Non-Euclidean Astronomy, Probability-Statistics, Etc.

[osher]

I teach and do research in college mathematics/statistics including mathematical physics. I have been especially impressed over the years by the greater acceptance of new ideas by astronomers and astrophysicists compared to physicists, mathematicians, engineers/computer scientists and most other scientists/quantitative specialists.

I think that it has to do with the Non-Euclidean historical idea, and so I'll say a few words about it. The Ancient Greeks were Euclideans, who thought that the world/universe was flat in the rather curious sense that if two line segments were drawn which could be checked as keeping the same distance apart for finite lengths, then the Greeks considered as an axiom that they would remain the same distance apart no matter how far they were extended. Since the Earth is a flattened sphere, this is not even true on the Earth except approximately at tiny distance scales.

The Non-Euclidean geometers dropped this PARALLEL POSTULATE, and this eventually led to general relativity and string/brane/duality physics and so on with a little help from some other ideas.

Mathematics NEVER extended the Non-Euclidean idea to branches other than geometry, and this tendency of math/sciences/engineering practitioners NOT to examine their axioms and assumptions and even definitions and emphases/interests is actually an unscientific bias with very serious consequences.

Non-Euclidean Astronomy, Probability-Statistics, Topology, Algebra, and so on would simply change one or more axioms or postulates or theoretical assumptions/principles or even foci of interest and EXPLORE the results as part of a competing theory. My wife Marleen and I did this with probability-statistics in a 1980-81 UC Berkeley philosophy symposium, and we've continued that ever since.

We made a few mistakes like publishing in more obscure philosophy journals, contacting prominent Euclidean-type mathematicians and scientists and engineers who had not the faintest interest in really different ideas [an astronomical waste of time, so to speak]. I wasted prodigious time on the internet, thinking that somebody would surely be interested. Finally, in 2000, my paper which also related probability-statistics to mathematical physics and fuzzy multivalued logics was published in B.N. Kursunuglu et. al. [Editors] Quantum Gravity, Generalized Theory of Gravitation, and Superstring Theory-Based Unification, Kluwer Academic N.Y. 2000, 89-97. Thinking that this indicated a breakthrough, I delivered another paper the next year at the Quantum Gravity conference, and was later informed that there was not enough interest in logical approaches to physics.

The SCIENTIFIC-MATHEMATICAL METHODS which include mathematics and engineering in their quantitative aspects need to clarify their own implicit axioms and assumptions. BUREAUCRACY cannot be part of their methods or their environments, since bureaucracy selects ideas and theories by their social or cultural ties to the scientist rather than for their merit or even their quality of being alternatives that need to be considered. Popularity as judged by the sheer NUMBER of people in the field who believe in some theory can also not be part of the scientific-mathematical methods based on the fact that Creative Geniuses have so often been in disagreement with their contemporaries and with past trends [Mozart, Beethoven, Socrates, Pierre De Fermat, Sir Isaac Newton, Galileo, Leonardo Da Vinci, Einstein, Kurt Godel, George Cantor, etc.].

Worse even than BUREAUCRACY itself but often part of it is the scientific view that only one master theory can be right and that also other theories will fall by the wayside - I call it Law of the Jungle Competition as opposed to Sports Teams Competition. The former kills of the competitors, the latter keeps them competing for motivation and interest and recognition that today's evidence may not equal the total evidence accumulated tomorrow or at some other future time - and also because there may be more than one picture of the world as for example the Schrodinger and Heisenberg quantum mechanics pictures, the Euclidean and Non-Euclidean pictures, etc.

Bad Science thus has at least two aspects - bad from WITHIN the mainstream and bad from WITHOUT the mainstream by people who do not learn the best of the past before they rebel against the past or who sometimes are even charlatans.

Osher Doctorow

[Zathras]

Mathematics NEVER extended the Non-Euclidean idea to branches other than geometry, and this tendency of math/sciences/engineering practitioners NOT to examine their axioms and assumptions and even definitions and emphases/interests is actually an unscientific bias with very serious consequences.

