The Nature of Reality – Unorganized Thoughts

So I was having a conversation about Vicki about a few things, and I kinda had this epiphany, which I will share here. I’ll start by citing Heisenburg¹ and his divide between a “subjective” reality, grounded in an ethical dimension, and an “objective” one, grounded in a mathematical, scientific dimension. Note that the “subjective” reality may not always be subjective – we can all agree that the Norway massacre was bad. Nor is the “objective” reality always objective, because of the nature of data and its biases and uncertainties.

Now as usual I’m going to start from a quantum mechanical perspective, that is, the world can be described by specific states represented by solutions to the Schrodinger wave equation. Likewise, we can also think of the world as a linear superposition of states. There is nothing wrong with this since you an either analyze a state by looking at its constituents (elements) or looking at it as a whole (a kind of a Gestaltic wholism). Indeed, quantum mechanics (QM) serves as a good proxy for the workings of the world in general. What QM tells us reflects what is readily obvious in front of our eyes. We cannot know everything; the reality is the superposition and the constituents, but our observation (read: bias, analysis, etc.) often collapses this into one single element.

But for now we consider specific states rather than the superposition. And we use this idea to look into the nature of the “objective” and “subjective” reality. I will expand Heisenburg’s definition, and I will expand the meaning of objective to include the idea of a Fact. Meanwhile, I shall expand the meaning of subjective to include the idea of an Opinion. Now granted, this has ambiguity too… the statement “God loves us” is Fact to a Christian, while Opinion to an Atheist. So I will define Fact as something which is truely true, as in what really is true from an omnipotent perspective. So if the Christian God actually does exist as described in the Hebrew Bible, “God loves us” is Fact. Now Opinion I will define as a *variable* qualifier or label to something that is truely true. (If God does not exist, “God loves us” is neither Opinion nor Fact, as it would not be part of any reality.) Note that objective labels (“there are 4 ducks in this pond”) that are absolutely true are Fact.

Examples: “I think this banana is sweet” is Fact because you actually do think that. “This banana is sweet” is Opinion in that you’re labeling the banana as sweet or not sweet.

Facts describe the objective reality. As such, in the QM perspective, Facts are synonymous to States. There is an infinite amount of states, because we can describe the reality around us with an infinite amount of Facts. Opinions describe the subjective reality. Now here’s the kick. Reality is by definition independent of what you or me say. So I’m going to turn my entire post upside down by saying that there is no such thing as subjective reality! (Well, as we have it now – I’ll change up its definition later.)

But there is such a thing as a subjective and objective qualifier. Let’s go back to the “banana is sweet” thing. It’s objective in the context of what is really sweet – perhaps quantified by sucrose concentrations or XXX ion activation in taste buds. In a more general twist, a statement such as “banana is good” can be dissected into objective facts. Perhaps the quality of something can be objectified by its effects, but that’s beyond the point. C.S. Lewis argues in Mere Christianity that some things are universally good or bad. We just disagree on the more minor issues. Other times, we are oblivious to what is really good or bad – female genital mutilation, for example. Our subjective qualifier clouds the more objective qualifier, and the lack of universality among humans doesn’t change the objective qualifier.

The objective qualifier, then, is what I’m going to redefine as the “subjective” reality. It’s the part of reality that is still there, but is often disputed. “There is a lamp on my desk” is Fact. “This lamp looks good” can be objectively weighed against other lamps based on geometry, compositional qualities, and balance. That is the subjective reality, or the objective qualifier.

Objective qualifiers in the QM world represent, ultimately, energy levels. Each state can occupy at most one energy level, but these are discrete at the most generalized level – the level of good and evil. The two cannot mix. In QM, energy levels represent eigenvalues of the Hamiltonian operator in the Schrodinger equation. Each state, or [non-superposition] solution to the Schrodinger equations, spans eigenspaces of those eigenvalues. Together, these sum of all the eigenspaces of all the eigenvalues of the Hamiltonian make up reality. And so, each Fact can occupy an objective qualifier level. Perhaps “this lamp is on my desk” would occupy the eigenvalue “Neutral”. “80 people were killed by a madman in Norway” would occupy the eigenvalue “Evil”. Something like that. Eigenspaces may have arbitrary amounts of dimensions depending on the number of linearly independent eigenfunction solutions that form the basis of that set (this is called degeneracy). In reality, the eigenspaces are infinite-dimensional – there are an infinite amount of evil and good things in this world, and the degeneracies are infinite.

There are some other little things I have to mention. Some things can be more good or more bad or more sweet than others. The eigenfunctions that form a basis set in an eigenspace may have different lengths, which we will interpret here as the amount of good or bad or sweet in a Fact. In QM, the functions are orthonormal. However, normalization is achieved by changing the weights on each of the basis functions, and is a separate consideration, and so we can ignore this discrepancy.

