Cosmology

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Originally Posted by desiresjab

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.

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It makes sense that the factors of 2 would be important here.

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Originally Posted by desiresjab

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.

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The idea of two unequal primes acting as units makes sense.

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Originally Posted by desiresjab

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.

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I think I remember reading something like that in Derbyshire's text. I think ultimately the ratio of the number of primes in the two sets are supposed to converge to 1 implying they have the same number, but initially the 4n + 3 set has more. I couldn't find the page to reference it.

It just so happens it would be possible to devise extremely far from square rectangles in which either p-1 or q-1 was loaded with factors of 2. We assume p and q are both odd primes, as usual, with one of them a 4n+1 prime and the other a 4n+3, to keep matters clear, and to put the burden of factors of 2 entirely on the minus one of the 4n+1 prime.

It takes two dvisions by 2 to divide the rectangle into quadrants, and one more division by 2 to diagonally divide the bottom left and top right quadrants as in Eisenstein's diagram.

The p-1 or q-1 of all 4n+3 primes have only one factor of 2 to give. As long as the 4n+1 prime provides two factors of 2, which it must at least do by its nature, all three divisons can take place preserving the possibility of triangles WAY and YAX containing the same number of lattice points.

We do not really expect it, though, in the case of a very eccentric rectangle ABCD. Quite the opposite. We expect there not to be the same number of lattice points in WAY and YAX.

My conjecture about twin primes seems to involve both concepts--low eccentricity and factors of 2. So far, I am too dumb to prove it. I cannot even say if it is provable, or if it has already been proven.

Circumstantial support for the conjecture also lies in the fact that in all cases where p has more than two factors of 2, the diagonal will be dividing columns of even numbers. If WAY and YAX differ in the number of their lattice points they must differ by at least two, which is harder geometrically to do for a low eccentricity rectangle than to differ by one, it seems to me. I don't see how it could happen. I say it cannot.

I can close in on it logically but so far cannot find a mathematical method to prove it.

Even numbers of course can be partitioned into two evens or two odds in a variety of ways. 12 is 11+1 and 10+2, also 9+3 and 8+4, etc.

To acheive an eccentric additive partition in an Eisenstein count of lattice points would seem to requires high eccentricity on the part of the rectangle ABCD, i.e., less squareness. Right?

The eccentricity is why for the eccentric Eisenstein rectangle 5X13 (two 4n+1 primes), the count for WAXY is 12 lattice points yet a division into YAX and WAY through a diagonal division yields 7+5, allowing the QR to be negative, since 5 and 7 are both odd, all that is required.

The above result begs the question of whether there are two 4n+1 primes vastly far out the number line which are only four units apart, giving their rectangle profoundly low eccentricity, yet which somehow acheive a difference of two in their count of lattice points for YAX and WAY? Or are they all forced due to extremely low eccentricity to always be even, even, and furthermore identical in value?

This is another perspective to the conjecture.

A key thing to consider is that if you do choose two 4n+1 primes far out the number line separated by only four units, one of them is forced to have only two factors of 2. Evennes follows a pattern in the even numbers. Here is a list read left to right for the even numbers 2, 4, 6, 8...., showing how many factors of two each even number contains.

1, 2, 1, 3, 1, 2, 1, 4 1, 2, 1, 3, 1, 2, 1, 5 1, 2, 1, 3, 1, 2, 1, 4

1, 2, 1, 3, 1, 2, 1, 6 1, 2, 1, 3, 1, 2, 1, 4 1, 2, 1, 3, 1, 2, 1, 5 ...


My guess is that if the minus ones of two "twin" 4n+1 primes far out the number line can indeed acheive an eccentric partition (anything other than a split down the middle) it must be because as we can see from the list, that one of them has a mere two factors of 2. The other one can be slightly richer in 2's, or vastly richer. Now I am wondering if the size of that difference plays any part, if, of course, an eccentric partition under any condition can occur at all.

I don't think anyone else is going to do it, so it must be up to me. Surely there are a few sets of twin 4n+1's far enough out for low eccentricity yet within range of calculational investigation using some available tools.

