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Logicism: Mathematics as a Subset of Logic

According to the philosophy of mathematics, logicism is a based program on the theses that mathematics is a continuation of logic, most or all of mathematics is reducible to logic, and most or all of mathematics may be modeled in logic, all of which are true for some coherent meaning of the word "logic." One of the researchers, identified only as Dedekind, was able to develop a model that satisfied the axioms that characterize the real numbers by employing a particular set of rational numbers in his research. As a result of this and other considerations, he concluded that algebra, arithmetic, and analysis could be reduced to the natural numbers with a "logic" of categories. As a result, logicism is the idea that mathematics is attributable to logic and, as a result, is nothing more than a subset of logic. Mathematics can be known in advance, according to logicism, but our understanding of mathematics is simply an extension of our understanding of logic in broad, and is therefore analytic, requiring no particular capacity of mathematical intuition on the side of the observer. The fundamental basis of mathematics, according to this viewpoint, is logic, and all mathematical propositions are required to be logical truths. This theory argued that if one accepts logic, one is obligated to accept arithmetic and that one does not require any additional rationale for accepting, and that one does not need to employ any specified technique of evidence to accept arithmetic, (Shapiro, 2017).

Intuitionism is predicated on the notion that mathematics is a product of the human imagination. To comprehend the validity of a mathematical statement, one must first establish a mental model that shows that the statement is correct. Communication amongst mathematicians is only an aid in constructing the same mental model in various brains. In regards to providing far-reaching ramifications for the everyday routine of mathematics, this perspective on mathematics has the additional consequence that the notion of the infinite regress is no longer true. Indeed, there are statements, such as the Riemann assumption, in which there is now neither proof of the assertion nor evidence of the negation of the statement. Because understanding the negation of an assertion under intuitionism indicates that one can demonstrate that the statement is false, this suggests that both statements are false intuitionistically, at least for the time being. This temporal dependence of intuitionism is critical: assertions can become demonstrable over time and thus can become intuitionistically valid even if they were not previously validated in the intuitionist sense, (Lubarsky,R.F; Richman ; Schuster,P, 2018). 

The instrumentalist view of mathematics holds that mathematics is a buildup of rules, facts, and skills that are to be applied in the pursuit of some external goal or objective. As a result, mathematics is a collection of unrelated rules and facts that serve a practical purpose. Being a scientific discipline, instrumentalism is a philosophical perspective that holds that a scientific concept or theory's value is ascertained not by if it is true or corresponds to actuality in some way, but by the degree to which it assists in making precise numerical predictions or resolving conceptual issues. This is the perspective that theories should be viewed predominantly as skills to solve practical issues rather than as comprehensive characterizations of the physical world. A common implication is that instrumentalists often question whether it is even reasonable to learn of logical terms as correlating to the outer world in the first place. In this sense, instrumentalism is completely contradictory to scientific realism, which either holds that the purpose of theories is not simply to garner reliable predictions, but also to accurately describe the world, (Lubarsky,R.F; Richman ; Schuster,P, 2018).

Intuitionism: Mathematics as a Product of Human Imagination

The public believes that maths teachers must demonstrate how and where to solve problems to their students. Others believe that when children discover how to solve difficulties on their own, they learn more quickly and effectively. According to some studies, kids are better at addressing real-world problems once they have memorized their arithmetic knowledge, which they do first. Numerous studies show that teaching math concepts through the use of word problems helps youngsters retain more of what they've learned more effectively. Some educators place a great emphasis on teachers adhering to a predefined curriculum plan, while others believe that teachers should constantly alter the breadth and sequence of topics they teach based on their students' comprehension and willingness to study to achieve the optimal learning outcomes. Many experts have increasingly argued that teachers' expertise and perceptions influence their instructional practice, that the majority of studies on teacher beliefs used the qualitative analysis or small samples, and that most of them involved prospective teachers rather than practicing teachers, as has been demonstrated. This serves to corroborate the notion that the majority of teachers will deliver the mathematical ideas following their perspective of what mathematics is, (Leng, 2018). 