Finally – and apologies for the post’s length – we talk about subjective qualifiers, which do not consist of reality. Imagine what happens in a computer program. There is an objective bunch of executable code somewhere in the computer’s memory, but this is all abstracted in the higher levels by stuff like objects. In OOP languages, these objects can be referred to by a variable. These variables refer, to the object itself. You can change or swap references all you want, but the object itself remains unchanged. Likewise, subjective qualifiers act as references to the eigenfunctions of reality. We do not know the workings of reality itself, just as for beginner programmers, you do not know what goes on in object instantiation. A legit computer program will swap references and variables all the time, just as people change subjective qualifiers. The nature of reality remains however independent of the references you put on it. We see pointers, not the eigenvalues themselves.

¹See Physics and Beyond: Encounters and Conversations, by Werner Heisenburg, pp 82-84.

“Guessing” a Function

This is my physics professor’s favorite line.

So I am going to do this while I am still stationary in LeConte, before I go out and about and delay this to time t=infinity.

A few weeks ago, I got obsessed with an idea, which was to track the views/likes/dislikes of popular YouTube videos with time, and also the U-Ratio/U-Value (=ln(likes/dislikes), see post here). The motive was to figure out how YT views fit to mathematical relations. I was going to continue doing this, but a friend said YouTube keeps stats so I didn’t have to track.

Well, I couldn’t find those stats, so my interest has been piqued again. As such, let’s start. My belief is that while guessing the function is hard, guessing the rate of CHANGE of the function is easy. Let the function of views versus time be Y(t), in the spirit of YT HAHAHA. And the time derivative would be dY/dt.

Whatever Y(t) and dY/dt is, it will have to follow a set of principles derived from intuition, prior observations, and a little thinking. From these, we can start formulating guesses. They are as follows:

1) After a very long time, dY/dt approaches zero as people forget about the video. But dY/dt never hits zero, and Y(t) remains strictly increasing, because new people will still stumble upon the video. Nor will Y ever approach any asymptote; there is no limit on how many views a given video can get.
–> The only function that satisfies these conditions is the logarithm, with the derivative proportional to 1/t.

2) The number of views varies between night and day. More people would view the video during daytime than during nighttime; thus the rate of change of views dY/dt is larger during the day. dY/dt would thus be periodic with a period of 24 hours, as would Y(t).
–> A sinusoidal or cosinusoidal wave would work here. Since dY/dt is not zero at t=0, cos(t) would be a preferred form for dY/dt. To prevent possible problems with negative values of dY/dt, we take the square of the cosine… it doesn’t affect anything since the cosine outputs a dimensionless number and we can adjust the period as needed.

3) The rate of change of views is never really zero for finite times, so a pure cosine-squared does not work (since its minima are at zero). Additionally, the relative minima in the periodic wave varies with time — near the beginning, the nightly minima are greater since the video has more momentum.
–> Thus, we must add a correction to whatever we have for dY/dt — a shift that introduces the time-dependence of the minima. Since the rate of change of views is follows 1/t, this shift will also be of that form.

4) We must introduce a horizontal shift to the dY/dt to avoid discontinuous derivatives for t [greater than or equal to] 0.

Now we incorporate all components of the guesses.

For (1) and (2): multiplying one function by another function stretches one by the other for all ranges other than zero. So for dY/dt, we can take the cosine-squared and the 1/t and multiply the two, because we want a periodic function “stretched” by the overall trend of the video having a high popularity initially followed by decreasing interest as time increases. We don’t want the minima to be at zero though, so….

…For (3): we take whatever we have and append something of the form 1/t to the end of it. This shifts all the minima by such.

Finally, for (4) we insert (t+b) for t’s in the denominator, where b is a positive constant, to relegate the unrealistic divide-by-zeroes to a trivial quadrant.

And so, here’s the function!

A is the amplitude of oscillation (mostly constant for all videos, perhaps dependent on the spread of views across time zones to dampen the oscillation), and ω relates to the period (fixed to make the period 24 hr). The ratio of constants c/b relates to the initial momentum of the video — the magnitude of the initial shift.

A plot of dY/dt, w/o the horizontal shift:

(Source of graph)

We can find Y(t) given that the initial condition of Y(0)=0. However, it is a little complicated, and involves the function Ci(x) – the cosine integral. Nevertheless, a quick Wolfram Alpha integration and subsequent plot indicates a logarithm that oscillates – pretty much what we want.

What Can I Do with a Grid… —-> Part 2

As promised, I have more to say about the grid above my head.

#1 and #2 can be found here.

3) A Fourier Series for the Grid

So I came to think, if I can fit a sine wave to a grid, can I superpose sine waves in a Fourier Series fashion to form the grid itself? I figure it might be possible, but a better idea would be to consider it from a differential calculus perspective.

