Understanding the Un-Understandable

Nietzsche once said that humans are the “as-yet-undermined-animal” (Beyond Good and Evil, Part 3). Now think about that for a second. What he implies is that every other being on Earth has its life cut out for it. We can, to a certain degree, predict what a squirrel does every day. Or a tiger. Or a table. They have purposes, so to speak — they have their destiny. We, on the other hand, have a large degree of autonomy; we make our own destinies.. But we use our autonomy in varied ways — some good, some questionable, and some terrible. And so we wonder about each other.

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The Intertwining of Mathematics and Morality. Part 1 — Separate Logics

I had this sorta inceptive idea one day. It starts with the idea of the free market juggernaut. It’s an enormously wonderful idea. It’s beyond successful, having laid the foundations for our industrial, scientific, and technological luxuries today. And yet, there’s a fly in the ointment. The more prosperous an economy gets as a whole, the more individual people get screwed. The bigger the gap in poor and wealthy; the more companies can get away with screwing people with dangerous chemicals, outsourcing of jobs, and pollution of our environment. You see, there has to regulation of the economy, because in the end, economies are people…. not numbers. If people get screwed, the economy takes a hit, which is not something numbers alone can tell us. If there is enough gap between rich and poor, there will be civil unrest. As far as I know, no free market model explicitly takes that into account. Economic theories forget about right and wrong, and focus on the numbers.

So that’s the thing right there. People vs. numbers. The idea of right and wrong. People have the understanding of right and wrong. Numbers don’t.

1 / 6+ billion is essentially zero in the number world. It is everything to a person who just lost his/her loved one.

Numbers are cold, hard facts. But the idea of right and wrong — it transcends numbers. Doing the right thing may cost you. That’s numbers right there. But it’s the right thing to do. There seems to be a different logic at work here. The foundation of science is rooted in mathematical logic. The foundation of morality, this idea of right and wrong, is quite different. And the funny thing is, in this whole entire universe, throughout all of space-time, only us, who have been on this tiny blue dot for a blink of an eye… only we possess this different logic. At the same time, only we possess the number logic too. So who’s to say one logic is more valid than another?

I hate knocking on mathematics here. I love the intrinsic beauty in mathematics, and its broad applications in describing the world. But it can’t touch morality. Morality is just one of those really weird things. It’s there, and we all know it’s there, and it makes a difference, as we saw in the free market. But all that we have worshiped in the past couple of centuries — the scientific method, the grounding of our understanding in mathematical logic — it doesn’t lay a hand on morality. The *moral* of the story: numbers do not span the entire space of reality. If we can build up a theory of everything with numbers, can we build up a theory of everything with morality? Or do we need both? Does one simply have more applicability in one domain, whereas the other in another domain? Questions..

Maybe math is the tool to understand the outside, the universe… whereas morality is the tool to understand ourselves. I dunno.

You see, this is important, because if math doesn’t explain ALL of the universe, you’re missing out if you’re saying science is the end-all-be-all. Religion has been universal because humans couldn’t explain what was outside. It still is universal because, while we’ve extensively used mathematical logic to understand what’s outside of us, we haven’t used moral logic to understand what we’re made of.

And in the end, both logics only exist on one tiny blue dot in the middle of a vast universe. We don’t know where either logic comes from.

Food for thought.

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.

The Nature… of God?

For this post I will be making two key assumptions…

1) Special Relativity (hereafter, SR) is valid. SR operates under the tenet that the speed of light c is absolute, and thus every speed < c is reference frame dependent.
2) The Christian God, as described in the Bible, is valid.

Now with that said, let’s get on to the post.

Taking the tenets of SR, the following relationship can be derived:

“…where Δt is the time interval between two co-local events (i.e. happening at the same place) for an observer in some inertial frame (e.g. ticks on his clock) – this is known as the proper time, Δt ’ is the time interval between those same events, as measured by another observer, inertially moving with velocity v with respect to the former observer, [and] v is the relative velocity between the observer and the moving clock…” (Wikipedia)

This is known as time dilation. Simply stated, if I were travelling very very fast (close to c), my clock will start to tick slower than a stationary observer on Earth. Time gets extended.

Another relationship can be derived, as follows:

“where
L is the proper length (the length of the object in its rest frame),
L’ is the length observed by an observer in relative motion with respect to the object,
v is the relative velocity between the observer and the moving object,
c is the speed of light,” (Wikipedia)

This is known as length contraction… if I travel fast, space itself will contract around me.

But here comes the million dollar question – what if I was traveling at c? Forget for a moment that this is impossible, and imagine a photon looking down on Earth. Now let’s say the photon measures a time of three seconds. How much time would have elapsed on Earth? What about five seconds? No, wait, the denominator goes to zero! Likewise, let’s wonder if the photon measures a piece of space. No, forget about that — forget about the Earth — the whole UNIVERSE will have collapsed to a singularity! The photon knows of no time, nor does it know of space, because these things don’t make sense to a photon from the relativistic perspective. Instead, it sees all infinity of space, and all infinity of time – at any single “moment”.

