Spatial Intuition Research Articles

Geometric Generalisation of Surrogate Model-Based Optimisation to Combinatorial and Program Spaces

Surrogate models (SMs) can profitably be employed, often in conjunction with evolutionary algorithms, in optimisation in which it is expensive to test candidate solutions. The spatial intuition behind SMs makes them naturally suited to continuous problems, and the only combinatorial problems that have been previously addressed are those with solutions that can be encoded as integer vectors. We show how radial basis functions can provide a generalised SM for combinatorial problems which have a geometric solution representation, through the conversion of that representation to a different metric space. This approach allows an SM to be cast in a natural way for the problem at hand, without ad hoc adaptation to a specific representation. We test this adaptation process on problems involving binary strings, permutations, and tree-based genetic programs.

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.

Spatial processes and Galois/concept lattices

There is a fair amount of interest in the use of Galois or Concept Lattices in the social sciences as a way of representing complex data structures; it does not seem to be widely appreciated that these structures are homologous to those arising from a discrete noncompensatory multidimensional response process, as originally outlined by Coombs. Here I formalize the spatial intuition underlying Coombs’s proposal, and show how, for any set of data, we may recreate the set of all possible minimal spatial representations.

DIGITAL RECONSTRUCTION OF THE CHURCH OF SAN ILDEFONSO AT ZAMORA (SPAIN) USING ORTHOWARE

Abstract. This paper describes the surveying process performed on the Church of San Ildefonso in Zamora (Spain) using Orthoware©, an innovative photogrammetric software tool specially designed for the digital reconstruction of cultural heritage. This software is the result of intense R&D at Metria Digital after several years producing heritage true orthoimages, plans and full color 3D models in a commercial environment. Orthoware allows true measurements and produces high quality true orthoimages and full color 3D models starting out from conventional digital photographs. This is a progressive tool designed for non-specialist users, providing intuitive and quick methods to visually diagnose the quality of the results. The Church of San Ildefonso in Zamora was built over a previously existing temple whose Romanesque construction dates back to the 11th century, although it is nowadays hidden among extensions and remodelings carried out up until 18th century. With a length of more than 30 meters, the original floor plan of the Church consists on three naves and three apses, although only one nave is visible now, covered by groin vault, and one semicircular apse partially hidden by the current building. The south front maintains the greater part of its Romanesque origins, in spite of being hidden and higher than the present ground level. At the feet of the temple a tower rises, whose first stage is Romanesque but which has been altered by numerous Baroque elements. The objective of the photogrammetric reconstruction was the integral survey of the monument, including its four façades, interior faces and roofs for the production of some true-orthoimages, cross-sections, longitudinal-sections and ground plans at scale 1/50 with an accuracy of 10 mm for the drawings and a pixel size below 10 mm for the synthetic imagery. The usual photogrammetric workflow for producing true-orthoimages and digital 3D models in cultural heritage depends to a great extent on finding and matching homologous features amongst various images. This is a time-consuming, repetitive and blind recognition process requiring some spatial intuition and experience from the user. Orthoware makes use of automatic image analysis and computer vision techniques to improve this processing. Its innovative graphic user interface allows full interactivity and corrections or refinements to be performed at any time providing intuitive diagnostics from partial results. This flow involves very intensive calculations achieved through a software development specific for GPU (Graphic Processing Unit), using OpenGL (Open Graphics Library) and DirectX. Therefore, a combination of manual and automatic techniques makes up the ideal tool for a production environment. The workflow presented herein allow users to reduce orthoimage production time from months to just a few weeks.

Kant on geometry and spatial intuition

I use recent work on Kant and diagrammatic reasoning to develop a reconsideration of central aspects of Kant’s philosophy of geometry and its relation to spatial intuition. In particular, I reconsider in this light the relations between geometrical concepts and their schemata, and the relationship between pure and empirical intuition. I argue that diagrammatic interpretations of Kant’s theory of geometrical intuition can, at best, capture only part of what Kant’s conception involves and that, for example, they cannot explain why Kant takes geometrical constructions in the style of Euclid to provide us with an a priori framework for physical space. I attempt, along the way, to shed new light on the relationship between Kant’s theory of space and the debate between Newton and Leibniz to which he was reacting, and also on the role of geometry and spatial intuition in the transcendental deduction of the categories.

Spatial Intuition in Elementary Arithmetic: A Neurocomputational Account

Elementary arithmetic (e.g., addition, subtraction) in humans has been shown to exhibit spatial properties. Its exact nature has remained elusive, however. To address this issue, we combine two earlier models for parietal cortex: A model we recently proposed on number-space interactions and a modeling framework of parietal cortex that implements radial basis functions for performing spatial transformations. Together, they provide us with a framework in which elementary arithmetic is based on evolutionarily more basic spatial transformations, thus providing the first implemented instance of Dehaene and Cohen's recycling hypothesis.