Consider a single pulse of the wave from x=0 to x=L. For the equivalent square, the derivative is +1 from 0 < x < L/2, -1 from L/2 < x < L. For the sine wave the range from -1 to +1 remains the same, but the derivative is continuous and it smoothly decreases from +1 to -1 instead of suddenly jumping. So we consider the Fourier Series for the derivative. It would have to start off with a single sinusoidal wave. But as terms are added, the derivative from 0 < x < L/2 would approach a constant 1, while the function from L/2 < x < L would approach a constant -1. Meanwhile, the limit would approach a nonexistence at x=L/2, and the derivative would cease to be continuous at that point.

This function that I am describing above – the derivative of the grid – is well-known. It’s called the square wave: Link. Its Fourier Series is given as follows:

Taking the integral of the Series yields the [top half of] the grid, more or less minus a few details here and there.

4) Integration; Storm Prediction Center Watch Boxes

So speaking of integration, the illustration provided in the previous post of the bed schematic lends itself to the area problem. How can I add up the squares represented by the grid itself to best approximate the area of the region of the bed NOT covered by the bed sheet? There may be an analytical method to this, but for now I am reduced down to guessing, which I do quite frequently when I’m lying down in my bunk all by myself. Of course, if I had more squares, an appropriate sum of the area of the squares approaches the area of the “naked” region itself.

This may be a rudimentary method, but the national Storm Prediction Center (SPC) does kinda the same thing in its issuing of weather watch boxes. These “boxes” are parallelograms delineated by a set of coordinates. Here’s an old prototypical example of an SPC watch box:

Now note that, along the boundaries of the watch, it’s difficult to say whether you are in the watch itself or not – unless you use an approximation method. Instead of using squares and saying “are you in this square or not”, the SPC uses current county outlines to “integrate” the watch box region. [Okay, they’re not interested in integrating, but I just like using that analogy.] It’s a little more precise than using a linear extrapolation between coordinate points on a spherical topology, and a lot more straightforward to the public than using arbitrary squares.

The problem is, the approximation’s going to be off if you’re using awkwardly shaped “differential elements” where counties just don’t jive very well with the parallelogram. Or if your threat area doesn’t fit well with the parallelogram. So recently, SPC has shifted philosophy by placing the county outline first, and keeping the parallelogram as a simple proxy used to make Weather Channel graphics more spiffy. But in reality, the parallelogram has been largely discarded in favor of county outlines.

Still, in day to day rests on my bed, I imagine the oval of the bed not covered by the sheet as a watch box. The grid represents my counties. It’s fun to think about. Terrible approximations, just like SPC watches used to be.

5) Drawing Tornadoes

Find two arbitrary points on the grid. From each point, follow their gridlines down towards where they converge. Stop following the lines anywhere before that convergence point, but do not cross or go over that convergence point. Draw a horizontal line across the two “top” points and across the two “bottom” points where you stopped.

Congrats, you drew a tornado!

[Insert illustration here, later.]

What Can I Do With a Grid Above My Head? –An Exploration of my Obsessions

In my new apartment, I sleep on the lower bunk where the top bunk is held up by a grid base. So every night before I sleep (or every afternoon before I nap), I have a great view of that grid above me. I have pictures of it, but they suck, so I will simply draw this on PS:

The blue is the bed sheet, and the brown is the bottom of the bed itself. The gray lines form what is supposed to be a square grid… apologies for the bad artistic skills.

So, with that said, what can I DO with this?

1. Poke the bed in between grid lines. ‘Nuff said.

2. Draw Sine Waves.

If you don’t believe me, consider this: an entire square grid can be formed from specified tangent lines to a sinusoid and their normals. The converse also works.

Let’s use the simplest sine function with wavelength 2π to represent the sinuosidal wave. At any arbitrary point on the grid, let’s say x=0, the slope of the wave is

Let the wavelength be λ. Halfway through a period, at x=λ/2, the slope is -1 since cos(π) = -1.  At x=λ, the slope is back to 1. The upshot here is that tangent lines at half-wavelengths, starting from a point where the slope is unity, oscillates between +/- 1. This diagram sums it up quite well:

Here comes the kicker. The slopes of these tangent lines are normal. The normals of these tangent lines are also normal. Everything is normal! Furthermore, by symmetry, the lengths of the both the tangent and the normal lines between where they cross each other and where they cross the x-axis are the same. They form squares.

Now let’s try to relate the wavelength of the wave to the length of each square. From the illustration above, a half-wavelength will cover the diagonal of the square. Let the side of each square have length L. Thus,

In this case of a 2π-periodic wave, L = π√2 / 2. But let’s generalize it for any L and any λ.

First we observe that for this to work, the slopes MUST be +/- 1 at the midpoints between extrema. Otherwise the square breaks down into some other ugly quadrilateral because tangents at half-wavelengths won’t be normal to each other. Mathematics wise, we observe that for

Since the wavenumber k = 2π/λ,

and so, after substituting L for λ and rearranging,

And so, given the side length of an arbitrary square in a regular grid, a sine wave may be fitted to the grid, as we can find A and k = 1/A from the above equations.

Since this is getting long, I will save a Part 2 for later. 🙂

[Note: Graphs generated with http://rechneronline.de/function-graphs/]