Geometric For Science (Revised Edition): An O...
Download File >>>>> https://bltlly.com/2tDzFF
The theme of symmetry in geometry is nearly as old as the science of geometry itself.[76] Symmetric shapes such as the circle, regular polygons and platonic solids held deep significance for many ancient philosophers[77] and were investigated in detail before the time of Euclid.[40] Symmetric patterns occur in nature and were artistically rendered in a multitude of forms, including the graphics of Leonardo da Vinci, M. C. Escher, and others.[78] In the second half of the 19th century, the relationship between symmetry and geometry came under intense scrutiny. Felix Klein's Erlangen program proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation group, determines what geometry is.[79] Symmetry in classical Euclidean geometry is represented by congruences and rigid motions, whereas in projective geometry an analogous role is played by collineations, geometric transformations that take straight lines into straight lines.[80] However it was in the new geometries of Bolyai and Lobachevsky, Riemann, Clifford and Klein, and Sophus Lie that Klein's idea to 'define a geometry via its symmetry group' found its inspiration.[81] Both discrete and continuous symmetries play prominent roles in geometry, the former in topology and geometric group theory,[82][83] the latter in Lie theory and Riemannian geometry.[84][85]
Leibniz approaches the challenge by distinguishing abstract and concrete things as subjects of our ideas. While only God can have a priori knowledge of the complete notions of concrete things or individuals, we can at least have a priori knowledge of abstracta as, for example, geometrical figures because they are finite in their properties. Also, what is true for one kind of abstracta, as for example a triangle, is true of all members of that kind, for example, for all triangles. In contrast, because concrete things or individuals have infinitely many properties and are the only member of their kind, we as finite beings cannot reach their complete concepts and have to rely on empirical knowledge too when it comes to individuals (Loemker 331-8; A II, 2, N. 14). This distinction, closely related to the distinction of necessary and contingent truths, allowed Leibniz to distinguish human and divine knowledge by a qualitative criterion. Moreover, it also provided a criterion to distinguish contingent from necessary knowledge, thereby paving the path for human and divine freedom. This solution gave Leibniz sufficient confidence to present at least the headings of his Discourse on Metaphysics to the Jansenist theologian and Cartesian Arnauld in 1686, with the long sec. 13 being especially provocative in respect to free will. Clearly, at this time, Leibniz had worked out his new metaphysics (based, however, on the problematic new Geometrical Method), which would make modern science compatible with Christian dogmatics and especially allow for free will by a softened determinism.
All these thinkers extended the Geometrical Method beyond mathematics, claiming its value for the investigation of realia, of real things instead of mere geometrical figures. Such extension of the Geometrical Method to real things was done with the goal to produce certainty of knowledge, a certainty guaranteed by the necessity of geometrical demonstrations. But if it would indeed lead to necessary demonstrations about nature, politics, and ethics, it would introduce necessitarianism into natural, social, and moral sciences, and space would not be left for miracles and, even worse, for free will. This can be seen in the cases of Hobbes and Spinoza, who both were strict determinists. In contrast, it was precisely the recognition of this threat of determinism or necessitarianism implied in the Geometrical Method that led Henry More very early to his criticism of Descartes and since the 1660s to his massive rejection of Cartesianism (More 1711, 58). Besides the theological concern about human haughtiness, it was the threat of necessitarianism that was the true source of the lasting protest against the Geometrical Method throughout the 17th and 18th centuries.
Like core knowledge of number, core geometrical knowledge seems to be a universal capability of the human mind. Geometric and spatial thinking are important in and of themselves, because they connect mathematics with the physical world, and play an important role in modeling phenomena whose origins are not necessarily physical (i.e. networks or graphs). They are also important because they support the development of number and arithmetic concepts and skills. Thus, geometry is essential for all grade levels for many reasons: its mathematical content, its roles in physical sciences, engineering, and many other subjects, and its strong aesthetic connections.
Some members of the geometric algebra represent geometric objects in Rn. Other members represent geometric operations on the geometric objects. Geometric algebra and its extension to geometric calculus unify, simplify, and generalize vast areas of mathematics that involve geometric ideas,including linear algebra, multivariable calculus, real analysis, complex analysis, and euclidean, noneuclidean, and projective geometry. They provide a unified mathematical language for physics (classical and quantum mechanics, electrodynamics, relativity), the geometrical aspects of computer science (e.g., graphics, robotics, computer vision), and engineering.
Csaba D. Toth is a professor of mathematics at Cal State Northridge, located in the city of Los Angeles, and a visiting professor of computer science at Tufts University in the Boston metro area. He is the author of more than 90 papers in discrete and computational geometry. His main research interests are in hierarchical subdivisions in low-dimensional spaces, topological graph theory, and geometric optimization. 781b155fdc