The list above is called the ruler function. Believe it or not, that discontinuous thing has even been re-tooled for calculus.

To get off the math, Does anyone believe there is such a thing as the Nature of Reality, or is that another comfortable metaphor?

Is what we perceive reality? Does this include people hallucinating, too?

Or in the night imagining some fear
How easy is a bush supposed a bear
--Shakespeare

How much is objective reality, and how much is subjective reality?

Whatever flames upon the night
Man's own resinous heart has fed
--Yeats

I am much more at home in the fuzzy world of reality than in the artificial world of statistical "accuracy".

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Originally Posted by desiresjab

It just so happens it would be possible to devise extremely far from square rectangles in which either p-1 or q-1 was loaded with factors of 2. We assume p and q are both odd primes, as usual, with one of them a 4n+1 prime and the other a 4n+3, to keep matters clear, and to put the burden of factors of 2 entirely on the minus one of the 4n+1 prime.

It takes two dvisions by 2 to divide the rectangle into quadrants, and one more division by 2 to diagonally divide the bottom left and top right quadrants as in Eisenstein's diagram.

The p-1 or q-1 of all 4n+3 primes have only one factor of 2 to give. As long as the 4n+1 prime provides two factors of 2, which it must at least do by its nature, all three divisons can take place preserving the possibility of triangles WAY and YAX containing the same number of lattice points.

We do not really expect it, though, in the case of a very eccentric rectangle ABCD. Quite the opposite. We expect there not to be the same number of lattice points in WAY and YAX.

My conjecture about twin primes seems to involve both concepts--low eccentricity and factors of 2. So far, I am too dumb to prove it. I cannot even say if it is provable, or if it has already been proven.

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Prior to proving the conjecture, just stating it is important.

There appears to be more to the conjecture than that twin primes have the same number of lattice points in Eisenstein's diagram. How would you state your conjecture more generally? Are there other pairs of primes, beside twins, that should have the same number of lattice points as twin primes would? Could I assume pairs of primes, one of the form 4n + 1 and the other of the form 4m + 3 have the same number of lattice points?

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Originally Posted by desiresjab

To get off the math, Does anyone believe there is such a thing as the Nature of Reality, or is that another comfortable metaphor?

Is what we perceive reality? Does this include people hallucinating, too?

Or in the night imagining some fear
How easy is a bush supposed a bear
--Shakespeare

How much is objective reality, and how much is subjective reality?

Whatever flames upon the night
Man's own resinous heart has fed
--Yeats

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I agree with Dreamwoven that statistical accuracy seems artificial. That even goes for the probability in the quantum wave equations. All that statistics does is makes things look objective, but that is because it is no longer talking about the particular reality that is in front of us.

Part of the problem of reality is that we think and hope it is unconscious and objective. However, everything, including quantum reality, may be participating in various forms of subjectivity that seem foreign to our own. There may be nothing that does not participate in some form of subjectivity.

It is convenient to find ways to objectify reality. Some people rely on sacred texts. Some people rely on mathematics. Both of these work to some extent in the sense that they justify a belief that reality isn't totally chaotic.

Does the indeterminism of quantum reality imply that quantum reality shares in a form of subjectivity of its own since it appears to be making choices when we ask it questions? I think it does. What difference does that make? Probably not much. We will still be looking for patterns we can rely on. We will still be looking to better understand the universe around us. The only difference is we should have a greater respect for the reality we participate in. It is not dead. It is not objective and it cannot be completely objectified through statistics, mathematics or a set of sacred texts.

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Originally Posted by YesNo

I agree with Dreamwoven that statistical accuracy seems artificial. That even goes for the probability in the quantum wave equations. All that statistics does is makes things look objective, but that is because it is no longer talking about the particular reality that is in front of us.

Part of the problem of reality is that we think and hope it is unconscious and objective. However, everything, including quantum reality, may be participating in various forms of subjectivity that seem foreign to our own. There may be nothing that does not participate in some form of subjectivity.

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It takes a large pair of "mays" for what I have highlighted in red, mister.

