Cosmology

Great post. Nice amount for me to think about.

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Originally Posted by YesNo

Your questions make me wonder what is meant by a minimal polynomial. I think it means that given a root, such as √5 or (1+√5)/2, the minimal polynomial of that root is a polynomial with rational coefficients with minimal degree. If we have square roots of non-square integers, these would have a second degree polynomial as their minimal polynomials. Rational numbers would have first degree polynomials as their minimal polynomial. One can always multiply that polynomial by some other linear polynomial, say x-7, to get a larger degree polynomial. The minimal polynomial is associated with specific algebraic numbers, the roots of that polynomial. (The term of the minimal polynomial with the largest degree should have 1 as its coefficient. This guarantees uniqueness of that polynomial. It can be found since the coefficients of the minimal polynomial are over a field such as the rationals, Q.)

What one has with Q(√5) are all of the algebraic numbers that can be written as a + b√5 with a and b being rational numbers. All the rational numbers are in this field because b could be 0.

What the discriminant being congruent to 1 or 3 mod 4 is supposed to tell us is whether there exist algebraic integers in Q(√5) that have a 2 in the denominator or not. The 2 won't cancel in all cases if 5 is congruent to 1 mod 4.

What does it mean to be an algebraic integer rather than just another algebraic number? The minimal polynomial has integer coefficients and the coefficient of the largest non-zero term is 1.

As an example, consider (1+√5)/2.

This is an algebraic number in Q(√5) because (1/2)+(1/2)√5 is of the form a + b√5 where a and b are rational numbers, in this case both rational numbers are 1/2.

To find its minimal polynomial, I used the idea that if (1+√5)/2 is a root then so is (1-√5)/2. (That might be worth trying to prove, but I can't think of the proof at the moment.) If r and s are roots of a quadratic polynomial, then (x - r)(x - s) = x² - (r+s)x + rs. So, to get the middle term I add the two roots (1+√5)/2 and (1-√5)/2. I get 1 and then subtract it. To get the unit term I multiply those two roots to get -1. So the minimal polynomial is x²-x-1. Using the quadratic formula, I check that (1+√5)/2 is a root of that polynomial.

Is it an algebraic integer? Yes. The coefficients of its minimal polynomial are all integers and the highest term has coefficient of 1.

So Q(√5) has algebraic integers that have a 2 in the denominators as the determinant tells us to expect. That means the ring of integers of Q(√5) cannot be completely represented by Z[√5]. There are algebraic integers in Q(√5) that don't have this 2 in the denominator (such as all the rational integers and √5), but we are only interested in knowing if some of them need that 2 in the denominator.

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Are you saying...well, exactly what are you calling an algebraic integer--x²-x-1, or one of its roots? Do those roots with a 2 in the denominator become classified as algebraic integers whenever they are the roots of monic polynomials?

Even when you do a great job I still have dumb questions, you see.

The algebraic integers are the roots of monic polynomials with integer coefficients, that is with coefficients in Z. Algebraic numbers in general are defined in a similar way. They are the roots of monic polynomials with rational coefficients, that is with coefficients in Q.

For quadratic number fields sometimes those roots have a 2 in the denominator such as (1-√5)/2 which comes from the quadratic formula with the 2a in the denominator.

Yes, very good. Your explanations are quite acceptable. I take it, then, that an algebraic number with a denominator of 2 does not become an algebraic integer just because it happens to be the root of a monic polynomial with integer coefficients. It is still only an algebraic number, but an important one now on equal footing with an algebraic integer that is a root because it is a root itself.

The denominator of these roots tells us whether the root is an algebraic integer or merely a "rootish," algebraic number.

Multiplying the roots together as you did one can see some action in the denominator. However, working with the quadratic formula in the usual high school fashion, nothing seems to reveal itself with regard to 4n. At that level I do not detect anything about that which we are speaking. I already know there will be an unreduced 2 in the denominator at the end, or there will not be, and I know why it will or will not be there.

I feel I am getting pretty close, but there are still small pieces here and there I do not have in place yet.

When you start out with a field extension you already can see what "type," the root will be, whether it will be a 4n+1 or a 4n+3 number, for instance. When you start out with the quadratic formula you are trying to determine what the roots are. Once you find a root you could always assume a field extension was made earlier. I do not see anything discouraging me from looking at it this way.

Unless I am making a field extension, I have to use the quadratic formula to find roots, and therefore will not know the relationship of my roots to 4n in advance, or not until the formula has worked its root-finding magic.

At least that is my current view of the whole situation. Some of it is bound to be deficient, I suppose, downright incorrect, or short-sighted.

And, by God, I believe I know (at least I hope I am right) that every algebraic integer is the root of some minimum polynomial with integer coefficients. But of course every root is not reciprocatively an algebraic integer (because we know of the existence of some monic polynomials with integer coefficients whose roots are yet of the form a/2 + br/2).

