(with tongue placed loosely in cheek)
I’ve decided to invent my own paradox, The Maher Paradox. The Maher Paradox states:
All so-called paradoxes are, on closer inspection, not in fact paradoxes.
Which I know isn’t really a paradox, but then many so called paradoxes aren’t paradoxes to begin with and are even less so on closer inspection. Maybe a more accurate title might be, The Maher Paradox Conjecture.
First, the obvious question to ask: what is a paradox?
At its most basic level, a paradox is a statement which seems to contradict itself, as in the sentence, I always lie. If the statement is correct then I can’t be lying, rendering the proposition untrue as I have told the truth about always lying and therefore can’t always lie. But this means the statement is in its own way a truthful statement about my lying in that I am true to my word in saying that I am always lying. By making a statement that is in fact true, I have lied about always lying and therefore the statement is in an important way correct. Which is again a paradox. Oh what a tangled web we weave when first we learn to paradox.
At a deeper level, paradoxes are mathematical, physical or philosophical statements that say something about the absurdity of the world or the absurdity of competing theories for the way the world works.
Schrödinger’s Cat, for instance, is a paradox. Erwin Schrödinger created his infamous cat in a box that is both alive and dead until you observe it as a pastiche of the idea of quantum superposition, where a quantum system can be in a super-state of possible configurations until it is ‘observed’ and the system then collapses into one state or another. It was meant to be a criticism of quantum theory, but was repurposed and became an illustration for how quantum theory works. I’m sure it is to Schrödinger’s eternal chagrin that his eponymous cat has become synonymous with quantum mechanics and science communication the world over. If only he was spinning in his grave, that would be a form of perpetual motion; which is also a paradox.
The thing about these second kinds of paradox is they are often idealised mathematical models for a world that actually have no basis in reality. Though we should also allow that they are often used as reductio ad absurdum, pushing a theory to its logical limits.
The prime example of this is Russell’s Paradox. Russell’s Paradox is an important innovation in the development of set theory, but the quasi-real world example that Bertrand Russell used to illustrate his theory employs Ancient Greek Barbers:
Russell’s Paradox: A barber shaves only those men who do not shave themselves.
The paradox then arrives from asking, who shaves the barber? If he doesn’t shave himself then he should be the one to shave himself, but then he would be shaving himself, so he can’t shave himself, but then he isn’t shaving himself so he can shave himself, so then… and this is where we find ourselves stuck in a feedback loop of infinite regress.
Again, this has a much more technical implication in considering the set of all sets that are not members of sets themselves, which is beyond the scope of this discussion (and, incidentally, the expertise of the writer). The real world manifestation of Russell’s Paradox is easy to unpick. Few communities only have one barber and who exists in a vacuum. Two barbers who shave each other is enough to render the paradox unparadoxical.
Indeed, few barber’s or other hair cutting establishments have only one barber, with many hairdressers working freelance at one particular location. I’m sure this would have been true even in the ancient world. As to the set of all sets that are not members of themselves, this can be lumped in with considerations of infinity, to which we return later.
The same considerations can be made about Zeno’s Paradox. Zeno actually formulated a number of paradoxes, but here we are specifically interested in his paradox of motion, which is what most people understand by Zeno’s Paradox. Achilles and the Tortoise is often how Zeno’s Paradox is framed, but I want to use another common example for simplification of explanation.
There is a frog on a lily pad at the centre of a pond. In order to reach the bank of the shore, the frog must hop halfway across the pond. Then it must jump half the remaining distance, which, is to say, a quarter. Then another half of the distance left, which is an eighth, and then a sixteenth and a thirty-secondth. In this way, the frog will never reach the bank, as it will be jumping forever through smaller and smaller intervals.
Of course, if frogs really moved like this, the entire frog population would have gone extinct a long time ago. This is not how real world objects behave. The frog will reach the bank in a handful of jumps. Maybe even a single jump. Frogs are also amphibious and can swim for shore. Achilles will in fact catch up with the tortoise in no time, no matter what length of a headstart he gives it (within reason).
These are all clever examples to get us thinking about mathematical objects which tend towards infinity or zero, but they have little real world analogue. In learning about calculus, we learn about the idea of a mathematical gap tending towards zero, but it is important that the limit never actually reach zero. If it did, calculus would in fact become useless, because any number multiplied by zero is zero.
