The latest draft (posted 05/14/22) of this short, concise work of proof, theory, and metatheory provides summary meta-proofs and verification of the work and results presented in the Theory and Metatheory of Atemporal Primacy and Riemann, Metatheory, and Proof. In this version, several new and revised definitions of terms were added to subsection SS.1; and many corrected equations, theorems, metatheorems, proofs, and explanations are included in the main text. The body of the text is approximately 18 pages, with 3 (...) sections; 1, an Introduction (with a 108 page listing of key terms & definitions), sect. 2, the Results, sect. 3, Discussion (commentary & predictions), and sect. 4, Works Cited. As much as possible, the style is intended for readability and understanding by very bright children (with some interest & knowledge of maths, etc.) and very interested nonprofessionals. The results of this project also enable upgrades of number theory, set theory, proof theory, metamathematics, the foundations of science, and quantum mechanics theory (etc.). (shrink)

This paper is trying to show the significance for science history studies of the internet´s access to rare books each time more easier; it was choiced the work of one of the most important and influent mathematicians, Georg Friedrich Bernhard Riemann (1826-1866), “Oeuvres Mathématiques”, composed with published memories during Riemann´s life, and posthumously published memories and fragments. It was been analised and commented the posthumous memory about the hypothesis which serve to geometry basis, fundamental on modern sciences, (...) as Riemann´s space conception, and without it, it couldn´t be possible the exact formulation of Relativity Theory or fundamental advances on mathematics like the solving of some differential equations. It was been trying to show how the reading of an original text of the important mathematician can be rich and full of teaching through the return to his thinking origin. (shrink)

I investigate the role of geometric intuition in Frege’s early mathematical works and the significance of his view of the role of intuition in geometry to properly understanding the aims of his logicist project. I critically evaluate the interpretations of Mark Wilson, Jamie Tappenden, and Michael Dummett. The final analysis that I provide clarifies the relationship of Frege’s restricted logicist project to dominant trends in German mathematical research, in particular to Weierstrassian arithmetization and to the Riemannian conceptual/geometrical tradition at (...) Göttingen. Concurring with Tappenden, I hold that Frege’s logicism should not be understood as a continuing a project of reductionist arithmetization. However, Frege does not quite take up the Riemannian banner either. His logicism supports a hierarchical understanding of the structure of mathematical knowledge, according to which arithmetic is applicable to geometry but not vice versa because the former is more general, as revealed by the strictly logical nature of its objects in comparison to the intuitional nature of geometric objects. I suggest, in particular, that Frege intended that foundational work would show the use of geometric intuition in complex analysis, a source of error for Riemann that Weierstrass was proud to have uncovered, to be inessential. (shrink)

While I was working about some basic physical phenomena, I discovered some geometric relations that also interest mathematics. In this work, I applied the rules I have been proven to P=NP? problem over impossibility of perpendicularity in the universe. It also brings out extremely interesting results out like imaginary numbers which are known as real numbers currently. Also it seems that Euclidean Geometry is impossible. The actual geometry is Riemann Geometry and complex numbers are real.

Einstein field equations was originally derived by Einstein in 1915 in respect with canonical formalism of Riemann geometry,i.e. by using the classical sufficiently smooth metric tensor, smooth Riemann curvature tensor, smooth Ricci tensor,smooth scalar curvature, etc.. However have soon been found singular solutions of the Einstein field equations with degenerate and singular metric tensor and singular Riemann curvature tensor. These degenerate and singular solutions of the Einstein field equations was formally accepted by main part of scientific (...) community beyond rigorous canonical formalism of Riemannian geometry. (shrink)

The nature of time is variously understood and varied expressions of time available are critically discussed. The nature of time formation, its structure and textures are presented taking examples from natural sciences and Indian spirituality. The physics and mathematics used to evolve the concept of time are chronologically presented. The mathematical allusion and physical illusion associated with the concept of theories of relativity are analyzed. The mathematical conjectures responsible for evolution of theories of relativity are pronounced. The missing physical reality (...) in theories of relativity in relation to appearance of contraction of length and dilation of time is given comparing these mathematical illusions associated; with the optical illusion of appearance of a scale as bent in water in a beaker to stress the point of illusion - mental and physical respectively. The physical and mathematical nature of time, its measurement and passage and the complex mathematical nature of physical and psychological times are presented. (shrink)

