I shall simply have to plead ignorance on this one, but I was surprised nevertheless that I hadn't seen a competing explanation of the universe quite like this before. "How can it be that I've not heard a peep about something so ostensibly groundbreaking?" I wondered. Well, I haven't yet busied myself with reading any sort of refutation of this theory, and it's even harder yet to find follow up on the massive potential of such a description of the universe as this. As it stands, however, I can't help but predict that it was unable catch a lot of traction with cosmologists, but I'm wondering if anyone out there is/was familiar with this and can provide further information?
As an aside, what do we think about this idea, metaphorical plot holes and all? Clearly it doesn't address some of the protracted and lingering complexities that the BBT does, and yet it explains other core issues that the BBT does not. My interest has been piqued, but as much as I'd love to see big bang cosmology fall to the superfluous wayside - thus silencing men like William Lane Craig momentarily - I don't think I'll get too excited just yet.
There are easy to understand models of both elliptic and hyperbolic geometry. For eliptical geometry the sphere provides a model: the lines are great circles, which are geodesics on the sphere and any two great circles intersect.
The easiest model for hyperbolic geometry is the one given by Poincaré. The space of points is the interior of a fixed circle and the lines are the arcs of circles which meet the fixed circle orthogonally. The other common example is the pseudosphere, a surface of constant negative curvature obtained by rotating the tractrix around its asymptote.
Gauss was the first mathematician to consider non-euclidean geometries, Bolyai was the first to use the term non-euclidean, and Lobachevsky was the first to publish. It was immediately understood that this "solved" the problem of the parallel postulate.
Can the hyperbolic plane be isometrically embedded into R^4?
This would be a surface in 4-dimensional space, where the metric comes from the Euclidean metric in R^4, and the metric on the surface has constant negative curvature.
Apparently it's been proved you can't isometrically embed the hyperbolic plane into R^3, but you can into R^5.
you can't embed the hyperbolic plane in R^3. There is an embedding in R^6, but I have not heard that it can be done in R^5.
It might be just an immersion of the hyperbolic plane into R^5, in general the hyperbolic space H^n can be immersed into R^(4n-3). Apparently this was proved by Henke but I don't have the paper.
I think the point you are making is satisfied by the well known models of hyperbolic geometry—that it can be modeled within euclidean geometry and therefore if euclidean geometry is consistent, the so is hyperbolic geometry.
That does two things: 1) eliminates the possibility that hyperbolic geometry is inconsistent, and 2) shows that the parallel postulate is independent of the others.
Philosophy is the self-correction by consciousness of its own initial excess of subjectivity—Alfred North Whitehead
Hm-mm, Alfred North Whitehead was a philosopher and mathematician.
I don't know his mathematics so I'll leave it alone.
His philosophy is fair game. I agree that we homo sapiens start our lives with subjectivity in excess; I'm not confident that we self-correct and see an excess of self-congratulation in Whitehead's words above.
About philosophers, an unidentified critic once said that wherever they start their search for knowledge, they stop searching when they reach themselves.
If Godel's Theorem can be extended to philosophy, one might conclude that philosophers aren't able to search past themselves and will leave some subjectivity uncorrected.
Whitehead was Bertrand Russell's teacher and co-author with him of Principia Mathematica, the huge two volume logic book that tried to derive mathematics from logic.
Whitehead's philosophy, summed up in his book, Process and Reality, is in opposition to philosophical tradition because its base is what he called actual entities rather than objects. It has not had much influence in philosophy until recently when some have taken up "process philosophy" as a basis for "process physics" and revived Whitehead's thinking. It remains unclear whether this will prove of any value.
Whitehead's comment is parallel to Clive Bell's remark that the first step toward civilization is the correction of instinct by reason.
Dr. Clark, this is the first I've heard of Clive Bell.
About his view that the first step toward civilization is the correction of instinct by reason, I remain skeptical.
Upon retiring from computer manufacturing I became a student again, this time in the fine arts and the social, or soft, sciences.
Fashions come and fashions go in the social sciences and some years ago I heard of a hypothesis that, unlike non-human animals, humans have no instincts, and that the instincts had "evolved" to become the emotions.
As was my custom, I suspended judgment and thought no more of it until I saw your mention of Bell's remark.
I first heard of Rene Descartes' I think, therefore I am in the mid 1950s in an Intro to Philosophy course. As the professor went on with his lecture, I knew that sticking a pin into an arm would persuade me that I am.
Several years in hardball politics clarified my views and to Descartes' words I added, in a poem I won't lay on you now, He didn't feel; he only half-was.
I read that poem to other poets and segued to my next one with "Now that I have destroyed the Western intellectual tradition,...."
An overstatement, yes, but fun nonetheless.
Between anthropocentrism and a tendency to self-congratulate, we humans do need to continue developing our reason. As Whitehead surmised, we do have an initial excess of subjectivity.
Clive Bell was a member of the Bloomsbury group and a writer on art and literature in the 1920's. He married Vanessa Stephen, the sister of Virginia Woolf. The full quote, from his essay Civilization goes like this:
The first step towards civilization is the correcting of instinct by reason; the second, the deliberate rejction of immediate satisfactions with a view to obtaining subtler.
Bell was not using the word instinct in its modern sense as a term of art in the behavioral sciences, meaning unlearned behavior. That usage is only a half century old and Bell was writing earlier.
As the professor went on with his lecture, I knew that sticking a pin into an arm would persuade me that I am.
Perhaps your introductory professor gave you a mistaken impression of what Descartes actually said in the Discourse. Despite the exceptional clarity of his writing many readers misinterpret Descartes.
Mathew, for a few moments after reading your post I saw no similarity between a philosopher stopping a search when he reaches himself and continuing a search after he'd found what he was seeking.
I was about to conclude that the former searcher satisfies egocentric needs and the latter searcher satisfies pragmatic needs, but then realized that you'd said more, that the latter searcher stops when he finds what resembles, or embodies, truth.
What can be more egocentric and more pragmatic than seeing truth where one is?
I here Tyson Degrasse? not sure his exact name.. has a new Cosmos / Sagan . fyi 411
supposed to be mind blowing! yes!