I found myself driving home today after (materially) returning to work for the first time in a number of weeks, the pandemic still being what it is and weighing heavily upon all of our lives and lifestyles in the catastrophic way it has (and – at time of writing – continues to). Making my way through the afternoon traffic, exhausted after an earlier-than-usual-and consequently-sleep-deprived start to my day, and – of all things – wondering what the probability might be of me having an accident today, any day at all. My mind wandered, as it does, to the extreme and exceptional case of “if I were to drive home often enough, or enough times, I would eventually have an accident” as a matter of statistical probability and as an intransigent likelihood of numbers.
“What, though,” I quietly wondered, ” if I was to drive home not for a mere large number of times, or even for a hundred years, a million, a billion, but an infinite number of times?” What then? Infinity is not what you think it is, in fact – if anything should ever prove to more unintelligible (or at least – beyond intuitive comprehension) than an infinite sequence, I should truly like to know what it might be. Eternity being the first and most obvious counter-suggestion but being that this temporal entity merely (!) represents an infinitely-extended sequence of moments, an infinitely long duration – to suggest it as an equally unintelligible concept is only to paint the problem a different semantic colour as to drive home an infinite number of times, not only should I need a plausibly infinite number of vehicles but it should have to take me an infinite duration to complete – or more strictly speaking – to not complete because an infinite sequence is of necessity that which is foundationally and constitutively incomplete, in-complete-able.
Infinity is a concept so vast that it can contain anything at all, including itself. It is, for instance, a mathematical fact that in any Universe that is infinite in spatial extension, anything that can happen – that is, that is allowed or possible under the “laws of physics” – will happen and furthermore will happen an infinite number of times. This means that all possible variations of matter, energy, sequence, pattern, truth, falsity, concept, experience, and even of you (and all possible variations in time, space or alternative biological or historical configuration) that can happen, will happen and will do so an infinite number of times is a strange conceptual space, indeed. This sentence, you reading it, the thoughts you are having, the place where you currently exist and the physical and psychological sensations you are experiencing there, all the possible paths and historical possibilities that led up to this moment for you and all the possible variations of material and psychological state including all those possible unexpected or strange ones that remain permissible under the laws of physics. In an infinite Universe these all exist – perhaps not all at once or all in the same place (what would that even mean?) but all there, somewhere and repeated an infinite number of times.
Of course, by “permissible under the laws of physics” we may be opening up yet another barrel of ultimately and utterly unintelligible (logical) worms as this is only a sensible statement in as much as we can consider that the laws of physics are in any sense irrevocably (or at least contingently) complete or self-consistent, which they are not. Further to this is the mischievously brain-bonking fact that under certain – if contentious, but science always is argued which is what makes it interesting – cosmological models the Universe must indeed assume some kind of infinite spatial extension. (I recommend Max Tegmark’s excellent book “Our Mathematical Universe” for some numerical and material insights from physics into what “an infinite Universe” actually might – or does – mean and just how it is possible that we may have arrived in one.)
It may (or may not) have not helped much that while I was driving home I was listening to a podcast from the Santa Fe Institute (Exponentials, Economics and Ecology, with David Krakauer) which was at that moment exploring the conceptual implications orbiting a topic of cities being complex “hyper-objects” but, for whatever reason – good or bad – there I was staring at the immobile rear end of the car in front of me and stopped at busy traffic lights when it occurred to me that in any infinite sequence of possible events that those things that do not or probabilistically can not occur are as likely as those that do. It seems as though I had found a way out of what had previously appeared as an inevitable fate of (eventual, given enough time and enough journeys) car accidents, and not only of car accidents of varying degrees of severity and improbability across that entire spectrum of variables permissible under the (apparent or at least agreed-upon and consensus-reality validating) constraint of physical possibilities under the laws of physics, but an infinite number of them.
