Article about nothing

As a child, I was fascinated by the story, probably known to many readers, about "soup on a nail." My grandmother (XNUMXst century of birth) told me this in the version “The Cossack came and asked for water, because he has a nail and he will cook soup on it.” The curious hostess gave him a pot of water… and we know what happened next: “the soup should be salty, daitye, granny, salt”, then he washed the meat “to improve the taste” and so on. In the end, he threw away the "boiled" nail.

So this article was supposed to be about the emptiness of space - and this is about the landing of a European apparatus on the comet 67P / Churyumov-Gerasimenko on November 12, 2014. But while writing, I succumbed to a long-standing habit, I'm still a mathematician. How is it with Likeс Zero mathematics?

How does Nothing exist?

It cannot be said that Nothing exists. It exists at least as a philosophical, mathematical, religious and perfectly colloquial concept. Zero is an ordinary number, zero degrees on a thermometer is also a temperature, and a zero balance in a bank is an unpleasant but common occurrence. Note that there is no zero year in the chronology, and this is because zero was introduced into mathematics only in the late Middle Ages, later than the chronology proposed by the monk Dionysius (XNUMXth century).

Oddly enough, we really could do without this zero and, therefore, without negative numbers. In one of the textbooks on logic, I found an exercise: draw or say how you imagine the absence of fish. Amazing, isn't it? Anyone can draw a fish, but not one?

Now briefly basic mathematics course. Granting the existence privilege to the empty set marked with a crossed-out circle ∅ is a necessary procedure analogous to adding zero to the set of numbers. The empty set is the only set that does not contain any elements. Such collections:

But there are no two different empty sets. The empty set is included in every other set:

Indeed, the rules of mathematical logic say that set A is contained in set B if and only if the sentence:

it entails

In the case of the empty set ∅, the proposition is always false, and thus, according to the laws of logic, the implication is generally true. Everything stems from a lie (“here I will grow a cactus if you move to the next class ...”). So, since the empty set is contained in each of the others, then if they were two different ones, each of them would be contained in the other. However, if two sets are contained within each other, they are equal. That's why: there is only one empty set!

The postulate of the existence of an empty set does not contradict any laws of mathematics, so why not bring it to life? The philosophical principle calledOccam's razor» An order to exclude unnecessary concepts, but just right the concept of an empty set is very useful in mathematics. Note that the empty set has a dimension of -1 (minus one) - zero-dimensional elements are points and their sparse systems, one-dimensional elements are lines, and we talked about very complex mathematical elements with fractal dimension in the chapter on fractals.

It is interesting that the whole building of mathematics: numbers, numbers, functions, operators, integrals, differentials, equations ... can be derived from one concept - an empty set! It is enough to assume that there is an empty set, the newly created elements can be combined into sets to be able to build all the math. This is how the German logician Gottlob Frege constructed the natural numbers. Zero is a class of sets whose elements are in mutual correspondence with the elements of the empty set. One is a class of sets whose elements are in mutual correspondence with the elements of a set whose only element is the empty set. Two is a class of sets whose elements are one-to-one with the elements of the set consisting of the empty set and the set whose only element is the empty set... and so on. At first glance, this seems to be something very complicated, but in fact it is not.

Blue spread over me

smelled of heather and smelled of mint...

Wojciech Mlynarski, Harvest Girl

It is hard to imagine

Nothing is hard to imagine. In Stanisław Lem's story "How the World Was Saved", the designer Trurl built a machine that would do everything starting with a letter. When Klapaucius ordered it to be built Nic, the machine began to remove various objects from the world - with the ultimate goal of removing everything. By the time the frightened Klapaucius stopped the car, galleys, yews, hanging, hacks, rhymes, beaters, poufs, grinders, skewers, philidrons and frosts had disappeared from the world forever. And indeed, they disappeared forever ...

Józef Tischner wrote very well about nothingness in his History of Mountain Philosophy. During my last vacation, I decided to experience this nothingness, namely, I went to the peat bogs between Nowy Targ and Jabłonka in Podhale. This area is even called Pustachia. You go, you go, but the road does not decrease - of course, on our modest, Polish scale. One day I took a bus in the Canadian province of Saskatchewan. Outside was a cornfield. I took a nap for half an hour. When I woke up, we were driving through the same cornfield... But wait, is this empty? In a sense, the absence of change is just emptiness.

We are accustomed to the constant presence of various objects around us, and from Something you cannot run away even with your eyes closed. “I think, therefore I am,” said Descartes. If I have already thought something, then I exist, which means that there is at least something in the world (namely, I). Does what I thought exist? This can be discussed, but in modern quantum mechanics, the Heisenberg principle is known: each observation perturbs the state of the observed object. Until we see it Nic it does not exist, and when we start looking, the object ceases to be Like and it becomes Something. It's getting absurd anthropic principle: There is no point in asking what the world would be like if we didn't exist. The world is what it seems to us. Perhaps other beings will see the Earth as angular?

