In this course you are introduced to methods of writing proofs which are rigorous, readable, and elegant. Mathematical communication, both written and spoken, is emphasized throughout the course. You will also explore proof-relevant mathematics by interacting with the Lean proof assistant.
Values
Math 301 champions a particular set of values and standards about mathematics.
- Mathematics as a precisely defined language for communication
- After we have ceased to exist, the life has become extinct, the universe dark, and eventually, the entropy of the universe at its maximum, there is no humans, no consciousness, and no mathematics. There is nothing eternal about mathematics, it is only good so far as we exist. But, it is one of the best tools we have in communicating complicated ideas effectively. In this course we emphasise the value of mathematics as a precisely defined language for communication.
- Rigour is more important than cleverness
- In this course, rigour, clarity of thought, and expression are much more important than clever tricks, possibly in contrast to your experience with other maths courses.
- There is a history to mathematics and its development
- We will put every piece of mathematics we discuss in its historical context as much as possible.
Goals
Upon successful completion of this course you are able to:
- Communicate your mathematical ideas clearly and robustly by writing correct, clear and precise mathematical proofs.
- Communicate your mathematical ideas efficiently and concisely, in an appropriate level of detail.
- Formalize and digitize your proofs into computer codes.
- Interact with the proof assistant
Lean. - Discerning the assumptions and conclusions in reading mathematical theorems and knowing where to start and where to end in constructing your proofs.
- Accurately recall concepts and definitions and state and prove theorems in the mathematical areas covered.
- Accurately use standard mathematical notation and terminology in mathematical writing, including symbolic logic, sets and set operations, infinities, functions, metric spaces, etc.
- Identify techniques for proving a proposition based on its logical structure.
- Recognize and apply standard proof techniques, including proof by contraposition, proof by contradiction, weak and strong mathematical induction, and the well-ordering principle.
- Learn to simplify your proofs.
- Evaluate proofs of mathematical statements in terms of the features that make them effective or ineffective with regard to criteria of constructively and computability.
- Modeling novel mathematical problems using the tools you learn in this course and finding solutions to those problems in your model.