Các Cơ Sở Của Số Học

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Published date: 25/05/2022 Publisher: Thu Hien Dich Truong Author: Gottlob Frege Translator: Huynh Duy Thanh

Gottlob Frege (holy name: Friedrich Ludwig Gottlob Frege) was born in Wismar, on the Baltic coast of Germany, on August 8, 1848. His father, Karl Alexander (1809 -1866), was the founder, rector of a private girls' school, and his mother, Auguste Bialloblotzky (died 1878), was the school's teacher and later principal. He attended the Gymnasium in Wismar from 1864 to 1869, and after passing the Abitur exam in the spring of 1869, he went on to study at the University of Jena. He studied here for four semesters, enrolling in chemistry, mathematics and philosophy, before transferring to the University of Göttingen (perhaps on the advice of his mentor in Jena, Ernst Abbe), where he studied 5 semesters, enrolled in physics, math and philosophy of religion. He obtained his doctorate in 1873 with the thesis "On a Geometrical Representation of Imaginary Form in Plane" (On the geometric representation of virtual forms in the plane), which further developed the work of Gauss, one of the mathematicians validated complex numbers by showing how they could be represented by a point on the plane, and showing how imaginary parts could be represented geometrically.

Although this is a purely geometric work, Frege's thesis shows a forward direction in his thought. For he was clearly interested in how results in one area could extend into another. The reason for doing so is because of the arithmetic behind it, the subject includes both the intuitive and the non-intuitive; and this is what Frege emphasizes throughout his later work. Indeed, it provided the starting point for his next work, the Habil thesis needed to become a university lecturer, submitted in early 1874 to the University of Jena and included in the faculty application. your. Titled "Methods of Calculation based on an Extension of the Concept of Magnitude", it is in this work that Frege's fundamental concerns are found. Frege argues that what underlies the concept of magnitude, and makes it intuitively independent, are the general properties of addition, and that addition itself is "the subject of these fundamental propositions to from which the whole of arithmetic sprang forth from a single seed". All other calculation methods are derived from addition ‒, for example, repeated addition produces multiplication. What is involved, Frege argues, is the iterative application of an operation, which can be represented by a suitable function, such that the value of the function given a given argument from it can become the argument. of that function. For example, plus 1 can be represented by the following function f(x) = x+1, plus 2 equals ff(x), and so on. doubled is g(x)=x+x, multiplied by four is gg (x), and so on. Frege continues to examine the relationships between many different types of mathematical functions; but the key point to note here is the central role of the proposed function concept in the necessity theory of magnitude, making it possible to connect the different fields of arithmetic. It was Frege's development of function theory that played an important role in the logical transformation that he later contributed to.

After completing his Habil thesis, and the necessary oral exam, i.e. public debate and mock lecture, in May 1874, Frege was appointed to the position of Privatdozent (unpaid lecturer) at the University of Jena, where he taught for the rest of his career. During his first few years, his teaching responsibilities were heavy, and he published only four short works, three of which were book reviews, before publishing his first book, Begriffsschrift, in 1879 Although this work is celebrated today for ushering in the age of modern logic, it is clear from the Preface that Frege's aim in developing a stronger theory of logic was not to focus on improving traditional logic. system which aims to give arithmetic the strongest possible bases. Since the strongest bases are found to be logical ones, the task becomes determining how much arithmetic range can be established from logic. After explaining his theory of logic in Parts I, II, and III, Frege did indeed succeed in showing that mathematical induction can be analyzed in terms of pure logic, and this result certainly did. encouraged him to attempt to establish all arithmetic from logic. Logical programming, as it is called today, preoccupied him for the next quarter of a century.

But why does one find it necessary to provide a basis for arithmetic? The answer lies in the developments in mathematics in the nineteenth century. In the work of Gauss, Lobatchevsky, Bolyai, and Riemann, the first non-Euclidean geometries were constructed by replacing Euclid's notorious V axiom, the parallel axiom, with another. Many people assumed that contradictions would appear sooner or later, but when double elliptic geometry was proven applicable to spheres, the 'line' being interpreted as the great circle, it was found that the figure non-Euclidean learning is consistent if Euclidean geometry is

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