Mathematics is the essence of science and our daily life

Mathematics is a formal science that starts from axioms and after logical reasoning, studying properties, abstract structures, and relationships between abstract entities such as numbers, geometric shapes, symbols, glyphs, or symbols in general.

Mathematics is a group of formal languages that can be used as a tool for posing problems in specific contexts. For example, the following statement can be expressed in two ways: X is greater than Y and Y is greater than Z, or in a simplified form it can be said that X > Y > Z. This is why mathematics is just a pidgin with a tool for each specific problem (at, For example, 2 x 2 = 4).

The natural sciences have used mathematics extensively to explain various observable phenomena, as expressed by Eugene Paul Wiener:

The enormous usefulness of mathematics in the natural sciences is something that borders on the mysterious, and there is no explanation for it. It is not at all natural that "laws of nature" exist, much less that man is capable of discovering them. The miracle of how appropriate the language of mathematics is to the formulation of the laws of physics is a wonderful gift that we neither understand nor deserve.

Galileo Galilei, along the same lines, had expressed it thus:

Mathematics is the language in which God wrote the universe.

Through abstraction and the use of logic in reasoning, mathematics has evolved based on calculation and measurement, along with the systematic study of the shape and movement of physical objects. Mathematics, from its inception, has had a practical purpose.

Explanations based on logic first appeared with Hellenic mathematics, especially Euclid's Elements. Mathematics continued to develop, with continual interruptions, until in the Renaissance mathematical innovations interacted with new scientific discoveries. As a consequence, there was an acceleration in research that continues to the present day.

Today, mathematics is used around the world as an essential tool in many fields, including the natural sciences, the applied sciences, the humanities, medicine, the social sciences, and even disciplines that seemingly do not exist. they are linked to it, like music (for example, in matters of harmonic resonance).

Applied mathematics, a branch that aims to apply its knowledge to other fields, inspires and benefits from new mathematical discoveries and sometimes leads to the development of new disciplines. Scientists are also involved in pure mathematics, regardless of its application, although its practical applications are often discovered over time.

Introduction

the science of the origin of the words

Mathematics requires effort in teaching or learning, referring to areas of knowledge that can only be understood after being taught in them, such as astronomy. Mathematical art will be opposed in this music, the art of music, which would be an art, such as poetry, rhetoric, and the like, can be directly appreciated, "one can understand without being instructed".

Although the term was already in use by the Pythagoreans in the 6th century BC. Its meaning reached its most technical and least "mathematical study" in the time of Aristotle (4th century BC).

The most commonly used form is plural mathematics (whose acronym is "companions"), which has the same meaning as the singular and comes from the Latin form (Cicero), based on the plural, used by Aristotle and meaning, roughly, "all things are mathematical". However, some authors use the singular form of the term; This is the case of Bourbaki, in The Elements treatise, he highlights the unification of this field provided by the modern axiomatic view, although he also uses the plural form as in (Elements of the History of Mathematics) 1969, which may indicate that it was Bourbaki who finally achieved the unification mathematics.

Similarly, in his book (1948) he raised the issue in the section «Mathematics, singular or plural» where he defends the conceptual exclusivity of mathematics even though he uses the plural form in the mentioned writing.

Some definitions of mathematics.

Establishing clear and precise definitions is the basis, but defining them was difficult, some definitions of famous thinkers are presented:

• René Descartes: Mathematics is the science of system and measurement, of beautiful chains of reasoning, all simple and easy.
• David Hilbert: Mathematical analysis is a symphony of infinity. Is a system of formulas that can be proved.
• Benjamin Pierce: Is the science that draws necessary conclusions.
• Bertrand Russell: Possesses not only truth but a certain poisonous beauty. A cold and severe beauty is like a sculptural beauty.
• Ibo Bonilla: Practicing it deconstructs the rhythms of the universe. It is the science of constructing a studied reality. the set of its elements, proportions, relationships, and development patterns in ideal conditions for a specific region.
• John David Barrow: At the end of the day, math is the name we give to the set of all possible patterns and interrelationships. Some of these patterns are between shapes, others are sequences of numbers, and others are more abstract relationships between structures. The essence of mathematics lies in the relationship between quantities and qualities.

Epistemology and controversy about mathematics as a science

The cognitive and scientific character of mathematics has been widely discussed. In practice, they are used to study quantitative relationships, structures, geometric relationships, and variable volumes.

Mathematicians look for patterns, formulate new conjectures, and attempt to reach mathematical truth through rigorous deduction. These allow them to establish the appropriate axioms and definitions for that purpose.

Some classical definitions restrict mathematics to reasoning about quantities, although only a part of current mathematics uses numbers, predominating the logical analysis of non-quantitative abstract constructions.

There is some discussion about whether mathematical objects, such as numbers and points, really exist or simply come from the human imagination. Mathematician Benjamin Peirce defined it as:

the science that draws the necessary conclusions.