I'm not sure if I'm missing something here. For example, in algebra, people have gone through all the axioms of the number system to see what happens if they are discarded. What happens if commutivity in a ring is discarded, or if the axiom of the existence of an identity in the ring is discarded, for example. There are a few bedrock axioms that are almost never messed with (Axiom of Choice or Axiom of Non-contradiction), but I don't see how you could do much of anything without these axioms. What am I missing here?

[informant]

Geometry is algebra in math today! This makes no sense to me.

P.S. The Axiom of Choice can -- and has on occasion -- been put aside.

[Zathras]

On 2003-01-10 17:00, osher wrote:
. . .
Worse even than BUREAUCRACY itself but often part of it is the scientific view that only one master theory can be right and that also other theories will fall by the wayside - I call it Law of the Jungle Competition as opposed to Sports Teams Competition. The former kills of the competitors, the latter keeps them competing for motivation and interest and recognition that today's evidence may not equal the total evidence accumulated tomorrow or at some other future time - and also because there may be more than one picture of the world as for example the Schrodinger and Heisenberg quantum mechanics pictures, the Euclidean and Non-Euclidean pictures, etc.
. . .

Why do you see this killing off of theories as a problem? If the two theories give different results, then an expiriment can be conducted to determine the correct one. If they give equivalent results (Heisenberg v. Schrodinger), than there is no such killing off, unless and until some other theory or evidence comes up.

[osher]

I just lost a long reply by pressing the wrong key, so let me make this brief.

1. Informant, call or access any large university department of mathematics and ask whether they have separate geometry and algebra branches. If, on the extreme unlikelihood that you have managed to reach a fluke, try about 25 large universities. Then return to this forum and tell me that algebra = geometry.

2. Zathras, over 2000 years of progress in science were delayed because the Ancient Greeks refused to change the Parallel Postulate. Do you think it was because the Ancient Greeks had no idea of experimental evidence even in a very rough primitive sense, or because they didn't question axioms [question-mark - my question-mark and parentheses keys among others are out]. To figure this out, suppose that the Greeks had allowed an alternative theory that the world is actually a flattened sphere or flattened oblate ball to be more precise. Do you think that this would not have stimulated experiments to determine which theory is correct - in fact, it would have ADVANCED the scientific method.

But let's get closer to home. In probability statistics, the ONLY school allowed in the mainstream that describes the probable influence or dependence of one event/variable/process on another is the [Bayesian] Conditional Probability school which divides y/x where x, y are two probabities and y is less than or equal to x. In 1980, my wife Marleen and I introduced a second probable influence, 1 plus y - x, which replaces division by subtraction [the 1 keeps the result a probability and can be derived without actually selecting it arbitrarily, which I will explain if I have time]. We keep y less than or equal to x. The new probable influence, unlike the conditional one, is defined even when x = 0 and does not blow up near x = 0, which means that it is excellent for describing RARE EVENTS. The old one is not. This is what my wife Marleen and I batted out heads against the mainstream in attempting to explain for 23 years more or less. We explained it even clearer than here. NOT ONE PERSON WAS INTERESTED IN DOING RESEARCH IN THIS.

In pure algebra so-called, consider the operations of addition-subtraction vs multiplication-division. All of these certainly exist in a ring with multiplicative inverse and a field. NOBODY before me asked whether addition-subtraction could refer to different mathematical or physical scenarios from multiplication-division, and everybody simply assumed that the fact that they relate via the distributive law a[b plus c] = ab plus ac etc. is all we have to know about their relationships. The axiom here is much, much more subtle than flat vs bent space so to speak. But it is just as important. In fact, you can even reformulate the above probability-statistics example in terms of this. While you are doing that, I have done considerable work on the Jacobson Radical and its star product x o y = x plus y - xy in this whole issue. Take a look at the Jacobson Radical and the star product on the internet as keywords, and also quasiregular, circle composition product, Radical, etc.

