Chaos Eliminates the Need for the Multiverse

Original author: Noson S. Yanofsky
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Scientists are exploring the universe and seeing an amazing structure. Fantastic complex objects and processes are found in it. Each event in the universe follows the exact laws of nature, ideally expressed in the language of mathematics. These laws seem to us to be fine-tuned so that life can appear, and, in particular, intelligent life. What are these laws of nature and how do we find them?

The universe is so well structured and ordered that we compare it with the most complex and accurate inventions of our time. In the XVIII and XIX centuries, the Universe was compared with perfectly working watches. Philosophers then discussed the Watchmaker. In the XX and XXI centuries, the most complex object is a computer. The universe is compared to a perfectly functioning supercomputer . Researchers are wondering: how was this computer programmed?

How can this whole structure be explained? Why do laws seem ideally tuned to the emergence of life, and why are they expressed in such a precise mathematical language? Is the universe really as structured as it seems?

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One of the answers to these questions is Platonism (or its cousin, realism ). It is a belief that the laws of nature are objective and have always existed. They have the exact and ideal form that exists in the world of Plato. These laws work perfectly, and they formed the universe that we are observing. They not only exist in this world, but also live next to ideal mathematics. This should help explain why laws are written in the language of mathematics.

Platonism leaves much to be desired, and much more. The main problem of Platonism is metaphysics, not science. But even if we accept it, there will still be many questions. Why are there laws in the platonic world that give rise to intelligent life in the Universe, and not some others? How did this platonic attic come about? Why does the physical universe follow ephemeral rules? How do scientists and mathematicians gain access to Plato's treasure chest in the form of exact ideals?

The multiverse is another answer that has recently become very fashionable. This theory is an attempt to explain why there are laws in our universe that give life. The believer in the Multiverse asserts that our universe is only one of many. Each universe has its own set of rules and its possible structures that correspond to them. Physicists promoting the multiverse theory believe that the laws in every universe are random. In our Universe, we see structures suitable for life because we are fortunate enough to live in one of the few universes in which there are such laws. And although the Multiverse explains some of the structures we see, some questions remain open. Instead of asking why the Universe has such structures, we can ask why the Multiverse has such structures. One more problem: if the Multiverse will answer some of our questions, then who said that it exists? Since most believe that we have no connection with other universes, the question of the existence of the Multiverse remains in the field of metaphysics.

There is another, more interesting, explanation of the structure of the laws of nature. Instead of saying that the Universe is very structured, say that the Universe is chaotic and for the most part there is no structure in it. And the reason we see structures is because scientists work like a sieve; they focus only on those phenomena that have a structure and that can be predicted. They do not consider all phenomena. Instead, they choose only those that they can handle.

Some people will say that science studies all physical phenomena. This is not true. Who will win the next presidential election and move to the White House is a physical question, but not a single scientist will seek an exact answer to it. Whether the computer will stop working or not with certain input data is a physical question, and yet from Alan Turing we learn that it is impossible to answer it. Scientists have classified the general textures and heights of various types of clouds, but are generally not at all interested in the exact shape of the cloud. Although its form is a physical phenomenon, scientists do not even try to study it. Science does not study all phenomena. Science studies predictable physical phenomena. It is almost a tautology: science predicts predictable phenomena.

Scientists described a criterion for the phenomena that they decided to study: it is called symmetry. Symmetry is a property according to which, despite the change of something, some unchanged part remains. When we say that the face has symmetry, we mean that if you reflect the left side and replace it with the right, it will look the same. When physicists use the word "symmetry", they discuss sets of physical phenomena. A set of phenomena has symmetry if, after some change, it remains the same. The most obvious example is location symmetry. This means that if you conduct the same experiment in two different places, the results should be the same. The symmetry of time means that the results of experiments should not depend on when the experiment was conducted. There are many other types of symmetry.