Gravitation is interpreted to be an “ontomathematical” force or interaction rather than an only physical one. That approach restores Newton’s original design of universal gravitation in the framework of “The Mathematical Principles of Natural Philosophy”, which allows for Einstein’s special and general relativity to be also reinterpreted ontomathematically. The entanglement theory of quantum gravitation is inherently involved also ontomathematically by virtue of the consideration of the qubit Hilbert space after entanglement as the Fourier counterpart of pseudo-Riemannian space. Gravitation can be (...) also interpreted as purely mathematical or logical “force” or “interaction” as a corollary from its ontomathematical (rather than physical) realization. The ontomathematical approach to gravitation is implicit in general relativity equating it to operators in pseudo-Riemannian space obeying the Einstein field equation and also well-known by the “geometrization of physics”. Quantum mechanics shares the same by the separable complex Hilbert space and defining “physical quantity” by the Hermitian operators on it. One can interpret special Minkowski space involved by special relativity and the qubit Hilbert space of quantum information as Fourier counterparts immediately noticing that general relativity means gravitation as the Fourier counterpart of non-Hermitian operators implying non-unitarity and the violation of energy conservation and thus destroying Pauli’s particle paradigm. Since the Standard model obeys it, this explains the impossibility of “quantum gravitation” in any framework conservatively generalizing the Standard model so that it would include gravitation along with electromagnetic, weak, and strong interactions. Einstein’s geometrization of gravitation can be continued into a purely mathematical theory of it following Euclid’s realization for geometry to be exhaustively built in a deductive and axiomatic way as well as Riemann’s parametrization of all the class of Euclidean and non-Euclidean geometries by “space curvature”, then being generalized to Minkowski space as the operators on pseudo-Riemannian space as the Einstein field equation means gravitation. The transition from mathematical gravitation to logical one can rely on the historical lesson of the pair of Lobachevski’s and Riemann’s approaches now “reversely”, i.e., from the latter to the former. Logical gravitation is linkable to Hegel’s dialectical logic and ontological dialectics abandoning their interpretations as a new zero logic substituting classical propionyl logic. The approach of ontomathematics generalizing that of ontology, traceable even to Aristotle’s reformation of Plato’s doctrine, needs Hegel’s doctrine to be formalized as a first-order logic naturally containing Boolean algebra, isomorphic to both classical propositional logic and set theory being the class of all first-order logics, as a sub-logic along with Peano arithmetic as another sub-logic. The first-order logic at issue is called Hilbert arithmetic and elaborated in detail in other papers. It allows for both self-foundation of mathematics to be internally proved as complete and furthermore, quantum mechanics reinterpreted as quantum information to be included by the qubit Hilbert space interpretable in turn as a dual and physical counterpart of Hilbert arithmetic in a narrow sense, that is, both counterparts constitute Hilbert arithmetic in a wide sense, being mathematical and physical simultaneously and thus overcoming the Cartesian dualism of “body” gapped from “mind” by an abyss. Then, the proper philosophical interpretation of gravitation to be the fundamental ontomathematical force or interaction overcomes the ridiculous belief of the Big Bang wrongly alleged to be a scientific theory. Ontomathematical gravitation suggests an omnipresent and omnitemporal medium of “God’s” creation “ex nihilo” following only the natural necessity of quantum-information conservation particularly and locally manifested as energy conservation. (shrink)

"There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact" Mark Twain-Life on the Mississippi -/- This is a lovely book full of fascinating info on the evolution of physics and cosmology. Its main theme is how the idea of higher dimensional geometry created by Riemann, recently extended to 24 dimensions by string theory, has revolutionized our understanding of the universe. Everyone knows that Riemann created multidimensional (...) geometry in 1854 but it is amazing to learn that he also was a physicist who believed that it held the key to explaining the fundamental laws of physics. Maxwell´s equations did not exist then and Riemann´s untimely death at age 39 prevented his pursuit of these ideas. Both he and his British translator Clifford believed that magnetic and electric fields resulted from the bending of space in the 4th dimension-more than 50 years before Einstein! The fourth dimension became a standard subject in the popular media for the next 50 years with several stories by HG Wells using it and even Lenin wrote about it. The American mathematician Hinton had widely publicized his idea that light is a vibration in the 4th spatial dimension. Amazingly, physicists and most mathematicians forgot about it and when Einstein was looking for the math needed to encompass general relativity 60 years later, he had never heard of Riemannian geometry. He spent 3 years trying to find the equations for general relativity and only after a math friend told him about Riemann was he able to complete his work. Riemann´s equations with four dimensional metric tensors describing every point in space were incorporated almost unchanged into relativity. And on and on it goes. Since this review I have written a great deal on the language games of math and science, uncertainty, incompleteness, the limits of computation etc., so those interested should find them useful since this volume like most science frequently wanders across the line into philosophy (scientism). -/- Those wishing a comprehensive up to date framework for human behavior from the modern two systems view may consult my book ‘The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle’ 2nd ed (2019). Those interested in more of my writings may see ‘Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 3rd ed (2019), The Logical Structure of Human Behavior (2019), and Suicidal Utopian Delusions in the 21st Century 4th ed (2019). (shrink)

The work provides comprehensively definitive, unconditional proofs of Riemann's hypothesis, Goldbach's conjecture, the 'twin primes' conjecture, the Collatz conjecture, the Newcomb-Benford theorem, and the Quine-Putnam Indispensability thesis. The proofs validate holonomic metamathematics, meta-ontology, new number theory, new proof theory, new philosophy of logic, and unconditional disproof of the P/NP problem. The proofs, metatheory, and definitions are also confirmed and verified with graphic proof of intrinsic enabling and sustaining principles of reality.