Finally, I chose the Schrodinger-Heisenberg comparison without realizing that somebody could misinterpret that. They give different PICTURES or INTERPRETATIONS in quantum theory even though they are mathematically isomorphic [an isomorphism which was proven much later by mathematicians].

A much better example would be the idea that space is nothing but the relationships between objects or matter. Leibniz was quite fond of this notion. The Ether school then went in the opposite direction and endowed space with substance. Einstein vascillated somewhat on the question and was probably of both minds but definitely disliked the ether idea as it was formally stated. He is credited by T. Y. Cao of Boston University [see his 1997 volume by Cambridge University Press, England] with first discovering the idea of a SUBSTANTIVE FIELD THEORY in physics, namely the gravitational field theory, but only his successors Stephen Hawking and Sir Roger Penrose and a few quantum people of exceptional mathematical talent like Paul Dirac of Cambridge University [later U. Tallahassee after he retired from the Chair of Sir Isaac Newton] realized that the vacuum aspect of space and the black hole's spatiotemporal aspects and the particle/antiparticle duality in the vacuum endowed space[-time] with something of its old 'DISCREDITED' ETHER-LIKE aspect as something real apart from the mass or source of the field. There are numerous other examples of ideas that appeared to be overwhelmingly contra-indicated by experiment which came back eventually in a slightly different form. Killing off those ideas because of experiments which may later be superceded by other experiments is a very, very bad idea. In fact, it would definitely qualify as BAD SCIENCE.

Hopefully, some of this will be comprehensible to the members of this forum. I had considered posting to the Non-Mainstream forum but I noticed that this has twice the membership roughly. However, I reserve the right to switch if I have to keep justifying Non-Mainstream ideas.

Osher Doctorow

[Zathras]

. . .
2. Zathras, over 2000 years of progress in science were delayed because the Ancient Greeks refused to change the Parallel Postulate. Do you think it was because the Ancient Greeks had no idea of experimental evidence even in a very rough primitive sense, or because they didn't question axioms [question-mark - my question-mark and parentheses keys among others are out]. To figure this out, suppose that the Greeks had allowed an alternative theory that the world is actually a flattened sphere or flattened oblate ball to be more precise. Do you think that this would not have stimulated experiments to determine which theory is correct - in fact, it would have ADVANCED the scientific method.
. . .

No, I don't think science was delayed because of a refusal to reconsider the parallel postulate. You don't need non-euclidean geometry to explain newton's mechanics (or lagrangian mechanics for that matter), electromagnetism, or statistical mechanics. There was certainly no expirimental evidence for a need to change the parallel postulate. No matter how much expirimental ability the greeks had, they simply had no reason to question the parallel postulate.

That's how science works: you develop the theory and only change the axioms when the expirimental evidence says so. It hasn't worked the other way around (the way you propose), mainly because we need the phenomonology to guide our choice of axioms. If the axioms fit the phenomonology, there is no need to adjust the axioms. Translation: if it ain't broke, don't fix it.

[osher]

