MIK-1018 What Is Mathematics? – Foundations of Mathematical Thought - 2.5 stp

This course explores mathematics from within: its objects, its methods, and its internal logic. What distinguishes mathematical thinking? What makes a proof convincing? How do abstraction and structure shape modern mathematics?

We begin with classical conceptions of number, geometry, and proof, and follow the development of foundational ideas that shaped mathematics into a rigorous and axiomatic discipline. The course examines the rise of set theory, the formalization of mathematical reasoning, and the emergence of structural viewpoints that define contemporary mathematics.

Central topics include the axiomatic method, the nature of proof, the concept of infinity, formal systems, computability, and Gödel’s incompleteness theorems. Through these developments, students gain insight into both the power and the limitations of formal mathematical reasoning.

Rather than treating mathematics as a collection of results, the course presents it as a coherent intellectual enterprise grounded in precise definitions, logical deduction, and the search for structure.

General university admissions certification or prior learning and work experience qualifications.

Application code 9199.

Knowledge

The student:

• has broad knowledge of the major foundational approaches to mathematics, including Platonism, formalism, logicism, and intuitionism

• understands key historical developments that shaped modern mathematics, from classical conceptions of number and geometry to the formal and axiomatic turn of the 19th and 20th centuries

• has insight into the emergence of set theory, the axiomatic method, and structural viewpoints in mathematics

• understands the significance of Gödel’s incompleteness theorems, computability theory, and the inherent limitations of formal systems

• is familiar with central concepts such as proof, abstraction, infinity, formalization, and mathematical structure

Skills and General Competence

The student:

• can analyze and discuss the foundational assumptions underlying mathematical definitions, proofs, and theories

• can critically assess different approaches to the foundations of mathematics and their implications for mathematical practice

• can reflect on mathematics as a coherent intellectual enterprise grounded in logical reasoning and structural thinking

• can connect foundational perspectives to contemporary developments in mathematics, computation, and related scientific fields

This is an asynchronous, digital micro course. There are no scheduled lectures or meetings. All course materials and resources are available digitally via Canvas. The course can be taken at any point during the semester. The course is offered both in the spring and fall.

There will be approximately 12 hours of digital lectures and modules and approximately 50 hours of self study and problem solving.