The work provides comprehensively definitive, unconditional proofs of Riemann's hypothesis, Goldbach's conjecture, the 'twin primes' conjecture, the Collatz conjecture, the Newcomb-Benford theorem, and the Quine-Putnam Indispensability thesis. The proofs validate holonomic metamathematics, meta-ontology, new number theory, new proof theory, new philosophy of logic, and unconditional disproof of the P/NP problem. The proofs, metatheory, and definitions are also confirmed and verified with graphic proof of intrinsic enabling and sustaining principles of reality.

Geometry, etymologically the “science of measuring the Earth”, is a mathematical formalization of space. Just as formal concepts of number may be rooted in an evolutionary ancient system for perceiving numerical quantity, the fathers of geometry may have been inspired by their perception of space. Is the spatial content of formal Euclidean geometry universally present in the way humans perceive space, or is Euclidean geometry a mental construction, specific to those who have received appropriate instruction? The (...) spatial content of the formal theories of geometry may depart from spatial perception for two reasons: first, because in geometry, only some of the features of spatial figures are theoretically relevant; and second, because some geometric concepts go beyond any possible perceptual experience. Focusing in turn on these two aspects of geometry, we will present several lines of research on US adults and children from the age of three years, and participants from an Amazonian culture, the Mundurucu. Almost all the aspects of geometry tested proved to be shared between these two cultures. Nevertheless, some aspects involve a process of mental construction where explicit instruction seem to play a role in the US, but that can still take place in the absence of instruction in geometry. (shrink)

A look at the dynamical concept of space and space-generating processes to be found in Kant, J.F. Herbart and the mathematician Bernhard Riemann's philosophical writings.

Does geometry constitues a core set of intuitions present in all humans, regarless of their language or schooling ? We used two non verbal tests to probe the conceptual primitives of geometry in the Munduruku, an isolated Amazonian indigene group. Our results provide evidence for geometrical intuitions in the absence of schooling, experience with graphic symbols or maps, or a rich language of geometrical terms.

The notion of linguistic geometry is defined in this book. It is pertinent to keep in the record that linguistic geometry differs from classical geometry. Many basic or fundamental concepts and notions of classical geometry are not true or extendable in the case of linguistic geometry. Hence, for simple illustration, facts like two distinct points in classical geometry always define a line passing through them; this is generally not true in linguistic geometry. Suppose (...) we have two linguistic points as tall and light we cannot connect them, or technically, there is no line between them. However, let's take, for instance, two linguistic points, tall and very short, associated with the linguistic variable height of a person. We have a directed line joining from the linguistic point very short to the linguistic point tall. In this case, it is important to note that the direction is essential when the linguistic variable is a person's height. The other way line, from tall to very short, has no meaning. So in linguistic geometry, in general, we may not have a linguistic line; granted, we have a line, but we may not have it in both directions; the line may be directed. The linguistic distance is very far. So, the linguistic line directed or otherwise exists if and only if they are comparable. Hence the very concept of extending the line infinitely does not exist. Likewise, we cannot say as in classical geometry; three noncollinear points determine the plane in linguistic geometry. Further, we do not have the notion of the linguistic area of well-defined figures like a triangle, quadrilateral or any polygon as in the case of classical geometry. The best part of linguistic geometry is that we can define the new notion of linguistic social information geometric networks analogous to social information networks. This will be a boon to non-mathematics researchers in socio-sciences in other fields where natural languages can replace mathematics. (shrink)

In the article, an argument is given that Euclidean geometry is a priori in the same way that numbers are a priori, the result of modelling, not the world, but our activities in the world.

in this paper we return to Marshall Clagett’s view about the existence of an ancient Greek geometry of motion. It can be read in two ways. As a basic presentation of ancient Greek geometry of motion, followed by some aspects of its further development in landmark works by Galileo and Newton. Conversely, it can be read as a basic presentation of aspects of Galileo’s and Newton’s mathematics that can be considered as developments of a geometry of motion (...) that was first conceived by ancient Greek mathematicians. (shrink)

Some recently popular accounts of perception account for the phenomenal character of perceptual experience in terms of the qualities of objects. My concern in this paper is with naturalistic versions of such a phenomenal externalist view. Focusing on visual spatial perception, I argue that naturalistic phenomenal externalism conflicts with a number of scientific facts about the geometrical characteristics of visual spatial experience.

We cannot imagine two straight lines intersecting at two points even though they may do so. In this case our abilities to imagine depend upon our abilities to visualise.