The words 'if it ain't broke, don't fix it' reflect the mentality of concrete-oriented engineers. Let me explain something about Bad Science, since I love to debunk Bad Science but also to praise Non-Mainstream Good Science. One of the worst problems with Ingenious Imitators, who constitute something near 99 percent of scientists and mathematicians as opposed to Creative Geniuses, is in my 64 years mostly spent in mathematics-science a serious imbalance between concreteness and abstractness. Engineers, to whom we owe the Nazi era and the Cold War, specialize in concreteness to the point of absurdity with a very few exceptions who are mostly Non-Mainstream refugees from the Mainstream Engineering current. Astrophysicists tend to balance the abstract and concrete rather well, and similarly for mathematicians specializing in differential equations and analysis [partial, ordinary, functional, nonsmooth, etc.] except for probability-statistics as a branch-outgrowth of analysis [in the latter case, the Materialist influence from statistics-hungry bureaucrats appears to have captured the mainstream]. Geometers tend to side with differential equations and analysis people but there is a large minority who are so abstract that they wouldn't recognize a space if it fell on them. Algebraists tend to go EXTREMELY in the direction of pure abstraction. Number theorists tend to be not only concrete-oriented but FINITE-oriented, and computer scientists take their cue from number theorists - Pierre De Fermat, who founded modern number theory, was an outstanding exception, and there are several others in number theory who have been Creative Geniuses in almost every generation. Computer Scientists tend to be almost indistinguishable from engineers. Most mathematics jobs outside academia, and many inside, are now for computer scientists, although about half of computer science jobs are for engineer computer scientists specifically. We have to thank computer scientists and engineers for inventing the ULTIMATE FIX IT AND IT'S BROKEN REPAIR - the merging-downsizing of corporations with replacement of jobs by computers and the non-follow-up of fired employees.

Every time a computer scientist so-called improves things, roughly speaking many people lose jobs. Their job is to FIX IT AND BREAK IT - break people's jobs so that business will be less expensive without people.

But this is only the tip of the iceberg. The President of the United States, George Bush, believes in sequential invasion strategies - invade one nation at a time or no nation depending on the way the Materialist current is blowing. If he were a mathematician of any competence in non-mainstream game theory, which John Nash of A Beautiful Mind was not really fully into, he would notice the much better SIMULTANEOUS INVASION STRATEGY. Instead of invading Afghanistan and then waiting while Afghan terrorists run away to Pakistan and sneak back when convenient, and then taking a year to decide whether to invade Iraq which is still undecided and looking like a second year [which gives Saddam a chance to move his weapons to adjacent Islamic friendly nations at the first hint of inspection], and waiting forever while Saudi Arabia funds Islamic terrorists and gives free rein to its violent Wahhabi Islam movement, and allowing the anti-Israel terrorists to move freely from one Islamic nation to another, the Simultaneous Strategy says the following.

Since we are the self-proclaimed Greatest Superpower on Earth, and since the Al Queda can attack us and our friends simultaneously, surely we can invade a few Non-Superpower nations like Iraq, Syria, Libya, Saudi Arabia [what are they going to do - throw oil at us], Lebanon, Iran, Western Pakistan, Yemen, Somalia, and let Israel invade so-called Palestine. Then we have no more 9-11s, no more Afghan murders of our troops, no more Copycat Snipers, and we even get to Support Our front line allies' troops as well as our own.

But the don't fix the unbroken pipe theory of Bush and Powell holds that something isn't broken until the water floods you out of your home or apartment. There's no OUNCE OF PREVENTION IS WORTH A TON OF REPAIR - actually, plumbers don't usually believe in that either, and that's where the pipe analogy comes from somewhere in one's ancestral background. These guys make sure to draw the line very conservatively - 3000 dead is barely passing to require Afghanistan, but not for repairing the supporters and sympathizer nations of Afghanistan which are all the nations that I mentioned [Pakistan's few top military leaders allegedly support us, while most of their population support the terrorists and hide them, and the Saudi friendship with Bush is purely Materialistic money- and power- friendship, hopefully not sensation-materialism although there is surely time for that eventually].

I can't spend too much time teaching you plumbing and non-plumbing, so pardon me if I go on to somebody else, perhaps permanently.

Osher Doctorow

[DStahl]

Ho, ho! Osher, I tend to agree with your politics!

But I tend not to accept statements like "One of the worst problems with Ingenious Imitators, who constitute something near 99 percent of scientists and mathematicians as opposed to Creative Geniuses..." as much more than spleen-venting. It's virtually always false to characterize 99% of any group as "imitators" or the like, in my opinion.

There's no indication that only creative geniuses can do good science. Much knowledge is accumulated incrementally, and perhaps that purpose is served as well by careful, well-supported work as by strokes of genius. Again, just my opinion.

