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There are countless things in this world that remain unclear, simply because life is full of complexity. However, we can find comfort in knowing that there are fundamental ideas that anyone can grasp. But, as is often the case, when we dive deeper into these seemingly simple ideas, we realize we’ve uncovered a massive and intricate subject.
10. The 300-Page Proof of ‘1+1=2’
The equation 1+1=2 is likely the first math concept most of us encounter, as addition and subtraction are some of the most basic ideas in mathematics. If you have one apple and someone gives you another, you end up with two apples. Similarly, if you start with two apples and someone takes one, you’re left with just one. This simple truth crosses all barriers, from language to culture. Yet, here’s the surprising part: the proof for 1+1=2 spans over 300 pages and wasn’t definitively established until the 20th century.
As Stephen Fry explains in this helpful video, Bertrand Russell, in the early 1900s, sought to definitively prove the validity of mathematics. He began with the most basic concept we know and set out to demonstrate that 1+1=2. What seemed like a simple task turned into a challenge, requiring the mathematician and philosopher 372 pages of complex equations. This monumental work was published as Principia Mathematica in three volumes, which you can dive into if you have a few weeks to spare in the near future.
9. The Concept of ‘Almost Surely’ Is a Mathematical Headache
If we were to say that an event is almost certain to happen, how would you explain that to a child? Perhaps you’d call it nearly guaranteed, but then you’d be left to clarify what 'nearly' means in this context, which would only add to the confusion. It’s a tricky concept because the idea of something being “almost sure” is inherently vague.
Fortunately, this idea is well defined in the realm of statistical mathematics, though it’s a bit daunting at first glance. To borrow from an online mathematics textbook:
“In probability theory, a property is said to hold almost surely if it holds for all sample points, except possibly for some sample points forming a subset of a zero-probability event.”
In simpler terms, this means that even when an event has a 100% chance of happening, it might not actually happen. For instance, if you flipped a coin a million times, statistically, the likelihood of getting at least one heads is practically one. However, there remains an almost nonexistent chance that the coin could land on tails every time. So, while the odds are virtually certain, it's still impossible to state that the event will definitely occur.
8. Explaining the Word ‘The’ Is Surprisingly Complex
The word “the” is among the most frequently used words in the English language. It’s so common that many of us never pause to consider how peculiar it really is.
As explored here, it’s one of the hardest words to explain to non-native English speakers because it has such a broad range of uses, some of which seem oddly confusing when examined closely. To quote:
“Why do we say, ‘I love the ballet,’ but not ‘I love the cable TV?’ Why do we say, ‘I have the flu,’ but not ‘I have the headache?’ Why do we say, ‘winter is the coldest season,’ and not ‘winter is coldest season?’”
Think about it—‘the’ is used in countless scenarios, pointing to various ideas, objects, and concepts. It can indicate everything from a specific item to an abstract idea, and native speakers can effortlessly sense when it’s used wrongly without even realizing it.
As highlighted in the article linked above, the dictionary outlines nearly twenty different ways the word can be correctly used, which makes finding a clear-cut definition even more challenging. Don’t believe us? Try defining it yourself in the comments and share your thoughts.
7. No One Agrees on a Single Theory About How Bikes Function
Bicycles have been around for over a century, and during that time, we’ve perfected travel across land, sea, and air, while making incredible advancements in space exploration. We now have planes that can cross the globe in just a few hours, so by now, you’d think we would have the simple bicycle fully understood. But surprisingly, that’s not the case.
As highlighted in this article, there has been ongoing debate among scientists about how bicycles work, particularly how they manage to stay upright, almost since they were first introduced. For many years, the dominant theory suggested that the spinning wheels' gyroscopic force helped keep the bike steady. However, when scientists created a special bike with devices to eliminate the gyroscopic forces, it still remained upright, and no one could explain why.
Some theories suggest that the bike’s design allows it to steer into a fall, correcting its balance, but these are still mere hypotheses. And since bicycle dynamics isn’t a particularly popular field of study, it’s unlikely that we’ll have a concrete answer anytime soon.
6. How Long Is a Piece of String? The Answer Is Elusive
If someone hands you a piece of string and asks you how long it is, you'd probably think that answering the question would be a simple, albeit somewhat strange task. But what if they asked you to specify the exact length of that string? This was the challenge comedian Alan Davies took on for a BBC special called *How Long is a Piece of String?* by asking a group of scientists the seemingly straightforward question.