Why must people complain that mathematical accuracy seems artificial, rather than accept that currently if there is any road toward this kind of knowledge it begins and ends somewhere with a guy tabulating statistics, and that he is, after all, tabulating in a language we created from the nature of what cannot be otherwise?

The point is, either someone has a better approach, or they do not. Of course, what good is it to us right now if it is not objectively presented? I admit there may be better ways. Will those ways please step forward?

The nice thing is that only means will the right interpreter step forward, the right philosopher, at whatever stage we are stalled. Someone once said:

Psychology is a body of theory awaiting phenomena; parapsychology is a body of phenomena awaiting theory.

Do not expect numbers to write an equation for the Sermon on the Mount or the best way of life. But if someday a deep theorem paves the way to verifiable astral travel, do not be surprised either.

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Originally Posted by YesNo

Does the indeterminism of quantum reality imply that quantum reality shares in a form of subjectivity of its own since it appears to be making choices when we ask it questions?

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As Socrates did of those who leaned too heavily toward or against the existence of gods, I would call the above presumptuous at this stage, in fact throughout, from imply to of its own to making choices.

I feel imply is far too strong, it means something will follow or is a necessary extension or consequence of. At any rate, slang usuage of the word is not fit for philosophical discourse where precision is needed, I am hopeful you will agree.

To say quantum reality participates in a subjectivity of its own is open to literally anything, you must admit. I propose that subjectivity in the quantum world is statistics, and an objective grappling with what we can only call randomness looks very different when "experienced" from the quantum perspective.

Choice is a very interpretive choice of words when talking about electrons and their ilk. I think it is a purely semantic convenience.

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Originally Posted by desiresjab

It takes a large pair of "mays" for what I have highlighted in red, mister.

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I use "may" to not sound too dogmatic. Replace it with "is" if you like.

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Originally Posted by desiresjab

The point is, either someone has a better approach, or they do not. Of course, what good is it to us right now if it is not objectively presented? I admit there may be better ways. Will those ways please step forward?

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For most people, for their immediate problems, a better approach is through some form of meditation which involves their subjectivity.

Edit: It occurred to me that another way, another better approach to knowledge that matters, is through "middle-way" ethics. Again this is a subjective and not an objective approach to knowledge. These approaches, meditation and ethics, subjective thought and intentional action, are not available to deterministic reality (computers, robots, etc) nor to any hypothetical, random, unconscious reality such as zombies.

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Originally Posted by desiresjab

Do not expect numbers to write an equation for the Sermon on the Mount or the best way of life. But if someday a deep theorem paves the way to verifiable astral travel, do not be surprised either.

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I don't expect numbers to do that. That would be one reason why numbers are inadequate to answer most problems people actually have.

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Originally Posted by desiresjab

As Socrates did of those who leaned too heavily toward or against the existence of gods, I would call the above presumptuous at this stage, in fact throughout, from imply to of its own to making choices.

I feel imply is far too strong, it means something will follow or is a necessary extension or consequence of. At any rate, slang usuage of the word is not fit for philosophical discourse where precision is needed, I am hopeful you will agree.

To say quantum reality participates in a subjectivity of its own is open to literally anything, you must admit. I propose that subjectivity in the quantum world is statistics, and an objective grappling with what we can only call randomness looks very different when "experienced" from the quantum perspective.

Choice is a very interpretive choice of words when talking about electrons and their ilk. I think it is a purely semantic convenience.

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My use of "choice" and "subjectivity", especially when discussing reality that seems very different from our own, requires definitions which can be accepted or rejected. I accept the following definitions.

If we test something and it gives an answer that is neither deterministic nor random, I define that answer as a "choice".

If we detect a choice, I define whatever reality made that choice as having a "subjectivity" allowing the choice to be made.

These definitions of "choice" and "subjectivity" could be applied to our own choices and subjectivity which is why I use those specific words and do not make up other words.

I am interested in seeing what has convinced you of quantum consciousness. I cannot read everything I would like to--I simply write too much--so I sometimes must rely on shortened syntheses from trusted associates. I do not know the details of these particular experiments, though I am familiar with the rudiments of wave-particle-slits experimentation. How did they set the experiment up in a way that led to your convictions, if that is not too strong a word?

