Buzz
Numerous mysteries elude our understanding; personally, I am an endless fountain of ignorance. Yet, there's a crucial distinction between what we don't know and what is inherently unknowable. For instance, while Shakespeare's exact birth date remains a mystery (though his baptism date is recorded), it's not impossible that future discoveries, such as a long-lost document, could reveal it. Thus, his birth date is unknown, not unknowable. This list explores 10 things that are fundamentally unknowable—topics that are not just currently unknown but will forever remain beyond human comprehension.
Many of these concepts are rooted in mathematics. I've aimed to simplify them as much as possible, given my limited mathematical expertise. My goal is to make them accessible enough for even a non-expert like myself to grasp.
10. Sets and More Sets
Unknowable Thing: What’s in a set of sets that don’t contain themselves?
To tackle several of these items, a bit of mathematics is required! This paradox, uncovered by Bertrand Russell in 1901, holds the top spot because it fundamentally introduces the concept of the 'unknowable.'
Let’s begin with the notion of a set. A set is simply a group of objects—for instance, the set of positive even numbers includes 2, 4, 6, 8… or the set of prime numbers comprising 2, 3, 5, 7, 11… So far, so straightforward.
Can sets include other sets? Absolutely—you can have a set of sets that contain other sets, and naturally, that set would include itself. In fact, sets can be divided into two categories: those that contain themselves and those that don’t.
Now, imagine a set (let’s call it S) that consists of sets which do not contain themselves. Does S include itself? If it does, it shouldn’t belong to the set, but if it doesn’t, then it should. This creates a paradox where S is endlessly oscillating in and out of itself.
This paradox sparked significant distress among mathematicians. Picture someone demonstrating that a number could be both even and odd simultaneously—it’s equally troubling. The paradox has been circumvented, primarily by redefining the foundations of set theory.
9. Graham’s Number
It’s often said that the challenge with humanity’s understanding of the universe lies in our brains being accustomed to small numbers, short distances, and brief time spans. Graham’s number is so enormous that it can overwhelm most minds; it’s truly colossal. To provide context, let’s examine some other famously large numbers:
Many are familiar with a googol—a number so vast it’s 10^100, represented as a 1 followed by 100 zeros. For most practical purposes, it’s considered an enormous figure.
There are far larger numbers in existence; a googolplex, for instance, is written as 1 followed by a googol zeros. Mathematician Stanley Skewes has defined numbers that dwarf even a googolplex in size.
To provide perspective, even the smallest of these, the googol, vastly exceeds the number of particles in the observable universe, which is approximately 10^87.
Graham’s number, however, completely overshadows these 'small' numbers. It was introduced by Ronald Graham in his highly complex research on multi-dimensional hypercubes, serving as an upper bound for one of the solutions. To put it simply, Graham’s number is so immense that the entire universe lacks the capacity to physically represent it, even if each digit were as tiny as an electron. It’s incomparably larger than Skewes’ numbers.
One fascinating aspect of Graham’s number is that its final digits can be calculated, and it’s known to end with a 7.
8. Smallest Integer
Unknowable Thing: What’s the smallest positive integer not definable in under eleven words?
This question delves into the philosophy of mathematics. To clarify, an integer is a whole number (like 1, 2, 3, etc.), and smaller integers can easily be described in words:
“The square of 2” = 4 “One more than 4” = 5
...and so on. Now, imagine the vast number of eleven-word sentences possible. While there are many, the English language has a finite vocabulary (around 750,000 words), meaning there’s a finite number of eleven-word sentences. Eventually, you’d exhaust these combinations, leaving some integers indefinable in under eleven words. However, the phrase 'The smallest positive integer not definable in under eleven words' itself uses only ten words, creating a paradox.
This is known as Berry’s paradox, and it’s essentially a linguistic trick—shifting subtly from naming numbers to describing them. Despite this, no one can actually identify such a number!
7. Software
Unknowable Thing: Will a computer program ever stop?
During my Pure Mathematics classes in school, a frequent complaint was that the material seemed 'useless.' The teacher’s response was always, 'You’re learning this because it’s on the syllabus.' The Turing Halting problem might sound like a purely academic exercise with no real-world application. However, it played a crucial role in the creation of digital computers.
