Another batch of paradoxes and thought experiments
This collection will take you much less time to read than to reflect on the paradoxes presented in it. Some of the problems are contradictory only at first glance, others, even after hundreds of years of intense mental work on them by the greatest mathematicians, philosophers and economists, seem insoluble. Who knows, maybe it’s you who will be able to formulate a solution to one of these problems, which will become, as they say, textbook and will be included in all textbooks.
1. The paradox of value
The phenomenon, also known as the diamond and water paradox or the Smith paradox (named after Adam Smith, the classical economist who is believed to be the first to formulate this paradox), is that while water as a resource is much more useful than pieces of crystal carbon, which we call diamonds, the price of the latter on the international market is incomparably higher than the cost of water.
Adam Smith
From the point of view of survival, humanity really needs water much more than diamonds, but its reserves, of course, are more than those of diamonds, so experts say that there is nothing strange in the price difference - after all, we are talking about the cost per unit of each resource, and it is largely determined by this a factor like marginal utility.
With a continuous act of consumption of a resource, its marginal utility and, as a result, the value inevitably falls - this pattern was discovered in the 19th century by the Prussian economist Hermann Heinrich Gossen. In simple terms, if a person is consistently offered three glasses of water, he will drink the first one, wash the water from the second, and the third will go to the floor.
Promotional video:
Most of humanity does not experience an acute need for water - to get enough of it, you just have to turn on the water tap, but not everyone has diamonds, which is why they are so expensive.
2. The paradox of the murdered grandfather
This paradox was suggested in 1943 by the French science fiction writer Rene Barzhavel in his book The Careless Traveler (original Le Voyageur Imprudent).
Rene Barzhavel
Suppose you managed to invent a time machine, and you went to the past on it. What happens if you meet your grandfather there and kill him before he met your grandmother? Probably, not everyone will like this bloodthirsty scenario, so, say, you prevent the meeting in another way, for example, take him to the other end of the world, where he will never know about its existence, the paradox does not disappear from this.
If the meeting does not take place, your mother or father will not be born, will not be able to conceive you, and accordingly you will not invent a time machine and go back in time, so grandfather will be able to marry grandmother without hindrance, they will have one of your parents, and so on. - the paradox is obvious.
The story of the grandfather who was killed in the past is often cited by scientists as proof of the fundamental impossibility of time travel, but some experts say that under certain conditions the paradox is quite solvable. For example, by killing his grandfather, the time traveler will create an alternate version of reality in which he will never be born.
In addition, many suggest that even having fallen into the past, a person will not be able to influence him, as this will lead to a change in the future, of which he is a part. For example, an attempt to murder a grandfather is deliberately doomed to failure - after all, if a grandson exists, then his grandfather, one way or another, survived the assassination attempt.
3. Ship Theseus
The name of the paradox was given by one of the Greek myths describing the exploits of the legendary Theseus, one of the Athenian kings. According to legend, the Athenians kept the ship on which Theseus returned to Athens from the island of Crete for several hundred years. Of course, the ship gradually deteriorated, and the carpenters replaced the rotten boards with new ones, as a result of which not a piece of old wood remained in it. The best minds in the world, including prominent philosophers like Thomas Hobbes and John Locke, have pondered for centuries whether theseus could be considered to have been on this ship.
Thus, the essence of the paradox is as follows: if you replace all parts of the object with new ones, can it be the same object? In addition, the question arises - if you assemble exactly the same object from the old parts, which of the two will be "the same"? Representatives of different philosophical schools gave directly opposite answers to these questions, but some contradictions in possible solutions to Theseus' paradox still exist.
By the way, given that the cells of our body are almost completely renewed every seven years, can we assume that in the mirror we see the same person as seven years ago?
4. Galileo's paradox
The phenomenon discovered by Galileo Galilei demonstrates the contradictory properties of infinite sets. A brief formulation of the paradox is as follows: there are as many natural numbers as there are squares, that is, the number of elements of an infinite set 1, 2, 3, 4 … is equal to the number of elements of an infinite set 1, 4, 9, 16 …
At first glance, there is no contradiction here, but the same Galileo in his work "Two Sciences" asserts: some numbers are exact squares (that is, you can extract a whole square root from them), while others are not, therefore, exact squares together with ordinary numbers there must be more than one exact square. Meanwhile, earlier in "Sciences" there is a postulate that there are as many squares of natural numbers as there are natural numbers themselves, and these two statements are directly opposite to each other.
Galileo himself believed that the paradox can be solved only in relation to finite sets, but Georg Cantor, one of the German mathematicians of the 19th century, developed his theory of sets, according to which Galileo's second postulate (about the same number of elements) is also true for infinite sets. For this, Cantor introduced the concept of cardinality, which coincided in the calculations for both infinite sets.
5. The paradox of frugality
The most famous formulation of a curious economic phenomenon described by Waddill Ketchings and William Foster is: "The more we save for a rainy day, the sooner it will come." To understand the essence of the contradiction contained in this phenomenon, a little economic theory.
William Foster
If during an economic downturn, a large part of the population begins to save their savings, the aggregate demand for goods decreases, which in turn leads to a decrease in earnings and, as a consequence, to a drop in the overall level of savings and a reduction in savings. Simply put, there is a kind of vicious circle where consumers spend less money, but thereby worsen their well-being.
In a way, the paradox of frugality is analogous to the problem in game theory called the prisoner's dilemma: actions that are beneficial to each participant in a situation individually are harmful to them as a whole.