This work examines the unique way in which Benedict de Spinoza combines two significant philosophical principles: that real existence requires causal power and that geometrical objects display exceptionally clearly how things have properties in virtue of their essences. Valtteri Viljanen argues that underlying Spinoza's psychology and ethics is a compelling metaphysical theory according to which each and every genuine thing is an entity of power endowed with an internal structure akin to that of geometrical objects. This allows Spinoza to offer (...) a theory of existence and of action - human and non-human alike - as dynamic striving that takes place with the same kind of necessity and intelligibility that pertain to geometry. This fresh and original study will interest a wide range of readers in Spinoza studies and early modern philosophy more generally. (shrink)

Evolution and geometry generate complexity in similar ways. Evolution drives natural selection while geometry may capture the logic of this selection and express it visually, in terms of specific generic properties representing some kind of advantage. Geometry is ideally suited for expressing the logic of evolutionary selection for symmetry, which is found in the shape curves of vein systems and other natural objects such as leaves, cell membranes, or tunnel systems built by ants. The topology and (...) class='Hi'>geometry of symmetry is controlled by numerical parameters, which act in analogy with a biological organism’s DNA. The introductory part of this paper reviews findings from experiments illustrating the critical role of two-dimensional (2D) design parameters, affine geometry and shape symmetry for visual or tactile shape sensation and perception-based decision making in populations of experts and non-experts. It will be shown that 2D fractal symmetry, referred to herein as the “symmetry of things in a thing”, results from principles very similar to those of affine projection. Results from experiments on aesthetic and visual preference judgments in response to 2D fractal trees with varying degrees of asymmetry are presented. In a first experiment (psychophysical scaling procedure), non-expert observers had to rate (on a scale from 0 to 10) the perceived beauty of a random series of 2D fractal trees with varying degrees of fractal symmetry. In a second experiment (two-alternative forced choice procedure), they had to express their preference for one of two shapes from the series. The shape pairs were presented successively in random order. Results show that the smallest possible fractal deviation from “symmetry of things in a thing” significantly reduces the perceived attractiveness of such shapes. The potential of future studies where different levels of complexity of fractal patterns are weighed against different degrees of symmetry is pointed out in the conclusion. (shrink)

Why does ‘ethnic cleansing’ occur? Why does the rise of nationalist feeling in Europe and of Black separatist movements in the United States often go hand in hand with an upsurge of anti-Semitism? Why do some mixings of distinct religious and ethnic groups succeed, where others (for example in Northern Ireland, or in Bosnia) fail so catastrophically? Why do phrases like ‘balkanisation’, ‘dismemberment’, ‘mutilation’, ‘violation of the motherland’ occur so often in warmongering rhetoric? All of these questions are, it will (...) turn out, connected. To understand how they are connected we we will need to examine how human beings acquire a relationship to specific chunks of land, a relationship that is emotionally so strong that they are prepared to die – or kill – to protect that land for themselves or to win it back from others. Territoriality, the biologically rooted predisposition to defend core areas of home ranges against intruders, is a near-universal phenomenon amongst animals of all species. But the ways in which defended territories are conceived and demarcated differ widely from species to species and from group to group. By coming to an understanding of the geometry of these differences we can come to understand also some of the factors which give rise to interethnic conflict. (shrink)

This paper examines Hobbes’s criticisms of Robert Boyle’s air-pump experiments in light of Hobbes’s account in _De Corpore_ and _De Homine_ of the relationship of natural philosophy to geometry. I argue that Hobbes’s criticisms rely upon his understanding of what counts as “true physics.” Instead of seeing Hobbes as defending natural philosophy as “a causal enterprise … [that] as such, secured total and irrevocable assent,” 1 I argue that, in his disagreement with Boyle, Hobbes relied upon his understanding of (...) natural philosophy as a mixed mathematical science. In a mixed mathematical science one can mix facts from experience with causal principles borrowed from geometry. Hobbes’s harsh criticisms of Boyle’s philosophy, especially in the _Dialogus Physicus, sive De natura aeris_, should thus be understood as Hobbes advancing his view of the proper relationship of natural philosophy to geometry in terms of mixing principles from geometry with facts from experience. Understood in this light, Hobbes need not be taken to reject or diminish the importance of experiment/experience; nor should Hobbes’s criticisms in _Dialogus Physicus_ be understood as rejecting experimenting as ignoble and not befitting a philosopher. Instead, Hobbes’s viewpoint is that experiment/experience must be understood within its proper place – it establishes the ‘that’ for a mixed mathematical science explanation. (shrink)

Kant argued that Euclidean geometry is synthesized on the basis of an a priori intuition of space. This proposal inspired much behavioral research probing whether spatial navigation in humans and animals conforms to the predictions of Euclidean geometry. However, Euclidean geometry also includes concepts that transcend the perceptible, such as objects that are infinitely small or infinitely large, or statements of necessity and impossibility. We tested the hypothesis that certain aspects of nonperceptible Euclidian geometry map onto (...) intuitions of space that are present in all humans, even in the absence of formal mathematical education. Our tests probed intuitions of points, lines, and surfaces in participants from an indigene group in the Amazon, the Mundurucu, as well as adults and age-matched children controls from the United States and France and younger US children without education in geometry. The responses of Mundurucu adults and children converged with that of mathematically educated adults and children and revealed an intuitive understanding of essential properties of Euclidean geometry. For instance, on a surface described to them as perfectly planar, the Mundurucu's estimations of the internal angles of triangles added up to ∼180 degrees, and when asked explicitly, they stated that there exists one single parallel line to any given line through a given point. These intuitions were also partially in place in the group of younger US participants. We conclude that, during childhood, humans develop geometrical intuitions that spontaneously accord with the principles of Euclidean geometry, even in the absence of training in mathematics. (shrink)

I explore some ways in which one might base an account of the fundamental metaphysics of geometry on the mathematical theory of Linear Structures recently developed by Tim Maudlin (2010). Having considered some of the challenges facing this approach, Idevelop an alternative approach, according to which the fundamental ontology includes concrete entities structurally isomorphic to functions from space-time points to real numbers.