I don't know much about quantum physics, but I did try to understand it when discussing "many worlds" a few years ago on these forums. However, I don't think one has to know much about it to get the relevant points.

The key to the consciousness idea is "indeterminacy" which implies both that something is not completely determined and also not random as a coin toss. I'll admit that is my idea. Most people I've read who talk about quantum consciousness are referring to the consciousness of the observer, not what is observed.

In addition I am interested in the non-individualistic nature of these quantum particles and the non-local behavior of particles that are entangled. This assumes that space and time are determined by local properties that Einstein posited. Until I read Canales, I assumed they were true. Now I'm not sure.

The double slit experiments that impress me are those showing what happens on the detection screen when one or two slits are open. Those form the base cases. It makes it look as if the slits are determining what happens. Then one tries to know which slit a quantum particle actually went through. However, just knowing that changes the wave pattern on the detection screen making it look as if two single slits were used and not a double slit, but that is after the fact of having gone through the double slit and not two single slits. So passing through a double slit is not all that is affecting what the result on the detection screen will be. The non-individualistic nature of the process is seen by passing the quantum particles one by one through the double slit without knowing which slit they went through. The pattern then becomes the same wave pattern on the detection screen which is not a random pattern implying that each individual particle (if one can continue thinking of them in this way) went through both slits at the same time and interfered with itself.

There is an additional question of just what is it that is going through those slits and arriving at the detection screen. Is it worth continuing to use the "particle" metaphor if the particle is required to go through both slits at the same time and interfere with itself? Is it even worth calling it a "wave" since one can detect which slit it went through and break the wave pattern.

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Originally Posted by YesNo

The key to the consciousness idea is "indeterminacy" which implies both that something is not completely determined and also not random as a coin toss. I'll admit that is my idea. Most people I've read who talk about quantum consciousness are referring to the consciousness of the observer, not what is observed.

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It harkens back to "cosmic consciousness."

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Originally Posted by YesNo

In addition I am interested in the non-individualistic nature of these quantum particles and the non-local behavior of particles that are entangled. This assumes that space and time are determined by local properties that Einstein posited. Until I read Canales, I assumed they were true. Now I'm not sure.

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Entangled particles seem determined. The state of one can always be known by checking the other.

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Originally Posted by YesNo

The double slit experiments that impress me are those showing what happens on the detection screen when one or two slits are open. Those form the base cases. It makes it look as if the slits are determining what happens. Then one tries to know which slit a quantum particle actually went through. However, just knowing that changes the wave pattern on the detection screen making it look as if two single slits were used and not a double slit, but that is after the fact of having gone through the double slit and not two single slits. So passing through a double slit is not all that is affecting what the result on the detection screen will be. The non-individualistic nature of the process is seen by passing the quantum particles one by one through the double slit without knowing which slit they went through. The pattern then becomes the same wave pattern on the detection screen which is not a random pattern implying that each individual particle (if one can continue thinking of them in this way) went through both slits at the same time and interfered with itself.

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Sure, that is all very impressive and mysterious. It means there is an awful lot we cannot explain. But nothing in it comples me to believe quantum particles are conscious.

We have an outside light. On top of it is a sensor which detects light or darkness to determine when to turn the light on. Under your beliefs I would have to call the sensor conscious. When the sensor gets too covered in bird sh*t the light mistakenly stays on all the time. You would still call it conscioiusness, I guess. Is sensitivity to change consciousness?

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Originally Posted by YesNo

There is an additional question of just what is it that is going through those slits and arriving at the detection screen. Is it worth continuing to use the "particle" metaphor if the particle is required to go through both slits at the same time and interfere with itself? Is it even worth calling it a "wave" since one can detect which slit it went through and break the wave pattern.

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Wavicles.

This is a bit like the latest astronomy post on "multiverses". We can't confirm that our universe is the only universe, but we build theories on the assumption that it is the only one. You might like to look at that post from space.com.