A positron (such a positive electron) is a hole in space, "there is no electron." In the process of annihilation, the electron jumps into this hole and “nothing happens” – there is no hole, no electron. I will skip a lot of jokes about holes in Swiss cheese (“the more I have, the less there ...”). The famous composer John Cage had already used his ideas to such an extent that he composed (?) a piece of music (?) in which the orchestra sits motionless for 4 minutes 33 seconds and, of course, does not play anything. “Four minutes and thirty-three seconds is two hundred and seventy-three, 273, and minus 273 degrees is absolute zero, at which all movement stops,” the composer (?) explained.

Filter to zero, nothing, nick, nick, nothing, zero!

Jerzy in Andrzej Wajda's film Over the Years,

days go by"

How about everyone?

Many people (from simple farmers to prominent philosophers) wondered about the phenomenon of existence. In mathematics, the situation is simple: there is something that is consistent.

She disappeared into the marginal fields

In cornflowers, weeds and lion's mouths ...

Well, things like this happen

Especially in the harvest, and during the harvest

especially…

Wojciech Mlynarski, Harvest Girl

Everything is at the other extreme of Nothing. In mathematics, we know that Everything doesn't exist. Just a far too inaccurate notion that his existence would be free of controversy. This can be understood by the example of the old paradox: "If God is omnipotent, then create a stone to pick up?" The mathematical proof that there cannot be sets of all sets is based on the theorem singer-Bershtein, which says that "an infinite number" (mathematical: cardinal number) the set of all members of a given set is greater than the number of elements of this set.

If a set has elements, then it has 2ⁿ subsets; for example, when = 3 and the set consists of {1, 2, 3} then the following subsets exist:

• three two-element sets: each of them is missing one of the numbers 1, 2, 3,
• one empty set,
• three one-element sets,
• the whole set {1,2,3}

– only eight, 2³And readers who have recently graduated from school, I would like to recall the corresponding formula:

Each of the Newtonian symbols in this formula determines the number of k-element sets in the -element set.

In mathematics, binomial coefficients appear in many other places, such as in interesting formulas for reduced multiplication:

and from their exact form, their interdependence is much more interesting.

It is difficult to understand what - as far as logic and mathematics are concerned - is, and what Everything is not. Arguments for non-existence Just the same as that of Winnie the Pooh, who politely asked his guest, Tiger, do Tigers like honey, acorns and thistles at all? “Tigers like everything,” answered the one from which Kubus concluded that if they like everything, then they also like to sleep on the floor, therefore, he, Vinnie, can return to bed.

Another argument Russell's paradox. There is a barber in town who shaves all the men who don't shave themselves. Does he shave himself? Both answers contradict the condition put forward that they kill those, and only those, who do not do it themselves.

Looking for a collection of all collections

In conclusion, I will give a clever, but most mathematical proof that there is no set of all sets (not to be confused with it).

First, we will show that for any non-empty set X, it is impossible to find a mutually unique function that maps this set to the set of its subsets P(X). So let's assume that this function exists. Let's denote it by the traditional f. What is f from x? This is a collection. Does xf belong to x? This is unknown. Either you have to or you don't. But for some x it must still be such that it does not belong to f of x. Well, then consider the set of all x for which x does not belong to f(x). Denote it (this set) by A. It corresponds to some element a of the set X. Does a belong to A? Let's assume you should. But A is a set containing only those elements of x that do not belong to f(x) ... Well, maybe it does not belong to A? But the set A contains all the elements of this property, and hence also A. End of the proof.

Therefore, if there were a set of all sets, it would itself be a subset of itself, which is impossible according to the previous reasoning.

Phew, I don't think many readers have seen this proof. Rather, I brought it up to show what mathematicians had to do at the end of the nineteenth century, when they began to study the foundations of their own science. It turned out that problems lie where no one expected them. Moreover, for the whole of mathematics, these reasoning about the bases do not matter: no matter what happens in the cellars - the whole building of mathematics stands on a solid rock.

Meanwhile, at the top...

We note one more morality from the stories of Stanislav Lem. In one of his travels, Iyon Tichi reached a planet whose inhabitants, after a long evolution, finally reached the highest stage of development. They are all strong, they can do anything, they have everything at their fingertips… and they do nothing. They lay down on the sand and pour it between their fingers. “If everything is possible, it’s not worth it,” they explain to the shocked Ijon. May this not happen to our European civilization...