Using as a starting point recent and apparently incompatible conclusions by Saunders and Knox, I revisit the question of the correct spacetime setting for Newtonian physics. I argue that understood correctly, these two versions of Newtonian physics make the same claims both about the background geometry required to define the theory, and about the inertial structure of the theory. In doing so I illustrate and explore in detail the view—espoused by Knox, and also by Brown —that inertial structure is (...) defined by the dynamics governing subsystems of a larger system. This clarifies some interesting features of Newtonian physics, notably the distinction between using the theory to model subsystems of a larger whole and using it to model complete universes, and the scale-relativity of spacetime structure. _1_ Introduction _2_ Newtonian Mechanics and Galilean Spacetime _3_ Vector Relationism and Maxwellian Spacetime _4_ Recovering the Galilei Group: Dynamics of Subsystems _5_ Knox on Inertial Structure _6_ Connections on Maxwellian Spacetime _7_ Knox on Newtonian Gravity _8_ Vector Relationism and Newton–Cartan Theory _9_ Inertial Structure in Newton–Cartan Gravity _10_ Reconciling Knox and Saunders _11_ Conclusions. (shrink)

We discuss the relationship between logic, geometry and probability theory under the light of a novel approach to quantum probabilities which generalizes the method developed by R. T. Cox to the quantum logical approach to physical theories.

This paper examines explanations that turn on non-local geometrical facts about the space of possible configurations a system can occupy. I argue that it makes sense to contrast such explanations from "geometry of motion" with causal explanations. I also explore how my analysis of these explanations cuts across the distinction between kinematics and dynamics.

In the past decade, three philosophers in particular have recently explored the relation between desert and intrinsic value. Fred Feldman argues that consequentialism need not give much weight – or indeed any weight at all – to the happiness of persons who undeservedly experience pleasure. He defends the claim that the intrinsic value of a state of affairs is determined by the “fit” between the amount of well-being that a person receives and the amount of well-being that the person deserves. (...) Shelly Kagan uses a similar claim to motivate the view that equality is not intrinsically valuable. Thomas Hurka argues that desert is a third-order value, which is a function of the relation between the second-order value of having a virtuous or vicious character and the first-order value of experiencing pleasure or pain. In this paper, we sketch a theory of desert as fittingness and defend a general account of the relation between desert, well-being, and intrinsic value. We then discuss various applications of our “geometry of desert,” including a solution to the problem of the Repugnant Conclusion. (shrink)

In the brain the relations between free neurons and the conditioned ones establish the constraints for the informational neural processes. These constraints reflect the systemenvironment state, i.e. the dynamics of homeocognitive activities. The constraints allow us to define the cost function in the phase space of free neurons so as to trace the trajectories of the possible configurations at minimal cost while respecting the constraints imposed. Since the space of the free states is a manifold or a non orthogonal space, (...) the minimum distance is not a straight line but a geodesic. The minimum condition is expressed by a set of ordinary differential equation ( ODE ) that in general are not linear. In the brain there is not an algorithm or a physical field that regulates the computation, then we must consider an emergent process coming out of the neural collective behavior triggered by synaptic variability. We define the neural computation as the study of the classes of trajectories on a manifold geometry defined under suitable constraints. The cost function supervises pseudo equilibrium thermodynamics effects that manage the computational activities from beginning to end and realizes an optimal control through constraints and geodetics. The task of this work is to establish a connection between the geometry of neural computation and cost functions. To illustrate the essential mathematical aspects we will use as toy model a Network Resistor with Adaptive Memory (Memristors).The information geometry here defined is an analog computation, therefore it does not suffer the limits of the Turing computation and it seems to respond to the demand for a greater biological plausibility. The model of brain optimal control proposed here can be a good foundation for implementing the concept of "intentionality",according to the suggestion of W. Freeman. Indeed, the geodesic in the brain states can produce suitable behavior to realize wanted functions and invariants as neural expressionsof cognitive intentions. (shrink)

