m. moniri

In recent years, many efforts have been made to use mathematics to solve problems outside the field of mathematics. These fields go far beyond the topics traditionally studied in applied mathematics departments. Even parts of mathematics that until recently were too far-fetched to have concrete applications are now successfully used in various fields. The main question of this article is why mathematics is so useful. This is an old question about mathematics, and although it has been discussed a lot, it does not seem to have a concluding answer. Here we examine some features of mathematics that seem to play an important role in all its applications. We will also examine the effect of accepting some philosophical views on this issue. Among these philosophical views, realism and nominalism, Quine's naturalism, and Penelope Maddy's mathematical naturalism can be mentioned. The history of two important areas of mathematics, namely abstract algebra and formal logic, have been examined in detail as examples.

Is arithmetic determinate and definitive? In other words, are there reasons for the correctness or falsity of each arithmetic sentence? For example, is there evidence that Goldbach’s conjecture is true or false, even if we don’t know about it? At first glance, the answer seems to be clearly yes. But how can you be sure? What does Gödel’s incompleteness theorem say about this? What is the relationship between independence of some axioms of set theory, such as the axiom of choice and the continuum hypothesis, with these questions? In this article, we will examine these questions. In addition, we will examine the impact of the presence of unconventional facilities such as computing machines that are able to perform an infinite number of instructions in a finite time, as well as proof systems equipped with infinite rules on the answers to the above questions.