The response, ironically, was “it depends,” because the precise definition of length varies depending on who you ask. Mathematicians explained to the comedian that a piece of string could, in theory, be infinitely long. On the other hand, physicists pointed out that due to subatomic physics and the phenomenon where atoms can be in two places at once, measuring the string's length is impossible.
5. Yawning
Yawning is an intriguing and puzzling phenomenon. Just talking about it can cause some people to do it (you might be yawning right now). There's really no other bodily function quite like it.
Some of you might be familiar with the long-held theory that yawning serves to keep us alert by encouraging our bodies to inhale a large quantity of oxygen. This theory makes sense, as we tend to yawn when we’re tired or bored—situations where an extra burst of energy would be helpful.
However, experiments have conclusively disproven this idea over time. In fact, there is no widely accepted explanation for why we yawn, despite the fact that everyone does it. One commonly believed theory suggests that yawning cools the brain, as experiments have shown that the brain's temperature is one of the few things that changes during a yawn.
As for why yawning is contagious, no one has the answer to that either.
1. We Appreciate Things for Reasons Beyond Simple Enjoyment
Enjoyment is a peculiar thing because it’s entirely subjective—what one person finds delightful, another might strongly dislike. You might assume that we enjoy things because they make us feel good in some way, but scientists have definitively shown that this is only part of the equation.
For instance, people can be convinced they love a certain food or wine simply by being told it’s incredibly expensive. The same goes for objects—people will naturally opt for a more expensive product over a cheaper alternative, just due to the price. Enjoyment barely factors into the decision. This phenomenon is known in marketing as the 'Chivas Regal effect,' named after the scotch that saw its sales soar after they merely raised the price of their product.
To further illustrate, consider the famous experiment in which wine experts were deceived into thinking a cheap bottle of wine was an exquisite vintage, just by changing the labels. Their enjoyment didn’t stem from a deep appreciation of wine—it was based solely on the belief that they were drinking fine wine. Which, to be honest, is much simpler.
4. The Concepts of Left and Right Have Long Baffled Philosophers
How would you describe the idea of left and right to someone who has never heard those terms? Would you reference your relative position to a familiar, unchanging landmark? Or perhaps you’d try a more abstract approach, maybe using the rotation of the Earth or something similarly stable and constant. But what if you were speaking to an alien from a planet with a different rotation, or one who couldn’t see? This age-old question has intrigued philosophers for years because, without a universal reference point, defining left and right becomes an incredibly difficult task.
Take German philosopher Immanuel Kant, for example. He once said, 'Imagine the first thing created was a human hand, then it must necessarily be either a right hand or a left hand.'
Yet, with only one hand, it’s impossible to identify which one it is without a second one for comparison. Think about it—right and left hands are distinctly different, but if you were to describe them, your descriptions would be identical because, in essence, they are the same. But they’re not the same because, as Kant noted, a left hand won’t fit into a right-handed glove, demonstrating a clear difference between the two. However, explaining this difference is nearly impossible without the presence of both hands.
If you think this topic is getting a bit too complicated, consider the fact that there is actually a 400-page book dedicated to the philosophy of right and left, called The Philosophy Of Right And Left. To put that in perspective, that’s more pages than it took to prove that 1+1=2.
10. The Proof for ‘1+1=2’ Is 300 Pages Long
The equation 1+1=2 is likely the first math concept most of us encounter. Addition and subtraction are some of the easiest ideas in mathematics. For instance, if you have one apple and someone gives you another, you end up with two apples. Similarly, if you have two apples and one is taken away, you're left with one apple. This is a basic, universal truth that transcends all languages and cultures. What makes the following statement so astonishing is that the proof of 1+1=2 spans over 300 pages and wasn't conclusively established until the 20th century.
As Stephen Fry illustrates in this handy clip, Bertrand Russell embarked on a monumental task in the early 20th century. He aimed to demonstrate that mathematics was fundamentally sound, starting with the simplest premise: 1+1=2. But what seemed like an obvious truth turned into a gargantuan challenge, requiring Russell a staggering 372 pages of complex calculations. His comprehensive solution was published in Principia Mathematica, a three-volume work, which we invite you to explore—assuming you're not too busy for the next few weeks.
