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CONSIDER a few

physicallycompletely indistinguishable twins who are, however,mentallydifferent people. Here “A” is represented in more than one instance, and the identity A=A is violated.The law of logic A is A is a consequence of our understanding of the stationarity of the world. It is impossible to imagine a reality where this law does not work. Our world is not real in the classical sense of the word.

The dialectical law of anisotropic pair avoidance is discovered. That is, nothing in the world is identical with itself, A + A = A growing old. You are not who you once were, you are getting old.

None at all. All our ideas about the world that we can spread from person to person are just simplified models based on basic entities (for example, a point, a straight line, an atom, etc.) and set rules for interaction between basic entities (the same logic, for example). Some may be better at describing objective reality, and some may be worse, but they are still just models.

In the universe (in the World), everything can be counted and measured. Therefore, the answer to this question can be given by natural integers. 1 is 1 whatever we take: a leaf from a tree; a planet; a photon; a window, etc. – all that will be ONE. It's the same with the other number-two or seven.

Numbers are the basis of the universe. (A. F. Losev devoted several of his works to the number.)

This is called in formal logic the” law ” (or concept) of

identity.At the heart of the universe is (if you believe philosophers and “common sense”) the “law of similarity”: “

” (instead of a triple colon, insert any “verb”: known, generated, attracted, treated, understood, etc.). The extremely abstract and ideal formalized expression of which seems to be the “law of identity”.like … likeSuch a “wonderful” reality (where there is no identity) does not even need to bother to imagine (it is always around you live!). Any “knowledge” without a knowing subject “does not work” (it does not exist at all without it), and therefore the “law of identity” does not work outside the knowing subject. As the ancients said, “you

” (complete ideal identity is possible only in completely abstract and formal “concepts” such as mathematics or geometry, and in real life there is only “similarity”, because everything flows and everything changes). But you, as a cognizing subject, have every right to think otherwise (there is no one “who is right” to judge “objectively”).can't step into the same water twiceIn short: yes, such a reality can be imagined. Moreover, we live in it, according to physicists.

Currently, there are detailed logical systems that reject the “A is A” law – the so-called “Schrodinger logics”. They are based on the idea of Erwin Schrodinger – one of the pioneers of quantum mechanics and a Nobel laureate – that in particle physics, talking about the identity of objects is often meaningless.

We know that quantum mechanics is often called the “killer” of the fundamental laws of classical logic – for example, the law of non-contradiction (a particle can be in two states that are incompatible from the classical point of view, so that a statement about it can be true and false at the same time), or the law of the excluded third (in addition to truth and falsity, statements about elementary particles often have to be assigned a third meaning – “uncertainty”). But the law of identity, it seems, should work even in the microcosm: how can a particle not be equal to itself?

In fact, we are somewhat deceived by the phrase “itself” – it already implies the identity of the particle in advance. It is as if we are able to decide separately whether there is one particle in front of us, and then check whether it is equal to itself. To get rid of the illusion of evidence, it would be useful to ask: on what grounds do we judge the identity or difference of something?

Traditionally, in matters of identification/distinction, we rely on the famous Leibnizian principle of “identity of the indistinguishable”: if objects have all their properties coincide (that is, they cannot be distinguished by any signs), then these objects are identical. All the usual logic and mathematics treat the concepts of identity and indistinguishability as synonymous. But as soon as we notice that indistinguishability is an epistemological concept (it describes the ability of the knowing subject to distinguish something), and identity is an ontological concept (it describes the characteristic of the object itself), we realize that they do not have to coincide. You never know what objects we can't distinguish – what does it matter to them? Should they “stick together” just because we can't tell them apart?

Let's return to elementary particles. What do we know about them? That their description is possible only in terms of quantum states. They have no properties other than quantum states. So, it turns out that in the subatomic world there are particles for which indistinguishability does not imply identity – these are bosons. An unlimited number of identical bosons can simultaneously exist in a single quantum state. (For fermions, by the way, the opposite is true – there can be no more than one fermion in the same quantum state. In other words, fermions behave like law-abiding citizens of the Leibniz universe, and bosons behave like hooligans who systematically violate the principle of identity of the indistinguishable.)

The bad news is that we have to pay a price for this boson hooliganism – first of all, by abandoning classical set theory. But almost all our logic and mathematics are built on it! By the way, for the first time the idea of the need to abandon classical set theory in describing the quantum world was expressed by our Russian mathematician Yuri Manin in 1976.

The good news is that we have a rough idea of how to do this. In 1980, the Brazilian logician Newton Da Costa developed quasi-set theory, where “quasi-set” refers to a collection of objects that are indistinguishable but not identical. Actually, semantics for Schrodinger logics is usually built on the basis of this theory. And if these logics have clear semantics, then we can really imagine how to reason about a reality in which the law “A is A”does not work.

A = A is not a “law of the universe”. This is the law of thinking. And not thinking at all, but only thinking about statements.

To understand its meaning, one must first understand the meaning of equivalence. A = B if and only if, when substituting in any statement containing the name A, we substitute B, the truth of the statement will not change. Example: “first positive even number”= “2”. If the statement “if you add one to the first positive even number, you get three” is replaced with the statement “if you add one to 2, you get three”, then the truth of the statement does not change (both statements are true). If the statement “the first positive even number is greater than a thousand” is replaced by the statement “2 is greater than a thousand”, the truth value also does not change (both statements are false). And the same applies to all statements where “the first positive even number is greater than a thousand” and “2”appear.

So, if the name A is replaced by itself in any statement, then the statement remains just as true or just as false.

That, in fact, is the whole point of this “basic law”.