I might think that the subject of a science makes some difference in the culture of ideas amongst its practitioners. An astronomer can't experiment directly on a blue giant star like a physicist can with electrons and positrons, nor can he watch a single star from birth to death like a botanist can watch a calypso orchid. Botany was, in the beginning, a science of collection and nomenclature; astronomy was in the beginning the observation of untouchable, mysterious objects. Physics searches for rules that are always true everywhere and one electron is exactly like another. Astronomy tries to find explanations for complex, distant objects.

Just some thoughts.


<font size=-1>[ This Message was edited by: DStahl on 2003-01-11 02:29 ]</font>

[informant]

On 2003-01-10 21:10, osher wrote:

1. Informant, call or access any large university department of mathematics and ask whether they have separate geometry and algebra branches. If, on the extreme unlikelihood that you have managed to reach a fluke, try about 25 large universities. Then return to this forum and tell me that algebra = geometry.

What do you mean, "branches"?
I'm sure that Algebra and Geometry are two separate courses. They ought to be. But that's just a pragmatic way to subdivide math into parts that are digestable by the students. It's an administrative/pedagogical division.
From a stricktly mathematical point of view, though -- which I'm sure is your concern here --, geometry today is essentially linear algebra.

About osher's other claims...

Popularity as judged by the sheer NUMBER of people in the field who believe in some theory can also not be part of the scientific-mathematical methods based on the fact that Creative Geniuses have so often been in disagreement with their contemporaries and with past trends [Mozart, Beethoven, Socrates, Pierre De Fermat, Sir Isaac Newton, Galileo, Leonardo Da Vinci, Einstein, Kurt Godel, George Cantor, etc.].

Ah, but mathematics, physics and engineering aren’t creative arts. Unlike the arts, science can’t rely solely on opinion; it must also submit itself to the burden of proof.

There are numerous other examples of ideas that appeared to be overwhelmingly contra-indicated by experiment which came back eventually in a slightly different form. Killing off those ideas because of experiments which may later be superceded by other experiments is a very, very bad idea. In fact, it would definitely qualify as BAD SCIENCE.

So, we should accept any theory, no matter how often it has been disproved by experiments, because one day a new empirical context may appear in which a similar theory is appropriate…
Now, where did Occam leave his Razor?… [img]/phpBB/images/smiles/icon_smile.gif[/img]

Engineers, to whom we owe the Nazi era and the Cold War, specialize in concreteness to the point of absurdity with a very few exceptions who are mostly Non-Mainstream refugees from the Mainstream Engineering current.

Care to substantiate that accusation with some proof?
Or is proof too “concrete” a thing for your liking?

There is some more nonsense in your posts, but at this time I will settle for asking: how does it all relate to astronomy? You did notice that this is an astronomy forum…



<font size=-1>[ This Message was edited by: informant on 2003-01-11 08:01 ]</font>

[informant]

Actually, this thread gives me an opportunity to ask an astronomy-related question that I’ve wondered about. I’ve heard that indeed Einstein’s theory of Relativity questions the intuitively plausible notion that the geometry of the universe is Euclidean. This seems to have something to do with the “curvature” of space, a concept which I have trouble grasping.
From what I’ve heard, though, recent astronomical findings indicate that the universe has zero curvature – and therefore an Euclidean geometry.
I may be very off track, since there’s a lot here that I don’t really know. But I wonder if any of the posters who are more versed in astrophysics/relativity could explain this.

[cable]

Osher,
welcome on board.
Here we have humor but no bureaucracy [img]/phpBB/images/smiles/icon_smile.gif[/img]

There are numerous other examples of ideas that appeared to be overwhelmingly contra-indicated by experiment which came back eventually in a slightly different form. Killing off those ideas because of experiments which may later be superceded by other experiments is a very, very bad idea. In fact, it would definitely qualify as BAD SCIENCE.

u may come up with a very bright math model, accepted by all mathematicians ...
if u model is about physics/astrophysics oissues, then it MUST be validated by physical experiment.
eg. a very logical math model describing gravity with a particle called "graviton" , won't be accepted until we detect the graviton.
at the other hand, the strange/bizarre (to our mind) math model for QM, is accepted kuz it's supported by experiment.

