Premise: Analog and Digital systems
If you attended the lecture on electrical physics in your school, you might recall hearing your teacher telling you something about two essential types of systems, an important distinction if you happen to stumble into the topic of electronics viz. that a system can be either analogue or digital.
As trivial as this distinction may seem for someone who doesn't intend to pursue a career in applied electronics, these two words, digital and analogue happen to hold a lot of meaning outside of this little bubble of scientific jargon.
That our understanding of the basic nature of the universe might rest upon this trivial distinction might seem like an overstatement at this point, but what follows might sway you into believing it very well might be true.
For starters, what do these two words mean?
Let us look at an analogue system.
An analogue system, operates across a wide spectrum of states. Say for example, a system which provides variable voltages. If such a system operates between a voltage of, say, 1 and 100, it can have any number of states between these two devised limits. A system of this sort has the probability of being in any one of the states between these two numbers at any point in time. It could be 2, could be 32.5, 94.659, basically any number between 1 and 100 be it rational, irrational, whole, terminating or non-terminating.
This gives you a rough sketch of a purely analogue system.
We then move on towards digital systems.
A digital system, as opposed to an analogue one, can only have one of two states at an instant of time. Take for example one of the most basic types of digital systems, an electric switch. At a given instant, the said switch can be either on or off. Nothing in between. Digital systems find a plethora of practical applications in boolean algebra and the inner workings of computing systems, but that is not where we aim to dabble right now.
Alright. Great.
What does all of this have to do with the our understanding of the universe and all that stuff?
Here's where we take a step back.
A huge one.
Zeno's Paradox
Back in ancient Greece a Mathematician/Philosopher (yes, you could be both at the same time back then), Zeno of Elea, in support of his tutor/mentor (I don't know how their relationship worked, and honestly don't care) Parmenides' doctrine, put forth, a series of Paradoxes, out of which only nine survive.
Parmenides' doctrine had one simple conjecture, that all motion was in some form, an illusion, and that everything was in fact static. Zeno, might have taken that a tad bit too seriously as we'll soon see.
For a man born in 490 BC, Zeno was a genius, because some of his paradoxes still remain potentially unsolvable even today, where mathematics stands leaps and bounds above what it used to be during the golden age of Greece.
Although most of his paradoxes are very interesting to look at and incite questions whose roots go as deep as the frontiers of modern physics, we will look at just one, the infamous "Achilles and the tortoise".
Aristotle, born almost a decade after Zeno reflected upon the paradox thus,
"In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead."
Lets put this into easier words shall we.
The paradox, simulates a scenario much akin to the famous fable of the hare and the tortoise, which we all know always ends with the moral "the slow and steady wins the race", although, quite frankly, in the practical world, that is usually not the case.
However, the scenario differs from the story in two important ways. One of them of course , is a distinction without a difference, that the hare in this case has been raplaced by Achilles, a well built (and quite frankly handsome) young man. And the second one is this.
Achilles, being a gentleman (or perhaps too cocky), allows the cute little tortoise a hundred meter headstart, in an attempt to bring the blatantly unfair game slightly towards the fair side.
Let us assume now, that the finish line is an arbitrarily large distance away, just enough to make the 100 meter headstart a negligible problem for Achilles. Let us also assume, that the tortoise is fairly slower than Achilles walking at a speed of around 1 meter per second (I'll be quick to point out, just as Brian Clegg was,that the tortoise, was quite definitely on steroids), Achilles, on the other hand, has a speed of 10 meters per second, and honestly, he isn't even trying.
The odds, quite evidently are in favour of my dude Achilles and in a practical scenario (not involving any sleeping or slacking smh), he will most definitely always win.
But here's the thing.
When we bring the maths in, he doesn't quite seem too.
It sounds stupid, but trust me, it won't in a while.
Let our hypothetical race begin, and for our ease of understanding, we'll analyze it, second by second.
At T minus zero seconds (at the exact instant the game begins) the tortoise is ofcourse, quite a long way away (a hundred meters to be precise).
A second passes. The tortoise, running at a meter a second, is 101 meters from the starting line, Achilles, strides forward by ten meters, the distance between them now is 91 meters.
