A Real Monkey Will Never Write Shakespeare’s Hamlet
The infinite monkey theorem may suggest that given an infinite universe, a real monkey will literally write Hamlet with a typewriter. Really?
In an infinite universe, there must be an infinite number of earths, humans, and consequently, infinitely many copies of you and me. As a matter of course, an infinite number of real monkeys must exist in such a universe too. These monkeys are going to do infinitely many things, and one of them is randomly writing Shakespeare’s Hamlet with some random typewriter, isn’t it? The answer may depend on context and definition as theory and reality collide. The implications for theoretical physics may be revealing.
An infinite universe may inhibit crazy phenomena, but perhaps not as crazy as you may expect. It seems that the existence of an infinite number of copies of you might be much more plausible than a real monkey literally typing Shakespeare’s Hamlet with a typewriter. Why is that?
Let’s start with the probably easy case (with emphasis on “probably” because when it comes to the number infinity, seemingly trivial answers may turn out to be tricky at best, unsolvable paradoxical conundrums at worst). The probably easy case is on the theoretical spectrum of definition of the infinite monkey theorem, and by the way, this is actually what comes closest to what the original infinite monkey theorem actually states.
For that probably easy case of the theorem, we assume there is an infinite universe of whatever kind: no matter it is infinitely vast by space, exists an infinite long time, or is infinite in quantity in the sense of a multiverse consisting of infinitely many universes, or even a combination of those kinds of variants. Consequently, we assume this somehow infinite universe provides us with an infinite supply of real monkeys. We do not require a true monkey to literally write Hamlet with a typewriter on A4 paper. Instead, it is meant that an unconscious monkey’s random behavior somehow sequentially encodes Shakespeare’s Hamlet. It does not really matter whether the spatio-temporal coordinates of the roaming monkey or the monkey’s grunts encode Hamlet. Seriously, Hamlet could even be encoded in a sequence of monkey gesture ass wiggles. Experimental physicists will not shy away from analyzing years of video recordings of monkey ass wiggles if it is about proving a theory. All that matters is that the random sequence the monkey anyhow produces will generate Hamlet in any encoded form (no matter how abstract it may be encoded). Assuming that there’s an infinite number of monkeys, and further assuming that each monkey does produce a true random sequence then it seems comprehensible, or at least logical, that Shakespeare’s Hamlet will be encoded by a monkey somewhere sometime in the universe.
In its most theoretical form, the infinite monkey theorem tells us nothing else other than that given a true random process, anything will be encoded in an infinitely long sequence of random bits. The monkey and Hamlet are only metaphors used to make it a catchy theorem for academics. It sounds noble to put a monkey, usually an interesting animal for science, and Hamlet, a piece of world literature, into one theorem. (And I’m nearly certain nobody thinks about a sequence of truly random ass wiggles.)
But there are a bunch of assumptions of purity in the aforementioned case, and the more we “relax” these assumptions, i.e. the more we move away from the theoretical case to reality, the more things start to get tricky! To put it casually, one might say that the theoretical aspects of the number infinity collide with the practical reality of the biology of monkeys. Can we understand the practical case and its potential implications about our (infinite or finite) universe? Let’s be as practical as it can get in our world of thought experiments. Because now, if we are talking about a monkey writing Hamlet with a typewriter, we are dead serious about that. We expect the job to be done by a real monkey using a real typewriter and producing a pile of real A4 paper pad with Hamlet on it. (The odds may change if we alternatively allow a laptop keyboard with a text editor and auto-correction feature, or a huge pile of banana leaves and a chalk pen, but in principle, it’s the same – except for if the auto-correction has an AI copilot detecting the intention to write Hamlet after the monkey typed the first sentence randomly...)
Imagine a monkey sitting somewhere in the universe on some planet at an alien-made table, holding an alien-made pen, and writing alien-language letters. Note that even though it is an alien-monkey, it is still a mere monkey. Therefore, the monkey is a creature with subpar intelligence compared to another alien species on that planet. This means that the table, typewriter, and paper have been manufactured by an alien species that might dominate this Earth-like planet. The monkey just randomly ended up there, in the right place, sitting before the typewriter and doing “the right thing”, which is typing Hamlet. But this seems more comical rather than realistic, wouldn’t you agree?
It is most certainly impossible that a monkey will ever literally write Hamlet somewhere in an infinite universe. The reason is that a monkey is not evolutionarily designed to type a text with a typewriter. If it writes a text for a longer time, it is, by definition, not a monkey. If it is a monkey, it will never sit and write for longer than a few minutes. Even though, out of an infinite number of monkeys on infinite earths, each with humans, an infinite number of monkeys will end up as someone’s pet, and although many such pet monkeys may randomly sit at someone’s home office desk, they will never sit there for weeks or months and type something like Shakespeare. A monkey would hold a pen for a maximum of half an hour or so. A monkey can type one letter, two, three, but after 100 words, the probability doesn’t just go to 0.00000000…1, it goes to exactly 0 because it would be biologically infeasible. Due to the sake of evolution, you won’t see a herd of unicorns inline skating in the wild on some earth-like planet, even though it would not violate the fundamental laws of physics.
Not everything possible by the laws of physics must necessarily arise in an infinite universe, at least not without preceding evolution.
Inline skating animals may happen in an alien circus, but not in the wild. An animal doing something extraordinary may happen once in a while, but there are clear boundaries of the extent of extraordinarity, because it must make biological sense. Well, one might argue that there could be other environments that cause biological evolution to come up with inline skating unicorns in the wild. Maybe. But let’s refine our particular case of the theorem: will there be monkeys writing Hamlet on planet Earths that look exactly like our planet (i.e. planets with the same environment as our Earth)?
