Quote:General relativity can be a difficult subject to master, but its mathematical
and conceptual structure involves a fairly straight-forward extension of structures
that characterize 19th century physics. The fundamental physical laws
(Einstein's equations for general relativity) are expressed as partial differential
equations, a familiar if difficult mathematical subject. The state of the system
is determined by the set of fields satisfying these equations, and observable
quantities are functionals of these fields. The mathematics is just that of the
usual calculus: differential equations and their real-valued solutions.
In quantum mechanics, the state of a system is best thought of as a different
sort of mathematical object: a vector in a complex vector space, the so-called
state space. One can sometimes interpret this vector as a function, the wavefunction,
although this comes with the non-classical feature that wave-functions are complex-valued.
What's truly completely different is the treatment of observable
quantities, which correspond to self-adjoint linear operators on the state
space. This has no parallel in classical physics, and violates our intuitions about
how physics should work.
Quote:PBS: ""As for what it [quantum mechanics] means, I leave that to the philosophers. But personally, I think they have no idea what they are
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Quote:..and even physicists who do have that training can run off the rails when speculating in natural languages on what mathematics reveal. The "strong" anthropogenic argument that our observation of a photon not only causes it to collapse into a wave packet but also causes its emission in the first place is an example, IMHO, of physicists running off the rails.
"Scientists aren't perfect, just peer reviewed.""Eye of Sauron Observatory", featuring "Sauron's Other Eye", 16" dob, conical Royce mirror.
Quote:Few mathematicians study works from past centuries; compared with contemporary mathematics, such older writings seem to them almost like the work of children."
Quote:I'll get right on that. Can math not be explained using concepts that philosophers use?
Quote:Newtonian physics, which was of immense interest to philosophers, does not challenge our fundamental human perception/conception/construction of reality. So in that sense one might say that philosophers of the pre-Quantum, pre-Relativity era felt pretty confident about the whole thing.
But while QM and Relativity have also been of immense interest to many current philosophers of science, both QM and Relativity are so alien to our fundamental human perception/conception/construction of reality that their philosophical explorations rest on a far less firm ground. The main problem is that neither QM nor Relativity make much sense in the context of natural language conceptuality (which is the language of a philosopher); they only make sense in the language of mathematics. Most philosophers (if not all) do not really have the mathematical training to fully grasp QM and Relativity (I know that I don't), and even physicists who do have that training can run off the rails when speculating in natural languages on what mathematics reveal. The "strong" anthropogenic argument that our observation of a photon not only causes it to collapse into a wave packet but also causes its emission in the first place is an example, IMHO, of physicists running off the rails.
Quote: Things have certain identities and behave in certain definable ways (edit: No matter whether we're looking or not, no matter how it makes us feel), this should be true at all times and at all scales.
Quote:In physics, you are constrained by what is measureably true.
Quote:Same thing happens at large scales - we can use simple newtonian dynamics to predict orbits around the earth pretty well. They break down a little bit at the scale of the sun (the precession of Mercury does not match newtonian predictions), so we have to go up to relativity. At larger scales, we have to include the curvature and expansion of space. This is probably one of the core differences between philosophy and physics - in philosophy you can say what should be true. In physics, you are constrained by what is measureably true.
Quote:What is space?
Quote: Why is it correct to think of space as existing only within the universe?
Quote:In the geometry of spacetime, the shortest distance between 2 points is defined
by the path of light (or any electromagnetic wave). This is called the null,
or lightlike geodesic.
Consider 2 points, A & B, in a static spacetime. Light traveling from point A
to point B defines the geodesic connecting the points. In the time the light
travels from A to B, it undergoes X cycles of the light wave (if I recall the
frequency of visible light is in the hundreds of terahertz).
Now consider points A & B in an *expanding* spacetime manifold. Initially,
the separation is the same as above. As light travels from A to B, the manifold
expands. When the light reaches point B, it will have undergone exactly the
same number of wave cycles as in the above example above. How? As the manifold
expands the wavelength increases, so the frequency decreases. When light
arrives at point B, it has a longer wavelength and so appears to be redshifted.
From the point of reference of the light wave, it has undergone exactly the same
same number of wave cycles in both cases, no "new" space has been traversed.
As is often the case in physics, what can easily be expressed in mathematics
is difficult to explain using just words.
My alternate answer to where the expanding universe is going is that it
is going to some place out in the sun, right down Highway 61.