It is the same way that smaller and smaller decimal numbers divided by a whole number will tend closer and closer towards infinity (1/0.5 = 2, 1/0.25 = 4, 1/0.001 = 1000 etc.). This is because if you divide any number by a fraction, 1/½, 1/¼ etc., you in fact take the denominator of the fraction, move it to the top of the equation and multiply it by the number, which in our examples is always 1. 1/½ becomes (2x1)/1, which is to say, 2. As the denominator of the fraction doing the division becomes larger and larger, the result of the calculation becomes equally large.
The
implication here is that as the decimal number hits zero, the result of this
whole number division will be infinity, which is why if you try to divide any
number by zero on a calculating device, it will return an error message. Yet as
soon as the decimal number hits zero, the impossibly large number will reset.
The result of 1/0 isn’t infinity. It is in fact zero
We know this from calculating trigometric functions. The value of sine starts as 0 when an angle is 0. It then rises to 1 at 90O, returns to zero at 180O and drops to -1 at 270O, before returning to 0 at 360O. The cosine function cycles in similar fashion, but is 90O out of phase, such that it starts at 1, drops to 0 at 90O, drops further to -1 at 180O, before climbing back to 1 at 360O, through 0 at 270o.
The Tan function is mathematically identical to dividing the sine function by the cosine function. We start with: 0/1 at 0O, which is to say 0. The value of Tan then rises asymptotically (i.e. getting closer and closer to a number at smaller and smaller increments without ever actually reaching it) towards infinity. But at 90O, where sin = 1 is divided by cos = 0 (1/0), Tan becomes 0. A similar rise is seen between 90O and 270o, where the value of Tan rises from -∞ to ∞. It should be noted, however, that mathematicians set the value of Tan at 90O and 270o to zero for ease of calculation. However, they were right to do this, because the value of any number multiplied or divided by zero is in fact zero.
Leaving mathematics for awhile, another paradox that isn’t really a paradox to begin with is The Fermi Paradox. The Fermi Paradox asks why, if the universe is teeming with intelligent life, we haven’t found any of it yet. Many explanations are put forward to try and resolve this paradox, but few of them are very satisfactory because they seem to be bourne out of human arrogance. i.e. the question asked honestly is, if the universe is teeming with life, why wouldn’t it want to immediately contact us?
Yet there are any number of reasonable explanations that human beings perhaps don’t want to think about because they remind us of how utterly unimportant we are at a universal, or even a galactic scale. We have been broadcasting radio signals for barely a century. Travelling at the speed of light, those first broadcasts have reached out to a sphere of one hundred light years distance from the Earth and will probably be so weak that they have became lost in the background noise of cosmic radiation. Even then, a hundred light years is a tiny fraction of the way across even our own galaxy. The vast majority of visible stars in the night sky, even when seen from a location relatively free from light pollution, are within about four thousand light years from Earth. Which, again, is a tiny fraction of the volume of the Milky Way.
Moreover, the Milky Way is but one galaxy of two trillion that exist in the observable universe. Some astronomers think even this might be an underestimate, when you consider all of the galaxies that are too dim to be seen or are blocked out by other objects. If we think about the galactic disk of dust that makes up the central bulk of the Milky Way, it is blocking out a significant portion of the sky (including, importantly, much of the other half of the Milky Way, which could be teeming with life for all we know). There could then be twice as many galaxies than we know about in the observable universe, swelling the number to four trillion. And even the observable universe is thought to account for only around fifteen percent of the entire universe, the rest of which has expanded beyond the point at which its light can ever reach us. Meaning there might conservatively be as many as twenty trillion galaxies in the entire universe, each containing hundreds of billions of stars. At this point, the human brain’s ability to understand truly astronomical numbers has long since broken down.
At these scales we are dealing with probability and statistics. There will necessarily be many galaxies in the universe that are teeming with life and, conversely, those that are entirely sterile or home to a handful of technologically advanced civilizations during their entire lifespan. There is no reason why we couldn’t be living in one of these second kinds of galaxies and will never be able to communicate with another society within the Milky Way because they simply do not exist in the here and now. If we were able to peer inside that galaxy five over from our own, we might find a place similar to the world imagined in Star Trek, with a number of space faring civilizations coming together to form a federation of planets. Ten galaxies over in the opposite direction, there is a galaxy like Star Wars (not so very, very far away after all). In another galaxy within out local group there is a planet on which exist things we would recognise as dragons. Here we sit (maybe) on a desert island surrounded by rain forests on all sides. If only we had the skills, tools and material to build a boat. A device able to hold us in stasis for thousands of years would also be a help.