There are a plethora of compelling reasons to investigate the history of mathematics. Because it shows them how mathematics has evolved through time and in different places, it aids students in developing a deeper grasp of the mathematics they already have learned. It teaches students to think creatively and flexibly by allowing them to observe historical evidence there are multiple, totally valid ways to understand concepts and to perform operations at the same time. In an ideal world, a course in the history of mathematics would be included in each mathematics main plan. Other benefits of studying the history of mathematics entail the following: historical knowledge increases students' motivation and aids in the development of a positive attitude towards mathematics; visualizing the barriers encountered in the development of mathematics in the earlier days enables them to see the difficulties encountered in the present; solving problems from history aids in the order to develop students' mathematical thinking; history needs to bring out the human side of mathematical ability, and history helps bring out the human aspect of mathematical thinking and it helps students to create a positive attitude towards mathematics, (Cuomo, 2017) 

One of how it may be implemented in the classroom is by permitting the lesson to be planned in such a manner that the students can visit a museum to witness the influence that mathematics has on our society and our culture on a particular day. For instance, the artwork of Henry Moore was inspired by the string models constructed by Theodore Olivier to educate descriptive geometry in the nineteenth century. Salvador Dali's paintings made extensive use of principles from higher-dimensional geometry. Another method is to allow students to tour an antiquarian book room and feel the excitement of getting a picture of a work by Euclid, Pascal, Newton, Euler, and other great thinkers in their hands. The ability to view images on the internet is a terrific first step, but the ability to physically see and smell the genuine thing, as well as, if feasible, touch it, is a system was successfully that can increase the students' desire to comprehend the data contained in the books. Finally, specific video animations can be shown to learners in the classroom to help them strengthen their mathematical comprehension, (Cuomo, 2017). 

Instrumentalism: Mathematics as a Collection of Skills

Student-centered learning is an excellent method of acquiring mathematical knowledge. One way to accomplish this is to empower students to take the initiative in the classroom by allowing them to share their work and conduct group discussions. Learners take responsibility for their studying, writing, and learning to build, test, and refine their critical thinking skills and knowledge. Students can also participate in discussions that are held accountable to the assignment, the studying, and the norms of reasoning that have been established. Students should be included in the life choice process regarding their education, (Goodman, 2016) 

The second method of embedding cultural responsiveness in content and process of instruction is to build on the various resources and perspectives of students in the classroom. This method is known as constructivism. The school system is encouraged to employ guiding principles in language acquisition and mathematics learning to improve academic success, (Goodman, 2016) 

Allowing learners to apply in dynamic, collaborative groupings to resolve issues and study texts to demonstrate comprehension of a task or idea from many viewpoints is the third method of instruction. Fourth, the school promotes the inclusion of all children by employing guiding principles such as dual-certified instructors, differentiated instruction, competent aides, and tailored learning plans to help students with special needs in general education classrooms, (Goodman, 2016) 

Student-centered learning can also be developed by encouraging students to continually build their thinking around ideas and concepts, as well as to be able to describe the processes and thoughts that they engaged in when dealing with a challenging task. Both students and teachers are capable of listening to and exchanging ideas to build on and apply fresh information as well as alternative methods to their grasp of an idea or notion that is becoming increasingly complex, (Goodman, 2016)

Studying, writing, and thinking habits can also be applied to a variety of genres and disciplines by students who have developed good habits of mind. Students can ask questions, pose inquiries, and investigate responses as a means of engaging in real-life scenarios and transferring their knowledge to other fields. They can apply their knowledge to a variety of situations and settings, (Goodman, 2016) 

The vocational and art interests can be implemented in a variety of ways, including academic curriculum, improving student engagements, encouraging students with a range of student learning to excel in studying, and improving students' academic growth, among other things. As well as valuing the health of children, schools should educate them in good ways to deal with life's obstacles and achieve wellness in their own lives. Students can also be evaluated on their participation in the process, their group work, and their final product, (Goodman, 2016). Additionally, all students should be stimulated and inspired to pursue on their own as a result of their previous experience with the COVID-19 epidemic. 

References 

Cuomo, S. (2017). Ancient Mathematics, Reviewed. London: Routledge. 

Goodman, M. K. (2016). An Introduction of Early Development of Mathematics. Hoboken: Wiley. 

Leng, M. (2018). Mathematics and Reality, Reviewed. Oxford University Press. 

Lubarsky,R.F; Richman ; Schuster,P. (2018). The Kripke Schema in metric topology. Mathematical Logic, Reviewed, 58(6): 498-501. 

Shapiro, S. (2017). Thinking About Mathematics. In The Philosophy of Mathematics, Reviewed. Oxford, UK: Oxford University Press.

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