Cosmology

[YesNo]

I finished the book a couple of days ago. I will probably have to read it again after it all settles.

The main problem in the book is whether time is really a succession of infinitesimal instants such as a point on a mathematical line or whether it is something with duration that we access through our subjectivity moving from a past to a future. For practical purposes, like synchronizing clocks, there is use-value in modeling time as a point on a mathematical line. That is not the question. The question is whether time really is a dimension of infinitesimal time-points linked to three dimensions of space-points called "spacetime". Einstein claims that spacetime is real. Bergson claims that the model has use value, but nonetheless it is a mathematical fiction falsified by our own experience of time.

One of the consequences of believing in spacetime is that the universe is then a "block" with four mathematical dimensions in which nothing happens. Both the future and the past are illusions. There is no "arrow of time". Nothing evolves. Why would that be the case? Because the mathematical equations used to model the universe don't change. Einstein promoted these equations from Lorentz' relativity model to reality itself. This allowed time to be reversible in Einstein's view of reality.

Belief in spacetime implies belief that light is the maximal speed and that it is a mathematical (and physical) constant. It is not just that it is convenient to view light as constant, but that light really has a constant speed, not only today but throughout the history of the universe. Indeed it is convenient to make that assumption. Around 1900 people were looking for something that didn't change on which they could base measurements of both length and duration. They couldn't find anything until some decades earlier it was discovered that light had a finite speed and then that it looked like we could not detect a difference in its speed.

Belief in spacetime is just one interpretation of relativity. Relativity itself predates Einstein's deterministic interpretation. Lorentz and Poincare had the mathematical model of special relativity prior to Einstein and neither Lorentz nor Poincare accepted Einstein's interpretation of it. What that means is that we can have relativity without being forced to accept Einstein's block universe determinism. What I wonder is to what extent this also applies to general relativity.

Finally there is the media problem with relativity. Do we (that is you and me as people hearing the scientific discussions) accept a scientific interpretation on rational grounds or because it has been promoted in the media with questionable rhetoric? Is science for us a political event? In the case of Einstein he went out of his way to promote his interpretation in the media and he used ad hominem arguments against those opposing his views suggesting that they were too stupid to understand him or that they were antisemitic. Now Bergson was also Jewish, so the debate wasn't about antisemitism. I doubt that the people who disagreed with Einstein were any more stupid than the people promoting Einstein's own interpretation.

As a conclusion, I think Einstein's deterministic block universe speculation has been falsified both by quantum physics and by the progression of living organisms from birth to death. Time, whatever it is, is not reversible. Hence Einstein's interpretation of Lorentz' original relativity theory is false. Nor is it needed to keep the benefits of relativity theory. Also, the politics involved in that discussion makes me wary of the politics involved in scientific discussions that we hear today.

[Dreamwoven]

This is an interesting philosophical argument. What do you mean by politics in your last sentence? Alternatively, what would not be a political argument that would be acceptable?

[YesNo]

One probably cannot avoid politics in discussions about science. On one level it involves which group of scientists get a bigger share of limited funding. On another it involves which view of reality will dominate our own common sense.

For example, I've been aware of the name Einstein since a child. I didn't even hear of Bohr until a few years ago when I started looking at quantum physics. I didn't know the name Bergson until I read this book. Although I heard the name Lorentz because of the mathematics grounding relativity, he appeared more as a footnote in the theory of relativity. It was his idea. Why is Einstein so front and center in my common sense? He didn't win his debate against Bohr. He shouldn't have won any debate against Bergson. That is probably not a "politics" in the strict sense of who will govern, but it affects who governs my common sense just as a political candidate might and the ad hominem rhetorical techniques to promote one person over the other seem similar.

[YesNo]

I am reading John Derbyshire's "Prime Obsession". He tried to explain the Riemanann Hypothesis in such simple terms that any one could understand it. I'll have to see if he was right. At one point when he was explaining the derivative he made this statement (page 108):

The steepness of the curve varies from point to point. At every point it has a definite numerical value, though, just as your automobile has a definite speed at any point while you are accelerating--namely, the speed you see if you glance at the speedometer.

That made me realize how naively we move from a mathematical model to reality. Does reality really have "points"?