On the other hand:

when the laws of mathematics refer to reality, they are not exact; when they are exact, they do not refer to reality.

The scientific character of mathematics has been discussed because its procedures and results have firmness and inevitability that do not exist in other disciplines such as physics, chemistry, or biology. Thus, mathematics would be tautological, infallible, and a priori, while others, such as geology or physiology, would be fallible and a posteriori. It is these characteristics that make it doubtful to place it in the same rank as the aforementioned disciplines. John Stuart Mill stated:

Logic does not observe or invent or discover, but it judges.

Thus, mathematicians can discover new procedures to solve integrals or theorems, but they are unable to discover an event that casts doubt on the Pythagorean Theorem or any other, as is constantly the case with the natural sciences.

Mathematics can be understood as a science; if so, its object and method should be indicated. However, Some believe that it is an official language, safe, efficient language, applicable to the understanding of nature, as Galileo indicated; In addition, many phenomena of a social nature, others of a biological or geological nature, can be studied through the application of differential equations, calculation of probabilities or set theory.

Precisely, the progress of physics and chemistry has required the invention of new concepts, instruments, and methods in mathematics, especially in real analysis, complex analysis, and matrix analysis.

History of mathematics

The history of mathematics in the field of study of investigations into the origins of discoveries in mathematics, the methods for developing its concepts, and, to some extent, also by the mathematicians involved. Its emergence in human history is closely linked to the development of the concept of numbers, a process that occurred gradually in early human societies.

Although they had a certain ability to estimate volumes and amounts, at first they had no idea how many. Thus, numbers greater than two or three have no nouns, so they used an equivalent expression for "many" to refer to a larger group.

The next step in this development is the emergence of something close to the concept of number, albeit a very basic one, not yet as an abstract entity, but as a property or characteristic of a definite group.

Subsequently, progress in the complexity of the social structure and its relationships was reflected in the development of mathematics. The problems to be solved became more and more difficult and it was no longer enough, as in primitive societies, to simply count things and inform others about the basic elements of the calculated set, but it became necessary to count larger and larger sets, to determine the time, to work with dates, enable the calculation of equations for barter. It is the time when names and numerical symbols appear.

Before the modern era and the worldwide spread of knowledge, written examples of new mathematical developments appeared in only a few places.

Traditionally, it was believed that mathematics, as a science, arose to make calculations in trade, measure the Earth, and predict astronomical events. These three needs can be related in one way or another to the broad subdivision in the study of structure, space, and change.

Egyptian and Babylonian mathematics were developed extensively by Hellenism, as methods were refined (particularly the introduction of mathematical precision in proofs) and matters relating to this science were expanded.

Many Greek and Arabic mathematical texts were translated into Latin, further developing mathematics in the Middle Ages. Since the Italian Renaissance, in the fifteenth century, new mathematical developments, interacting with contemporary scientific discoveries, have been growing exponentially today.

Formal, methodological, and aesthetic aspects

Inspiration, pure and applied mathematics, and aesthetics

It is very likely that the art of arithmetic was developed even before writing, and is associated primarily with accounting and property management, commerce, surveying, and later astronomy.

All sciences presently present problems that mathematicians study, and at the same time, new problems are emerging within mathematics itself. For example, physicist Richard Feynman proposed that the path be complimentary as a basis for quantum mechanics, combining mathematical reasoning with a physics approach, but a completely satisfactory definition from a mathematical point of view has yet to be achieved.

Similarly, string theory, a developing scientific theory that attempts to unite the four fundamental forces of physics, continues to inspire modern mathematics.

Some are only relevant to the field in which they were inspired and are applied to other problems in that field. However, those inspired by a specific field are often useful in many fields and are included in generally accepted mathematical concepts. The wonderful fact that even the purest ones usually have practical applications is what Eugene Wegener defined as follows:

The unreasonable effectiveness of mathematics in the natural sciences.

As in most fields of study, the explosion of knowledge in the scientific age has led to specialization. There is an important difference between pure mathematics and applied mathematics.

Most research mathematicians focus on only one of these areas, sometimes choosing when they begin their undergraduate studies. Many areas of applied mathematics have been merged with other fields traditionally outside of mathematics and have become independent disciplines, such as statistics, operations research, or computer science.

Those who have a penchant for it consider the aesthetic aspect to be the dominant determinant of most of them. Many scholars talk about its elegance, its inherent aesthetics, and its inner beauty. In general, one of its most valuable aspects is its simplicity. There is beauty in a simple and powerful proof, such as Euclid's proof that there are infinitely many prime numbers, and in elegant numerical analysis that speeds up arithmetic operations, such as the Fast Fourier Transform.

In his book The Mathematician's Apology, G. H. Hardy expressed his conviction that these aesthetic considerations, by themselves, are sufficient to justify the study of pure mathematics. Mathematicians often struggle to find proofs for particularly elegant theorems, eccentric mathematician Paul Erdos points out that It is finding proofs for the “Book” in which God wrote his favorite proofs.