9. The Definition of ‘Almost Surely’ Is a Mathematical Nightmare
When we say an event is 'almost sure' to happen, how would we explain that to a young child? Perhaps we’d call it 'practically guaranteed,' but this raises the question of what 'practically' really means in that context, which only complicates things further. The challenge lies in the inherent vagueness of the idea that something is 'almost sure' to occur—it's an ambiguous concept in and of itself.
Fortunately, the concept is grounded in statistical mathematics, which offers a clear explanation. Unfortunately, at first glance, it can seem overwhelmingly complex. As an online math textbook on the subject states:
“In probability theory, a property is said to hold almost surely if it holds for all sample points, except possibly for some sample points forming a subset of a zero-probability event.”
In simpler terms, this means that even when an event has a 100 percent likelihood of happening, it won’t necessarily occur. For instance, if you flipped a coin a million times, the probability of landing on heads at least once is effectively one. Yet, there remains an almost negligible chance that the coin could land on tails every single time. So while the event is practically guaranteed to happen, it’s still impossible to state with certainty that it will.
8. Defining the Word ‘The’ Is Really Difficult
The word 'the' is one of the most common terms in the English language. It's so frequently used that many of us might never pause to consider how peculiar it truly is.
As explained here, it's arguably one of the toughest words to describe to someone who isn’t a native English speaker, due to its vast range of uses, some of which seem rather strange when observed from an objective viewpoint. To illustrate:
‘Why do we say, ‘I love the ballet,’ but not ‘I love the cable TV?’ Why do we say, ‘I have the flu,’ but not ‘I have the headache?’ Why do we say, ‘winter is the coldest season,’ and not ‘winter is coldest season?’ ’
Consider this—we use 'the' in countless different contexts, referring to everything from specific objects to abstract ideas, effortlessly switching between them. Native speakers can instinctively sense when it’s misused without even thinking.
As mentioned in the referenced article, the dictionary itself identifies nearly twenty different ways this word can be properly used, making it incredibly hard to pinpoint a precise definition. Don’t believe us? Try defining it yourself in the comments and let us know how you fare.
7. There’s No Single, Widely Accepted Explanation for How Bicycles Operate
Bicycles have been around for more than a century. While we've made astounding strides in land, sea, and air travel, and even ventured into space, the seemingly simple bicycle remains a bit of a mystery. Despite having airplanes that can circle the globe in mere hours, we haven't quite unlocked the full secrets of the bicycle yet.
As detailed in this article, scientists have long been at odds over the precise mechanics of how bikes stay upright. This debate has persisted almost since the bicycle's invention. For a while, the dominant theory was that the spinning wheels' gyroscopic forces kept the bike balanced, but when a specially-designed bike was built to negate these forces, it still stayed upright, leaving experts at a loss for an explanation.
Some theorize that the design of the bike allows it to steer into a fall, correcting itself in the process. However, these are still just speculative ideas. Furthermore, since bicycle dynamics isn't a field that attracts much research funding, it seems unlikely that we'll uncover a concrete answer anytime soon.
6. The Length of a String: A Question with No Clear Answer
Imagine someone handing you a piece of string and asking how long it is. You’d probably think it’s a simple, if somewhat unusual question. But how would you respond if they wanted an exact measurement? This question was central to comedian Alan Davies' BBC TV special titled 'How Long is a Piece of String?', where he asked a group of scientists to give him a precise answer.
The answer turned out to be unexpectedly elusive. Mathematicians argued that a piece of string could theoretically have infinite length, while physicists claimed that the challenge of accurately measuring a string arises from subatomic physics. Since atoms can exist in multiple places simultaneously, determining the string’s exact length is, quite literally, impossible.
5. The Science of Yawning
Yawning is a strange and perplexing phenomenon. In fact, even talking about yawning can trigger the urge to do it (some of you might be yawning right now). There’s simply no other bodily function quite like it.
Some of you might recognize the age-old belief that yawning helps us stay awake by providing a boost of oxygen. This idea makes sense, especially since we tend to yawn when we’re feeling tired or bored—moments when extra energy would certainly come in handy.
However, years of experimentation have thoroughly debunked this theory. In reality, there’s no universally accepted explanation for why we yawn, even though it’s something everyone does. One widely supported theory suggests that yawning helps cool the brain, as studies show that during a yawn, the temperature of the brain itself is one of the few things that noticeably changes.
