Paradoxes can be found everywhere, from ecology to geometry and from logic to chemistry. Even the computer on which you are reading the article is full of paradoxes. Here are ten explanations for some rather fascinating paradoxes. Some of them are so strange that we simply cannot fully understand what the point is.
1. The Banach-Tarski paradox
Imagine that you are holding a ball in your hands. Now imagine that you started to tear this ball into pieces, and the pieces can be of any shape you like. Then put the pieces together so that you get two balls instead of one. How big will these balls be compared to the original ball?
According to set theory, the two resulting balls will be the same size and shape as the original ball. In addition, if we take into account that the balls have different volumes in this case, then any of the balls can be transformed in accordance with the other. This allows us to conclude that a pea can be divided into balls the size of the Sun.
The trick of the paradox is that you can break the balls into pieces of any shape. In practice, this cannot be done - the structure of the material and, ultimately, the size of the atoms impose some restrictions.
For it to be truly possible to break the ball the way you like, it must contain an infinite number of available zero-dimensional points. Then the ball of such points will be infinitely dense, and when you break it, the shapes of the pieces may turn out to be so complex that they will not have a certain volume. And you can collect these pieces, each of which contains an infinite number of points, into a new ball of any size. The new ball will still be composed of infinite points, and both balls will be equally infinitely dense.
Promotional video:
If you try to put the idea into practice, then nothing will work. But everything works out great when working with mathematical spheres - infinitely divisible number sets in three-dimensional space. The solved paradox is called the Banach-Tarski theorem and plays a huge role in mathematical set theory.
2. The Peto paradox
Obviously, whales are much larger than us, which means they have a lot more cells in their bodies. And every cell in the body can theoretically become malignant. Therefore, whales are much more likely to develop cancer than humans, right?
Not this way. The Peto Paradox, named after Oxford professor Richard Peto, argues that there is no correlation between animal size and cancer. Humans and whales have a similar chance of contracting cancer, but some breeds of tiny mice are much more likely.
Some biologists believe that the lack of correlation in the Peto paradox can be explained by the fact that larger animals are better at resisting tumors: the mechanism works in such a way as to prevent cell mutation during the division process.
3. The problem of the present
For something to physically exist, it must be present in our world for some time. There can be no object without length, width and height, and there can be no object without “duration” - an “instant” object, that is, one that does not exist for at least some amount of time does not exist at all.
According to universal nihilism, the past and the future do not take up time in the present. In addition, it is impossible to quantify the duration that we call "present time": any amount of time that you call "present time" can be divided into parts - past, present and future.
If the present lasts, say, a second, then this second can be divided into three parts: the first part will be the past, the second - the present, the third - the future. The third of a second, which we now call the present, can also be divided into three parts. You probably already got the idea - you can go on like this endlessly.
Thus, the present does not really exist because it does not last through time. Universal nihilism uses this argument to prove that nothing exists at all.
4. The Moravec paradox
When solving problems that require thoughtful reasoning, people have difficulty. On the other hand, basic motor and sensory functions such as walking are not difficult at all.
But if we talk about computers, the opposite is true: it is very easy for computers to solve the most complex logical problems like developing a chess strategy, but it is much more difficult to program a computer so that it can walk or reproduce human speech. This distinction between natural and artificial intelligence is known as the Moravec paradox.
Hans Moravek, a researcher in the Robotics Department at Carnegie Mellon University, explains this observation through the idea of reverse engineering our own brains. Reverse engineering is most difficult for tasks that humans do unconsciously, such as motor functions.
Since abstract thinking became a part of human behavior less than 100,000 years ago, our ability to solve abstract problems is conscious. Thus, it is much easier for us to create technology that emulates this behavior. On the other hand, we do not comprehend such actions as walking or talking, so it is more difficult for us to get artificial intelligence to do the same.
5. Benford's law
What is the chance that the random number will start with the number "1"? Or from the number "3"? Or with "7"? If you are a little familiar with the theory of probability, you can assume that the probability is one in nine, or about 11%.
If you look at the real numbers, you will notice that "9" is much less common than 11% of the time. There are also far fewer digits than expected, starting with "8", but a whopping 30% of numbers starting with the digit "1". This paradoxical picture manifests itself in all sorts of real cases, from population size to stock prices and river lengths.
Physicist Frank Benford first noted this phenomenon in 1938. He found that the frequency of occurrence of a digit as the first one drops as the digit increases from one to nine. That is, "1" appears as the first digit in about 30.1% of cases, "2" appears in about 17.6% of cases, "3" appears in about 12.5%, and so on until "9" appears in as the first digit in only 4.6% of cases.
To understand this, imagine that you are numbering lottery tickets sequentially. When you have numbered tickets from one to nine, there is an 11.1% chance of any number being first. When you add ticket # 10, the chance of a random number starting with "1" increases to 18.2%. You add tickets # 11 to # 19, and the chance that the ticket number starts with “1” continues to grow, reaching a maximum of 58%. Now you add ticket number 20 and continue to number the tickets. The chance that a number will start at "2" goes up, and the chance that it starts at "1" slowly decreases.