Phenomena chosen by scientists for research must have many different types of symmetries. When a physicist sees many phenomena, he must first determine whether they have symmetry. He conducts experiments in different places and at different times. If he achieves the same results, he then studies them in search of the root cause. If the experiments turned out to be asymmetric, he ignores them.

And although scientists such as Galileo and Newton saw symmetry in physical phenomena, the full power of symmetry was first investigated by Albert Einstein. He stated that the laws of physics should remain the same even if the experimenter moves at a speed close to the speed of light. Mindful of this symmetry, he was able to create the laws of the special theory of relativity. Einstein was the first to realize that symmetry is the defining characteristic of physics. Whoever has symmetry will have the law of nature. And the rest does not belong to science.

Shortly after Einstein showed the vital importance of symmetry to science, Emmy Noether proved a powerful theorem that determined the relationship between symmetry and conservation laws. It is connected with the constants of nature, the central part of modern physics. Again, in the presence of symmetry, there will be conservation laws and constants. A physicist must be a sieve and study phenomena with symmetry, allowing those phenomena that do not have symmetry to slip through your fingers.

There are several problems with this explanation of the structures existing in the Universe. For example, it seems that the phenomena we choose that have the laws of nature generate all other phenomena. All the laws of particle physics, gravity, and quantum theory have symmetries, and they are all studied by physicists. It seems that all phenomena come from these theories, even those that do not possess symmetry. So, although the definition of the next president of the United States is not part of the task of science, this phenomenon will be determined by sociology, which is defined by psychology, which is determined by neurobiology, which depends on chemistry, which depends on particle physics and quantum mechanics. To determine the winner of the election is too difficult for scientists, but the election results depend on the laws of physics, which are part of science.

Despite the fact that we could not explain the structure of the laws of nature, we believe that this is the best candidate for a solution. This is one of those solutions that do not include any kind of metaphysical principles or the existence of a multitude of invisible universes. We do not need to look beyond the limits of the Universe in search of the cause of the structure inside it. We need only look at how we view phenomena.

Before we continue, it must be pointed out that our solution has a property in common with a solution related to the Multiverse. We postulated that for the most part the Universe is chaotic and there is no special structure in it. We just focus on a small number of existing structures. In the same way, one who believes in the Multiverse believes that most of it lacks the structure to form intelligent life. Only in a small number of selected universes can complex structures be found. And we, the population of this complex Universe, are focusing on these rare structures. Both solutions consist in concentrating on a small volume of the structure, which is part of a huge chaotic whole.

Hierarchy of Numerical Systems


The idea that we see structure only because we select a subset of phenomena is new and difficult to perceive. In mathematics, there is a similar situation, which is much easier to understand. We will concentrate on one important example, in which the selection process is clearly visible. First, we need to take a short tour of several numerical systems and their properties.

Consider real numbers. In one of the high schools, the teacher draws a line of real numbers on the blackboard and claims that it has all the numbers you will ever need. If we take two real numbers, we can add, subtract, multiply and divide them. They consist of a numerical system used in all aspects of science. Real numbers have one important property: they are ordered. of any two different real numbers, one will be larger than the other. Imagine a number line: from two different points, one will be to the right of the other. This property is so obvious that it is rarely talked about.

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Emmy Noether

And although real numbers seem like a finished picture, the story does not end there. Already in the sixteenth century, mathematicians began the search for more complex numerical systems. They began to work with the "imaginary" number i, whose property is such that its square is -1. This clearly contrasts with any real number whose square is always positive. They defined the imaginary number as the product of a real number and i. Mathematicians defined a complex number as the sum of the real and the imaginary. If r 1 and r 2 are real numbers, then r 1 + r 2i is a complex number. Since the complex number consists of two real numbers, we draw them on a two-dimensional plane. The line of real numbers is on the complex plane. This corresponds to the fact that each real number r 1 can be considered as a complex r 1 + 0i (the number itself plus the zero complex term).

