OFFICIAL SYLLABUS MATH 531-ALGEBRAIC CONTENT, PEDAGOGY, AND CONNECTIONS
... 2.1.4 The distributions of various types of real numbers 2.2.1 The complex numbers and the complex plane 2.2.2 The geometry of complex number arithmetic Chapter 3: Functions 3.1.1 What is a function? 3.1.2 Problem analysis: from equations to functions 3.2.1 Analyzing real functions 3.2.2 Composition ...
... 2.1.4 The distributions of various types of real numbers 2.2.1 The complex numbers and the complex plane 2.2.2 The geometry of complex number arithmetic Chapter 3: Functions 3.1.1 What is a function? 3.1.2 Problem analysis: from equations to functions 3.2.1 Analyzing real functions 3.2.2 Composition ...
Notes
... Brouwer famous for his fixed point theorem and for an approach to mathematics called intuitionism. He was not only a brilliant mathematician, he was good at “provoking” other top mathematicians, especially David Hilbert. He introduced several fundamental ideas into mathematics including a “computati ...
... Brouwer famous for his fixed point theorem and for an approach to mathematics called intuitionism. He was not only a brilliant mathematician, he was good at “provoking” other top mathematicians, especially David Hilbert. He introduced several fundamental ideas into mathematics including a “computati ...
PROBLEM SET 9 (1) There is always a bi
... (1) There is always a bi-infinite path, and in fact we can write down an explicit vertex in one. Label the path of length n with numbers {0, . . . , n} such that i ∼ j ⇔ |i − j| = 1, and consider the vertex in the ultraproduct corresponding to the sequence bn/2c of vertices in the factors. For any f ...
... (1) There is always a bi-infinite path, and in fact we can write down an explicit vertex in one. Label the path of length n with numbers {0, . . . , n} such that i ∼ j ⇔ |i − j| = 1, and consider the vertex in the ultraproduct corresponding to the sequence bn/2c of vertices in the factors. For any f ...
Non-standard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals. Non-standard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers.Non-standard analysis was originated in the early 1960s by the mathematician Abraham Robinson. He wrote:[...] the idea of infinitely small or infinitesimal quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the formative stages of the Differential and Integral Calculus. As for the objection [...] that the distance between two distinct real numbers cannot be infinitely small, Gottfried Wilhelm Leibniz argued that the theory of infinitesimals implies the introduction of ideal numbers which might be infinitely small or infinitely large compared with the real numbers but which were to possess the same properties as the latterRobinson argued that this law of continuity of Leibniz's is a precursor of the transfer principle. Robinson continued:However, neither he nor his disciples and successors were able to give a rational development leading up to a system of this sort. As a result, the theory of infinitesimals gradually fell into disrepute and was replaced eventually by the classical theory of limits.Robinson continues:It is shown in this book that Leibniz's ideas can be fully vindicated and that they lead to a novel and fruitful approach to classical Analysis and to many other branches of mathematics. The key to our method is provided by the detailed analysis of the relation between mathematical languages and mathematical structures which lies at the bottom of contemporary model theory.In 1973, intuitionist Arend Heyting praised non-standard analysis as ""a standard model of important mathematical research"".