So what does this have to do with God?

The Bible says many things about God. He is all-knowing, all-powerful. Paraphrasing, he knows the number of hairs on our head and [something else I forgot] before we are even born. He knows each of us by name, on a personal level. He listens to each and every one of our prayers. He knows what we are going to do before we even do it. Now a lot of people have a beef with this – and for good reason; it doesn’t sound very logical that God can do all this at the same time. C.S. Lewis provides a rebuttal: God is out of time (167-171). God knows of no time, and He knows of no space either – He knows our destiny before we can act on it, and He sees the order of the universe and the cries of our hearts all at the same time. Now this sounds familiar…

Wait. Am I implying God is a photon?!

Well, not really, because photons can’t do things like love us and stuff. BUT what we can assume from this, is that many inconsistencies about God can be accounted for by believing that God is a massless being traveling at the speed of light. Massless, because SR proves that massive particles can’t travel at the speed of light, but more importantly, because it creates a distinction between what we are made of – mass, and what God is made of – not mass. Theologically, we can then recognize that the Creator and the created are separate things.

Another thing is that SR effects are symmetric between reference frames. If infinite time and zero space on the Earth is seen from a photon in its frame, the photon of speed c will appear to us in our rest frame as something that goes through infinitely many lives every second, but can never be observed to have a finite dimension*. For those who ask why God never reveals Himself to us – He can try, but if He is traveling at c, we will never be able to see Him.

I can’t say much beyond “science can reconciliate Christian notions of God if He travels at the speed of light”. It’s a pretty random hypothesis, anyway. We can even argue that God is energy if we want, or even better, that He is outside of our Universe. There’s plenty of hypotheses out there, and thankfully we’ll never truly know the inner workings of God. But it’s something cool to think about.

——–

*If SR is correct, we could never be able to observe massless particles such as photons traveling at c, because they would be size zero in our reference frame. I don’t know if I’m missing something and we’ve actually observed and measured photons, or if I’m right and “seeing” a photon is physically impossible. I suppose that, if photons are a quantum construct, they don’t actually exist in reality except to make bundles of energy mathematically and logically easier to handle. This is an entirely different discussion though.

The Nature of Light

I had this epiphany long ago, but it took this line from Surely You’re Joking, Mr. Feynman to remind me of it.

They gave out dark glasses that you can watch [the first atomic bomb test] with. Dark glasses! Twenty miles away, you couldn’t see a damn thing through dark glasses. So I figured the only thing that could really hurt your eyes (bright light can never hurt your eyes) is ultraviolet light. I got behind a truck windshield, because the ultraviolet can’t go through class, so that would be safe, and so I could see the damn thing. (134)

And I was thinking about the bold part, and I came to realize he was right. You see things when light of some wavelength enters your eye and hits a molecule called retinal (the aldehyde form of the most common form of Vitamin A). The 11-cis form of retinal is isomerized to its most stable all-trans form, and the conformational change ultimately leads to the nerve impulse that registers as vision.

The key to realizing what Feynman says is that retinal absorbs at a specific wavelength. If you think about the photoelectric effect, electrons in a metal are “knocked out” by light only of a certain minimum energy or above. Likewise for a certain piece of light to be registered by retinal photoisomerization, it must have a certain amount of minimum required energy. Isomerization has a certain activation barrier.

As it turns out the intensity of light has nothing to do with its energy. From studies done of the photoelectric effect, it was determined that the frequency of light is what determines its energy, and that these come in discrete “lumps” (as Feynman puts it), known as photons. Each photon has a set frequency – a set energy. The intensity, on the other hand, is determined by the probability of finding a photon. A brighter light is simply more photons of the same energy. So it cannot possibly hurt your eye, because you’re still seeing photons of energies that your body already knows how to deal with – just many more than you’re used to.

But that wasn’t my epiphany. Let me explain what exactly is.

In church we are taught to bring light in a world of darkness. And there’s this nice, optimistic belief that while we can light up the darkness, the darkness cannot smother the light. Now this is great and all – but is it true? We investigate with the photon idea.

In [my favorite interpretation of] quantum theory, we see that classical light waves are actually propagating probability distributions of photons. Classical physics teaches us that the intensity of a classical wave is proportional to the square of its amplitude, which itself is proportional to the wave’s energy. If we allow energies to be quantized in discrete particles, and the wave to be propagating in an arbitrary space, then we can see how amplitude becomes related to the number of particles per unit space. We can imagine how this in turn is related to the probability of finding a particle. So with this established…

In darkness, the intensity is zero, so the probability of finding a light particle – a photon – is equivalently zero. From here I deduce that if we add light to darkness, we add photons, increasing that probability to a nonzero number. Yay we can do this! If I want to shine my light onto others, I simply add some of my unique photons. That’s quite nice.

