Relationship between logic and mathematics. Logic and mathematics have shown a historical correlation that continues to this day. Without the use of logic, mathematics would not be possible, and with it, we could imagine the number of virtues that society today could not enjoy society due to the absence of this science.
But how does this relationship happen? At what point do mathematics and logic go hand in hand and converge on a new phenomenon? To get the answer, we must do a small review of these two concepts to know how is the relationship between logic and mathematics. Let’s start by understanding a little bit about logic.
Logic: reasoning and its ability to organize
Logic is a branch of philosophy, being a formal science whose material object is human reasoning. Logic has been able to conceptualize the organizing character of human thought, that is, its ability to establish connections between the elements of reality to generate rational conclusions about reality.
In other words, logic studies the way humans construct rational thought. For this, processes such as inference, prediction, demonstration, etc. are analyzed. In this way, the contributions of logic have managed to diversify in a splendid way, serving as a precedent in the development of hundreds of disciplines.
Law, science, psychology, physics, language and mathematics are just one of the branches that we can highlight, they have enjoyed great contributions thanks to logic. Science can thank him for part of his triumph in the scientific method, thanks to logical principles. that govern the rigorous character in the interpretation of results, or also, in the search for truth itself.
The objective of logic has been, precisely, to understand the way in which we access valid knowledge through our reasoning, this being one of the most unique characteristics of humans. It is here that we can make a link with mathematics and its relationship with logic.
How does logic relate to mathematics?
To begin, we can refer to the example of a theorem, which is nothing more than a proposition that shows us that the relationship between two or more variables will produce a certain result. The validity and construction of theorems is possible thanks to logic, that is, the ability to deduce, establish and interpret relationships between elements.
Through logic, we can create relationships between numbers, which in turn allows us to deduce a result through the calculation itself. In this sense, logic operates intrinsically in mathematical processes, precisely in resolution processes. Otherwise, we would not be able to build such relationships, and, at the same time, solve them.
Many of these ideas acquire greater boom in the twentieth century, because logic had not been treated from a mathematical approach with due importance until this date. It was shown that, by means of increasingly abstract studies in logic and proof theory, mathematical objects could also be analyzed within logic. Here, the mathematical logic.
If we wanted to define in a few words the previous section, we could talk about the birth of mathematical logic. It is also called logistic logic or symbolic logic, and is responsible for studying logic on mathematics, through methodology from logic applied to mathematics.
Through logical reasoning, relationships are established between variables, applying different mechanisms to demonstrate and infer the results from logic. With this, logical analyses are made to argue how mathematical propositions can be established.
In this way, the inference itself is studied, which is possible through subareas of mathematical logic such as propositional logic. This is how mathematical reasoning processes are studied, in order to understand how demonstrations are carried out from mathematical analysis.
For the study of mathematical objects, mathematical logic emphasizes the study of other mathematical concepts such as algorithms, sets, and proofs. This, in order to operate according to mathematical taxonomy, outlining its mechanisms according to the properties of its object of study, in this case, the mathematical elements.
The development of mathematical logic has given rise to some theories such as set theory, which is responsible for studying the relationship between two or more sets, this being one more form of mathematical analysis that allows to interpret the way in which the elements of each set can be related.
Logic has diversified mathematics
Mathematical logic makes it possible for an axiom to be proved without, effectively, being proved. That is, it is not necessary to demonstrate what can be deduced by means of logic, of the understanding of the relations of the elements, in this case, of the mathematical type.
Axiomatizing mathematical theories and theorems has been thanks to mathematical logic, which allows the development of mechanisms in computation and formal systems respectively. Mathematical logic is a method applied especially to mathematics, in order to establish logical systems for the analysis of its elements and understand their results.
The relationship between logic and mathematics is essential, because in the second, we show a strong need to adopt valid reasoning, which allows establishing reasonable relationships to obtain reliable, true results. This is nothing more than talking about the important role that logic plays in mathematics.
We could not enjoy a science as formal as mathematics without the faculty provided by logic, thus accessing the valid knowledge that, historically, has been the basis for the development of multiple disciplines and areas of knowledge, authors of great contributions to society itself. This is how logic and mathematics have to relate.
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