The point is the Fermi Paradox isn’t a paradox, but a statement about the enduring arrogance of humanity and its need to always place itself at the centre of things. We used to believe Earth was the centre of the universe. Then we replaced Earth with the sun. When galaxies were discovered, we placed the Milky Way at the centre, until we then found out about the Big Bang and realised that the universe in fact has no centre. The Fermi Paradox is in its way a further attempt to place Earth back at the centre by assuming that any advanced civilization would immediately want to make contact with us before anything or anyone else. The implication is then that because they haven’t made contact, they must not exist,, This then makes humanity unique in the universe and finally replaces Earth at the centre of things.
The Fermi Paradox isn’t a paradox and there are any number of answers to the question that require no convoluted solutions or explanations rooted in solipsistic navel gazing. The Milky Way might be relatively deserted at the present epoch. Or maybe the half we can’t see on the other side of the galactic disc is a hotbed of intelligent, space faring life that will always remain invisible to us. Or communication between star systems remains impractical due to the distances involved and the power required to direct a broadcast at one particular star system. Or maybe there is a way to communicate which requires no electromagnetic communication (radio waves, etc.), which prevents anyone else listening in (Ursula Le Guin and Liu Cixin, amongst other sci-fi writers, include such devices in their novels). Or maybe, just maybe, humanity just isn’t that interesting. Maybe we have been visited many times by other civilisations, but they always move on after scanning us because worlds like ours are ten a penny. Whatever the solution, it certainly isn’t a paradox.
Another thorny issue in the general area of paradoxes concerns infinity. Human beings have been struggling with the concept of infinity for thousands of years and it seems to have driven us mad in the process. One thinks of the Total Perspective Vortex in the Hitchhiker’s Guide to Galaxy; a punishment device in which its victims are shown the unimaginable scale of the universe compared to themselves, an invisible dot on an invisible dot, next to a sign, saying, You are here. The experience fries the mind. Thinking about infinity has sent more than one mathematician into the mouth of madness.
The sciences contain two schools of thought about infinity. Physicists encounter infinity and assume something must be wrong with their calculations. At the edge of the big bang or at the centre of a black hole, the equations break down and the theorists conclude something must be missing from their calculations, or there is something we don’t know about these extremes, where general relativity and quantum mechanics interact. Knowing more, say through some kind of Grand Unified Theory, would make the infinities disappear.
Pure mathematicians, on the other hand, invent different types of infinity, which is a paradox. One of the classic examples is the idea of an infinite integer number line. In other words, a two dimensional line of all the whole numbers stretching from plus infinity on one side to minus infinity on the other.
But then we can say that between each whole number is a half number. Between 0 and 1 is 0.5. And between 1 and 2 is 1.5. On an infinite integer number line, there must also be an infinite number of half numbers. Moreover, between 0 and 0.5 is 0.25. And between 0 and 0.25 is 0.125. And between 0 and 0.125... well, you get the idea. So there are, so the theory goes, an infinite number of subdivisions on an already infinite number line. An infinite number of infinities. An invisible dot on an invisible dot.
Except an infinite integer number line is a paradox and The Maher Paradox Conjecture states that all paradoxes are not paradoxes. The problem with declaring some finite thing infinite is that we are comparing apples with oranges, only it is much more complicated (and you don’t even get a fruit salad at the end of it). To explain why this is so, it is useful to think about the work of Descartes.
Rene Descartes (1596-1650) was a genius. In many ways, he was the last person to be both a great mathematician and a great philosopher. The two disciplines separated not long after his death and have continued to be estranged from one another for much of the last four hundred years (though Bertrand Russell arguably filled a similar niche). Descartes gave us both Cartesian geometry (at its simplest, graphs containing a Y and an X axis), and also Cogito Ergo Sum. I think therefore I am.
Most people have some level of awareness of Descartes’s most famous conjecture. I think, therefore I am, remains a masterful piece of reasoning from first principles and real world evidence. Unfortunately, Descartes was subject to, if not directly influenced by, the religious dogma of his day, and when he turned his attention to the existence of God, the same reason failed him.