I am just musing out loud to see if I am correct or incorrect on a few ideas. Jigsaw puzzles are completed at exponential acceleration as time increases.

An algebraic number I think could be defined as the root of a monic polynomial where the coefficients come from the rational numbers, Q. The reason to use a monic polynomial is to avoid having many of these minimal polynomials. For example, x - 7 = 0 gives the same root, 7, as 2x - 14 = 0 does. Since the rational numbers form a field, they have inverses so we can divide by the coefficient of the term with the highest power of x and make it 1. For example, 2x - 3 = 0 can be written as x - (3/2) = 0 and the root 3/2 is an algebraic number. In this case it is a rational number as well.

If it turns out that this minimal, monic polynomial has all integer coefficients, then one defines that root as being not only an algebraic number, but also an algebraic integer no matter what it looks like. That means (1-√5)/2, even though there is a 2 in the denominator, is also an algebraic integer. It is the root of a minimal, monic polynomial with integer coefficients. So it is an algebraic integer.

If we are looking at quadratic algebraic number fields, that is fields where the rationals Q are extended by the square root of a non-square integer, such as √5, then checking whether 5 is congruent to 1 or 3 mod 4 will tell us if the algebraic integers could have a 2 in the denominator or not.

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Originally Posted by YesNo

An algebraic number I think could be defined as the root of a monic polynomial where the coefficients come from the rational numbers, Q. The reason to use a monic polynomial is to avoid having many of these minimal polynomials. For example, x - 7 = 0 gives the same root, 7, as 2x - 14 = 0 does. Since the rational numbers form a field, they have inverses so we can divide by the coefficient of the term with the highest power of x and make it 1. For example, 2x - 3 = 0 can be written as x - (3/2) = 0 and the root 3/2 is an algebraic number. In this case it is a rational number as well.

If it turns out that this minimal, monic polynomial has all integer coefficients, then one defines that root as being not only an algebraic number, but also an algebraic integer no matter what it looks like. That means (1-√5)/2, even though there is a 2 in the denominator, is also an algebraic integer. It is the root of a minimal, monic polynomial with integer coefficients. So it is an algebraic integer.

If we are looking at quadratic algebraic number fields, that is fields where the rationals Q are extended by the square root of a non-square integer, such as √5, then checking whether 5 is congruent to 1 or 3 mod 4 will tell us if the algebraic integers could have a 2 in the denominator or not.

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Ah, in red is what I have been driving at and harping on, I don't know why. It was an instinct or the unconscious memory of something I read. It seems now that these numbers with 2 in the denominator can indeed be classified as algebraic integers as long as they are the roots of monic polynomials with integer (no, rational) coefficients.

Hmmm...I still don't know if it is rational coefficients or integer ones.

Well, lad, the only thing we have not done is the long, 19th century algebraic manipulations where these ideas came from. I don't know if we need to do that. Between your forced didacticism and my own efforts understanding seems to have arrived.

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Now we must ask: have we moved an inch, cosmologically speaking? What do ideals (and for that matter quadratic reciprocity) have to do with cosmology?

Well, do not forget, the deeper structure which we believe silently rules the universe and ourselves is what we hoped to catch a glimpse of by delving into exotic maths. Whether we have done that is a matter for debate, perhaps.

Even when numbers look exhausted and capable of no more order, enough genius is always able to find more structure nested in them. Ideals demonstrate this. I deals did not capture the ultimate order. Ideals could (ideally) apply in only some of the cases where unique factorization is not possible among the integers, polynomials et al. The theory made inroads, it did not settle all matters once and for all; it pointed a way forward.

We will continue to discover deeper and less accessible structures within numbers themselves, which will eventually connect to our own consciousness, I believe. Our consciousness likely hails from some deep structure we have barely glimpsed. Someday a connection will be made between us and the arithmetic structure we keep unraveling, is my one trusted belief.

I don't know if there is any cosmological significance in this, but there may be. I have enjoyed thinking about it and I did read most of Dedekind's book finally after having forgotten I bought it long ago.

Symmetry is supposed to be related to cosmological ideas. I forget how at the moment. That might be a place to continue pursuing the relationship between cosmology and mathematics. Then, of course, there is also a study of tensors and the Lorentz transformation for special relativity. Here is Einstein's "The Meaning of Relativity" which I have read parts of in the past: http://www.gutenberg.org/files/36276...68a2f9e44ff27b

One place where I think mathematics leads cosmologists astray is in a belief in constants. For example, is the speed of light really a constant? Is big G, the gravitational constant, really constant? It is convenient for the mathematics that they are constant, but I don't know how we would be able to tell.

The orchestration of coordinated activity of microtubules in brain neurons is one of the initial steps to understanding consciousness as a mathematical phenomenon, i.e., one which can be explained and predicted using mathematical tools some of which might not yet exist.

