Mathematics is an ancient fundamental science, which idealises and notes properties of real or fictional objects down in a formal language. Maths emerged from the basic operations, such as counting, measuring, comparing and describing the shapes of real objects.  This is the science about keeping order everywhere. It is used to make precise definitions of contents the humanities and natural sciences. It can be said that mathematics is the basis of the basics, the universal language, which is possible to describe just about everything. Galileo Galilei once said:  ‘The grand book of the universe is written in the language of mathematics’. Immanuel Kant added: ‘In every department of physical science there is only so much science, properly so-called, as there is mathematics’. And the German mathematican and logician David Hilbert summarised: ‘Mathematics is the foundation of all exact knowledge of natural phenomena’. But where is all that maths we are talking about? If you look around, you will probably see a few figures here and there: house numbers, numbers of pages in the book, or dates in calendar. However, these are just symbols humans made up. Mathematicians research abstract structures which are far more diverse. For instance, if a stone is throwed and watched closely it could be seen that it has a particular movement shape. The flight of any object would have exactly the same trajectory, the upside-down parabola. When we surveil how planets move along their orbits in space, we discover another repetitive shape, the ellipse. Hence, everything that is inside the universe, including people, is a part of mathematical structure. And the more we explore this vast system made of numbers, equations and patterns, the better we could understand nature itself. At the same time, we do not invent mathematical structures, we just discover them and develop markings for their description.

Well-known maths examines numbers, structure, space and differences (arithmetic, algebra, geometry and analysis respectively). In addition to these basic operations, logic, set theory and measurement uncertainty are presented. Mathematics is one of the few areas of knowledge where precise patterns exist, such as Fibonacci sequence, Kleiber’s law and the law of large numbers among other things.

In our environment there are plenty of mathematical patterns that are functional. One of which is Fibonacci sequence. This is a set of numbers where every element is a sum of two previous ones:0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 and so on, formally the sequence is endless. But where does the nature here come from? If one counts the location of leaves and branches on stems of numerous plants or the number of petals of flowers, you will end up with numbers 3, 5, 8, 13, 21, 34 or 55. The ratio of human body parts is based on a sequence as well. Every bone of an index finger from tip to base is larger than the previous one, which correlates to numbers 2, 3, 5 and 8.

Why are the Fibonacci numbers so important though? In this sequence a harmony close to “golden ratio”(“divina proportione”) is hidden and equal to 1.61803398875… If you divide one number to a previous one you will get:

1/1 = 1

2/1 = 2

3/2 = 1.5

5/3 = 1.6666

8/5 = 1.6

13/8 = 1.625

21/13 = 1.61538462

34/21 = 1.61904762

As the series continues, the golden ratio approaches number 1.618, becoming closer and closer, but unable to reach this definition.

When describing Fibonacci sequence visually, the Archimedean spiral arises:

It is a smooth line guided through the corners of rectangles, increasing the intervals of which is always even. Every rectangle has the ratio of the lengths of the sides approaching “the golden one”: 1,618:1 and has the following distinction: If a square has been cut, the same rectangle emerges, but smaller size though, and on and on it goes the infinite number of times.

The Universe is full of such spiral constructions. It can be found in the form of DNA, flowers, hurricanes, snails’ shells, tree cones, fingerprints and even galaxies — everywhere, where nature requires filling the space efficiently and evenly.

Probably the most shocking mathematical pattern is that the hearts of all mammals beat approximately the same number of times during their lifetime, around 1,5 billion times, even though small animals (e.g. mice) live just for a few years, while large (e.g. whales) can live for ages. This systematic pattern follows a simple mathematical formula: the metabolic rate is proportional to the body mass to the power of ¾.

