Retrospect the whole history of physics, we could see that lots of branches eventually are formulated using the language of geometry. For example, Newton uses geometry to set the foundation of classical mechanics; Kepler gets the law of celestial motions. In modern physics, Einstein unifies the gravity to the curvature using Riemann geometry. Electromagnetism has been reformulated in geometric language. Also, Yang-Mills theory explore the geometry of particle structures. What makes geometry so special and important?
For Democritus both space and matter were independent entities. Parmenides and Plato viewed matter as the property of space and viewed space alone as automomous and substantial. Aristotle conceive space as a collection of places and conceived time as intrinsically linked to change and movement.
Descartes’s program to reduce physics to geometry and geometry to algebra define space and time in his analytical geometry. Every mechanism could be expressed in this realm of extension and matter is pure extension.
In the tradition of Aristotle and Spinoza, Leibniz found the Cartesian physics unsatisfactory. Leibniz posited that only matter, and no space (vacuum, void) is fundamental.
For Leibniz, space and time are “relative beings” that are parasitic upon the existence of things which we ordinarily regard as their occupants. If there were no material bodies, there would be no space, and if there were no events or processes, there would be no time. What is primitive and irreducible for Leibniz are the spatial relations These spatial relations relate co-existing objects and do not relate regions of space, because such things do not exist. For Leibniz the concept of space as a spatial container is a false concept.
For Leibniz space is an abstraction from distances between physical objects in time, which is made up by the human mind and thus only ideal. A place is where a physical object has been at some time t. Without physical objects and instants of time there are no places and thus no space.
Space and time, for Leibniz, are the relations of order, of coexistence and succession, among phenomena and qualities. For logical reasons of this position, things must be unextended and events must take no time at all to occur; time, then, will be the order of succession of instantaneous events, and space the order of readiness to interact through the proximity of unextended qualities.
When Leibniz realized that extension was not essential to matter, he posited that force alone is; extension, then, is the order of relations among points endowed with force.
Joseph Agassi traced the influence of Leibniz on the history of physics and philosophy and how his program about relative space and time has partially be realized in the theory of general relativity.
http://www.tau.ac.il/~agass/joseph-papers/Leibniz.pdf
I am of opinion that the objects that one can visualize have a certain view. A picture shows definitely many details than an algebraized symbol-system
There can be no physics if there is no matter/energy and there can be no matter if there is no space. Where there is matter there must be space and where there is space there must be geometry.
A matter is only fully characterized when both the "identity" and "position" of its constituents are known...
position is always relative... thus you always need geometry!
Mathematics is simply a language for physicists primiraly and nowadays for others through all the available softwares, which are actually mathematically dependent!
For Democritus both space and matter were independent entities. Parmenides and Plato viewed matter as the property of space and viewed space alone as automomous and substantial. Aristotle conceive space as a collection of places and conceived time as intrinsically linked to change and movement.
Descartes’s program to reduce physics to geometry and geometry to algebra define space and time in his analytical geometry. Every mechanism could be expressed in this realm of extension and matter is pure extension.
In the tradition of Aristotle and Spinoza, Leibniz found the Cartesian physics unsatisfactory. Leibniz posited that only matter, and no space (vacuum, void) is fundamental.
For Leibniz, space and time are “relative beings” that are parasitic upon the existence of things which we ordinarily regard as their occupants. If there were no material bodies, there would be no space, and if there were no events or processes, there would be no time. What is primitive and irreducible for Leibniz are the spatial relations These spatial relations relate co-existing objects and do not relate regions of space, because such things do not exist. For Leibniz the concept of space as a spatial container is a false concept.
For Leibniz space is an abstraction from distances between physical objects in time, which is made up by the human mind and thus only ideal. A place is where a physical object has been at some time t. Without physical objects and instants of time there are no places and thus no space.
Space and time, for Leibniz, are the relations of order, of coexistence and succession, among phenomena and qualities. For logical reasons of this position, things must be unextended and events must take no time at all to occur; time, then, will be the order of succession of instantaneous events, and space the order of readiness to interact through the proximity of unextended qualities.
When Leibniz realized that extension was not essential to matter, he posited that force alone is; extension, then, is the order of relations among points endowed with force.
Joseph Agassi traced the influence of Leibniz on the history of physics and philosophy and how his program about relative space and time has partially be realized in the theory of general relativity.
http://www.tau.ac.il/~agass/joseph-papers/Leibniz.pdf
It depends on your favorite view.
Somebody could tell you that there always exists an isomorphism between any algebraic problem to some geometrical 'object'.
Someone else (N. Chatzidakis, 1872-1942, professor of Mathematics) could argue that:
ἀεὶ ὁ θεὸς ὁ μέγας γεωμετρεῖ * (3,14159)
[Aeì ho theòs ho mégas geōmetreî ]
["Always the great God applies geometry"]
i.e. the geometry is prime and everything else are simple projections on it.
Who knows?
As humans’ understanding of geometries evolve, so does the understanding of the basic physical laws that govern the universe. Physics can be broken into several ”specialized” areas, where the theories are approximately true. In classical mechanics, attributed mostly to Isaac Newton, speeds are much less than the speed of light. Special relativity, developed by Albert Einstein, applies when objects approach the speed of light, but space is flat and objects move at a constant velocity (1). In general relativity, objects are allowed to approach the speed of light and have non-constant velocities; space is curved.
