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A search for logic in mathematics While (logic) may simply refer to correct logical thinking in everyday life, it is also one of the oldest and most established branches of mathematics, and people often confuse the boundaries between logic and philosophy, and through this article we will solve for you all Ambiguous about logic in mathematics.

Introduction to logic in mathematics

At the beginning of our research, we will begin by defining logic in mathematics, to move on to clarifying the differences and all the confusion between mathematical and philosophical logic. Mathematics in our daily life.

A Greek mathematician who defined geometry with 6 letters

Research on logic in mathematics

Logic is related to theoretical computer science through computing theory, proof theory, algebra, number theory and algebraic geometry through model theory, analysis, and ergonomics theory through infinite harmonics and set theory. Foundational issues in mathematics.[1]

What is logic in mathematics?

Mathematical logic or symbolic logic is one of the classes of mathematics that relates to both the basics of mathematics and theoretical computer science in addition to philosophical logic, and logic is the study of truth and how to obtain universal facts through mathematical deduction, which is the basic language of mathematics and the basic principle of proof.[1]

The difference between philosophical logic and mathematical logic

The difference between philosophical logic and mathematical logic can be made by several points, which are the following:[2]

• In philosophical reasoning: propositions are simple statements that can be either true or false. Also, your proposals don’t have to be complicated, they can be as short as all yellow squares or Judy likes all things pink, your proposal is any statement that can be categorized as true or false.
• In mathematical logic: the propositions usually include mathematical symbols. In geometry, for example, you can have a proposition that says line AB is the bisector of the CD lines with the corresponding mathematical symbol for the lines instead of the word line. In algebra also, your proposition can be as simple as x = 2, depending on the type of the math you work with; You can have a combination of words with mathematical symbols or all mathematical symbols, what matters most is that your logical proposition can be described as either true or false.

History of Mathematical Logic

Mathematical reasoning and reasoning go back thousands of years, especially to the era of ancient Egyptian architects and Babylonian astronomers, but also to the development of logical thinking independently in India and China, and centuries later various groups of Greek mathematicians and philosophers were debating the nature of truth in an attempt to develop a formal system of logic and mathematical deduction, carried with it the ideas of Plato, Aristotle and many others throughout the Middle Ages and revived by scholars such as St. Thomas Aquinas and many Arab mathematicians, and among these scholars was Gottfried Leibniz, who was one of the first mathematicians to use a symbolic language for logic on Similar to what we use today, since then logic has become closely intertwined with concepts such as axioms, proof, and infinity or sets of numbers.[2]

Aristotle preferred to lay the foundations of mathematical logic

The idea of ​​logic was a major achievement of Aristotle in his efforts to produce correct laws of mathematical reasoning, and Aristotle was able to codify and organize these laws into a separate field of study. who asserts that a statement must be either true or false, and the key to his thinking was that Aristotle used mathematical examples taken from contemporary texts at the time to illustrate his principles, and although logic is derived from mathematics, logic was eventually considered an independent study of Mathematics but it is applicable to all types of thinking.[3]

Prospects of modern mathematical logic

The science of modern logic extends to include much broader horizons than was included in Aristotle’s logic, as the modern logicians developed theories and methods aimed at dealing with deductive issues in a completely different way from absolute induction. The eminent ones, and then the British philosopher Bertrand Russell, continued in this field, these logicians used arithmetic methods and methods that use symbols in contrast to the traditional logicals, and today logic is used mainly to test the validity of cases, as well as its important uses in the field of work with many devices Such as computers and electrical circuits, and until a logician tests the integrity of a case, he first analyzes its phrases, formulates and expresses them in the form of mathematical symbols. In mathematics, every letter or symbol is used in the case symbols for a word or for an entire phrase in many cases. Logic is a phrase such as: “the wise Socrates” in the form of “hs.” The logician then applies the rules of deduction sometimes, or the rules of inference, to determine the equations. The new T that can be deduced from the original premises, and the logical scientist continues to deduce the equations until he reaches a conclusion.[3]

Mechanism of modern symbolic mathematical logic

Modern symbolic logic is a development, modernization and correction of traditional logic, and this type of logic is based on deriving logical laws from a minority of principles (axioms and laws) and in a very accurate and complete way, in a more precise sense, it is a deductive system that starts from certain premises and ends up with the necessary theories about them, Relying on special rules, and using only the symbolic logical language, its appearance is due to Leibniz first and then George Paul II between (1815 and 1864), which was developed by Frege, Wittgenstein, Carnap and others, and it is sometimes called symbolic, mathematical, deductive, theoretical or algebra of logic. It is also called logarithmic logic or the logo seqa, and these tenses are fabricated according to the purpose of this name, and by the hands of Russell and Whitehead this logic was completed, and the symbolic mathematical logic was the result of their attempt to mix logic and mathematics as they developed mathematical logic to reach the symbolic logic, which is based It is based on the use of two types of symbols, which are constants and variables, and consists of four main sections: [4]

