1.2.1 Endless Search
1.2.1.1 Alexander von Humboldt said language is the “infinite use of finite means” (Humboldt 2014, 91), and Darwin echoed back that “from so simple a beginning endless forms most beautiful and most wonderful have been, and are being, evolved” (Darwin 2012, 138). Both point towards infinite possibilities emerging from simple bits.
We started with just two kinds of bits, A and B, and saw how they grew to a combinatorial space of more than two million configurations after only 19 pulses of time. Life is built from just 6 bits (carbon, hydrogen, nitrogen, oxygen, phosphorus, and sulfur). There are 20 bits (amino acids) that build all proteins, and 4 bits (adenine, guanine, cytosine, and thymine) that encode DNA. The English language starts with 26 bits. From so simple a beginning.
1.2.1.2 When we think of big numbers, we think astronomical—so how big is that? Physics tells us there are roughly 1080 atoms in the universe. We talk of megabytes and gigabytes, so let’s call the sum of all the atoms in the universe a cosmic byte. It’s strange to write down the number of atoms in the universe in just a few characters: 1080. But there they are: all the atoms in the entire universe.
1.2.1.3 And yet a cosmic byte is an infinitesimal fraction of the combinatorial spaces we are talking about. Take language: start with 26 letters and add 4 punctuation marks (period, comma, question mark, exclamation point). Ignoring capitalization and all other symbols, we have 30 bits. Now replace our As and Bs with an alphabet soup of 30 bits. At T10 we already have 3010 possible combinations. Shakespeare’s Hamlet is roughly 130,000 characters long. The combinatorial space of strings of that length is 30130,000. No monkey is going to clank out Hamlet on a typewriter within the lifetime of our universe (Lloyd 2007, 59).
1.2.1.4 Now turn inward, to the brain. We have roughly 86 billion neurons with thousands-fold more connections (Seth 2022, 23). The combinatorial space of possible connectivity patterns is unimaginably large—again dwarfing our cosmic byte. As Carl Sagan put it, the brain is “a very big place in a very small place” (The Library of Consciousness 2026).
1.2.1.5 Now look at proteins, which we’ll return to later. An average protein is ~200 amino acids long, built from just 20 bits. The combinatorial space is 20200 ≃ 10260. Even if the universe tried one new combination every Planck second from the Big Bang to now, it would not come close to exploring all possibilities (Kauffman 2000, 144–145). Life cannot depend on random sampling for discovery—the problem must ultimately be broken down.
1.2.1.6 Across language, biology, and the mind, the combinatorial spaces are physically unsearchable. There is a search threshold: a point beyond which brute-force exploration is impossible given finite time and resources.
If we want a formal definition, we might say that, for some given computational capacity c, time t, and space s, the threshold is the point at which c cannot iterate or randomly search s/2 in time t—that is, it has less than a 50% chance of finding an answer.
Practically, that threshold is certainly far smaller than a protein, a novel, or a mind. All things do not exist everywhere (at least in this universe).
1.2.2 Halting Problems
1.2.2.1 Each string in our system results from combinations and negations. But there is noise at each step. So while there may be regularity between steps, there will always be some unknowability in what comes next. And since the adjacent possible is defined by what can happen next, it is impossible to predict what the future adjacent possible will be.
1.2.2.2 But there may be a deeper unknowability at work. Strange things happen when you introduce self-reference and negation together—they lead to limits on knowability.
The liar’s paradox is only a paradox because the liar speaks about themself. Turing’s proof of the Halting Problem—the question of whether a computer program will execute in a finite amount of time—turns on a program that negates the prediction made about its own behavior (Turing 1937). Gödel’s incompleteness theorem relies on a sentence that asserts its own unprovability (Gödel 2022). For sufficiently expressive formal systems, you cannot have both completeness and consistency.
Imagine you not being you—it makes you feel a bit incomplete.
1.2.2.3 We haven’t proven that our system has undecidable properties, but we have the pieces in place. Once our system has the resources to be Turing complete (1.1.6.1), we’ve brought in the Halting Problem. There will be questions about the long-term behavior of strings—whether a given process will stabilize, loop, or negate itself—that are not merely hard to answer but undecidable in principle.
It would be curious and beautiful if the fundamental pieces of a complex system gave rise to undecidability by definition.
1.2.2.4 Each way in which our system is unknowable relates to time. Once we step down a path, branches of our combinatory space become unreachable—there isn’t enough time to search the combinatorial space at these scales. Our first few choices set us down a path that leaves vast areas unexplorable. Noise is tied to the second law of thermodynamics, which introduced an arrow of time. Recursion requires taking the output of the previous step.
The relationship between time, thermodynamics, biology, and complexity is increasingly being formalized. Biologists now talk about a biological arrow of time that arises because “this undecidability is manifested through three factors: self-reference, infinite computation, and negation” (Prokopenko et al. 2024, 12).
Physicists like Carlo Rovelli further propose that the best way to understand how time actually works is through the lens of entropy and, by extension, heat (Rovelli 2026)—in essence, a kind of thermal time.
Discussing Assembly Theory, Sara Imari Walker and Leroy Cronin explicitly relate time to an emergent property of a large number of objects (Walker and Cronin 2023).
Time is intertwined with knowability, entropy, and complexity. We will return to this idea in section 1.4, and it will become increasingly important.
“Time for you and time for me, And time yet for a hundred indecisions, And for a hundred visions and revisions, Before the taking of a toast and tea” (Eliot 1963, 4).
1.2.2.5 So the strings in our system are unknowable in layered ways: because the spaces they inhabit are too large to search; because each step in their progression is noisy; and, possibly, because the logic of self-reference places limits on what can be known at all.
We started with the simple and the finite and ended with the infinite and the unknowable.