In other words, if the size of one animal is twice as big as the other (10 kg and 5 kg or 1000 kg and 500 kg), it can be assumed that the metabolic rate of the first animal should also be twice as high. Nevertheless, the metabolic rate does not double; in fact, it increases by just about 75%, which correlates to a tremendous energy saving — 25 % for each size doubling. Thus, a pattern exists: The larger entity, the less energy is required to be produced for each cell to sustain life activity. For instance, cells of an elephant consume just about one tenth of the energy needed for rat’s cells and run ten times less intensively. It leads to a respective reduction of level of cells damage and metabolic processes wearout, which leads to the base of elephants’ longevity and gives us the possibility to understand the aging process.

This mathematical pattern, defined by a Swiss biologist Max Kleiber, is applicable for almost every mammal, bird, shellfish, bacteria, plant or cell. It is no less surprising that similar laws apply to all physiological quantities and life processes: heart rate, rate of evolution, height of trees, length of genome, amount of gray matter in the brain, life expectancy, and even the rate of growth of an organism.

Therefore, almost all of the physiological characteristics and events of a life cycle of any entity are defined primarily by its size. Because of the fact that cells of large organisms have to work slower than cells of the smaller ones, the pace of life systematically declines with the size growth: large mammals live longer, growing up takes longer, heart beats slower and cells work less intensively than the ones of small entities do. If the weight of an organism doubles, so does the life expectancy of an animal, and the duration of the growing up period by approximately 25%, the rate of processes in its organism slows down in the same proportion, though.

As any biological organism grows, high-complexity structures emerge, which require merging immense number of components and efficient maintenance. In living systems, this problem is solved by developing fractal-like (self-repeating) network structures. For example, the circulatory system is based on such a self-similar structure. Thanks to the branching of large arteries into tiny capillaries, blood carrying oxygen reaches all parts of the body. The lungs also have a fractal structure. The problem of maximising the transfer of oxygen into the bloodstream is solved by branching the bronchi into alveoli and tubular membranes with air channels.

In other words, by highlighting a small part of a structure, that has fractal properties, if viewed in some magnification, one can identify that it is similar to all system as a whole; by selecting an even smaller part from the already cut piece and enlarging it, we will again see that it is similar to the original arrangement. Such structure lets nature create easily a complex multiscale formation, save energy and fill the space as much as possible.

So many of us do not even suspect that almost all daily events in our lives are the result of the combined influence of a large number of small factors. For instance, commuting time depends on traffic, signalling, transportation, people and unforeseen circumstances. But when going to work regularly, an average time arises with some minor deviations. The law of large numbers is the mathematical formulation of this phenomenon. It postulates that as the number of trials increases, when the same experiment is repeated a large number of times, the arithmetic mean of the results approaches the expected value. Imagine a coin. One side of it has heads, while the other has tails. Every time you toss a coin there is a 50% chance you end up with heads. The more times the experiment is repeated, the closer the average number will be to 50%. While making a decision in daily life, people unintentionally use the law of large numbers very often. For example, when a person decides to spend winter holidays by the sea, they have a clear vision of what weather is to be expected.  The results of long-term meteorological observations help predict the air and water temperature, which will not differ much from the mean data at this time of year. Another interesting application of this mathematical law in our lives is the use of placebo (any imitation of medical intervention). There is a 50% chance that it will work and the same negative probability — this is the average value. The more experiments with placebo drugs, the closer the result will be to the average mean. Recent studies by Canadian scientists have shown that every year the effectiveness of tested medicines in relation to placebo, the effect of which extends to an increasing number of people, decreases. Scientists say that many common medicines today would not pass clinical trials. Nearly all methods of statistical observation are based on the law of large numbers. This is not just a fact from the field of probability theory, but a phenomenon that we encounter almost every day in our lives.

As science develops, mathematical methods enhance and penetrate different fields of knowledge, new instruments for describing phenomena are developed and hidden patterns are discovered. The most vivid examples are considered to be Gödel’s incompleteness theorem, non-Euclidean geometry of curved space and fuzzy logic.