Developable Surfaces: Their History and Application

Developable surfaces form a very small subset of all possible surfaces and were for centuries studied only in passing, but the discovery of differential calculus in the seventeenth century meant that their properties could be studied in greater depth. Here we show that the generating principles of developable surfaces were also at the core of their study by Monge. In a historical context, from the beginning of the study of developable surfaces, to the contributions Monge made to the field, it can be seen that the nature of developable surfaces is closely related to the spatial intuition and treatment of space as defined by Monge through his descriptive geometry, which played a major role in developing an international language of geometrical communication for architecture and engineering. The use of developable surfaces in the architecture of Frank Gehry is mentioned, in particular in relation to his fascination with ‘movement’ and its role in architectural design.

Abstraction and Intuition in Peano's Axiomatizations of Geometry

Peano's axiomatizations of geometry are abstract and non-intuitive in character, whereas Peano stresses his appeal to concrete spatial intuition in the choice of the axioms. This poses the problem of understanding the interrelationship between abstraction and intuition in his geometrical works. In this article I argue that axiomatization is, for Peano, a methodology to restructure geometry and isolate its organizing principles. The restructuring produces a more abstract presentation of geometry, which does not contradict its intuitive content but only puts it into a particular form.

TWO APPLICATIONS OF ART TO GEOMETRY

Geometry and art exploit the same source of human pleasure: the exercise of our spatial intuition. It is not surprising, then, that interconnections between them abound. Applications of geometry to art, of which we shall indicate a few, go back at least to Alberti’s De Pictura (1435). But although geometry started out, as it so often does, as a most courteous suitor in its relationship with art, it was soon to be affectionately rewarded. We shall study two of these rewards.

Lax proper maps of locales

We give a proof of localic Priestley duality. Our approach is based on lax proper maps of locales, which provide a vehicle for presenting the Priestley version of full Stone duality constructively and preserve spatial intuitions.

Acquiring new spatial intuitions: Learning to reason about rotations

There are certain simple rotations of objects that most people cannot reason about accurately. Reliable gaps in the understanding of a fundamental physical domain raise the question of how learning to reason in that domain might proceed. Using virtual reality techniques, this project investigated the nature of learning to reason across the domain of simple rotations. Learning consisted of the acquisition of spatial intuitions: there was encoding of useful spatiotemporal information in specific problem types and a gradual accumulation of this understanding across the domain. This pattern of learning through the accumulation of intuitions is especially interesting for rotational motion, in which an elegant domain-wide kinematics is available to support insightful learning. Individual ability to reason about rotations correlated highly with mastery motivation, skill in fluid reasoning, and skill in reasoning about spatial transformations. Thus, general cognitive advantages aided the understanding of individual rotations without guaranteeing immediate generalization across the domain.

Kant's "Argument from Geometry"

Kant's 'argument from geometry' is usually interpreted to be a regressive transcendental argument in support of the claim that we have a pure intuition of space. In this paper I defend an alternative interpretation of this argument according to which it is rather a progressive synthetic argument meant to identify and establish the essential role of pure spatial intuition in geometric cognition. In the course of reinterpreting the 'argument from geometry' I reassess the arguments of the Aesthetic and illustrate the origin of Kant's view of the role of pure intuition in geometry.

Space, Number and Structure: A Tale of Two Debates

Abstract Around the turn of the century, Poincare and Hilbert each published an account of geometry that took the discipline to be an implicit definition of its concepts. The terms ‘point’, ‘line’, and ‘plane’ can be applied to any system of objects that satisfies the axioms. Each mathematician found spirited opposition from a different logicist—Russell against Poincare' and Frege against Hilbert— who maintained the dying view that geometry essentially concerns space or spatial intuition. The debates illustrate the emerging idea of mathematics as the science of structure. The article closes with some remarks on structuralist and nonstructuralist themes in Frege's own development of arithmetic.

Friedrich albert langes bewundernswerte logische studien

Friedrich Albert Lange (1828‐1875) author of a famous History of Materialism and Critique of Its Present Significance(1866, English transi.I–III 1877–79, repr.1925 with introduction by Bertrand Russell), was also interested in the epistemological foundations of formal logic.Part I of his intended two‐volume Logische Studienwas published posthumously in 1877 by Hermann Cohen“head”of the Marburg school of neo‐Kantianism.Lange, departing from Kant, claims that spatial intuition is the source of the apodeictic character not only of the truths of mathematics, but also of the truths of logic.He aims at showing this by basing validity and invalidity of syllogistic inferences on an interpretation of the standard forms (of proposition in assertoric syllogistic) with the help of the five kinds of possible relations (in fact what is known today as the Gergonne‐Euler relations) between extensions of concepts given to us as areas in a plane, i.e.in space.Generality is achieved by considering all possible variations wi...

On the Consistency Problem for Set Theory: An Essay on the Cantorian Foundations of Classical Mathematics (I)

On the Consistency Problem for Set Theory: An Essay on the Cantorian Foundations of Classical Mathematics (I)

In Search of the Perfect Scalene Triangle

Geometry seems to afford more surprises than other branches of mathematics, since we have so many more spatial intuitions than numerical ones. We find it easy to deal with figures and drawings, as opposed to the rather more difficult business of juggling numbers mentally. However, many times our intuitions about geometrical figures lead to curious errors. We shall now look at an extremely simple ease and show how a small amount of orderly thinking can give a clear, general result in a situation where our natural feelings seem to go wrong.