Zeno made the assumption that reality did have points and from there concluded that no motion could occur. I think Zeno was right. If space and time were mathematical lines with points, nothing could happen. I don't know if Zeno thought his conclusion implied that no motion really occurred or whether he was trying to show that reality could not be made out of mathematical points.

It was not just philosophers who questioned mathematically continuous reality. Quantum physics started with rejecting a view that energy was equitably distributed across an infinite number of frequencies in the black body problem. Planck got around the problem by saying that energy was quantized and not continuous. Then things started to work.

Given Zeno and Planck, it is safe to say that reality and a mathematical continuum need not mix except at a level of approximation where one isn't looking too closely.

[Dreamwoven]

It was when the discussion became mathematical that I lost the thread.

[YesNo]

Cosmology today is full of mathematics. Rather than having conscious Gods who act as agents, like ourselves, such as Zeus or Thor or Brahma or Yahweh we have today unconscious mathematical equations with t representing time that has been objectified into a line of points.

Mathematics is a god with a lower "g" because that god is unconscious. But we don't think of these modern cosmological stories as myths and literature because we are caught in their enchantment, or bedevilment, depending on one's perspective.

[desiresjab]

I picked up Prime Obsession a few months ago and have been proceeding by jerks and starts until I am about 3/4 through. It always helps to get more things straight about these topics that are so difficult one may not approach them directly. One learns never to be surprised to find that work by Euler led to major developments by Reimann, or that Poincare anticipated giants not only in relativity but fractal geometry as well, which is a first cousin of chaos theory. Like Reimann, Poincare possessed high gifts in both math and physics. But he got to live longer. He was also a communicator whereas Reimann is often described as painfully shy.

Anyway, I know nothing of Bergson. From reading what you wrote I am unsure if his work was mainly philosophical or whether he muddied his hands with mathematics. I could check Wiki-pejia.

I am wondering how much impact a philosophical interpretation can have. It can alter the administration of things but cannot alter any mathematical truths. I believe philosophical interpretations are very important to the development of theories nonetheless. They can nudge toward particular investigations if the philosophical theory happens to be correct.

Anything that might reduce the time involved for grappling with old arguments is all right with me. As we discussed earlier and now again, issues from 1916 and 1922 are still being seriously debated in physics.

Mathematics, philosophy and physics are thoroughly bound up with each other now, because so many discoveries in math and physics bring heavy baggage to the table and renew the call for philosophy. Popularizers are indespensable for 99.99999% of people. Only the ones able to dig all the way to the roots are technically self reliant. Those are the people like Gauss, Einstein, Reimann, Poincare and Tao et al, and their bright disciples.

The long and short of that is, unless one is extremely bright and undertook this journey from an early age, acquiring all the right tools to investigate relativity or the Reimann conjecture, one's understanding will be a popularized understanding bereft of the technical details required for a truer grip.

Philosophy itself is changing as the need to interpret an explosion of physical theories presents new challenges. Technically versed writers operating one or two levels of abstraction above the "machine language" of folk like the prestigious list above, interpret the "message" for the interested masses, who are yet several more levels of abstraction above. Though not a new face of philosophy, it is now necessarily a prominent one.

It seems my vacation has made me talkative.

Einstein was a great image. The wild hair alone set him apart. His scientific disputes were more gentlemanly than most. But let us not forget, it was Einstein and not the others who made a verifiable prediction on the transit of Mercury. Predictions verifiable through measurement or experiment are of extreme value in scientific accomplishment.

The above merely by way of offering an explanation of why Einstein's interpretation of space-time might have prevailed philosophically over his anticipators and rivals.

All things are politicized, especially in our era. You can't get a right answer to anything, and that is typical.

[YesNo]

I am about half way through Prime Obsession and it is putting the Riemann Hypothesis in a perspective, both mathematical and historical, that makes sense. I'm glad to be reading it.

After reading Canales I don't think spacetime is anything more than a fiction. I don't know how the transit of Mercury fits into the creation of the brand name "Einstein", but I see "Einstein" as a marketing brand for an underlying cultural commodity. I don't think reality can be divided into points or instants. Planck's constant would be one argument against that and then there are Zeno's arguments that such a view would not allow motion to exist.

[desiresjab]

I am about half way through Prime Obsession and it is putting the Riemann Hypothesis in a perspective, both mathematical and historical, that makes sense. I'm glad to be reading it.