• The logic of cases: which uses judicial variables to study the value of the truth of the case as a unit without looking at its components, then searches for the links between them, relying on the constants of negation, kindness, separation, necessity and equivalence. And special deduction rules, and then he intends to prove all his theories, and he studies inference of its two types, by formulating it in the form of semantics of truth, so that he can test their validity with truth lists where he distinguished some incorrect cases of contrast, and also some corrupt measurements that he intended to correct.
• The logic of predicates: which studies propositions with consideration of their components, using boundary variables, and symbols for the walls of propositions, relying on the previous logical constants, to examine and develop with these tools the topics of traditional logic.
• Logic of categories: This logic studies issues as the connection of two categories, a category indicated by the subject and another indicated by the predicate, and formulates all inferences in the form of algebraic equations, to test their validity either by forms of geometry or by arithmetic proof. between the categories, and ends with building a deductive system consisting of premises of basic ideas such as the zero category, the universal category, the complementary category, and some special definitions and the deductions taken from algebraic operations on the categories. The variables of this format are categorical variables, in a more precise sense, each of them denotes a class, in addition to containing special symbols for the constants.
• The logic of relationships: which examines relationships through the initial ideas on which they are based. It also focuses on the processes of collecting and multiplying relationships, robbing the relationship and reversing the relationship, identity and embedding between relationships. It depends on variables that indicate relationships, and its constants are the previous logical constants: negation, containment, implication, meeting, multiplication and necessity, so it is completed as a precise deductive system.

Mathematical logical operations

Mathematics usually involves combining true or hypothetically true statements in different ways to produce or prove new true statements. From the English word true meaning true or if the hypothesis is false and symbolized by (F) and this first letter of the English word False meaning false, for example:[4]

• In 1492 Columbus sailed in the blue ocean, which is correct if we write (T).
• As for the phrase Napoleon won at Waterloo, this phrase is wrong if we write (F).

In general, we mean by formula that it is a statement that may include some variables which are either true or false whenever we assign certain values ​​to each of the variables, where the use of logic in mathematics is about mixing the specific language used in logic with the specific symbols used in mathematics.

The importance of mathematics in everyday life

Mathematics is very useful in daily life. Mathematics can help us to do many important things that have become a necessities of life. Here are some of the daily tasks in which mathematics is an integral part:[5]

• Money management: People will learn some skills in algebra class that will help them get money, and one of the important skills they will learn is how to calculate interest and compound interest, and they can also use this skill to manage their money, besides all this, this skill will also help them to make a better choice. Bank Account In deciding which credit card is best to own, people who borrow also need to understand the interest. It will also help them discover the best ways to save and invest money.
• Recreational sports: Geometry and trigonometry can help people who want to improve their skills in sports, as it can help them find the best way to hit the ball, make a basket or run around the track, basic knowledge of mathematics also helps in tracking sports results.
• Home Decoration and Reconstruction: Calculating areas is an important skill, as this will be useful for people in re-designing future homes and apartments, and for developing engineering thinking in all people.
• Knowledge in cooking: where people use mathematical knowledge when cooking without their knowledge of it, for example it is very common to use half or twice a recipe in food preparation, in this case people use proportions to make correct calculations for each ingredient, for example if a recipe requires 2 /3 cup of flour, the cook should calculate the amount of 1/2 or double 2/3 of the cup, then the cook should calculate the amount using standard measures used in baking such as 1/3 cup, 1/2 cup, or 1 cup.
• Helping people shop: People will use math when buying different items from the market. For example, when buying a new computer, people will need to know which store offers the best price or best financing. Math is also useful in finding the best deal for food by comparing names. For the same type of item, stores often have sales that give a percentage of the original price. Overall, it’s helpful for people to know how to figure out savings. This math skill is very useful because it helps us calculate discounts so we can buy an item at the best offered price.

The importance of logic in mathematics

The rules of logic give precise meaning to mathematical data, as they use these rules to distinguish between valid and invalid mathematical arguments. Aside from its importance in understanding mathematical thinking, logic has many applications in computer science, ranging from designing digital circuits to building computer programs and validating Programs, and logic is concerned with forms of thinking Because reasoning is involved in most intellectual activities, logic is relevant to a wide range of pursuits. The study of mathematical logic is essential for students of computer science, as well as of great value to students of mathematics and others who use mathematical proofs, for example students of linguistics, In the process of inference one makes conclusions, in inference one uses a set of statements and premises in order to justify another statement the conclusion, and the most reliable types of inferences are deductive inferences, where the conclusion must be true if the premises are so, remember the elementary geometry: Assuming that the assumptions are true here This theorem must be proven to be true in a purely logical way, like the Pythagorean theorem. They are used in geometric and other mathematical proofs of right triangles. [6]

Who is the one who sees philosophy as the science of being with what exists?

Conclusion of a research on logic in mathematics

At the end of this research, it must be noted that logic in mathematics is one of the very important aspects of the subject is the development of methods for the systematic study of the ways in which reasons can support conclusions, and therefore the development of logic has coincided with the development of philosophy throughout its history and in many cultures of the world, and work on Logic exists among the ancient Greeks and in classical Indian philosophy in medieval Islamic and Western philosophy through the explosion of activism in the modern era.

Thus, we have come to the end of this article, a research on logic in mathematics, in which we began by defining logic in mathematics, and moved to clarifying the differences and deciphering the confusion between mathematical and philosophical logic. Modern mathematical logic, its horizons and mechanism, and we ended our research with the importance of mathematics in our daily life.