This is where assembly theory (AT) comes into play. Assembly theory has recently been presented in a paper officially published in October 2023 in Nature. It is not only a recent concept but also heavily debated and could be described as controversial (some saying it’s a scam, others saying its revolutionary). In a nutshell, assembly theory is a framework to measure and quantify complexity in a particular way. Measuring is often the key to developing, creating or understanding something, and therefore, this may be onto something promising. You can think of assembly theory as a theory that might be capable of explaining the evolution of complexity in the universe, and maybe the evolution of the universe itself. More specifically, it may be powerful enough to explain the origin and evolution of life, including the emergence of intelligent life, in the universe. Assembly theory could be grasped as something more fundamental than the theory of biological evolution, as it precedes biological evolution. The theory simply states that in order for certain complex objects to arise, certain predecessor objects of slightly lower complexity must exist. Casually speaking, the universe cannot create something “super advanced” (such as an iPhone) out of a soup of “ultra primitive” objects (such as elementary atoms) instantaneously, in one go. In contrast, according to assembly theory, a certain chain of causality is required to assemble the iPhone (1. primitive life, 2. intelligent life, 3. science & technology, 4. hardware factory). But where is the link to our infinite monkey theorem, or, more precisely, the practical infinite monkey problem?
The practical monkey theorem exemplifies that for complex things to happen, an evolutionary predecessor is required. Shakespeare’s Hamlet is a very complex object, whose existence required a complex process of evolution and a higher intelligent creature (human with certain IQ) to create it. Even if you have an infinite number of monkeys loitering in jungles and home offices, they will never ever write Hamlet.
An infinite quantity of low complexity objects (i.e. monkeys) cannot make up for a finite quantity of high complexity objects (i.e. humans).
This brings us directly to the necessary causality chain that the assembly theory incorporates. Referring to the terminology in the assembly theory paper, Shakespeare’s Hamlet being written by a monkey might be considered “assembly possible”, which means it is mathematically feasible, but not “assembly contingent”, which means it is causality-wise feasible. Mathematically feasible, or “assembly possible”, means that, for example, there cannot be a jet plane with a speed of 1000km/h turning a sharp angle of 90 degrees with a perfect curve radius of 1 meter while maintaining its speed exactly at 1000km/h during the maneuver. The laws of physics simply do not allow for this kind of dynamic to take place; it surpasses a fundamental theoretical threshold. Reality is not a computer game. How exactly the manifold boundary between “assembly possible” and “assembly contingent” of our universe looks like seems extremely hard to figure out (my guess: we probably will never get close to that). But the fact alone that we know the boundary to “assembly contingent” exists may help us further evolve our understanding of the universe.
While many would agree that Hamlet won’t be written by a monkey in reality, not all of them would also agree that Boltzmann’s brain can’t be created in reality. Is it possible that a fully developed brain can randomly appear in the infinite universe, and similarly, the origin of life could have happened? Exactly this is what the theory of the Boltzmann brain would answer with a “yes”. However, the point is that both of these problems are very similar, or even equivalent. For the practical infinite monkey theorem the reason is that you cannot model the monkey’s real behavior by probability distribution functions which asymptotically decay into infinity. I believe that this in turn implies that the Boltzmann brain does not and will never materialize in reality. You cannot expect any object in the universe to produce any other object of arbitrary complexity just because theoretical mathematics would allow that; you can’t take a monkey and say “produce me Hamlet I give you an infinite number of years, but please stay a monkey!”; analogously, you can’t take a cloud of stardust and say “produce me an iPhone or Boltzmann’s brain, I give you an infinite amount of years, but you must stay a dust cloud!”. For the case of the monkey, if requiring the real Hamlet written by a typewriter on paper, this goes against the monkey’s biology. In more general terms, the object cannot produce certain objects unless it itself is changed beforehand (i.e. monkey evolves into human). This, in turn, implies a chain of causality, and therefore a certain timeline is required. (A deeper reason for the necessity of causality might be fundamental limitations of the low dimensionality of space-time. The Boltzmann brain’s ingredients can’t all be readily available at one point in time in the right order, at least not without clumping chaotically together; conversely, building Boltzmann’s brain sequentially atom-by-atom might neither be a possible option as it requires all ingredients to come randomly from different sides, whose locations are also limited in space-time, continuously producing these ingredients. But a complex object already existing before docking to the brain in construction is relatively stable and actually helps to “circumvent” the fundamental space-time limitations; you don’t need to assemble on-site, you pre-assemble off-site, then do the final assembly on-site — similarly to the construction of a prefab house).
In summary, this means that in an infinite universe, there may be an infinite number of variants of Hamlet written by an infinite number of Shakespeares and Shakespeare-like humans, which may have evolved from monkeys, but not even a single Hamlet variant was ever written directly by a monkey with a typewriter. A monkey must evolve into a human-like being before writing any Shakespeare. An infinite universe doesn’t need to realize every possibility (no matter whether in such a universe a finite number of things repeat an infinite number of times, or an infinite number of different things happen). Just because mathematics says the universe must be crazy, it doesn’t mean it really is that crazy.
Remark: There is no contradiction with fundamental mathematics when claiming that the infinite universe doesn’t realize every possibility. The cardinality of infinity is not a constant in mathematics. There are different cardinalities for different infinities. For example, the natural numbers, which are infinitely many, have a strictly lower cardinality than the real numbers, which are also infinitely many.
I would appreciate any questions, remarks or corrections. Thank you. Last minor edits 19th of April 2024.