God, Descartes said, is infinite. There nothing you can do to add anything to him. God is the alpha and omega. He is all. He is everything. Then Descartes states, apropos of nothing, that God is also infinitely good. But more than that. Descartes also says that God cannot be evil. He is beneficent and merciful. He is without fault or flaw.
Skipping over such obvious questions as, how can an infinite being be only male (a woke question to ask, I’m sure), or passing a cursory eye over the Bible to see how merciful and forgiving the Old Testament God truly is, we arrive at the idea of infinite goodness. Infinite goodness is another way of saying, all types of goodness. So what about evil? Good and evil are human concepts and have as little objective meaning as hot and cold, or light and dark. These are dualities, which are only important to sentient beings, with an arbitrary line of demarcation between them (when does a cold thing become hot and vice versa? – depends who you ask). Without one, there can be no understanding of the other.
As such, evil is a subset of good. Good is the absence of evil. Evil is the absence of good. The two are inseparable. An infinite amount of good would therefore also include an infinite amount of evil. If God were infinitely good, they would also need to be infinitely evil. Because, to be clear, there is a world of difference between having a lot of one quantity or substance and having an infinite amount of that one thing. At the point of infinity, any one finite thing spills out into everything else.
We can see this with the infinite integer number line. We can try to imagine a number line that stretches before and behind us for as far as we can see; for as far as our best telescopes can image. But an infinite integer line would merge to include all things that are not integers. An infinite amount of something includes all the things that are not that thing, because not being something is a subset of being that thing when viewed at an infinite scale.
An infinite integer number line also includes all irrational numbers (√2). All transcendental numbers (π and ex). And yes, all decimal numbers between the gaps. Also, every possible shade of red, every possible species of rose, every variation of Batman, Dr Who, Miss Marple, Romeo and Juliet, including anthropomorphised animals and alien species and an infinite number of things besides. Mixing a finite property like whole numbers or Shakespearean characters with the totality of infinity is a fruitless endeavour. Mathematics claims there are many different types of infinity, but there is in fact only one. Infinity is binary. There is infinity or there is not. And given that all possible events are not currently occurring simultaneously at all places in time and space probably means that infinity has never and will never exist.
One way to think about infinity, though it is probably as useless as every other attempt to visualise the invisibly infinite, is as a circle, inside of which sit all possibilities. All the mass of the multiverse, all its energy, all those shades of red and varieties of rose and all numbers, rational, irrational and transcendental, all people and all animals and alien species having a turn at being Batman, Bond, Dr Who, Hamlet, Desdemona, Anne Bennett, Daisy Buchanan, Milady, as well as everyone singing their versions of My Way, Summertime, Yesterday and Hallelujah to add to every other version that already exists, as well as an infinity of other possibilities besides.
This is an interesting thought experiment because of a curious factor of mathematical infinity. If you add any number to infinity, you still get infinity. ∞ + 1 = ∞: ∞ + ∞ = ∞ Yet if you think of infinity as a circle that contains all possibilities and things, you could argue that the reason why ∞ + 1 = ∞ is because ∞ already contains the concept of oneness. Nothing new is being added to infinity from outside circle (because there is nothing left out there) and so the total weight remains infinite. You could equally say that ∞ + Crayola Red = ∞. Or ∞ + Ian Watkin-Jenkins as James Bond = ∞. ∞ + Frensicllsenfordbablecox from Fornax 12 as Scarlet Witch = ∞. It would make as much sense.
Another paradox connected to this misunderstanding of infinity is the Hilbert Hotel. The Hilbert Hotel tries to say something about these paradoxical rules of addition when dealing with infinity. There is a hotel with an infinite number of rooms, each of which are occupied with an infinite amount of guests. But hold on a minute. We should be good enough at spotting non-paradoxical paradoxes at this point to see the problem already. If there are an infinite number of rooms, how can there be any energy left in the universe/multiverse to also have an infinite number of guests? Where has this new mater come from? If we have somehow found some extra matter and energy to create an infinite number of guests, can there really have been an infinite number of hotel rooms to begin with?
It gets worse, because Hilbert’s thought experiment continues by then having a new guest arrive at the hotel. Where has this person come from? Nothing remains outside the circle, so from what imagined realm has this guest arrived? “It don’t work”.