Benford's Law does not apply to all distributions of numbers. For example, sets of numbers whose range is limited (human height or weight) do not fall under the law. It also doesn't work with sets that are only of one or two orders.
However, the law covers many types of data. As a result, authorities can use the law to detect fraud: when the information provided does not follow Benford's law, the authorities can conclude that someone has fabricated the data.
6. C-paradox
Genes contain all the information needed to create and survive an organism. It goes without saying that complex organisms must have the most complex genomes, but this is not true.
Single-celled amoebae have genomes 100 times larger than humans, in fact, they have some of the largest genomes known. And in species very similar to each other, the genome can be radically different. This oddity is known as the C-paradox.
An interesting takeaway from the C-paradox is that the genome may be larger than necessary. If all the genomes in human DNA were to be used, then the number of mutations per generation would be incredibly high.
The genomes of many complex animals, such as humans and primates, include DNA that encodes nothing. This vast amount of unused DNA, which varies greatly from creature to creature, seems to be independent of anything, which creates the C-paradox.
7. An immortal ant on a rope
Imagine an ant crawling along a rubber rope one meter long at a speed of one centimeter per second. Also imagine that the rope stretches one kilometer every second. Will the ant ever make it to the end?
It seems logical that a normal ant is not capable of this, because the speed of its movement is much lower than the speed with which the rope stretches. However, the ant will eventually get to the opposite end.
Before the ant has even begun to move, 100% of the rope lies in front of it. A second later, the rope became much larger, but the ant also traveled some distance, and if you count in percentages, the distance that it must travel has decreased - it is already less than 100%, albeit not much.
Although the rope is constantly stretched, the small distance traveled by the ant also becomes larger. And while the overall rope lengthens at a constant rate, the ant's path gets slightly shorter every second. The ant also continues to move forward all the time at a constant speed. Thus, with every second the distance that he has already covered increases, and the distance that he must travel decreases. As a percentage, of course.
There is one condition for the problem to have a solution: the ant must be immortal. So, the ant will reach the end in 2.8 × 1043.429 seconds, which is slightly longer than the universe exists.
8. The paradox of ecological balance
The predator-prey model is an equation that describes the real ecological situation. For example, the model can determine how much the number of foxes and rabbits in the forest will change. Let's say that the grass that rabbits eat is growing in the forest. It can be assumed that such an outcome is favorable for rabbits, because with an abundance of grass they will reproduce well and increase their numbers.
The ecological balance paradox states that this is not so: at first, the number of rabbits will actually increase, but an increase in the rabbit population in a closed environment (forest) will lead to an increase in the fox population. Then the number of predators will increase so much that they will destroy all the prey first, and then they will die out themselves.
In practice, this paradox does not work for most animal species - if only because they do not live in a closed environment, so animal populations are stable. In addition, animals are able to evolve: for example, under new conditions, prey will have new defense mechanisms.
9. The newt paradox
Gather a group of friends and watch this video together. When done, have everyone give their opinion, whether the sound increases or decreases during all four tones. You will be surprised how different the answers will be.
To understand this paradox, you need to know a thing or two about musical notes. Each note has a certain pitch, which determines whether we hear a high or low sound. The note of the next higher octave sounds twice as high as the note of the previous octave. And each octave can be divided into two equal tritone intervals.
In the video, the newt separates each pair of sounds. In each pair, one sound is a mixture of the same notes from different octaves - for example, a combination of two C notes, where one sounds higher than the other. When the sound in a tritone transitions from one note to another (for example, a G sharp between two C's), you can reasonably interpret the note as being higher or lower than the previous one.
Another paradoxical property of newts is the feeling that the sound is constantly getting lower, although the pitch does not change. In our video, you can watch the effect for as long as ten minutes.
10. The Mpemba effect
Before you are two glasses of water, exactly the same in everything except one: the water temperature in the left glass is higher than in the right. Place both glasses in the freezer. In which glass will the water freeze faster? You can decide that in the right, in which the water was initially colder, but hot water will freeze faster than water at room temperature.
This strange effect is named after a Tanzanian student who observed it in 1986 when he froze milk to make ice cream. Some of the greatest thinkers - Aristotle, Francis Bacon, and René Descartes - have noted this phenomenon before, but have not been able to explain it. Aristotle, for example, hypothesized that a quality is enhanced in an environment opposite to this quality.
The Mpemba effect is possible due to several factors. There may be less water in a glass of hot water, since some of it will evaporate, and as a result, less water should freeze. Also, hot water contains less gas, which means that convection flows will occur more easily in such water, therefore, it will be easier for it to freeze.
Another theory is that the chemical bonds that hold water molecules together are weakened. A water molecule consists of two hydrogen atoms bonded to one oxygen atom. When the water heats up, the molecules move slightly away from each other, the bond between them weakens, and the molecules lose some energy - this allows hot water to cool faster than cold water.