If an infinite sequence of events, or for that matter of randomly-generated numbers or symbols, must contain all possible sequences and configurations of numbers or symbols under whatever counting system we are using, then there is quite plausibly a sequence of events in which I do not have an accident. On day one – no accident; on day two – no accident; on day three – no accident; and so on, and on, and on – forever. Oh yes, this might be unlikely, exquisitely and death-defyingly improbable and mind-bogglingly, bottom-burningly and “I got lost several paragraphs ago – what in the world are you even talking about?” unlikely but if an infinite sequence of symbols, numbers, configurations of all possible material states of affairs not only can but must contain every possible sequence of possible (or physically permissible) facts and actual events, then the sequence of events in which a specific event never happens is as likely as that in which it does occur.
To reduce this to numbers: an infinite sequence contains all possible other sequences. A random string of numbers that is infinite in length must contain all possible strings (i.e. patterns) of numbers. If one possible, however unlikely, string of numbers that might appear in the range of numbers between 1 and 10 is that in which a number 7 (for instance) never appears, then as one of a possible sequence of numbers considered over infinity it must appear, somewhere, in that infinite sequence.
Breaking this down even further – consider a fair coin toss and let Heads represent 0 and Tails represent 1. If an infinite number of coin tosses were made, then all possible sequences of 1 and 0 would (eventually) appear. being that an infinite sequence is quite literally and definitively a sequence without end, then at some point along that sequence a string of 10 zeroes would appear; somewhere else a string of 100 zeroes would appear; somewhere else a string of 1,000 zeroes would appear. It may be unlikely to find yourself (other than being monumentally bored by that time) experiencing a sequence of 1000, 10000 or even 100000 zeroes in a sequence but as a matter of inevitability these sequences would have to appear at some point in the sequence.
Can you see where this is going? Being that in an infinite sequence, any possible string of symbols that can happen will happen, then an infinite sequence of Heads (i.e. binary 0) will eventually occur. As the advertising informercials on television and YouTube might say: “but wait, there’s more!” – not only will an infinite sequence of one of two possibilities ( as used in this particular example) occur, but it will occur an infinite number of times.
Infinity is like this. It does not end. It does not end in ways that allow it to also contain itself and to also contain counter-intuitively unimaginable possibilities within itself. Beyond the materially impossible (or at the very least implausible) concept of any person driving home an infinite number of times, or of tossing a fair coin an infinite number of times, it occurs as a fact in any infinitely-extended sequence of possibilities that if driving home without having an accident is one of those possibilities, and if this eternally-recurring commute incurs that one of the possible sequences of configurations that must occur is that I never, ever (for an infinitely-extended sequence of possible states) have an accident, then one world exists in which I never, ever have an accident and my insurance premiums remain safe, and eternally so.
You may have noticed that randomness has snuck in the side door here – I started with infinite sequences and at some point (quite arbitrarily) started to assume, without substantive validation or logical justification that an infinite sequence of the kind that interests me is a random one. It might just be that random sequences are the most interesting among all possible sequences, for other (and here undeveloped) reasons, but as it turns out – randomness and material entropy is an irreducible fact of the physics that do seem to be in play in our own (finite or infinite) region of the Universe. Entropy is a favourite topic of mine and I shall return to that later, probably, but it is enough perhaps to state that in as far as Grand Intuitions go, my own intuition that entropy and or as randomness is the distributed, discontinuous presence of a mischievously indefinable infinity.
The degrees of freedom and the obscenely vast combinatorial and (associated) dimensional spaces of infinitely-extensible possibility inhabit a strange and mysterious Twilight Zone of possibility and probability and we are no more likely to explain this than we are to attain a Grand Unified Theory or Theory of Everything. That is what was going on in my mind as I drove home today. As I arrived home, I reversed into my driveway and, inadvertently and perhaps as a consequence of being so thoroughly distracted by my own thoughts, lightly collided with a large bushy plant and crushed a solar light that was sitting next to the driveway into thousands of glass shards. This was itself clearly one possibility among many and while not necessarily being an accident per se, this implies that perhaps I should just listen to commercial radio on my afternoon commutes rather than thinking about bizarre and unexpected, random enigmas of logic, hyper-objects, infinity and mathematics. Or not.