David Hilbert (1862-1943), who created the Hilbert Paradox, also imagined the Hilbert Curve, which is a geometric shape that can be replicated at smaller and smaller scales with more and more iterations until it, theoretically, can be replicated an infinite number of times inside a finite space. The problem with imagining any infinite thing inside finite space is that the limits of the real world will always prevent infinity from being reached.
If we were to trace out a Hilbert Curve using a pen or on a printer, at a small enough scale the line of the curve would be smaller than the width of a molecule of ink. If we could somehow continue the tracing using smaller and smaller wavelengths of light, through to gamma rays, we would at the very most hit the width of a single photon and be unable to progress any further. Even smaller orders of magnitude and we reach the Plank length, the smallest theorised physical length that can exist due to the limits imposed by quantum theory. A Hilbert Curve might theoretically tend towards infinity, but the limits of the physical world prevents it from actually happening and even by the time Hilbert’s curve banged up against Plank’s length, it would still be as far from infinity as when it started.
Infinity is probably impossible anyway, but it is almost certainly impossible in two dimensions or higher. Probably even then. Again, this bangs up against the limits of my intelligence and when I try to explain the following, I can’t quite illuminate what is dimly lit in the deep recesses of my mind. But it goes something like this:
It has to do with calculating the area of a larger and large two dimensional space. One simple way we learn at school is by multiplying the length of a space by its width. This is simplified when the length and width are the same, because then the area is equal to the length of one side, squared. A = x2. The area gets bigger and bigger the larger x is. 12 = 1; 22 = 4; 32 = 9; 42 = 16 etc. The numbers not only increase, but the space between them gets bigger and bigger: 25, 36, 49, 64, 81, 100. The gap between them in fact increases by an odd number each time, increasing by two from the previous gap= 1 – 4 = 3; 4 – 9 = 5; 9 – 16 = 7; 16 – 25 – 9.
Then, and this is where the fog creeps in, we can ask about the numbers that aren’t part of this results in this sequence. What about 2, 3, 5, 6, 7, 8, 10, 11, 12, 13 etc.? The point here is that in considering a two dimensional space that continues to increase in size, there are numbers that are excluded. If we were to expand this to three dimensions, where A = x3 ( and I’ll save you from having to consider cubes of a number sequence), even more numbers are excluded. So how can infinity exist in three dimensional space, when even a simple geometric sequence excludes numbers between the spaces? Infinity means everything and yet in more than one dimension, things are quickly excluded. Space is, for the most part, empty. The frog doesn’t make smaller and smaller jumps to reach the shore.
Here, I am sure fans of Zeno’s Paradox would argue that infinite regress is a factor, but again, the world does not behave like this. In theory, a handful of mating rabbits could produce enough offspring to entirely cover the surface of the Earth in a handful of years. And yet they don’t. Not once. Ever. There is always something that will prevent infinity from being reached because infinity is not a thing. It’s nice in theory, but so is democracy and cheap, affordable fusion power and they will probably never be truly achieved either.
I should state here that I love math(s). And mathematician(s). Mathematics created and has crafted the modern world since the time of the Industrial Revolution. Yet it has its limits (which is a kind of maths joke). Infinity is certainly at the edge of those limits and many of the paradoxes we have considered come about from confusion over infinity and considering it a real thing. Or considering it an infinite number of things. It is one or it is zero and it is almost certainly zero. Though there is also an argument to be had over whether there is ever really nothing. Zero is itself a kind of paradox in that by describing nothing it is in fact something. I will return to this idea at the end.
We should also, I suppose, talk about one of the most famous paradoxes, The Grandfather Paradox. In the Grandfather Paradox, you invent a time machine, go back in time and kill your own grandfather, who is now never born, so your father/mother doesn’t exist and now neither do you. So you can’t go back and kill your grandfather, because you don’t exist, so now your grandfather does exist and so do you, so you can go back and kill him. Then he doesn’t exist and neither do you… And on and on it goes, ad infinitum. Infinite regress.
The Grandfather Paradox is part of a group of Temporal Paradoxes, where future objects prevent past events, which prevent the future event affecting the past in the first place. Physicist, Joseph Polchinski, was one of the first to think about this kind of paradox. He used the example of a billiard ball that heads into a time machine, emerges a few seconds in the past and strikes its past self, deflecting it from the direction of time machine and preventing itself from traveling back in time in the first place. And, well, we know how this dance goes.