Before Gödel’s theorem was published in 1931, it was widely considered not just that everything was proved by mathematics, but the fact that in conceptual universe all truth could be proven. Gödel spoiled this dream and changed our perception of maths forever. The scientist showed that certain mathematical systems contain true statements that cannot be proven. And he did it with surprising agility, defining a mathematical statement, which was both truthful and impossible to prove. A simple example of this was the axiom of equality (X = X). This is supposed to be a true statement, but in fact we cannot back it up with mathematical proof, hence any adequate theory is incomplete or inconsistent. So, despite all of our efforts we are unable to bring all mathematics together to apply fixed rules. In contrast to how many procedures are noted down, there will always be true facts impossible to prove. Mathematical knowledge is destined to remain forever incomplete. Some scientists, primarily Sir Roger Penrose, a mathematician and physicist from the University of Oxford, used Gödel’s theorem to prove that human brain does not work like a computer, and an artificial intelligence is unreachable. The gist of the argument is that there is an element in mathematics which is entirely creative, and we have to consciously feel or judge the truth of a statement which is not proved by the available axioms, whereas the machine acts strictly logically. This is why our intelligence cannot be simulated by a computer.

The geometry of curved space, also known as non-Euclidean, has found wide application in modern science. Classical flat geometry is correct only on small scales, while when it comes to the scales of the solar system or any galaxy it is no longer able to work. So, all mankind lives on the surface of the Earth, which is almost a spherical object. One of the reasons why non-Euclidean geometry is difficult to accept is that it conflicts with our practical experience. It creates a cognitive dissonance as we perceive our world as flat, even though the Earth is spherical. It is easy to imagine a city in a shape of a grid with crossed straight streets. This perception is possible because the curvature of the earth is negligible in contrast to the sizes of our cities.  But imagine that you are located on the surface of the sphere which is so large that you are unable to state that it is round. A straight line on a sphere is equivalent to a line that goes straight around its circumference:

Mathematics operates by not just numbers, equations, and spatial structures, but logical operations as well. Every day we face plenty of problems, solutions to which require the ability to logical thinking. The truth is that most of the time we use fuzzy logic. Since the time of Aristotle, the basic rules of logic have been clear. All things either did or did not have defined properties: there were no half measures, and nothing could be both at the same time. The logic was two-valued – true or false. But during the 1930s a Polish mathematician Jan Łukasiewicz thought about statements that are both not true or false, like “tomorrow rain is expected”, which is impossible to check until the next day comes. Or a glass of warm water; it is neither cold nor hot. Such logic requires three definitions at least. Half a century later Berkeley-based mathematician Lotfi A. Zadeh suggested several categories of fuzzy logic: 1) completely true, 2) mostly true, 3) unsure whether true or not, 4) mostly false and 5) completely false. The term “fuzzy” referred to things that were unclear or vague. Since then, fuzzy logic has found applications in the most different areas. In the real world, we constantly encounter situations where we cannot determine whether a state is 100% true or false, in that case we use flexible thinking. In this way, we can take into account the inaccuracies and ambiguities of any situation. Nowadays your washing machine or an automatic gearbox inside your car likely has a fuzzy logic sensor. After all, we live in a world powered by uncertainty.

As reported by some scientists, mathematics is just a tool crafted to explain the reality around us. This point is rather debatable. According to astrophysicist Max Tegmark, the mathematical structure found in the natural world shows that mathematics does exist in reality, not just in the human mind. And one of the consequences of the mathematical nature of the universe is that scientists can theoretically predict the outcome of every observation. The very same mathematical law can govern many phenomena. For example, trigonometric functions apply to all wave motions: light, sound, radio waves, among other things. Even things we can see and touch, have mathematical proportions and patterns. Knowing patterns and examining the basics of natural structures, we are able not only understand the surrounding world but create and improve it as well. Because of that, mathematics is the core of all modern technologies. In fact, by listening to another person via the phone connection, you are listening to a mathematically synthesised speech, not their actual voice. Mathematisation of healthcare is proceeding rapidly. The equations of quantum mechanics are possible for everything from transistors and semiconductors to electron microscopy and magnetic resonance imaging. New technologies and techniques penetrate medicine worldwide. An active application of this exact science may become the decisive factor in accelerating the global revolution in the field of health and life extension.