of the objects we experience are necessary for judgments about magnitudes to be generally valid. I suggest that space is conceived by Helmholtz as one such structure. I will analyze his argument in its most detailed version, which is found in Helmholtz (Zahlen und Messen, erkenntnistheoretisch betrachtet 1887. In: Schriften zur Erkenntnistheorie. Springer, Berlin, 1921, 70–97). In support of my view, I will consider alternative formulations of the same argument by Ernst Cassirer and Otto Holder.

A Geometric Approach for the Sustainable Economic Growth of a Country: The Equilateral Triangular Model

Strategic perspectives of sustainable development are considered and discussed. The concept of environmental sustainability is addressed and its relation to sustainable development is discussed. By undertaking mathematical calculations and geometrical axioms, this study shows a different approach in regional development based on various settings between three fundamental concepts: Education + Research- Innovation + Entrepreneurship. The Triangle of Knowledge (ToK) is defined using the above three fundamental concepts. The Triangle of Knowledge refers to the interaction between education, research-innovation and entrepreneurship, which are key drivers for the Knowledge-Based-Society (KBS). This concept relates to the need to pay attention on an equal basis to the three factors on a systemic and continuous way. Particular contribution to the Triangle of Knowledge is attempted, through the introduction of Excellence within the proposed Equilateral Triangular Model Approach (ETMA). A new performance index is defined, the Excellence Performance Indicator (EPI) and used on this new proposed methodology of ETMA. A number of simulation runs using ETMA and the Triangle of Knowledge(ToK) is performed and a number of interesting results are obtained and discussed.

Geometric axioms for differentially closed fields with several commuting derivations

A geometric first-order axiomatization of differentially closed fields of characteristic zero with several commuting derivations, in the spirit of Pierce and Pillay (1998) [13], is formulated in terms of a relative notion of prolongation for Kolchin-closed sets.