When thinking about temporal paradoxes, I often marvel at the lack of imagination or artistic licence demonstrated by the framers of these thought experiments. I would like to blame the straight laced minds of scientists, but the man who developed the Grandfather Paradox, Rene Barjavel, was in fact a science fiction writer. Of course, the most obvious resolution to the Grandfather Paradox is that time travel is impossible and that is that. No paradox occurs. Which is the point of the Grandfather Paradox. Time travel creates paradoxes, ergo time travel must be impossible QED. But can’t we do better than that?
I love sci-fi, but one of my beefs with time travel stories is how they are always predicated on the idea of their being one, inviolable timeline that the heroes must restore to equilibrium. Star Trek, Stargate, Dr Who, even El Ministerio del Tiempo (The Ministry of Time, a romp though Spanish history, which includes nearly fifty years of fascist rule under General Franco), all must correct some wrinkle in the fabric of spacetime to return things to exactly how they were before.
Yet if we consider the Many Worlds Interpretation of Quantum Mechanics, for which no evidence so far exists, there is a possibly (slim, I admit), that for every quantum event that occurs, every other possibility also takes place, splitting reality into many parallel worlds. For instance: A sheet of glass is half silvered so that photons fired at its surface have an exactly 50/50 chance of passing through the glass or bouncing off it. The Many Worlds Interpretation of Quantum Mechanics states that each photon does both, bouncing off in one world and passing through in the other. If we scale this up to the macroscopic world, it means that for every decision or event that happens in a person life, every alternative is also played out in some parallel reality. Think Sliding Doors, if you must.
So let’s consider our patricidal time traveler. If every possibility is truly played out then there is already a reality in which the killer’s grandfather was killed and his mother/father was never born. Moreover, we could argue that there is a reality in which a time traveler came back in time and killed the man. The time traveler therefore creates no paradox, but fulfills a role. By killing his grandfather, he transitions into a world in which his grandfather was killed. The universe in which the man went on to have children and grandchildren is not prevented from existing, it merely plays out in parallel, but with the killer now stuck in the alternate realty in which he never existed and yet exists. He is also now, presumably, on the run from the police.
As to the billiard ball, I have a sense that as infinity is impossible and there is nothing to say the same event always has to play out the same way every time, my feeling is that the ball would bounce back and forth for a while, maybe for millions of years, but eventually the feedback loop would erode and the whole system would come to a stop. Like the way a camera pointed at a screen displaying the camera image creates a feedback loop of repeated images of itself, but is seen to be degraded at the rear, loosing colour and fidelity. Maybe a freak collision would send one or other of the billiard balls flying through the air, killing the scientist who started the experiment, preventing events from taking place in the first place.
I admit that time travel seems pretty unlikely and even if it is theoretically possible, it would require so much energy to achieve that no society, no matter how advanced, is ever going to be able to achieve it. However, I would suggest that the reason we don’t see time travelers popping back from the future to have a look at us is because we have yet to invent time travel. If we consider the theorised resolution to the Grandfather Paradox just discussed, it seems to me that anyone who travelled backwards in time would immediately enter a parallel dimension separate from our own with no hope of return. The mere displacement of air caused by a time travel device appearing in the past would be enough to change things, however slight. As soon as one started turning up to witness historical events, the drift would be even greater. And when you started changing things, like saving Diana or killing Hitler, you really would be adrift, like a raft caught in the gulf stream.
Which brings up another important question. Is time travel really possible even by travelling back in time? If your presence in the past affects the past, then it is not the past you know anymore. Kill Hitler and you change the world, and not necessarily for the better. Germany would still be rancid with anti-Semitism after the First World War and the Weimar Republic. Another leader might not make the mistakes Hitler did, while all of the other architects of the Holocaust might still be in power. Either the way, the world would be a totally different place, populated by totally different people. My grandparents, for instance, married at beginning of the war and then my grandfather went away with the Navy and my grandmother didn’t see him again until VE Day. My mother and aunt were born the following year. If they had lived together sooner than this then either my mother would never have been born, or she would have been born years earlier and not met my father.
There is almost a variation on Heisenberg’s Uncertainty Principle going on here. Just as a measurement of a particles position makes its momentum less knowable and vice versa, so witnessing events of the past changes the future. If you take a passive role in the past, the future continues as before, but you play no part in its development. Even betting on sporting events, a la Biff in Back to the Future 2, would probably affect the course of events in some way.
