30 October 2021
Beauty of Mathematical Proofonline course
This Workshop addresses the foundations of mathematics – and indeed the nature of the construct of mathematics itself. Perspectives that are not normally covered in the taught components of pre-university mathematics courses! The Workshop contains a detailed and “hands-on” overview of the types and constructions of mathematical proof.
This workshop is broken down into the following sections:
SECTION 1: Types of Mathematical Proofs
In this section, students will have the opportunity to explore and develop different types of mathematical proofs, including: direct proofs, proof by contradiction, the contrapositive, existence and uniqueness proofs, and of course, mathematical Induction. We explore the use of “visual proofs” and problem-solving methodologies in the construction of mathematical proofs. This section will be of especial interest to Further and Higher-Level Mathematics students, as well as those students registered for Mathematics Olympiads.
SECTION 2: Proof and Rigour
In this section, students will learn that mathematical proof is fundamentally a matter of rigour. And that interestingly the level of rigour used in mathematics has varied over the centuries, and paved the way to a resurgence of mathematical analysis later in the 19th century. The ancient Greeks for example, expected detailed arguments (consider the construction of Euclidean Geometry), but at the time of Newton (in the 17th century in his construction of calculus) and later Cauchy (in the early 19th century with his definition of limit and the infinitesimal) the analytical methods employed were less rigorous.
SECTION 3: 21st Century Mathematical Proofs
In the final section, students are exposed to techniques used in modern mathematics proofs. We begin with famous examples of the translation of a complex mathematical problem into different mathematical objects (sometimes from seemingly unrelated branches of mathematics). This then leads into an exploration of, and debate about, the use of so-called computer-assisted proofs. But not all 21st century proofs use sophisticated modern mathematical constructions, and in this unit we present to students a wonderful example of how a long-standing mathematical conjecture was proved in just a few pages using classical techniques. The section culminates with a presentation of how exciting developments in quantum physicists are attacking the Riemann Hypothesis and inspiring new mathematics.
Ian Tame is a Master postgraduate in Mathematics (with specialism stochastic processes). He has published mathematics articles (in English and German), possesses a postgraduate teaching qualification, and has over 25 years’ experience in education.
Students studying, or about to study mathematics, physics or philosophy at A-level or International Baccalaureate Diploma and other pre-university courses - especially those who aspire to study mathematics, science, computing or engineering-based disciplines at university; and current Undergraduate students studying Mathematics.
By the end of the course students will have acquired an overview of the different types of mathematical proofs, including techniques used in 21st century mathematical proofs. And students will have gained insights into the process of constructing a mathematical proof with an emphasis on the essential role of rigour.
EUR 99: plus 19% VAT