Jan 19, 2018 06:53 AM
(This post was last modified: Jan 19, 2018 07:34 AM by C C.)
https://aeon.co/essays/how-many-dimensio...to-reality
EXCERPT: . . . With three axes, we can describe forms in three-dimensional space. And again, every point is uniquely identified by three coordinates: it’s the necessary condition of three-ness that makes the space three-dimensional.
But why stop there? What if I add a fourth dimension? Let’s call it ‘p’. Now I can write an equation for something I claim is a sphere sitting in four-dimensional space: x2 + y2 + z2 + p2 = 1. I can’t draw this object for you, yet mathematically the addition of another dimension is a legitimate move. ‘Legitimate’ meaning there’s nothing logically inconsistent about doing so – there’s no reason I can’t.
A ‘dimension’ becomes a purely symbolic concept not necessarily linked to the material world at all
And I can keep on going, adding more dimensions. So I define a sphere in five-dimensional space with five coordinate axes (x, y, z, p, q) giving us the equation: x2 + y2 + z2+ p2 + q2 = 1. And one in six-dimensions: x2 + y2 + z2 + p2 + q2 + r2 = 1, and so on.
Although I might not be able to visualise higher-dimensional spheres, I can describe them symbolically, and one way of understanding the history of mathematics is as an unfolding realisation about what seemingly sensible things we can transcend. This is what Charles Dodgson, aka Lewis Carroll, was getting at when, in Through the Looking Glass, and What Alice Found There (1871), he had the White Queen assert her ability to believe ‘six impossible things before breakfast’.
Mathematically, I can describe a sphere in any number of dimensions I choose. All I have to do is keep adding new coordinate axes, what mathematicians call ‘degrees of freedom’. Conventionally, they are named x1, x2, x3, x4, x5, x6 et cetera. Just as any point on a Cartesian plane can be described by two (x, y) coordinates, so any point in a 17-dimensional space can be described by set of 17 coordinates (x1, x2, x3, x4, x5, x6 … x15, x16, x17). Surfaces like the spheres above, in such multidimensional spaces, are generically known as manifolds.
From the perspective of mathematics, a ‘dimension’ is nothing more than another coordinate axis (another degree of freedom), which ultimately becomes a purely symbolic concept not necessarily linked at all to the material world. In the 1860s, the pioneering logician Augustus De Morgan, whose work influenced Lewis Carroll, summed up the increasingly abstract view of this field by noting that mathematics is purely ‘the science of symbols’, and as such doesn’t have to relate to anything other than itself. Mathematics, in a sense, is logic let loose in the field of the imagination.
Unlike mathematicians, who are at liberty to play in the field of ideas, physics is bound to nature, and at least in principle, is allied with material things. Yet all this raises a liberating possibility, for if mathematics allows for more than three dimensions, and we think mathematics is useful for describing the world, how do we know that physical space is limited to three?
[...] The project of understanding the geometrical structure of space is one of the signature achievements of science, but it might be that physicists have reached the end of this road. For it turns out that, in a sense, Aristotle was right – there are indeed logical problems with the notion of extended space. For all the extraordinary successes of relativity, we know that its description of space cannot be the final one because at the quantum level it breaks down. For the past half-century, physicists have been trying without success to unite their understanding of space at the cosmological scale with what they observe at the quantum scale, and increasingly it seems that such a synthesis could require radical new physics.
[...] A view is emerging among some theoretical physicists that space might in fact be an emergent phenomenon created by something more fundamental, in much the same way that temperature emerges as a macroscopic property resulting from the motion of molecules. As Dijkgraaf put it: ‘The present point of view thinks of space-time not as a starting point, but as an end point, as a natural structure that emerges out of the complexity of quantum information.’
A leading proponent of new ways of thinking about space is the cosmologist Sean Carroll at Caltech, who recently said that classical space isn’t ‘a fundamental part of reality’s architecture’, and argued that we are wrong to assign such special status to its four or 10 or 11 dimensions. Where Dijkgraaf makes an analogy with temperature, Carroll invites us to consider ‘wetness’, an emergent phenomenon of lots of water molecules coming together. No individual water molecule is wet, only when you get a bunch of them together does wetness come into being as a quality. So, he says, space emerges from more basic things at the quantum level...