We know how to add, subtract, multiply and divide complex numbers. However, they have one unusual property. Unlike real numbers, complex numbers are not ordered. Which of the two complex numbers, say 3 + 7.2i and 6 - 4i, is greater, and which is less? There is no obvious answer. In principle, of course, complex numbers can be ordered, but this ordering will not correspond to their multiplication). The fact that complex numbers are not ordered means that we lose structure when we move from real to complex numbers.

But the story does not end with complex numbers. In the same way as complex numbers can be composed of material pairs, quaternions can be composed of complex pairs . Let c 1 = r 1 + r 2i and c 2 = r 3 + r 4 i will be complex numbers. Then we can define the quaternion as q = c 1 + c 2 j, where j is a special number. It turns out that any quaternion can be written as

r 1 + r 2 i + r 3 j + r 4 k

where i, j and k are special numbers similar to complex ones (ijk = -1 = i 2 = j 2 = k 2 ) So if complex numbers consist of two real numbers, then quaternions consist of four real numbers. Each complex number r 1 + r 2i can be considered as a special type of quaternion: r 1 + r 2 i + 0j + 0k. We can imagine quaternions as a four-dimensional space, the two-dimensional subset of which are complex numbers. It is quite difficult for us humans to visualize high-order spaces.

Quaternions are a complete numerical system. They can easily be added, subtracted, multiplied and divided. Like complex numbers, they cannot be ordered. And they have even less structure than complex numbers. If the multiplication of complex numbers is commutative, that is, for any complex numbers c 1 and c 2, c 1 c 2 = c 2 c 1, this is not true for all quaternions. This means that there are quaternions q 1 and q 2 such that q 1 q 2 is not equal to q 2 q 1 .

This process of doubling the number system using a new special number is called the Cayley-Dickson procedure in honor of mathematicians Arthur Cayley and Leonard Dickson. For a numerical system of a certain type, one can construct another numerical system with a dimension twice as large as the original. The new system is worse structured (it has fewer axioms) than the original.

Applying the Cayley-Dickson procedure to quaternions, we obtain a numerical system of octonions. This is an eight-dimensional number system. This means that each of the octoions can be written in eight real numbers, as

r 1 + r 2 i + r 3 j + r 4 k + r 5 l + r 6 m + r 7 n + r 8 p

Although these actions are rather complicated , we know how to add, subtract, multiply and divide octonions. Each quaternion can be written as a special type of octonion, in which the last four coefficients are equal to zero.

Like quaternions, octonions are neither ordered nor commutative. However, octonions are also not associative. All previous numerical systems considered were associative. This means that for any three elements, a, b and c, the two ways of multiplying them, a (bc) and (ab) c, are identical. However, this is not so for octonions. There are octonions o1, o2 and o3 such that o1 (o2o3) ≠ (o1o2) o3.

We can continue this doubling and get an even larger, 16-dimensional number system, called the sedenions. To describe it, 16 real numbers are required. Octonions are a special type of sedenions in which the last eight coefficients are zero. But researchers shy away from the Sedenions, because they lose an important property. Although they can be added, subtracted, and multiplied, there is no way to divide them. Most physicists believe that this is beyond the scope of “honest” mathematics. Even mathematicians find it difficult to handle them. You can create a 32-dimensional numerical system, and a 64-dimensional, and so on. But they are usually not talked about, because so far they have very few uses. We will concentrate on the octonions. All numerical systems can be summarized using this Venn diagram:

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Let us discuss the applicability of these numerical systems. Real numbers are used in all areas of physics. All quantities, measurements, lengths of physical objects or processes are given in the form of real numbers. Although complex numbers were formulated by mathematicians to help solve equations (i is the solution to the equation x 2 = -1), physicists began using complex numbers to discuss waves in the mid-19th century. In the 20th century, complex numbers became the basis for the study of quantum mechanics. Now complex numbers play an important role in various fields of physics. Quaternions appear in physics, but do not play important roles. Octonions, sedenions, and even larger numerical systems rarely appear in the physical literature.