A computational methodology called Grossone Infinity Computing introduced with the intention to allow one to work with infinities and infinitesimals numerically has been applied recently to a number of problems in numerical mathematics (optimization, numerical differentiation, numerical algorithms for solving ODEs, etc.). The possibility to use a specially developed computational device called the Infinity Computer (patented in USA and EU) for working with infinite and infinitesimal numbers numerically gives an additional advantage to this approach in comparison with traditional methodologies studying (...) infinities and infinitesimals only symbolically. The grossone methodology uses the Euclid’s Common Notion no. 5 ‘The whole is greater than the part’ and applies it to finite, infinite, and infinitesimal quantities and to finite and infinite sets and processes. It does not contradict Cantor’s and non-standard analysis views on infinity and can be considered as an applied development of their ideas. In this paper we consider infinite series and a particular attention is dedicated to divergent series with alternate signs. The Riemann series theorem states that conditionally convergent series can be rearranged in such a way that they either diverge or converge to an arbitrary real number. It is shown here that Riemann’s result is a consequence of the fact that symbol ∞ used traditionally does not allow us to express quantitatively the number of addends in the series, in other words, it just shows that the number of summands is infinite and does not allows us to count them. The usage of the grossone methodology allows us to see that (as it happens in the case where the number of addends is finite) rearrangements do not change the result for any sum with a fixed infinite number of summands. There are considered some traditional summation techniques such as Ramanujan summation producing results where to divergent series containing infinitely many positive integers negative results are assigned. It is shown that the careful counting of the number of addends in infinite series allows us to avoid this kind of results if grossone-based numerals are used. (shrink)

Explications of the reconstruction of Leibniz’s metaphysics that Deleuze undertakes in 'The Fold: Leibniz and the Baroque' focus predominantly on the role of the infinitesimal calculus developed by Leibniz.1 While not underestimat- ing the importance of the infinitesimal calculus and the law of continuity as reflected in the calculus of infinite series to any understanding of Leibniz’s metaphysics and to Deleuze’s reconstruction of it in The Fold, what I propose to examine in this paper is the role played by other (...) developments in mathematics that Deleuze draws upon, including those made by a number of Leibniz’s near contemporaries – the projective geometry that has its roots in the work of Desargues (1591–1661) and the ‘proto-topology’ that appears in the work of Du ̈rer (1471–1528) – and a number of the subsequent developments in these fields of mathematics. Deleuze brings this elaborate conjunction of material together in order to set up a mathematical idealization of the system that he considers to be implicit in Leibniz’s work. The result is a thoroughly mathematical explication of the structure of Leibniz’s metaphysics. What is provided in this paper is an exposition of the very mathematical underpinnings of this Deleuzian account of the structure of Leibniz’s metaphysics, which, I maintain, subtends the entire text of The Fold. (shrink)

Geometry defines entities that can be physically realized in space, and our knowledge of abstract geometry may therefore stem from our representations of the physical world. Here, we focus on Euclidean geometry, the geometry historically regarded as “natural”. We examine whether humans possess representations describing visual forms in the same way as Euclidean geometry – i.e., in terms of their shape and size. One hundred and twelve participants from the U.S. (age 3–34 years), and 25 (...) participants from the Amazon (age 5–67 years) were asked to locate geometric deviants in panels of 6 forms of variable orientation. Participants of all ages and from both cultures detected deviant forms defined in terms of shape or size, while only U.S. adults drew distinctions between mirror images (i.e. forms differing in “sense”). Moreover, irrelevant variations of sense did not disrupt the detection of a shape or size deviant, while irrelevant variations of shape or size did. At all ages and in both cultures, participants thus retained the same properties as Euclidean geometry in their analysis of visual forms, even in the absence of formal instruction in geometry. These findings show that representations of planar visual forms provide core intuitions on which humans’ knowledge in Euclidean geometry could possibly be grounded. (shrink)

Planar geometry was exploited for the computation of symmetric visual curves in the image plane, with consistent variations in local parameters such as sagitta, chordlength, and the curves’ height-to-width ratio, an indicator of the visual area covered by the curve, also called aspect ratio. Image representations of single curves (no local image context) were presented to human observers to measure their visual sensation of curvature magnitude elicited by a given curve. Nonlinear regression analysis was performed on both the individual (...) and the average data using two types of model: (1) a power function where. (shrink)

Kant was engaged in a lifelong struggle to achieve what he calls in the 1756 Physical Monadology (PM) a “marriage” of metaphysics and geometry (1:475). On one hand, this involved showing that metaphysics and geometry are complementary, despite the seemingly irreconcilable conflicts between these disciplines and between their respective advocates, the Leibnizian-Wolffians and the Newtonians. On the other hand, this involved defining the terms of their union, which meant among other things, articulating their respective roles in grounding Newtonian (...) natural science. In this paper, consider how Kant’s project of marrying metaphysics and geometry evolves from the pre-Critical to the Critical period and how key discussions in the Prolegomena are related to the lifelong marriage project. (shrink)

In this paper I present critical evaluations of Ayer and Putnam's views on the analyticity of geometry. By drawing on the historico-philosophical work of Michael Friedman on the relativized apriori; and Roberto Torretti on the foundations of geometry, I show how we can make sense of the assertion that pure geometry is analytic in Carnap's sense.