After reading Canales I don't think spacetime is anything more than a fiction. I don't know how the transit of Mercury fits into the creation of the brand name "Einstein", but I see "Einstein" as a marketing brand for an underlying cultural commodity. I don't think reality can be divided into points or instants. Planck's constant would be one argument against that and then there are Zeno's arguments that such a view would not allow motion to exist.

Only mathematics is not one of these useful fictions in my view, which is why I keep bringing it up. The Ptolemaic, Newtonian clock, big bang, space-time and multi-verse models are all fictions to me, wonderful fictions that advance our research and our journey. In the future their will be many more cosmic models, some will gain ascendancy for a while. I believe it is an infinite process. The most precious answers will always remain in question form.

Then again, Poincare among others considered Cantor's transfinite set theory to be a fiction. Whether this fiction has ever shown a useful or practical side, I am not sure. Cantor and others do show how transcendental numbers can be constructed. That is as close to a practical application as I know of. Modular arithmetic became the language of digital computer encryption. I think transfinite set theory is still only the language of itself, but I would not be shocked to learn I was wrong, either, because each difficult endeavor requires major effort to penetrate, and I haven't given enough to that argument.

Anyway, once we come to terms that mathematics is not one of these fictions, we are able to make a necessary separation. Mathematicians themselves proceed creatively, but the end result is the discovery of the obvious, the discovery of the only way things could ever have been with regard to the numbers.

The numbers can cause scientific theories to bloom or fade. An equals sign means eqaul cardinality, in the end. Whenever, finally, the numbers do not work out, the theory does not work out either, and goes away. String theory may be in the process of doing this.

Science without numbers cannot be precise. A pinch, a nubbin and a nip are not consistent unless they are standardized, as gram, milligram and microgram are.

Up to a point somtimes quite deep mathematics will support wonderful fictions then suddenly leave off its protection, exposing the fiction as a false route out of the maze. Euler once devised a formula that explicitly produced someting like the first forty-some-odd primes but thereafter could no longer be trusted.

Personally, I feel that if we knew everything about the nature and behavior of prime numbers, such knowledge would somehow provide ultimate answers to just about everything. That is why it is exciting to see the Reimann conjecture at the very front of mathematical research, since ultimately it is a conjecture about prime numbers. Prime numbers have always been a hot topic in mathematics, but it is good to see them clearly at the forefront and so much talent now concentrated on them. This can only lead to great things. Of course these coming answers will have big echoes in science and philsosophy.

[desiresjab]

Since I am having a lot of thoughts right now, I may as well go on with them. With regards to my conjecture involving twin primes, its extension to prime triplets is put in serious jeopardy by the simple observation that two 4n+3 primes that are almost unfathomably far out the number line and only four units apart, will produce something extraordiarily close to a square yet which a diagonal will neverhteless always cut into two unequal quantities of lattice points in the appropriate quadrant of Eisenstein's diagram. This closeness to a square is the main reason I made the conjecture in the first place, so in some sense that puts the whole conjecture at jeopardy. I need a way to attack the problem.

[Dreamwoven]

Only mathematics is not one of these useful fictions in my view, which is why I keep bringing it up. The Ptolemaic, Newtonian clock, big bang, space-time and multi-verse models are all fictions to me, wonderful fictions that advance our research and our journey. In the future their will be many more cosmic models, some will gain ascendancy for a while. I believe it is an infinite process. The most precious answers will always remain in question form.

Then again, Poincare among others considered Cantor's transfinite set theory to be a fiction. Whether this fiction has ever shown a useful or practical side, I am not sure. Cantor and others do show how transcendental numbers can be constructed. That is as close to a practical application as I know of. Modular arithmetic became the language of digital computer encryption. I think transfinite set theory is still only the language of itself, but I would not be shocked to learn I was wrong, either, because each difficult endeavor requires major effort to penetrate, and I haven't given enough to that argument.

Anyway, once we come to terms that mathematics is not one of these fictions, we are able to make a necessary separation. Mathematicians themselves proceed creatively, but the end result is the discovery of the obvious, the discovery of the only way things could ever have been with regard to the numbers.