As for why yawning is contagious? Well, that remains a mystery, too.
The distinction between left and right has baffled philosophers for centuries. These two directions are deeply ingrained in our understanding, yet their exact definition remains elusive. The challenge lies in pinpointing a fixed reference point from which to define them, which can be especially tricky when considering the vast variety of perspectives and experiences across different individuals and cultures. This conundrum continues to spark curiosity and debate.
Imagine trying to explain the concept of left and right to someone with no prior knowledge of these words. Do you use the position of a fixed landmark as your point of reference, or do you get creative and point to the Earth's rotation, a constant and unchanging force? Now, imagine you were speaking to an alien whose planet rotates differently than ours, or one who doesn't even have eyes. This is a classic philosophical puzzle that challenges the very definition of what left and right truly are, especially in the absence of a universal reference point.
Consider the thoughts of German philosopher Immanuel Kant, who proposed an intriguing idea: 'Let it be imagined that the first created thing were a human hand, then it must necessarily be either a right hand or a left hand.' This statement explores the notion that, without a second hand to compare it to, the distinction between the two becomes somewhat ambiguous. The idea challenges our ability to define right and left solely based on the inherent qualities of each hand, as it requires context to make sense.
However, even with a single hand, it’s difficult to explain its nature without the comparison to another. Think about it—while left and right hands are clearly distinct from each other, we struggle to articulate their differences in meaningful terms without referencing the other. Kant pointed out that a left hand won't fit into a right-handed glove, underscoring that while we can physically identify a difference, the essence of that difference is impossible to fully describe without context.
If you think we're making this too complicated, consider the existence of a 400-page book that delves into the philosophy of right and left, appropriately titled The Philosophy Of Right And Left. That’s more pages than were needed to prove that 1+1=2.
1. We Find Enjoyment in Things for Reasons Beyond Simple Enjoyment
Enjoyment is a strange concept, since it’s so subjective—one person might absolutely love a particular food, song, or movie, while another person could dislike it intensely. You might assume that we enjoy things simply because they make us feel good, but research has conclusively shown that this is only part of the explanation.
For instance, people can be tricked into thinking they love a certain food or wine simply by being told it’s very expensive. The same principle applies to products—many people will choose the pricier option over a cheaper one, driven only by the price difference. Enjoyment is rarely the main factor. This behavior is called the 'Chivas Regal effect' in marketing, named after the scotch brand that saw its sales soar after they just raised the price of their product.
To further make the point clear, there’s a well-known experiment where wine experts were tricked into thinking a cheap bottle of wine was a high-end vintage simply by switching the labels. Their enjoyment wasn’t driven by a deep love for wine—it was entirely based on the belief that they were drinking something special. In truth, that’s a much simpler explanation.
2. Some Mosquitoes Are Drawn to People Because of Their Clothing
If you’ve ever had the displeasure of being bitten by a mosquito, you’ve probably heard someone nearby offer a recycled explanation for why that insect chose you. Maybe they said you smelled nice, or that your blood type was irresistible, or maybe they even claimed your shirt made you look like an easy target. And no, we’re not being sarcastic—those are all things scientists believe can make you more appealing to mosquitoes.
As a recent Smithsonian article explains, 20 percent of people seem to have an inexplicable attraction for mosquitoes, and the reasons remain unclear. The simple explanation might be that something in a person’s blood attracts them, but it seems mosquitoes are actually drawn to a chemical signal emitted by the body. This signal is present in about 85 percent of us, which may also explain why some people appear invisible to mosquitoes—and it even reveals your blood type.
An even stranger theory suggests that mosquitoes are drawn to darker, more vivid colors. In fact, it's been hypothesized—and in some cases proven—that mosquitoes might bite people simply because they find their shirt appealing.
3. Rock-Paper-Scissors Is the Most Serious Game in the World
At first glance, rock-paper-scissors seems like the simplest game—it's the go-to method for settling any dispute because it’s just a matter of random chance, right?
Well, that’s not the case if you take into account the many research papers written on the subject. Psychologists have taken a keen interest in the game because of how closely it relates to subconscious human responses and game theory. As a result, there are numerous strategies designed to give players an advantage—including playing blindfolded to avoid being unconsciously influenced by an opponent’s body language.