Now let’s look at the reverse operation. Can darkness smother light? Darkness is defined as a lack of photons. If there already are photons, you have to actively remove the photons by hand; they won’t go away by themselves (that’s violating conservation of energy!). “Diffusion” (not sure how you can do that with photons “bound” by a probability wave, but for the sake of argument–) will reduce the probability towards zero, but never equivalently zero. You can smother light, but darkness by itself can never smother out light. On the other hand, light can kill darkness.

Optimistic thinking? Yes. Reality? Yes. Hooray! The upshot is, you see aspects of God all around you.

Even in quantum theory.

(And yes, I know that Einstein said that God does not play dice.)

Probability, and some Quantum Mechanics

Read God’s Debris, by Scott Adams (thanks to Charles L. for the recommendation). It’s a good read. I don’t agree with what it says because I do not believe in the same Theism but it offers up some interesting ideas. What interested me the most was its emphasis on probability.

Specifically, Adams claims that the world is governed by it. Now, this may be what we perceive the world as (a series of events each with a certain probability), but the root cause is not probability. Rather, it’s energetics. For instance, the equilibrium ratio of a state B to a state A of any reversible transformation can be tied back to how energetically favorable said process was. Examples include chemical reactions, diffusion, and heat transport… does G = -RT ln K ring a bell to anyone?

Nevertheless, where concepts of energetics get confusing, probability can work as an abstraction. The ‘classical’ example is quantum mechanics (excuse the pun there, for those who get it :P). By the Heisenburg Uncertainty Principle, one cannot know both the exact position and momentum of a particle at any time. So we use mathematical formulations called wavefunctions, equivalently orbitals, that map out the spatial probability distribution of an electron. One must keep in mind that orbitals are simply mathematical formulations. However, the three-dimensional shape and size, as well as the number of nodes of the orbital (in other words, where it changes “sign”), has immense implications on chemical behavior. The directionality of covalent bonds, ionization potentials, and chemical reactivity are just some examples of how important these mathematical functions are to the physical world.

But wait, you ask: how is the shape and size of an orbital related to its energy? Let’s take a look at where all this come from. It starts with one central equation:
This is, of course, the Schrodinger Equation. H is known as the “Hamiltonian operator” (related to total energy of the system), psi is the wavefunction, and E is an energy state.

Anyone with linear algebra experience should feel a lightbulb light up inside his/her head. The Schrodinger Equation looks very similar to this following, more generic equation:

…The definition of an eigenvalue lambda corresponding to an eigenvector v of a linear matrix transformation A. Relating back to the Schrodinger Equation, we can see that the wavefunction as an “eigenvector” of the Hamiltonian operator corresponding to the eigenvalue E. Thus a wavefunction is simply a mathematical solution that satisfies the Schrodinger equation, where E is an eigenvalue corresponding to an energy level. Only certain eigenvalues exist, so only certain energy levels exist – this is the notion of the quantization of energy that first separated quantum mechanics from classical mechanics.

That aside, we will now delve into the mathematics to further elucidate this connection between energy and probability. The Schrodinger Equation can be rewritten as a partial differential equation, which can be solved by a separation of variables after an appropriate change of variables to spherical coordinates.
(Note the similarities here to solving Fourier’s Heat Equation.) R is the radial term, related to the principle quantum number n, and this determines the “size” of the orbital. P is the azimuthal angular momentum term, related to the angular momentum quantum number l, and this determines the shape of the orbital. F is the magnetic quantum term, related to the magnetic quantum number ml, and this determines the spatial orientation (direction) of the orbital.

Recall that each of these components depends on the original Schrodinger equation – the energetics of it all. And these properties – the simple shape, size, and orientation of orbital probability distributions, determine all of chemistry and chemical bonding. Which explains a good amount of the natural world. Cool stuff! If you consider probability as an abstraction of energy, why yes, Adams is even right on the quantum mechanical level.

Sources
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydsch.html
Wikipedia articles: Quantum Number, Hamiltonian, Eigenvalues and Eigenvectors, Wavefunction, Atomic Orbital, Schrodinger Equation
God’s Debris

Epiphany

Today’s Sunday message by Pastor Ed was amazingly beautiful. His messages on gratitude and thanks are always among the best IMO, but today’s touched me a lot more, maybe because I was awake for once, lol. Or, maybe, because it hit me when I was at my weakest.

Through his message I came to realize that all the terrible, miserable, and least-fun things lately have been a byproduct of not circumstances but attitude. Maybe if I could see the glass half-full, and not focus on what is not there – what is empty and missing and not present – maybe then, I will enjoy my life more. For a fool who misses the great things in his life is like a fisherman who complained about the lack of shrimp in his net, when in fact he had a netted a bountiful harvest of scallops. I may not have the most friends, the best grades, the most likable personality, the strongest faith, the best research position, the greatest brain, the most time with my gf… but all the blessings I have – supportive parents, the best public university in the world, renewed peace with my roommates, heck, life itself – are truly miraculous in and of themselves.

May a spirit of thanksgiving be the charge for the rest of my semester, and my life.