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**We explain what mathematics is, what its history is like and what this science is for. Also, what are its characteristics and classification.**

When we speak of mathematics or mathematics, we refer to a set of formal languages that, starting from axioms and obeying logical reasoning, **serve to pose and solve problems in a precise** (unambiguous) way, within the framework of very specific contexts.

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This means that mathematics is **a set of formal laws** , that is, abstract, that concern objects in the mind of man , such as numbers, angles, geometric shapes, etc.

Mathematics **deals with the structure, order, relationship, accounting** , measurement or description of objects, but not what they are, what they are composed of, or the specific aspects of the universe .

The study of mathematics involves understanding **a number of complex reasoning systems** , combining axioms and theorems derived from them.

It is considered that, along with verbal language, mathematics is **one of the most powerful** , vast and complex mental tools created by humans.

Mathematics is, in effect, **a formal science** .

"Formal" means that **it deals with ideal objects** and not with real objects.

Numbers, geometric shapes, square roots, etc., **are not things that one can take or move around** , but mental tools.

As such they make sense in their own framework of operations, that is, in their specific context of understanding.

However, mathematics **is also an exact science** , insofar as it does not give rise to interpretation, subjectivity or doubt in its reasoning, but is handled in terms of accuracy.

**The result of a calculation operation** , for example, will always be the same if it is performed properly, regardless of who performs it, where, or for what purpose.

This means that **its results are replicable** , verifiable and always true, which allows it to pass the tests of the scientific method .

The word *mathematics ***comes from the ancient Greek word mathëmatiká** , which would translate something like "things that are learned."

This is because the ancients distinguished the “mathematical art” ( *mathëmatiké tékhnë* ), from other fields of knowledge, such as the “art of music” ( *mousikë tékhnë* ), because music could be appreciated despite not having been instructed, while mathematics does not; **to appreciate it you had to learn about it** .

However, what we understand as mathematics is much older in human history, since it **could have had the same temporal origin as writing** .

In fact, it is thought that the first attempts to take written notes corresponded to numbers and accounting, rather than words and senses.

This type of system **already existed in ancient Egypt and ancient Mesopotamia** , although the Greeks were the first to consider it a branch of philosophy .

The first Greek mathematicians **date from the 6th century BC. C.** and they were the so-called Pythagoreans, disciples of Pythagoras (c. 569 - c. 475 BC).

Later the mathematical study **would draw the attention of the great Greek philosopher Aristotle **(4th century BC), and later still of the Latin Cicero (106 - 43 BC).

During the Middle Ages it **was a field widely investigated by alchemists** and Islamic scholars, until its reappearance in the Renaissance , at the service of humanistic and scientific knowledge renewed in the West .

Mathematics is **a very powerful mental tool** .

It allows the human being to carry out a vast and complex series of operations that have a direct impact on real life, such as **the description and analysis of spaces** , quantities, relationships, shapes, proportions and certainty.

Without it, **it would not be possible to calculate, measure, or logically deduce** , things that we use every day in our lives without even stopping to think that we are applying the fundamentals of an extremely ancient science .

It is possible to recognize some 5000 branches of mathematics, which are traditionally grouped into four large “pure” mathematical fields:

**Quantity.**Where are the numbers: natural numbers, integers, real, rational, complex, etc.**Structure.**Where numbers and their relationships are used to describe and represent shapes and sets: algebra, number theory, combinatorics, graph theory, group theory, etc.**Space.**Where the numbers are at the service of the measurement of space and the calculation of the various possible relationships between spatial representations: geometry, trigonometry, differential geometry, topology, etc.**Change.**Where the numbers serve to express changing relationships, movements, displacements and change in general: calculus, vector calculus , dynamic systems, differential equations, chaos theory, etc.

In addition to the "pure" fields of mathematics or entirely formal, there are areas in which mathematics is dedicated to the **study of aspects of other areas of knowledge** , especially to the construction of tools for analysis and problem solving.

Some of them are:

**Statistics**Mathematics applied to probability and the ability to predict events on a percentage or proportional scale, in order to make informed decisions .**Mathematical models.**Numerical representations are used as a form of simulation of aspects of reality, to try to predict or understand in the abstract the relationships that exist in it. It is particularly useful in computing .**Financial mathematics.**Applied to the world of finance , mathematics lends its formal language to the expression of the economic and commercial relationships that make up this aspect of society .**Mathematical chemistry.**Chemistry uses it to express the proportional relationships that occur in the various and possible reactions of matter .

Mathematics **allows the written expression of numerous real-world relationships** , and opens the door to much more complex abstract formulations and calculations.

In human development, this **meant significant growth in their ability to abstract** and handle complex ideas .

It is a field of research that seems arid and detached from real life, but from which **gigantic advances have emerged in other sciences** , industries and technologies , otherwise they would lack a formal language to express their operations.

According to Chevallard, Bosch and Gascón, there are three types of operations that can be carried out with mathematics:

**Use familiar mathematics.**Take procedures invented by others and apply them to their own problems to solve them, using accumulated numerical and logical knowledge as tools.**Learn and teach math.**Faced with a complex problem, we can turn to experts in mathematics or books on it, to learn to use hitherto unknown methods and expand our reserve of numerical tools.**Create new math.**If there is no mathematical tool that can help us solve a specific problem, we can create one, taking as a starting point those we already know.

Practically **all the exact and social sciences** use mathematics to express their relationships and content.

From engineering, biology , chemistry, physics, astronomy and computing, in which this formal language is an indispensable base, to sociology , architecture , geography , psychology or graphic design, in which it plays a leading and determined role.

According to Howard Gardner's model of intelligences in his Theory of Multiple Intelligences, the ability to **use mathematics with ease and / or speed** usually involves an aspect of the human mind known as logical-mathematical or logical-formal intelligence.

It is supposed to be **essential in people with a scientific vocation** , and it is a type of intelligence that facilitates working with abstract concepts or complex arguments .

Among the most important mathematicians in history are:

- Pythagoras of Samos (570-495 BC)
- Euclid (c. 325 - c. 265 BC)
- Leonardo Pisano Bigollo (1170-1250)
- René Descartes (1596-1650)
- Leonhard Euler (1707-1783)
- Andrew Wiles (1953-)