An ideal constructor produces geometry from scratch, modelled through the bottom-up assembly of a graph-like lattice within a space that is defined, bootstrap-wise, by that lattice. Construction becomes the problem of assembling a homogeneous lattice in three-dimensional space; that becomes the problem of resolving geometrical frustration in quasicrystalline structure; achieved by reconceiving the lattice as a dynamical system. The resulting construction is presented as the introductory model sufficient to motivate the formal argument that it is a fundamental structure; based (...) on which, it is proposed that where mathematics’ numbers conventionally correspond to dimensionless points on the stateless number line, numbers more fundamentally correspond to an ordering of discrete objects constructed within the stateful number lattice. A second observation is that this fundamental lattice structure is helically configured with fractal character, which, as it relates to the geometry underlying spacetime, has relevance to questions in physics, particularly those involving wave-particle duality. (shrink)

Du Châtelet holds that mathematical representations play an explanatory role in natural science. Moreover, she writes that things proceed in nature as they do in geometry. How should we square these assertions with Du Châtelet’s idealism about mathematical objects, on which they are ‘fictions’ dependent on acts of abstraction? The question is especially pressing because some of her important interlocutors (Wolff, Maupertuis, and Voltaire) denied that mathematics informs us about the properties of material things. After situating Du Châtelet in (...) this debate, this chapter argues, first, that her account of the origins of mathematical objects is less subjectivist than it might seem. Mathematical objects are non-arbitrary, public entities. While mathematical objects are partly mind-dependent, so are material things. Mathematical objects can approximate the material. Second, it is argued that this moderate metaphysical position underlies Du Châtelet’s persistent claims that mathematics is required for certain kinds of general knowledge, including in natural science. The chapter concludes with an illustrative example: an analysis of Du Châtelet’s argument that matter is continuous. A key premise in the argument is that mathematical representations and material nature correspond. (shrink)

The purpose of this work is to address the relation existing between ancient Greek practical geometry and ancient Greek pure geometry. In the first part of the work, we will consider practical and pure geometry and how pure geometry can be seen, in some respects, as arising from an idealization of practical geometry. From an analysis of relevant extant texts, we will make explicit the idealizations at play in pure geometry in relation to practical (...) geometry, some of which are basically explicit in definitions, like that of segments in Euclid’s Elements. Then, we will address how in pure geometry we, so to speak, “refer back” to practical geometry. This occurs in two ways. One, in the propositions of pure geometry. The other, when applying pure geometry. In this case, geometrical objects can represent practical figures like, e.g., a practical segment. (shrink)

The role of conventions in the formulation of Thomas Reid’s theory of the geometry of vision, which he calls the “geometry of visibles”, is the subject of this investigation. In particular, we will examine the work of N. Daniels and R. Angell who have alleged that, respectively, Reid’s “geometry of visibles” and the geometry of the visual field are non-Euclidean. As will be demonstrated, however, the construction of any geometry of vision is subject to a (...) choice of conventions regarding the construction and assignment of its various properties, especially metric properties, and this fact undermines the claim for a unique non-Euclidean status for the geometry of vision. Finally, a suggestion is offered for trying to reconcile Reid’s direct realist theory of perception with his geometry of visibles. (shrink)

John Corcoran and George Boger. Aristotelian logic and Euclidean geometry. Bulletin of Symbolic Logic. 20 (2014) 131. -/- By an Aristotelian logic we mean any system of direct and indirect deductions, chains of reasoning linking conclusions to premises—complete syllogisms, to use Aristotle’s phrase—1) intended to show that their conclusions follow logically from their respective premises and 2) resembling those in Aristotle’s Prior Analytics. Such systems presuppose existence of cases where it is not obvious that the conclusion follows from the (...) premises: there must be something deductions can show. Corcoran calls a proposition that follows from given premises a hidden consequence of those premises if it is not obvious that the proposition follows from those premises. By a Euclidean geometry we mean an extended discourse beginning with basic premises—axioms, postulates, definitions—1) treating a universe of geometrical figures and 2) resembling Euclid’s Elements. There were Euclidean geometries before Euclid (fl. 300 BCE), even before Aristotle (384–322 BCE). Bochenski, Lukasiewicz, Patzig and others never new this or if they did they found it inconvenient to mention. Euclid shows no awareness of Aristotle. It is obvious today—as it should have been obvious in Euclid’s time, if anyone knew both—that Aristotle’s logic was insufficient for Euclid’s geometry: few if any geometrical theorems can be deduced from Euclid’s premises by means of Aristotle’s deductions. Aristotle’s writings don’t say whether his logic is sufficient for Euclidean geometry. But, there is not even one fully-presented example. However, Aristotle’s writings do make clear that he endorsed the goal of a sufficient system. Nevertheless, incredible as this is today, many logicians after Aristotle claimed that Aristotelian logics are sufficient for Euclidean geometries. This paper reviews and analyses such claims by Mill, Boole, De Morgan, Russell, Poincaré, and others. It also examines early contrary statements by Hintikka, Mueller, Smith, and others. Special attention is given to the argumentations pro or con and especially to their logical, epistemic, and ontological presuppositions. What methodology is necessary or sufficient to show that a given logic is adequate or inadequate to serve as the underlying logi of a given science. (shrink)