Another second. The tortoise is now 102 meters from the finish line, achilles is 20.
We are now in the position to establish that once 10 seconds have passed, Achilles will most certainly be at the point where the tortoise initially was. The tortoise on the other hand, will be 110 meters from the starting point, 10 meters ahead of his contender.
Another second, and Achilles is now only a meter behind his shelled rival.
Here's where things take a turn towards the interesting side.
We now look at the exact point where Achilles crosses the tortoise, but you'll be surprised once you realize....
The point doesn't actually arrive
11 seconds after the race starts, Achilles now stands at 110 meters from the starting line, at exactly the same point the tortoise was just a second ago, but the tortoise, being the steady bugger that he is, has already crossed that point by a meter and now stands at 111 meters from the starting point.
No problem, Achilles thinks, and reaches the 111 meter mark 0.1 seconds later, because 10 times 0.1 is 1 meter.
Huzzah! He thinks to himself.
But the tortoise grins back, because at that exact instant he still stands 0.1 meters ahead of him 111.1 meters from the starting point. Afterall, he didn't stop during that tenth of a second, did he?
Achilles, is starting to realize the problem now, but doesn't let it intimidate him, he takes yet another hundredth of a second to reach the 111.1 meter mark, but there the tortoise is, centimeters from his reach, still a hundredth of a meter ahead of him.
And it goes on, this endless dance between Achilles and the tortoise, Achilles, bound to lag behind the tortoise, for eternity.
But surely that can't be the case in reality. Ofcourse a body moving even slightly faster than another will eventually overtake it. It will, and it does.
But where's the problem in the maths?
Amusingly, there is neither a problem with the maths or with our conception of the practical scenario, but it might rather be with the disconnect between our mathematical model and the inner workings of spacetime.
And for that we'll use the analogue and digital distinction we looked into at the very start of this article/paper thingy, but there are some historical records that need to be set straight first.
Granularity over the years
Ever since we, as humans, have started pondering about the nature of the universe around us we have been drawn towards its essence. Back in the BCs, when we didn't have close to a clue about how we got here, what the stuff around us was made out of, how diseases spread etc etc. a really cool guy, who went by the name of Democritus, came up with an equally cool idea by the standards of his time, that all matter, is made out of an aggregation of a large number of indivisible things. He was partly right, namely, about matter being made out of smaller matter, but the indivisible part, he might not have gotten right. Nonetheless, Democritus' suggestion ended up becoming a prelude to the atomic theory, afterall, it was he, who suggested the word 'atomos' for those tiny little things he thought we couldn't divide.
Sadly though, we could.
Needless to say, Democritus' hypothesis ended up giving us insight into a very important fact about the universe. That matter, is "granular".
This was the first time we ended up realizing that no matter how seameless and continuous the macroscopic world may seem, it only appears so because of the intricate, harmonious and coordinated dynamics of the disconnected, microscopic world.
But that was not all we found out to be granular.
Eventually (and by eventually i mean more than two millenia later), Karl Ernst Ludwig Marx Planck (that's a mouthful), a German Physicist, born in Kiel, in 1858, and known to us now, simply as Max Planck grew up to spearhead a breakthrough in theoretical physics.
In 1900, Planck presented the famous quantum theory, which ended up solving a plethora of then unsolved physical problems, even the brightest of minds couldn't get their heads around. The photoelectric effect and the blackbody spectrum are good examples.
But what was so important about this theory?
Although quantum physics has ended up giving rise to a whole new generation of subjects -which I don't even have the capability to list, let alone explain- the subtlest of its points is this.
That just as we were mistaken about matter being continous, so were we about energy.
Energy too, in its essence, is granular.
Back in the day, when Newton was the coolest guy in town, and everything he did was the new big thing in the scientific community, physicists were clueless about this fact and honestly, they had every right to at the time. Electromagnetic radition, at the time was thought to be solely of wave nature and it was thought to be continous, its energy spread evenly across its wavefront.
