Euler and the Function Concept and Philosophy of Mathematics Essay

Euler and the Function Concept

In most cases, Euler has been identified as a calculator of genius and in fact, in the formal calculations together with transformation, in the elementary word sense, Euler still remains to be an exemplary calculator. On the other hand, he is known for creating the most recent and significant notions together with methods; this is the principal value in which certain cases can be clearly understood several years after his death. The purpose of function concept was to expand the number's concept with the aim of embracing real and to certain extents, complex numbers; creating symbolic algebra, for instance, Viete and Descartes; studying of motion as the main issue of science; creating links between geometry and algebra; introducing variable that had equations, which express how the variable is related; and introducing the identification or concept of variables' dependence and independence in equations. According to Euler, the variable quantity's function is referred to the analytical expression that is composed in a manner from the constant quantities or numbers and from variable; x is denoted as the variable quantity and all quantities rely in different ways on the x; the variables and the quantities can also rely on the function of the x; infinite series have made the concept of function to be very significant to mathematics; and it has encouraged the replacement of the curves, which were being used as the basic conceptual interest's object in the analysis.

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Philosophy of Mathematics

According to philosophy of mathematics, epistemology theory was developed to assist us in determining how people get to know mathematics, while ontology explain the sense in which the mathematical objects came it existence though it is an escape, according to Platonism, though mathematics is based on self-evident facts and it is more about non-material objects that exist independent without human thoughts: mathematical reality can only lie outside and the function of mathematicians is to find ways that can assist them in discovering or observing it in addition to being a theorems that can be used as a proof and at the same time give a vivid description of grandiloquently as the creations are important for observation; it is also a mental construction according to intuitionism, and that is why most human beings knowledge is known for starting with intuition, before proceeding to concepts and ending with the ideas: mathematics can be used to present how the most brilliant examples of how pure reasons have been successfully enlarging their domains without depending on past experience; mathematics cannot be more than what people have been thinking about. For empiricism, mathematics is known as the knowledge that is being supported with proofs, even with a received opinion that is true, if it cannot be earnestly and vigorously contested, it is likely to be held in a prejudiced manner, with little feelings or comprehension of mathematics' rational grounds. For empiricism, through inductive reasoning, mathematics can be established and this indicates that most theorems are likely to be fallible or there are moments in which they can be false; the theorems requires constant testing with a number of counter-examples that can be revised and corrected. According to social constructivism, mathematics is referred to a truth that is based on social agreements of the mathematicians: cultural and social values are important as far as determining mathematical knowledge development is concerned, though formalism believe that it is all about proving theorems, solving problems and knowing that everyone, especially mathematicians, know that they have answers, because they have mastered the ground rules.

Communication, Professionalism

As far as the history of mathematics is concerned, universities are assumed to be important platforms for mathematical research as well as education for the last few decades; accessing education can be restricted because of religion or ethnicity, person's gender, geographical isolation, especially from the center of power or knowledge and socio-economic circumstances; mathematical ability can improve the social status of an individual in addition to opening up career opportunities; and new mathematics are not only created by formal education. Through mathematics, it has been possible for the government projects to be in a position of producing geographical maps using precise information related to land valuation and ownership; government workers can use scientific and trigonometry instruments to come up with exact measurements; and application of applied mathematics opened up more opportunity for researchers and students to further their studies. Arithmetical questions are usually entertaining as well as delightful as other subjects; using mathematics, people have been counting millions of money each day, each month and each year; with the mathematical concepts, people have been working hard to claim priority in terms of proof and discovery; the work of several mathematicians have been recorded and stored in archives for future references; and mathematics have got a wider recognition, through in different mathematical communities, institutions (through serving the needs of the high school, college and undergraduate math teachers and students) as well as professions all over the world.