[cable]

From what I’ve heard, though, recent astronomical findings indicate that the universe has zero curvature – and therefore an Euclidean geometry.

some say it's quasi-euclidian: stars induce LOCAL (limited) curvatures.

Pr. Luminet suggests a strange model , a "crumpled space" :
http://www.unesco.org/courier/2001_05/uk/doss14.htm

[DStahl]

There's a really confusing (to me!) parallel use of the terms "flat" and "curved"--it's common to say that if omega is less than one the universe in spacetime is saddle-shaped and will expand forever; if omega is greater than one the universe is spherical in spacetime and will eventually collapse; and if, as seems to be expected, omega is one or almost exactly one then the universe is "flat."

But at the same time it's widely understood that the universe may well be finite but unbounded, in other words, a four-dimensional sphere or something, and on this picture the universe as a whole is curved back on itself and only appears flat and Euclidean on our puny observable scales. (The inflationary epoch may have expanded the universe so much that our entire horizon of observation, all 12 or so billion lightyears in every direction, is only an infitesimal bubble in its immensity...)

I suspect that the last "flat" is the kind your asking about, but if you're like me then the first kind keeps getting mixed up with it.

Added later: Well, poop. I did some poking around and it appears my message, above, is still confused. Can the spatial dimensions of the universe be finite but unbounded at any given time yet the overall spacetime still be either flat or negatively curved--ie, infinite and unbounded? Surely it must be so: we can speak of our universe at a time of B + t after the big bang as being the "size of a basketball," an emphatically finite size, yet such a universe could expand forever and even accelerate and thus have a negative overall curvature, right?

<font size=-1>[ This Message was edited by: DStahl on 2003-01-11 19:33 ]</font>

[JS Princeton]

DS and Info...

There are two types of curvature 3D and 4D. Most of the time, when we talk about a closed versus open universe we are talking 4D (including time)... usually. However one can also talk about 3D curvature which is entirely different. Suffice to say, the critical density (and therefore omega) uses a time derivative and therefore also depends on time (ie 4 dimensions). However, when we are talking about the universe's 3D shape we are referring to something called curvature (designated by the variable, "k") which gives us another three sets of open, closed, and flat cases (k positive, negative, or zero, respectively) that are INDEPENDENT of whether the universe in 4D is open, closed, or flat.

The two can be seen as related as we have a contribution to the energy density due to curvature (sometimes called Omega_k) since Einstein's Equations relate gravity (an energy, if you will) with curvature.

Hope this clears some things up.

[DStahl]

Yes. Using the spacetime map of the universe drawn by Gowan (and so kindly linked earlier by David Hall--thanks!) each of the concentric circles corresponds to the universe at a given time--ie a given radial distance from the initial singular point. The circular line representing the universe at that time is finite in length but unbounded, and if one imagines that the total circumference of our "present time" circle is some million times larger than the segment of it which lies within our observable universe, then it's obvious that the segment of the circle we can see will look just like a flat line--Euclidean space, if we expand it to 3-D.

But at the same time, we can imagine Gowan's map as being pasted onto a sphere, with the origin at the north pole and each concentric circle becoming a line of latitude. Then, if our present-day line were imagined to be the equator, the universe would have expanded in the past and would begin to contract as one moves into the future toward the south pole.

The evolution of the universe could then be summed up as a progression from polar bears to penguins.

I can also imagine that the initial singular point, instead of throwing off a circle-universe, threw off a hyperbola whose ends shot to infinity at the moment of creation. That would be an analogue of a spatially infinite, unbounded universe, I guess.

Thanks, JSPrinceton.