Alan Turing, an English mathematician and a prodigy from a young age, especially in mathematics, initially focused on theoretical aspects of computing machines. He aimed to represent mathematical statements numerically for processing by a hypothetical computing device. Turing conceptualized a universal computing machine, known today as a Turing Machine, purely as a theoretical construct without anticipating its physical realization.
Turing argued that any computer program would either run indefinitely or halt. He demonstrated that automatically predicting which outcome would occur is impossible. While one might suggest simply running the program to observe the result, consider the impracticality if the program halts only after 7 trillion years.
More on Turing: his reasoning is especially notable as it was developed in 1936, well before the advent of the first digital computer. By 1938, Turing was already engaged in code-breaking efforts at Bletchley Park, attempting to crack the German Enigma code. Recognizing the inefficiency of manual methods, Turing designed the Bombe, the initial decoding machine, which evolved into Colossus, the first programmable digital computer capable of testing numerous potential keys automatically. The significance of his contributions to cryptography was such that much of his work remained classified for decades, with some documents only recently disclosed, 60 years post their creation.
6. Does Not Compute
Unknowable Thing: Certain numbers exist that are beyond computation.
This intriguing concept was demonstrated by Alan Turing. To begin, the notion of 'infinity' isn't singular. Consider the count of positive whole numbers—it's infinite, as they never cease. Similarly, the quantity of positive even numbers is also infinite. Doubling any positive whole number yields an even number, indicating an equal infinite count.
Now, ponder the total of real numbers, which encompass fractions, irrational numbers like pi, and all whole numbers, whether positive or negative. The set of real numbers vastly outnumbers whole numbers. Between any two whole numbers lies an infinite array of real numbers, making the infinity of real numbers significantly larger than that of whole numbers.
With this understanding, one can logically deduce the following:
Imagine creating computer programs designed to produce real numbers, with each program corresponding to a unique real number.
You enumerate each program, assigning the first as '1', the second as '2', and so forth—utilizing the sequence of positive whole numbers for this purpose.
The issue arises because, despite your readiness to craft an endless array of programs, this infinity pales in comparison to the vastness of real numbers. Consequently, numerous—indeed, the majority of—real numbers remain unaccounted for and thus cannot be computed.
5. True or False?
Unknowable Thing: Within the realm of mathematics, there exist truths that elude proof—and their identities remain a mystery to us.
This mind-bending theorem was formulated by Kurt Gödel. The idea traces back to 1900 when David Hilbert presented 23 mathematical challenges he hoped would be resolved in the coming century. Among these was the quest to demonstrate the consistency of mathematics—a reassuring notion. However, Gödel disrupted this ambition in 1901 with his incompleteness theorem. While I won't delve into the specifics here—partly due to my limited grasp and partly because it took multiple lectures to even begin to comprehend—I recommend Wikipedia for those eager to explore further.
In essence, the theorem reveals that mathematics cannot be proven consistent solely through mathematics itself; a 'meta-language' is required. Additionally, Gödel demonstrated that certain mathematical truths exist which cannot be proven.
When I first encountered this theorem, it was hinted that Fermat’s Last Theorem might be one such 'unprovable truth.' However, this example was rendered moot when Andrew Wiles successfully proved it in 1995. That said, here are a couple of statements that might be true yet remain unprovable:
“No odd perfect number exists.”
A perfect number is defined as a positive whole number whose divisors sum up to the number itself. For instance, 6 is a perfect number—1 + 2 + 3 equals 1 * 2 * 3, which is 6.
28 stands as the next perfect number. These numbers are exceptionally rare, with only 41 discovered so far. The total number of perfect numbers remains a mystery, lying somewhere between 41 and infinity.
To date, all identified perfect numbers are even. However, it remains unknown whether an odd perfect number exists. If one does, it would be an extraordinarily large number, exceeding 10^1500—a 1 followed by 1500 zeros.
“Every even number can be expressed as the sum of two prime numbers.”
A prime number is divisible only by itself and 1. Interestingly, every even number tested so far has been the sum of two primes—for instance, 8 equals 5 plus 3, and 82 equals 51 plus 31. This holds true for numbers up to approximately 10^17. Additionally, as numbers grow larger, the likelihood of them being prime increases. Yet, the possibility remains that a sufficiently large even number might defy this pattern.
4. What’s Truth, man?
Within the realm of provability, we encounter Tarski’s undefinability theorem. To provide some context, here’s a glimpse into Tarski’s background.