The laws of mathematics we discover


The usual approach to considering these numerical systems is that real numbers are fundamental, and complex numbers, quaternions and octonions are weird larger sets that help mathematicians and physicists do something. Larger number systems are considered unimportant and uninteresting.

Let's turn this approach upside down. Instead of considering real numbers to be central and octonions to be a weird larger numerical system, imagine that octonions are fundamental, and other numerical systems are simply subsets of octonions. The only existing number system is the octonions. To paraphrase Leopold Kronecker: “God created the octonions, everything else is the work of man” [Kronecker said: “God created the whole numbers, everything else is the work of man” / approx. transl.]. The octonions contain all the numbers we need. (In this case, as was shown earlier, we can do the same trick with the Sedenions and even with 64-dimensional numerical systems. But we will convey our ideas with the help of octonions).

Let's see how we can deduce all the properties of a numerical system with which we are familiar. Although multiplication of octonions is not associative, if you need associative multiplication, you can take a special subset of octonions (we use the word "subset", but we need a special kind of subset suitable for operations in a numerical system. Such subsets are called subgroups, subfields, or "subsurface algebras with division "). So, if we select a subset of all octonions of the form

r 1 + r 2 i + r 3 j + r 4 k + 0l + 0m + 0n + 0p

then the multiplication will be associative (as in quaternions). If we go further and choose octonions of the form

r 1 + r 2i + 0j + 0k + 0l + 0m + 0n + 0p

then the multiplication will be commutative (like for complex numbers). If we continue to choose a subset of octonions of the form

r 1 + 0i + 0j + 0k + 0l + 0m + 0n + 0p

then we will get an ordered numerical system from them. All the necessary axioms “sit inside” the octonions.

There is nothing strange about this. If we have a structure, we can concentrate on a subset of special elements that satisfy certain requirements. Take any group. We can go through all its elements and choose such X that for all elements Y it will be true XY = YX. This subset is a commutative (abelian) group. In any group there is a subset that makes up the commutative group. We simply select those parts that satisfy the axiom, and ignore (put out the brackets) those that do not satisfy it. Our idea is that if the system has a certain structure, then special subsets of the system will satisfy more axioms than the original system.

This is similar to what we do in physics. We do not study all phenomena. We select those that satisfy the requirements of symmetry and predictability. In mathematics, we describe subsets using an axiom. In physics, we describe a selected subset of phenomena by the law of nature.

Mathematics for a subset chosen to satisfy the axiom is simpler than the mathematics of the whole set. This is because mathematicians work with axioms. They prove theorems and make models using axioms. When there are no such axioms, mathematics becomes more difficult or even impossible.

By analogy, a subset of phenomena is easier to explain by the law of nature, written in the language of mathematics. Conversely, when we observe a larger set of phenomena, it is more difficult to find the law of nature, and this mathematics becomes more difficult or even impossible.

Working in tandem and moving forward


There is an important analogy between physics and mathematics. In both areas, if we do not study the system as a whole, but look at particular subsets, we see more structure. In physics, we take a certain phenomenon (possessing symmetry), and ignore the rest. In mathematics, we consider certain subsets of structures and ignore the rest. These two bracketing operations work together.

The task of physics is to formulate a function from a set of observable physical phenomena leading to a mathematical structure:

observable physical phenomena → mathematical structure

That is, we must give a mathematical structure to the observed world. As physics advances and as we try to understand more and more observable physical phenomena, we need more and more large classes of mathematics. In terms of this function, if we want to increase the input of a function, we need to increase its output.

There are many examples of the expansion of physics and mathematics.

When physicists began working with quantum mechanics, they realized that ordered real numbers limit them too much. They needed a numerical system with fewer axioms. They discovered complex numbers.