Our understanding of these basic theories has been related to our understanding of the accompanying geometries (2). Newton’s work is based in the absolute flat space of Euclid. Einstein’s special relativity is set upon a geometry developed by Hermann Minkowski. The general theory of relativity is based upon curved space developed by Carl Gauss and Georgi Riemann. It is the progression of understanding of geometry that permitted an accompanying change in physics. The very properties of the geometries of Euclid, Minkowski, and Gauss and Riemann dictate the properties of physical laws. The marriage of geometry and physics is essential for the progression of humans’ understanding of physical laws..
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(1)-An object moving at a constant velocity moves at a constant speed in a straight line.
(2)-It is important to note that although in many cases the geometry preceded the physics chronologically, that was not always the case. Einstein developed his special theory before Minkowski had a geometry that helped explain it. However, it was Minkowski’s geometry that would help spur Einstein to his general theory. This paper (3) presents events in a pedagogical, rather than chronological, order:
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(3)-The Evolution of Geometry and its Marriage to Physics, Phillip M. Bitzer, University of New Orleans, December 10, 2003.
http://www.oocities.org/hannanphysics/DownloadFiles/PhysicsGeometry.pdf
Human mind takes pleasure in thinking that the world can be described in terms f what we can perceive.
Its not the geometry, I think, which is important. The most important and fundamental thing is "symmetries in nature" which are playing the role in Physics and are manifested in the form of Physical Laws. Geometry is just another way of expressing the symmetries. Since they are basically describing the same thing, they can be interlinked and can be used to simplify the arguments. That's all, I suppose.
Geometries are based on groups and groups are the tool for the expression of symmetries.
ONE of the mysteries of the universe is why it should speak the language of mathematics. Numbers and the relationships between them are, after all, just abstract reasoning. Yet mathematics has shown itself to be particularly adept at describing both the contents of the universe and the forces that act on them. Now comes a paper which argues that one branch of the subject—geometry—could form the basis of all the laws of physics.
Physicists are an overbearing bunch. They have long sought a “theory of everything”. Such an opus would unite the fundamental forces—gravity, electromagnetism and the two forces that become apparent only at the atomic scale—with the matter on which they act, in a single, overarching framework. It would describe the universe as it existed at the moment of its creation in the Big Bang.
The nearest thing they have to this—the Standard Model of particle physics—is messy in places and partial, because it omits gravity. Three decades of effort have been expended on string theory, which includes gravity but at the expense of having the universe inelegantly sprout hidden dimensions. Other potential avenues, such as loop quantum gravity, are also proving untidy. That a theory of everything might emerge from geometry would be neat, but it is a long shot.
Hi Mohammad, what is the paper arguing geometry forms all physical laws that you refer to?
Mohammad,
Why we can understand some aspects of nature with the mathematical language is the real question. I do not believe that the answer to this question is because nature has been written by the creator in this particular language (Galileo's idea), or because nature is fundamentally numbers (Pythagoras's idea). Or that nature is an ideal mathematical hierarchies of forms (Plato's idea). No, the answer is that we humans are the first animal on this planet which has invented abstract languages to communicate. In the process of improving communication we had to find ways to make communication unambiguous/objective. For that language naturally evolved logic, evolved numbers and counting procedures and measurement procedures and basic euclidean (flat 3D space) geometrical procedures. Trades and social organisation, building projects on a large scales requires these procedures to be precise. The logical rules withing language were extracted from language and made explicit and mathematical logic was invented. This logical procedures were then applied for expressing the geometrical procedures in terms of axiomatic theorems. The mathematical language was gradually invented as a reduction of natural language to a core self-consistent subset as a language of relations push by the necessity of removing all that is ambiguous. Geometry, the measurement of the earth, became later the language of physics because it is in itself a physics of the measurement of the earth. After completing a detailed geodetic survey of the kingdom of Hanover, Carl Friedrich Gauss (1777-1855) wrote in 1827 a paper called General Investigations of Curved Surfaces. In this paper, he used calculus to show that some of the basic notions of surface theory, including the notion of curvature, were intrinsic to the surface and did not depend on how the surface was situated in three-dimensional space. Differential geometry follows the basic intuitions of Gauss and Riemann of building up a complete geometrical system on the basis of concepts and axioms that only affect the immediate neihborhood of each point. From this concept, the notion of field arises. The points of interest in this field are not necessarily the origin but the ciritical points or singularities.
The discussions reveal only scientific development caused by understanding of geometry but not explaining the possible reason / reasons for that as the question demands for (i.e. What makes geometry so special and important?). I have one possible answer that is the Geometry helps scientists to set the initial / boundary conditions. And the boundary conditions play key role in developing theories..
Dear Professor Shu-Ch'ang Chang
I think the followings papers are:
http://www.pma.caltech.edu/Courses/ph136/yr2008/0401.2.K.pdf
http://www.math.oregonstate.edu/bridge/papers/dot+cross.pdf
I went to the string math conference in Stonybrook this week , for five days> In a word the geometry of the situation arises from the Stress energy tensor in the Einstein equations. Everything, including lie groups, and more comes from the complexity of space-time in early universe stress energy tensor dynamics.
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