Nobody had even the slightest of idea about there being any granules involved in these shenanigans. As it turns out there were. Planck was quick to point out that energy was not emitted continuously, but rather in a discontinuous manner, in the form of spurts or packets, or as he then named them "quanta" and as we now know them "photons". Energy, he said was granular, and although it may not seem that way, it surely is on a nanoscopic scale.
Today, in 2019 we are quite sure about the granularity of these two things, of matter and energy (which we came to know were basically two different states of the same thing, but you get the idea).
But what does this granularity have to do with Zeno's paradox and analogue and digital systems?
Enter "Planck's length"
Planck, you see was a smart man and before his death, left behind a whole system of units, called the Planck units. It is a system of units of measurement, devised solely on the basis of five physical constants (which, I'm going to copy and paste from Wikipedia now, thank you).
-the speed of light in a vacuum, c,
-the gravitational constant, G,
-the reduced Planck constant, ħ,
-the Coulomb constant, ke = 1/4πε0
-the Boltzmann constant, kB
There are precisely 5 basic Planck units, however, two of them are of interest here. The first one, is Planck length, the other is the Planck time.
The derivation of Planck length uses the Planck constant, the speed of light and the gravitational constant in such a way that it is dimensionally consistent with length, that is, both the left and right sides of the equation, upon simplification give the units of length.
The special thing about the Planck length?
It might be evidence of the granularity of space itself.
You see Planck length is believed to be the smallest length that can be measured and bear with me here, when i say it is not because we don't have the tools to measure anything smaller than this length, it is rather, because - get this - beneath this scale, the laws of the universe break the hell down.
Hilariously, we don't even have the tools to measure Planck length itself which comes out to be 1.616255(18)×10−35 meters. To put that into perspective that is around 1.6 x 10-23 picometers. The radius of a hydrogen atom is 53 pm, which comes to be about 3.3x10^24 times larger than the Planck length. Or as Brian Greene puts it, if an atom, was the size of the observable universe, a planck length would come out to be about the size of a tree.
The Planck scale aims to devise a lower limit to our universe, just as many thories devise an upper one. Whether or not quantum theory survives its adolescence, the Planck length seems to be quite a valid explanation for Zeno's apparently unsolvable paradox.
So here's the disconnect I was talking about in explaining the problem with the paradox.
Newtonian calculus, the one that we mostly use during our school physics course, uses infinitesimals in order to solve problems. Infinitesimals, are the exact opposites of infinites, they are teeny tiny values, so tiny, we cannot even begin to imagine them, but ever so slightly larger than zero. Infinitesimals, one could say are right at that border, where something, turns into nothing at all.
When we tried to solve Zeno's paradox, we made the assumption that the distance between the tortoise and Achilles could be divided into an infinite number of ifinitesimally small parts. We thought the distance was analogue, that there was a potentially infinite number of states between the two points. But if that were the case it would also take an infinite amount of time for Achilles to cross the tortoise. As it so happens, space at very small scales might only be finitely divisible, we may have been wrong about space, just as we were wrong about matter and energy and this shouldn't come as a surprise because we realized long ago, back when Einstein put forward his theory of general relativity, that space isn't "nothing", it is something in itself, because it has an inherent energy of its own.
Our movements, on very, very small scales, might be digital, that is to say, those teeny tiny particles at the tips of our fingers, and the tips of those teeny tiny particles themselves, might be jumping, ever so quickly, from one Planck volume to another, giving rise to this motion that seems so seameless on the macroscopic scale, because if it weren't, we wouldn't have been able to move in the first place.
This answer to Zeno's paradox seems likely, although the scientific community is split into two doctrines in relation to the answer to this argument. The other one, involves some sly mathematical reasoning by implementing limits, and honestly the former seems highly likely, although it is, as of yet, unproven.
But could these granules of space, these little pockets, be evidence for the existence of our world in a computer. Perhaps, a planck volume, is nothing but a voxel - the three dimensional counterpart to a pixel - and the lower limit to the universe is a direct reference towards the limits of the computer simulating us.
But this is all speculation.
Or perhaps, all of this is a lie and Parmenides was right.
That all of motion is an illusion.
And we are doomed to live that illusion, perpetually, every single day, for the rest of our lives.
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