When Albert Einstein wanted to describe general relativity, he realized that the mathematical structure of Euclidean space with its axiom about the plane (the fifth postulate of Euclidean) was too restrictive. He needed a curved, non-Euclidean space, to describe space-time in GR.

In quantum mechanics, it is known that in some systems the measurement of X, and then Y, will lead to results different from those obtained when we first measure Y and then X. For a mathematical description of this, you need to leave the cozy world of commutativity. They require a more general class of structures that do not imply commutativity.

When Boltzmann and Gibbs started talking about statistical mechanics, they realized that the laws they received were no longer deterministic. The experimental results were no longer divided by happened (p (X) = 1) or did not happen (p (X) = 0). Instead, probability mechanics is required for statistical mechanics. The chances of a certain experimental result are probability, and p (X) is an element of an infinite set [0, 1], and not from a bounded finite set {0, 1}.

When scientists started talking about the logic of quantum events, they realized that ordinary, distributive logic was too restrictive. They needed to form a more general class of logic in which the axiom of distributivity was no longer necessarily fulfilled. Now this is called quantum logic.

Paul DiracI realized this weakening of axioms 85 years ago when I wrote the following:
The continuous progress of physics requires for mathematics to formulate mathematics that continues to become more complex. This is natural and expected. But what scientists of the last century did not expect was a certain form, which the direction of complication of mathematics would take, namely, it was expected that mathematics would become more and more complex, but would be based on a constant basis of axioms and definitions. In fact, the modern development of physics requires mathematics, constantly shifting its foundations and becoming more abstract. Non-Euclidean geometry and non-commutative algebra, once considered the fiction and entertainment of thinkers, are now needed to describe the general facts of the physical world. It seems likely that this process of increasing abstraction will continue in the future, and the development of physics will be associated with the constant change and generalization of the axioms underlying mathematics, and not with the logical development of any of the mathematical schemes that are on a fixed basis [Dirac, PAM Quantized singularities in the electromagnetic field. Proceedings of the Royal Society 133, 60-72, (1931).].

With the development of physics and the discovery of an increasing number of phenomena, more and more large classes of mathematical structures are required with an ever smaller number of axioms with "increasing abstraction" and "generalization of axioms." Without a doubt, if Dirac were alive, he would write about the arrival of octonions or even sedenions in the world of necessary numerical systems.

To describe more phenomena, we need ever-larger classes of mathematical structures and fewer axioms. What is the logical conclusion of this trend? How far can it go? Physicists want to describe more and more phenomena in our universe. Suppose we want to describe all the phenomena of the universe. What kind of math will we need for this? How many axioms will a mathematical structure describing all the phenomena need? Of course, it is difficult to predict, but it is even more difficult not to speculate on this topic. One of the possible conclusions is that if we look at the whole Universe at once, and we do not bracket any subsets of phenomena, then we will need mathematics without any axioms at all. That is, in general, the Universe is free from structure and axioms are not needed to describe it. Total lawlessness! Mathematics without structure are just sets. It may, finally, eliminate all metaphysics associated with the laws of nature and mathematical structures. Only the way we study the Universe gives us the illusion of having a structure.

With this view of physics, we come to an even more complex question. These are future science projects. If the structure we see is illusory and comes from our way of studying certain phenomena, why do we see it? Instead of studying the laws of nature formulated by scientists, we need to study scientists and how they choose the laws of nature, subsets of phenomena and everything that is connected with them. What property of a person makes him such a good sieve? Instead of studying the Universe, we need to study the way we study it.

Nozon S. Yanovsky - Doctor of Mathematics from the Graduate School of the City University of New York. Professor of Computer Science at Brooklyn College, City University of New York. In addition to scientific work, he was a co-author of the book “Quantum Computing for Computer Specialists” and wrote “The External Limits of the Reasonable: What Science, Mathematics and Logic Can't Tell Us”.

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