The popularity of recreational mathematics is another sign that tells us about the fun that comes from solving math questions.

Blogging, language, and rigor

Most of the mathematical symbols in use today were not invented until the 18th century. Before that, mathematics was written in words, a laborious process that limited mathematical progress.

In the 18th century, Euler was responsible for many of the symbols in use today. Modern notation makes maths much easier for professionals, but difficult for beginners. Coding reduces math to the max and makes some codes contain a large amount of information.

Like musical notation, modern mathematical notation has a strict syntax and encodes information that would be difficult to write otherwise.

Mathematical language can also be difficult for beginners. Words like or just have more precise meanings than they do in everyday language. Also, words like open and body have very specific mathematical meanings. Mathematical terms, or mathematical language, include technical terms such as similarity or complementarity. The reason for the need to use notation and terminology is that mathematical language requires greater precision than everyday language. Mathematicians refer to this precision in language and logic as "rigor."

Rigor is an indispensable condition that must be met in mathematical proof. Mathematicians want their axioms to follow systematic thinking.

This avoids false theorems, based on fallible intuition, that have occurred several times in the history of this science.25 The level of accuracy expected in mathematics has varied over time: the Greeks sought detailed arguments, but sometimes Isaac Newton's methods were less rigorous.

Problems inherent in Newton's definitions led to a revival of rigorous analysis and formal proofs in the nineteenth century. Now, mathematicians continue to support each other with computer-aided proofs.

The axiom is traditionally interpreted as an "obvious truth", but this concept is problematic. In the formal world, an axiom is nothing more than a series of symbols, having intrinsic meaning only in the context of all formulas derived from an axiomatic system.

Mathematics as a science

Carl Friedrich Gauss referred to mathematics as the "queen of sciences", and in both the original Latin "centrum Regina" as well as the German Königen der Wissenschaften, the word "science" should be interpreted as (a field of) knowledge.

If science is the study of the physical world, then mathematics, or at least pure, is not a science.

Many philosophers believe that mathematics is not empirically falsifiable, and thus is not a science as defined by Karl Popper. However, In the 1930s, important work in mathematical logic showed that it was irreducible to logic, and Karl Popper concluded that most mathematical theories, such as physics and biology, are hypothetico-deductive theories. Therefore, pure mathematics has become closer to the natural sciences, whose hypotheses are merely guesses, and this is how it has been until now.

Other thinkers, notably Imre Lakatos, advocated a falsification version of mathematics itself.

An alternative view is that some scientific fields such as theoretical physics are mathematics with axioms that claim to correspond to reality.

Intuition and experimentation also play an important role in formulating conjectures in mathematics and other sciences. Empiricism continues to gain representation in it. Calculus and simulation play an increasing role in both science and mathematics and reducing objections to them does not use the scientific method.

In 2002, Stephen Wolfram, in his book A New Kind of Science, argued that computational mathematics deserved to be explored empirically as a scientific field.

The opinions of mathematicians on this issue vary widely. Many mathematicians call their field science to downplay its aesthetic profile and also to deny its history in the seven liberal arts.

Others feel that ignoring its connection to the sciences means ignoring the obvious relationship between mathematics and its applications in science and engineering, which has greatly fueled the development of mathematics.

Another topic of debate, which is somewhat related to the previous question, is whether mathematics was created like art or discovered like science. This is one of the many topics of interest to the philosophy of mathematics.

Sports awards are usually separated from their science equivalents. The most prestigious award in mathematics is the Fields Medal, established in 1936 and awarded every four years.

Often equivalent to the Nobel Prize in science. Other awards are the Wolf Prize in Mathematics, established in 1978, which recognizes the lifetime achievement of mathematicians, and the Abel Prize, another major international award, which was introduced in 2003. The latter two were awarded for excellent work, which can be innovative research. Or solve a pending problem in a specific domain.

German mathematician David Hilbert compiled a famous list of these 23 unsolved problems, called "Hilbert problems". This list has become very popular among mathematicians and at least nine of the problems have already been solved.

A new list of seven basic problems called Millennium Problems was published in 2000. The solution to each problem will be rewarded with a million dollars. Interestingly, only one (Riemann hypothesis) appears in both lists.

in conclusion

We find that mathematics constitutes an open, universal language that allows us a deep understanding of the universe around us and gives us powerful tools for solving problems and discovering patterns. It is not just a set of numbers and symbols, but rather a window to critical thinking and creativity, elevating us to new heights of knowledge and understanding.

Despite the complexity of mathematics, understanding it brings incomparable rewards, as correct answers to mathematical questions are moments of triumph and pride, enhancing self-confidence and logical thinking.

So, let us continue to explore the world of mathematics with curiosity and interest, and encourage everyone to use this powerful tool to achieve achievements and innovations in various fields of life. It is an exciting and useful journey that makes us more in-depth and interactive with the world around us.