Born to Jewish parents in pre-war Poland in 1901, Alfred Teitelbaum, later known as Tarski, was fortunate in many ways. Amidst the prevalent antisemitism of the time, he and his brother adopted the surname “Tarski” in 1923, a name they crafted to sound more Polish. They also converted from Judaism to Roman Catholicism, despite Alfred’s personal atheism.
During the late 1930s, Tarski sought several professorships in Poland but was unsuccessful. This rejection proved fortuitous. In 1939, he was invited to speak at a conference in the U.S., an opportunity he might have missed had he been newly appointed to a professorship. Tarski boarded the last ship departing Poland before the German invasion. Unaware he was fleeing, he left his children behind, expecting a quick return. They survived the war and reunited in 1946, though much of his extended family perished under German occupation.
Returning to the theorem, Tarski demonstrated that arithmetical truth cannot be defined within arithmetic itself. He further generalized this, showing that in any formal system, the concept of “truth” cannot be defined within that system.
Truth for a system can only be defined within a more powerful system, but this stronger system cannot define its own truth. This leads to an endless quest for truth, always requiring a yet stronger system, making ultimate truth unattainable.
3. Unknown Unknowables
Up to this point, we’ve discussed concepts that are inherently unknowable. Now, we turn to ideas that might be true but are impossible to verify. You might think examples are scarce, but consider this scenario:
Our universe is expanding, causing galaxies to move away from us at an accelerating rate. In approximately 2 trillion years, these galaxies will be so distant that they’ll no longer be observable. Their light will stretch into gamma rays with wavelengths exceeding the universe’s width. Future astronomers, living in that era, would have no way of knowing billions of galaxies once existed. Any claim about their existence would be met with skepticism and demands for evidence.
With this in mind, consider the present day—there could be truths about the universe that remain forever beyond our understanding. A sobering thought!
2. Chaitin’s Constant
Chaitin’s constant is a concept that mathematicians find logical but sounds bizarre to everyone else. It represents the probability that a randomly generated computer program will stop running. What makes it peculiar (among other things) is that each program has its own unique constant, resulting in an infinite number of values for this so-called “constant,” typically denoted by the Greek letter omega (Ω). Another intriguing aspect is that Ω is uncomputable—a frustrating limitation, as computing it could potentially resolve many unsolved mathematical problems.
1. Particle Particulars
Unknowable Thing: What is the exact location and speed of that particle?
We step away from the mind-bending realm of mathematics only to plunge into the even more perplexing domain of quantum physics. The uncertainty principle emerged from the study of subatomic particles, revolutionizing our understanding of the universe. In school, we were taught to visualize atoms as miniature solar systems, with a nucleus at the center and electrons orbiting like tiny marbles.
This model is entirely incorrect, and one of the breakthroughs that exposed its flaws was Heisenberg’s uncertainty principle. Werner Heisenberg, a German theoretical physicist, collaborated closely with Danish physicist Niels Bohr in the 1920s. Heisenberg’s reasoning can be summarized as follows:
To determine a particle’s position, I must observe it, which requires illuminating it with light. To illuminate it, I need to fire photons at it. However, when a photon strikes the particle, it alters the particle’s position. Thus, the act of measuring its position inherently changes it.
The principle states that it’s impossible to determine both the position and momentum of a particle at the same time. While this resembles the observer effect in experiments, where outcomes shift based on observation, the uncertainty principle is grounded in rigorous mathematics. It fundamentally altered our perception of the universe, particularly at the microscopic level. Electrons are no longer seen as fixed particles but as probability distributions—predicting their likely locations rather than pinpointing exact positions, as they could exist anywhere.
When first introduced, the uncertainty principle sparked significant controversy. Einstein famously objected, stating, “God does not play dice with the universe.” This period marked the divergence between quantum mechanics, which examines the smallest particles, and classical physics, which deals with larger objects and forces. This divide remains unresolved to this day.
+ Boring…
Unknowable Thing: Do uninteresting people truly exist?
It’s quite straightforward to argue that no one is truly uninteresting:
Imagine compiling a list of supposedly uninteresting individuals. Among them, the youngest would stand out simply by being the youngest—making them inherently interesting. Removing them from the list would leave a new youngest, who would also become interesting for the same reason. This process would continue until the list is empty. Therefore, if you ever think someone is uninteresting, you’ve likely misunderstood them.