If there is an area of discourse in which disagreement is virtually absent, it is mathematics. After all, mathematicians justify their claims with deductive proofs: arguments that entail their conclusions. But is mathematics really exceptional in this respect? Looking at the history and practice of mathematics, we soon realize that it is not. First, deductive arguments must start somewhere. How should we choose the starting points (i.e., the axioms)? Second, mathematicians, like the rest of us, are fallible. Their ability to (...) recognize whether a putative proof is correct is not infallible. In most cases, disagreement over the correctness of a putative proof is, however, evanescent. Once an error is spotted and communicated, the disagreement disappears. But this is not always the case. Sometimes it is recalcitrant; that is, it persists over time. In order to zoom in on this type of disagreement and explain its very possibility, we focus on a single case study: a decades-long (1921-1949) controversy between Federigo Enriques and Francesco Severi, two prominent exponents of the Italian school of algebraic geometry. We suggest that the instability of the mathematical community to which they belonged can be explained by the gap between an abstract criterion of rigor and local criteria of acceptability. It is this instability that made the existence of recalcitrant disagreement over putative proofs possible. We do not condemn speculative mathematics but rather its pretense of being rigorous mathematics. In this respect, we show that the overly self-confident Severi and the more intuitive, visionary Enriques had a completely different attitude. (shrink)

Against Thomas Mormann's argument that differential topology does not support Carnap's conventionalism in geometry we show their compatibility. However, Mormann's emphasis on the entanglement that characterizes topology and its associated metrics is not misplaced. It poses questions about limits of empirical inquiry. For Carnap, to pose a question is to give a statement with the task of deciding its truth. Mormann's point forces us to introduce more clarity to what it means to specify the task that decides between competing (...) hypotheses and in what way such a task may be both in practice and/or in principle impossible to carry out. (shrink)

We show how an epistemology informed by cognitive science promises to shed light on an ancient problem in the philosophy of mathematics: the problem of exactness. The problem of exactness arises because geometrical knowledge is thought to concern perfect geometrical forms, whereas the embodiment of such forms in the natural world may be imperfect. There thus arises an apparent mismatch between mathematical concepts and physical reality. We propose that the problem can be solved by emphasizing the ways in which the (...) brain can transform and organize its perceptual intake. It is not necessary for a geometrical form to be perfectly instantiated in order for perception of such a form to be the basis of a geometrical concept. (shrink)

This paper examines Helmholtz's attempt to use empirical psychology to refute certain of Kant's epistemological positions. Particularly, Helmholtz believed that his work in the psychology of visual perception showed Kant's doctrine of the a priori character of spatial intuition to be in error. Some of Helmholtz's arguments are effective, but this effectiveness derives from his arguments to show the possibility of obtaining evidence that the structure of physical space is non-Euclidean, and these arguments do not depend on his theory of (...) vision. Helmholtz's general attempt to provide an empirical account of the "inferences" of perception is regarded as a failure. (shrink)

The mathematical developments of the 19th century seemed to undermine Kant’s philosophy. Non-Euclidean geometries challenged Kant’s view that there is a spatial intuition rich enough to yield the truth of Euclidean geometry. Similarly, advancements in algebra challenged the view that temporal intuition provides a foundation for both it and arithmetic. Mathematics seemed increasingly detached from experience as well as its form; moreover, with advances in symbolic logic, mathematical inference also seemed independent of intuition. This paper considers various philosophical responses (...) to these changes, focusing on the idea of modifying Kant’s conception of intuition in order to accommodate the increasing abstractness of mathematics. It is argued that far from clinging to an outdated paradigm, programs based on new conceptions of intuition should be seen as motivated by important philosophical desiderata, such as the truth, apriority, distinctiveness and autonomy of mathematics. (shrink)

Typewritten notes on groups and geometry.

Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the (...) aforementioned areas. (shrink)

When beliefs are quantified as credences, they are related to each other in terms of closeness and accuracy. The “accuracy first” approach in formal epistemology wants to establish a normative account for credences based entirely on the alethic properties of the credence: how close it is to the truth. To pull off this project, there is a need for a scoring rule. There is widespread agreement about some constraints on this scoring rule, but not whether a unique scoring rule stands (...) above the rest. The Brier score equips credences with a structure similar to metric space and induces a “geometry of reason.” It enjoys great popularity in the current debate. I point out many of its weaknesses in this article. An alternative approach is to reject the geometry of reason and accept information theory in its stead. Information theory comes fully equipped with an axiomatic approach which covers probabilism, standard conditioning, and Jeffrey conditioning. It is not based on an underlying topology of a metric space, but uses a non-commutative divergence instead of a symmetric distance measure. I show that information theory, despite initial promise, also fails to accommodate basic epistemic intuitions; and speculate on its remediation. (shrink)