The numbers can cause scientific theories to bloom or fade. An equals sign means eqaul cardinality, in the end. Whenever, finally, the numbers do not work out, the theory does not work out either, and goes away. String theory may be in the process of doing this.

Science without numbers cannot be precise. A pinch, a nubbin and a nip are not consistent unless they are standardized, as gram, milligram and microgram are.

Up to a point somtimes quite deep mathematics will support wonderful fictions then suddenly leave off its protection, exposing the fiction as a false route out of the maze. Euler once devised a formula that explicitly produced someting like the first forty-some-odd primes but thereafter could no longer be trusted.

Personally, I feel that if we knew everything about the nature and behavior of prime numbers, such knowledge would somehow provide ultimate answers to just about everything. That is why it is exciting to see the Reimann conjecture at the very front of mathematical research, since ultimately it is a conjecture about prime numbers. Prime numbers have always been a hot topic in mathematics, but it is good to see them clearly at the forefront and so much talent now concentrated on them. This can only lead to great things. Of course these coming answers will have big echoes in science and philsosophy.

I basically agree with desiresjab on this.

[YesNo]

Since I am having a lot of thoughts right now, I may as well go on with them. With regards to my conjecture involving twin primes, its extension to prime triplets is put in serious jeopardy by the simple observation that two 4n+3 primes that are almost unfathomably far out the number line and only four units apart, will produce something extraordiarily close to a square yet which a diagonal will neverhteless always cut into two unequal quantities of lattice points in the appropriate quadrant of Eisenstein's diagram. This closeness to a square is the main reason I made the conjecture in the first place, so in some sense that puts the whole conjecture at jeopardy. I need a way to attack the problem.

I think you might be right about the twin primes and the number of lattice points, but all I have to go on are tests for twins below 100. I don't understand what you are saying about prime triplets. They would be numbers of the forms: p, p + 2 and p + 6 or p - 4, p and p + 2 where all of these are prime.

[YesNo]

Only mathematics is not one of these useful fictions in my view, which is why I keep bringing it up. The Ptolemaic, Newtonian clock, big bang, space-time and multi-verse models are all fictions to me, wonderful fictions that advance our research and our journey. In the future their will be many more cosmic models, some will gain ascendancy for a while. I believe it is an infinite process. The most precious answers will always remain in question form.

The problem is that mathematics need not apply to the universe. I view mathematics as a game that might have some practical value, but need not have any practical value. I see mathematics like the game of chess. Although chess has kings, queens, knights, bishops and pawns, we cannot expect those fictions to represent real kings, queens, knights, bishops or peasants any more than we can expect a mathematical structure to represent reality.

Then again, Poincare among others considered Cantor's transfinite set theory to be a fiction. Whether this fiction has ever shown a useful or practical side, I am not sure. Cantor and others do show how transcendental numbers can be constructed. That is as close to a practical application as I know of. Modular arithmetic became the language of digital computer encryption. I think transfinite set theory is still only the language of itself, but I would not be shocked to learn I was wrong, either, because each difficult endeavor requires major effort to penetrate, and I haven't given enough to that argument.

The universe has to be finite for life to exist in it. That would be suggested by Olber's paradox. Now there may be infinitely many universes and I suspect there are other universes than ours given the evidence that ours had a beginning, but outside of that possible infinity of universes, transfinite numbers have no use value.

Science without numbers cannot be precise. A pinch, a nubbin and a nip are not consistent unless they are standardized, as gram, milligram and microgram are.

I am not saying that numbers are not useful. All I am saying is that the universe does not go arbitrarily small which would be required if points actually existed. The physical justification for that is the need for Planck's constant.

Personally, I feel that if we knew everything about the nature and behavior of prime numbers, such knowledge would somehow provide ultimate answers to just about everything. That is why it is exciting to see the Reimann conjecture at the very front of mathematical research, since ultimately it is a conjecture about prime numbers. Prime numbers have always been a hot topic in mathematics, but it is good to see them clearly at the forefront and so much talent now concentrated on them. This can only lead to great things. Of course these coming answers will have big echoes in science and philsosophy.

The problem with mathematics is that we think its ability to perform an analysis step such as splitting composites into smaller primes and then doing a synthesis step of multiplying those primes to get the composite back again is something that might also work in physical reality. It might not. That is, mathematical reductionism, represented by the reduction of composites to primes, may only work well within mathematics.