6. The Pinocchio paradox
This is a subset of the philosophical problem known as the liar paradox. This paradox is simple in form, but by no means in content. It can be expressed in three words: "This statement is a lie", or even in two words - "I am lying." In the version with Pinocchio, the problem is formulated as follows: "My nose is growing now."
I think you understand the contradiction contained in this statement, but just in case, let's dot everything over it: if the phrase is correct, then the nose is really growing, but this means that at the moment the brainchild of Pope Carlo is lying, which cannot be, so as we have already found out that the statement is true. This means that the nose should not grow, but if this does not correspond to reality, the statement is still true, and this in turn indicates that Pinocchio is lying … And so on - the chain of mutually exclusive causes and effects can be continued indefinitely.
The paradox of the liar shows the contradiction between the statement in colloquial speech and formal logic. From the point of view of classical logic, the problem is insoluble, so the statement "I am lying" is not considered logical at all.
7. Russell's paradox
The paradox, which its discoverer, the famous British philosopher and mathematician Bertrand Russell, called nothing other than the barber's paradox, strictly speaking, can be considered one of the forms of the liar's paradox.
Suppose, as you walk past a hairdresser, you see an advertisement on it: “Do you shave yourself? If not, you are welcome to shave! I shave everyone who does not shave himself, and no one else! " It is natural to ask the question: how does a barber manage his own stubble if he shaves only those who do not shave on their own? If he himself does not shave his own beard, this is contrary to his boastful statement: "I shave all who do not shave themselves."
Of course, it is easiest to assume that the narrow-minded barber simply did not think about the contradiction contained in his signboard and forget about this problem, but trying to understand its essence is much more interesting, although this will require a short plunge into mathematical set theory.
Russell's paradox looks like this: “Let K be the set of all sets that do not contain themselves as a proper element. Does K contain itself as its own element? If yes, this refutes the statement that the sets in its composition "do not contain themselves as a proper element", if not, there is a contradiction with the fact that K is the set of all sets that do not contain themselves as a proper element, and therefore K must contain all possible elements, including yourself."
The problem arises due to the fact that Russell in his reasoning used the concept of "the set of all sets", which in itself is rather contradictory, and was guided by the laws of classical logic, which are not applicable in all cases (see paragraph six).
The discovery of the barber paradox provoked heated debates in various scientific circles, which have not subsided to this day. To "save" set theory, mathematicians have developed several systems of axioms, but there is no evidence of the consistency of these systems and, according to some scientists, there cannot be.
8. The birthday paradox
The crux of the problem is this: if there is a group of 23 or more people, the probability that two of them have the same birthday (day and month) is greater than 50%. For groups from 60 people, the chance is over 99%, but it reaches 100% only if there are at least 367 people in the group (taking into account leap years). This is evidenced by the Dirichlet principle, named after its discoverer, the German mathematician Peter Gustav Dirichlet.
Peter Gustav Dirichl
Strictly speaking, from a scientific point of view, this statement does not contradict logic and therefore is not a paradox, but it perfectly demonstrates the difference between the results of an intuitive approach and mathematical calculations, because at first glance, for such a small group, the probability of coincidence seems greatly overestimated.
If we consider each member of the group individually, estimating the likelihood of their birthday coinciding with someone else, for each person the chance is approximately 0.27%, so the total probability for all members of the group should be about 6.3% (23 / 365). But this is fundamentally wrong, because the number of possible options for choosing certain pairs of 23 people is much higher than the number of its members and is (23 * 22) / 2 = 253, based on the formula for calculating the so-called number of combinations from a given set. We will not delve into combinatorics, you can check the correctness of these calculations at your leisure.
For 253 variants of couples, the chance that the month and date of birth of the participants of one of them will be the same, as you probably guessed, is much more than 6.3%.
9. The problem of chicken and eggs
Surely, each of you at least once in your life was asked the question: "What appeared first - a chicken or an egg?" Experienced in zoology know the answer: birds were born from eggs long before the appearance of the order of chickens among them. It is worth noting that in the classical formulation it is just about a bird and an egg, but it also allows an easy solution: after all, for example, dinosaurs appeared before birds, and they also multiplied by laying eggs.
If we take into account all these subtleties, we can formulate the problem as follows: what appeared earlier - the first animal that lays eggs, or its own egg, because from somewhere a representative of a new species had to hatch.
The main problem is to establish a causal relationship between the phenomena of fuzzy volume. For a more complete understanding of this, check out the Principles of Fuzzy Logic - generalizations of classical logic and set theory.
To put it simply, the fact is that animals in the course of evolution have gone through countless intermediate stages - this also applies to the methods of breeding. At different evolutionary stages, they laid different objects that cannot be unequivocally identified as eggs, but have some similarities with them.
Probably, there is no objective solution to this problem, although, for example, the British philosopher Herbert Spencer proposed this option: "A chicken is just a way in which one egg produces another egg."
10. Cell disappearance
Unlike most of the other paradoxes of the collection, this playful "problem" does not contain contradictions, rather serves to train observation and makes you remember the basic laws of geometry.
If you are familiar with such tasks, you can skip watching the video - it contains its solution. We suggest everyone else not to climb, as they say, “to the end of the textbook,” but to think about it: the areas of the multi-colored figures are absolutely equal, but when they are rearranged, one of the cells “disappears” (or becomes “unnecessary” - depending on which variant of the position of the figures considered as initial). How can this be?
Hint: initially there is a little trick in the problem, which ensures its "paradoxicality", and if you manage to find it, everything will immediately fall into place, although the cell will still "disappear".