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Natural laws and elegant mathematics: problems and solutions

If mathematics can give us an elegant explanation of many physical phenomena, sometimes in real situations it is necessary to wade through thickets of numerical data




Since the time of Pythagoras, people believe in the special ability of beautiful mathematics to reveal to us all the secrets of the world. We used the famous article by Eugene Wigner " Unreasonable Effectiveness of Mathematics in the Natural Sciences " to discuss this topic with readers and solve several problems associated with it. The tasks were to demonstrate that, although mathematics is really very useful for creating idealized models and elegant explanations of many physical phenomena, in real situations it is sometimes necessary to wade through thickets of numerical data.

Scenario 1: Simplicity and uniformity


A) The object glides over a homogeneous surface, having an initial speed of 1. For each unit of distance, its speed decreases by 1/10 of the value that it had before starting to pass this particular segment. How far can an object travel before it stops completely? What is the general formula for calculating it?

This task has a catch. At first glance, it recalls the Zeno paradox of Achilles and the tortoise, which produces an infinite geometric progression 1 + 1/2 + 1/4 + 1/8 + ... for both time and distance. Although this sequence is infinite, it converges, and therefore it is possible to calculate its total amount (in this case, 2). Therefore, in this case, the finite distance is covered in a finite time [this can be argued: here we are not talking about a mathematical model, but about real motion, and therefore it makes no sense to limit the analysis of the paradox to mathematics - because Zenon just casts doubt on the applicability of the idealized to the real motion mathematical concepts / approx. transl.].

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B) The machine can move both forward and sideways, with the same ease. Its normal cruising speed in any direction is 1 unit on a smooth surface. The figure shows that she needs to overcome 10 terrain strips, each of which has a length of 10 units and a width of 1 unit. The length of the strips is perpendicular to the direction in which the machine needs to move. The machine is located in the middle of the first strip, which is smooth (smooth stripes are indicated by gray). After that, irregular (purple) and smooth stripes alternate.



However, irregularities on uneven stripes are not the same. Each strip consists of 10 square sections, which we can imagine in the form of artificial road irregularities. Roughnesses stand next to each other, the size of each of them is 1x1. Their properties vary. Irregularities can slow down the cruising speed of the machine by a value from 50% to 95%, and this value is changed in steps of 5%. Each of the uneven stripes is made up of irregularities of all 10 types, going in random order (the first purple strip shows one of the possible options for the distribution of irregularities). The machine can read the roughness of the area located directly in front of it (but only one), and can move sideways with its cruising speed equal to 1, so that, if desired, it moves another roughness that will not slow it down so much. To this, of course,time will pass, and if she moves sideways by a few squares, more time will be spent. After overcoming each magenta strip, cruising speed increases back to 1. What strategy should the car adopt for the fastest way through the entire territory? How long will it take?

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2:


Consider a hypothetical solid object in the shape of a rectangular triangle, with all its mass concentrated at the vertices. For simplicity, imagine that this object is two-dimensional - it has no thickness. Each vertex is a point with a mass of 1 unit, and the total mass of the object is 3 units. The base of the triangle is 4 units long, the vertical leg is 3 units, and the hypotenuse is 5 units. Imagine that the same triangle is located nearby, oriented exactly the same way, and the medians of both triangles (the segment connecting the middle of the hypotenuse with the opposite vertex) lie on one straight line, and the vertices of the right angles are 4 units apart. What will be the attraction acting on them? Does the law of gravitational attraction work if applied to two triangles as separate objects? What,if the triangles were located at a distance of 8 units of length from each other, and oriented in the same way? In such a situation, does the formula for gravitational attraction do better?

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So does the law of nature require elegant mathematics? And what makes elegant mathematics so capable and applicable in a wide range of problems? Among the advantages of mathematics,

one reader listed abstraction, built-in checks for consistency, continuity, work with infinities, feedback from physics and symmetry. Another cited the following story from life:

A few weeks ago I spoke with Don Lincoln, a physicist from Fermilab. I asked him: “Why is mathematics so good at describing the universe?” He replied that mathematical systems can be formulated in an infinite number of ways, so for any universe that has causal relationships, you can always find a mathematical platform that describes its physics.

Other readers have described similar observations. It seems to me that mathematics in itself is a huge set of laws and techniques that, due to their abstraction, can find application in many unrelated areas that have similar structure and dynamics, or mutual interaction of a different kind. We are also lucky to live in a universe in which elegant mathematics is useful. As one reader noted : “The mathematics and laws of Newton would be rather impractical if we lived in a universe with entropy close to maximum.”

But how fully can elegant mathematics describe nature? I will quote a comment from one of the readers in full:

One does not need to go deeper into biology to find that many mathematical representations become insufficiently complete: chemistry, materials science, condensed matter physics. For example, a water molecule cannot be described analytically using the tools of quantum mechanics for the same reason that the three-body problem is not available to us in celestial mechanics. Entire areas of science, such as thermodynamics and statistical mechanics, exist because some physical systems, such as an ice cube in water, are too complex to describe mathematically every molecule of water in ice and in capacity, not to mention Bose condensates -Einstein or superfluidity. Ohm's law is the electromagnetic version of the gross statistical sum, and Hooke's law and tension tensors are the elastic version of the gross statistical sum,and both of them refuse to work at a certain time or after a certain limit due to the dependence of the electric current on temperature and material, and the effects of irreversible deformation in physical bodies under heavy load.

The reason for the simplicity of most of the mathematics used in physics is that it is a crude statistical sum or a serious simplification of physical phenomena.

Why do we like elegant math so much?


One of the readers entitled his comment “elegance is the least energy expenditure of the brain” and wrote:

The feeling of “eureka!”, Reduction of high complexity to the simple principle of organizing neurons, or to the “mathematical law” in many problems is an example of simplification that gives that euphoric feeling energy saving. This principle may be related to the KISS (Keep It Simple, Stupid) principle, as well as to Einstein’s statement: “Everything needs to be reduced to the simplest form possible, but it’s not worth simplifying further.”

The realization by our brain that there are many phenomena connected with each other in nature gives rise to such symmetry that the self-organization of our neurons perceives as a way of storing information with minimal energy consumption.

As I wrote in my article on this subject, Occam's razor and pleasant sensation at the time of "Eureka!" are firmly registered in our brain and are manifestations of a unique cognitive-emotional connection that makes us rational. I suppose that every time in our head the quantity that I call “psychic entropy” decreases, we get a reward. This psychic entropy is not just compactness, but also the organization of connections invisible up to this point, and the feeling that everything is combined into a single whole. Evolution has made us intelligent, providing small internal rewards after solving each riddle - a very effective strategy.

So is Wigner right?


Yes and no. He was right that in mathematical descriptions of some physical problems there are abstract patterns and symmetries, and then mathematics shows all its power. However, there are such areas, both in physics and in other complex sciences, when this does not work. Perhaps Wigner was a little mystic, or a “patriot of mathematics,” and somewhat exaggerated the problem in his essay.

Source: https://habr.com/ru/post/undefined/

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