[desiresjab]

I think you might be right about the twin primes and the number of lattice points, but all I have to go on are tests for twins below 100. I don't understand what you are saying about prime triplets. They would be numbers of the forms: p, p + 2 and p + 6 or p - 4, p and p + 2 where all of these are prime.

What I now realize is that tremendous size and lying close together is not the key at all, is not what makes two primes behave a certain way in QR. The key is just as simple, however. The key is how many factors of 2 are involved in (p-1))(q-1), from the pure Eisensteinian perspective.

With two 4n+3 primes, p-1 and q-1 each have only a single factor of 2 to contribute. That is to say after dividing the total number of interior lattice points by 4, we come to an odd number, which of course cannot be divided evenly, so the two numbers have to have opposite characters when WAXY is divided once more by the diagonal. If only one more factor of 2 is available (which it always will be, as long as both primes are not 4n+3) to deal with this further division performed by the diagonal, then the diagonal will be dividing (apportioning) an even number of points in WAXY. If the power of 2 in the multiplication is only 2³, then WAXY is forced to produce negative exponents for both primes upon the further division by the diagonal. But if the power of 2 is 2⁴ or greater the exponents must always both be positive.

It appears that the nature of the exponents (and thereby reciprocity) depends only on the power of 2 in (p-1)(q-1), nothing else, in terms of Eisenstein's representation. The essence, the causal mechanism is none other than highly evenness. This was my original insight when I first started thinking about Brocard's problem and switched to QR, and before I actually understood what I was talking about. It has taken me this long to understand that my own insight was hitting the nail on the head squarely, to intend a pun.

With twin primes we are always guaranteed at least 2³. What we simply need in order to always be even, i.e. in one another's residue set, is a 4n+1 number wherein n itself carries at least one factor of 2. From knowing nothing more we can always state the character of both primes with respect to each other in QR.

* * * * *

QR does not work when p=q. Imagine an 11X11 square. φ is 110, not 100. This means you cannot even get four squares (not rectangles in this case) all with equal lattice points.

But what about a 17X17 square where there are plenty of 2's to go around? This case will provide four equal squares all right. But it is a dead end, a non-sequitur, because no number between 1 and 16 inclusive will ever square out to 17 (mod 17), and so forth for all primes.

A visibly cogent fact is that the line p=q on graphing paper is a 45 degree angle and is our diagonal, and goes through all the points (1,1), (2,2), (3,3),...(17,17). The method does not work on squares. It only works on rectangles. The diagonal hits eight lattice points in WAXY. 256 interior lattice points divided by four is 64 for our quadrant square, but eight of these cannot count because they hit lattice points, bringing WAXY down to 56 servicable points, and each small triangle to 28, indeed equal, but meaningless except perhaps for why it is meaningless. Only on rectangles where p≠q are there no lattice points on the diagonal. P and q respond identically but meaninglessly when p=q because they do not kick against one another rationing out squares under the other as modulus. At the moment I do not know how to subtract those eight extra points in the context of something meaningful, I just know eight would have to be subtracted in this particular case to somehow fictionally redirect the apparently nonsensical. This is all about finding the logic of why it is illogical for squares themselves.

Only on rectangles where p≠q are there no lattice points on the diagonal.

So where p=q, it would have to look like:

[(p-1)(q-1)]-[(p-1)/2].

This is

p²-2p+1-(p-1)/2, is equivalent to

2p²-4p+2-p-1=2p²-5p+1, which means nothing to me but the sense of the nonsense.

A modulus is about division and remainders, and division is about ratios, and QR is about two unequal primes acting as units for the other under the operation of squaring, spitting out squares as remainders. Pitcher and catcher. Then switch places while the other acts as divider and see which numbers its overlap spits out as squares.

* * * * *

An interesting note:

In Prime Obsession Derbyshire states that 4n+3 primes consistently out number 4n+1 primes. There may be one brief interlude where 4n+1 primes hold the lead, but then it reverts back to a 4n+3 lead, supposedly for good. If they will always hold the lead is probably unproven. I cannot remember, or if he said.

[desiresjab]

Further note:

φ(p)-φ(p-1) seems to be the proper calculation for subtracing eight which I was trying to arrive at above for a 17X17 square.