When there is enough "accord," the microtubules act in unison like a school of swallows banking and twisting at high speed without collisions.

I put accord in quotations. Call it a metaphor for consciousness. Scientists are busy constructing models of consciousness in 248 dimensional matrices compressed to 8 dimensions. They are looking for ways to make the swallows act in unison.

These are the initial baby steps. Where we get to is anyone's guess. My pal YesNo is quite convinced by the Searles argument that strong AI will never come about. Actually, I take YesNo more seriously than I take Searles. To me, Searles has constructed a semantic argument I do not feel compelled to even challenge. It is like one of those old semantic arguments by Kant or Spinoza that seem quaint and innocent enough these days to bring a smile to our lips at yesterday's children.

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There is this notion (unshakeable for most) that our own intelligence is real intelligence, and everything else, if it is not made of meat or DNA, is an artificial intelligence.

I beg to differ. Only if we came about entirely without the aid of any form of consciousness is our intelligence the "natural one," the real one. Otherwise, we ourselves are created, and therefore artificial.

If our intelligence (earthly) is therefore artificial, having been aided into existence or manifestation by any form of consciousness whatsoever, then I say the job of creating artificial intelligence is already a done deal, having been accomplished at least once to date.

By extrapolation I might contend that since YesNo and others believe in a consciousness that permeates individual atomic particles, the form of intelligence we represent can hardly have been attained without contact with any individual particles! This makes us and our intelligence artificial since we defined artificial as having been aided in any way by any form of consciousness, even the wee consciousness found in individual particles.

I did not see that you had come back to answer.

Yes, symmetry is a major idea. To really study it one should probably delve head first into group theory. Boy, I don't know if I am ready for that right now. I will go only where I have to out of intrigue. Dumb people have to limit their enterprises in some way. I do it by interest alone, letting nothing else interfere. There is always something in math I feel dumb for not knowing, and this provides my main drive. But the motivation is to follow structure deeper.

Would you draw any distinctions between constants from physics like the speed of light and a geometrical constant like pi or an arithmetical one like e? For it seems like a number which is its own derivative will not change under any circumstances I can predict.

The most that anyone can come up with is a model of reality. This is an objective map. It helps make predictions but just because we have a map does not mean that map IS reality. At most it is only a part of reality that we find interesting enough to want to make predictions about. In particular it does not contain our subjective perspective on reality.

The reason I reject AI (both strong and weak version) is because the AI computer is a deterministic-random machine. It is pure objectivity, like a table or chair. It cannot make a choice that is not part of an optimization process that can explain the decision. We can make such choices. A photon can make such "choices" as well. There is no programming or optimization underlying quantum indeterminism because there are no hidden variables to explain the indeterminism.

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Originally Posted by desiresjab

Would you draw any distinctions between constants from physics like the speed of light and a geometrical constant like pi or an arithmetical one like e? For it seems like a number which is its own derivative will not change under any circumstances I can predict.

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In mathematics, the constants such as pi or e do not change. They are not empirically derived. However, physical "constants" are empirically measured. They are useful up to a certain number of decimal places. We assume they are good for any number of decimal places and that they do not change with time. But how are we going to know that empirically?

We have grasped the essence of ideals. I feel my understanding of ideals is on a par with my grasp of QR. It is time to move on unless someone has a cogent remark about ideals at this point.

I do not have an area of math in mind to visit next that I feel relates to cosmology or the deeper understanding of structure in the universe. Exploring fancy counting methods from probablity theory would be interesting and fun but seems far away from cosmological pursuits to me. We only want math involved if it offers the possibility of deep glances at structure, not math merely for the sake of having it.

YesNo may have an area in mind that he feels is relevant.

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Belief #1

For the moment, I would like to turn back to what I feel is the surest idea in my philosophy.

For the universe--for all things--to come out of nothingness, is impossible. For nothingness means not only the tangible but the intangible as well. In true nothingness, there would be no existence of any kind. Even the potential for something to exist later is not permissible in nothingness, for that potential would be something which existed, though not tangible.

The forced result is that something had to always exist. There never was a time or a state in which pure nothingness prevailed. Pure nothingness cannot be, it is only a concept of the imagination.

It is logically undenaible that there is something eternal in the universe. Lacking a better name, that thing is existence itself, at the minimum, and perhaps God or consciousness, if we but knew the truth.

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Belief #2

In any possible universe with physics, 2 is the successor of 1. Universes which run backwards, universes which are p-adic, and all other apparent exceptions are easily remedied by a simple re-labling, e.g., the last event in a universe which runs backwards would become the first event under the re-labling, easily allaying the problem and casting it in its proper light as no more than a labling phenomenon. Under this belief it is forced that all universes will submit to our mathematical labling. No universe has a choice to refuse, for we are guaranteed to find mathematical lables that apply.