MORE: https://aeon.co/essays/how-many-dimensio...to-reality
EXCERPT: . . . With three axes, we can describe forms in three-dimensional space. And again, every point is uniquely identified by three coordinates: it’s the necessary condition of three-ness that makes the space three-dimensional.
But why stop there? What if I add a fourth dimension? Let’s call it ‘p’. Now I can write an equation for something I claim is a sphere sitting in four-dimensional space: x2 + y2 + z2 + p2 = 1. I can’t draw this object for you, yet mathematically the addition of another dimension is a legitimate move. ‘Legitimate’ meaning there’s nothing logically inconsistent about doing so – there’s no reason I can’t.
A ‘dimension’ becomes a purely symbolic concept not necessarily linked to the material world at all
And I can keep on going, adding more dimensions. So I define a sphere in five-dimensional space with five coordinate axes (x, y, z, p, q) giving us the equation: x2 + y2 + z2+ p2 + q2 = 1. And one in six-dimensions: x2 + y2 + z2 + p2 + q2 + r2 = 1, and so on.
Although I might not be able to visualise higher-dimensional spheres, I can describe them symbolically, and one way of understanding the history of mathematics is as an unfolding realisation about what seemingly sensible things we can transcend. This is what Charles Dodgson, aka Lewis Carroll, was getting at when, in Through the Looking Glass, and What Alice Found There (1871), he had the White Queen assert her ability to believe ‘six impossible things before breakfast’.
Mathematically, I can describe a sphere in any number of dimensions I choose. All I have to do is keep adding new coordinate axes, what mathematicians call ‘degrees of freedom’. Conventionally, they are named x1, x2, x3, x4, x5, x6 et cetera. Just as any point on a Cartesian plane can be described by two (x, y) coordinates, so any point in a 17-dimensional space can be described by set of 17 coordinates (x1, x2, x3, x4, x5, x6 … x15, x16, x17). Surfaces like the spheres above, in such multidimensional spaces, are generically known as manifolds.
From the perspective of mathematics, a ‘dimension’ is nothing more than another coordinate axis (another degree of freedom), which ultimately becomes a purely symbolic concept not necessarily linked at all to the material world. In the 1860s, the pioneering logician Augustus De Morgan, whose work influenced Lewis Carroll, summed up the increasingly abstract view of this field by noting that mathematics is purely ‘the science of symbols’, and as such doesn’t have to relate to anything other than itself. Mathematics, in a sense, is logic let loose in the field of the imagination.
Unlike mathematicians, who are at liberty to play in the field of ideas, physics is bound to nature, and at least in principle, is allied with material things. Yet all this raises a liberating possibility, for if mathematics allows for more than three dimensions, and we think mathematics is useful for describing the world, how do we know that physical space is limited to three?
[...] The project of understanding the geometrical structure of space is one of the signature achievements of science, but it might be that physicists have reached the end of this road. For it turns out that, in a sense, Aristotle was right – there are indeed logical problems with the notion of extended space. For all the extraordinary successes of relativity, we know that its description of space cannot be the final one because at the quantum level it breaks down. For the past half-century, physicists have been trying without success to unite their understanding of space at the cosmological scale with what they observe at the quantum scale, and increasingly it seems that such a synthesis could require radical new physics.
[...] A view is emerging among some theoretical physicists that space might in fact be an emergent phenomenon created by something more fundamental, in much the same way that temperature emerges as a macroscopic property resulting from the motion of molecules. As Dijkgraaf put it: ‘The present point of view thinks of space-time not as a starting point, but as an end point, as a natural structure that emerges out of the complexity of quantum information.’
A leading proponent of new ways of thinking about space is the cosmologist Sean Carroll at Caltech, who recently said that classical space isn’t ‘a fundamental part of reality’s architecture’, and argued that we are wrong to assign such special status to its four or 10 or 11 dimensions. Where Dijkgraaf makes an analogy with temperature, Carroll invites us to consider ‘wetness’, an emergent phenomenon of lots of water molecules coming together. No individual water molecule is wet, only when you get a bunch of them together does wetness come into being as a quality. So, he says, space emerges from more basic things at the quantum level...
MORE: https://aeon.co/essays/how-many-dimensio...to-reality