Yet even if you travelled back in time and remained in orbit, observing events from a distance, it is unlikely that events would proceed in exactly the same way for a second time. Time travel is essentially like creating a carbon copy of the universe and expecting it do everything the same way twice. Like Jeff Goldblum’s drop of water in Jurassic Park, the path of the water is unlikely to follow the same path twice. Maybe there are certain events, like World War Two, that we can’t avoid and are always likely to occur no matter what you do. The outcomes might be different, but the cause would be the same.
Perhaps the biggest objection to time travel is actually that it would involve making a copy of the universe and how do you make a copy of anything that big? But no matter who won the Second World War, the universe would continue just as it always has. Some events are unavoidable. A different outcome in 1945 wouldn’t prevent the Boxing Day Tsunami of 2004. Subjective reality would have to have some kind of locality, where sentient beings create their own separate version of the universe and then we really are getting into pseudo science and science fiction. It is interesting to ask if the universe existed before we built telescopes to observe it, but it is not a question we can ever answer and it is useless to try.
All of which is to say that there are any number of solutions to the Grandfather Paradox that prevent it from being a paradox. If I was going to write a fictional variation on the Grandfather Paradox (The Grandfather Paradox2), I’d make is so the Fates are so annoyed with the killer messing with causality, that every time he kills his grandfather he is born into a new family and then has to travel back in time to kill his new grandfather. Then he is born to a new family and the whole thing goes on and on in a variation of infinite regress, until he eventually shoots himself and the Fates are satisfied.
Paradoxes are paradoxical for a reason. They either show us the limits of our mathematical models and explanations for physical processes, or they demonstrate to us the limits of our understanding. Like physics’ version of infinity, we should recognise in a paradox a manifestation of some misunderstanding of the way the universe works and seek answers to resolve the paradox. Which, to be fair, is why many paradoxes are framed. But even where something like the Grandfather Paradox seems to disbar the prospect of time travel, we should find other solutions to disconnect between what we imagine would happen and what might actually happen. Whether or not time travel is possible, we should prepare for the possibility of paradox and look to proactively resolve it.
Other paradoxes seem to occur because we try to impose the concept of infinity on a finite universe. Yet infinity is not real. It is the imagined limit of our understanding (silly, arrogant humans); the boundary at which reason breaks down and the universe ceases to make sense. Once we loosen our grip on the concept of infinity, its paradoxes soon melt away.
One paradox remains and that is the universe itself. Is it forever or did it have a beginning? Here I am using a more general sense of the universe, more like the idea of a multiverse. We know the universe we inhabit had a beginning, although these days many scientists reject the idea of the universe started from a point of infinite energy (for reasons discussed above). It probably still started from a mass of highly dense energy, but occupied a finite space. So where did it come from?
It is almost an entirely pointless question, because there is no way to prove a theory one way or another. And even if we could show that our universe was spat out from another universe or ripped from the womb of a black hole, or was caused by two other universes interacting on the theorised brane of a multiverse, the question would still remain, then what created that universe/multiverse?
The question of the age of creation is a paradox, because if we accept that existence isn’t infinite, then what was there before everything was set in motion? Like the astronomers of old, we are tricked into invoking a Prime Mover: the force that set the heavens in motion. But then what set the Prime Mover in motion? Infinite Regress stalks us like Death itself.
Whatever the answer, it is one that is almost entirely unintelligible to sentient beings at our level of understanding. Is time meaningless outside of a universe such as ours? Even without time, wouldn’t there still be cause and effect? Maybe cause and effect evolved like everything else in existence. Still the question remains, how do you get something from nothing? Even by invoking God, you still have to explain their existence. The paradox remains.
Every rule has an exception. Of course, people misunderstand that particular phrase. A rule seems to have an exception. You investigate the exception, find a solution to it and the rule becomes all the more respectable. Like the particle experiment at CERN a few years ago that seemed to break the barrier on the speed of light. Particles appeared to be moving faster than light, but it turned out the detectors were faulty and the speed of light is safe (for now). The exception to the rule turned out not to be an exception at all.
If we ever
work out how the universe began, The Maher Paradox Conjecture will be proven
once and for all. I'm not sure I'll still be around to gloat.
