Introduction to Mathematical Proofs


Is there only one way to prove a mathematical statement? Do we always follow a deductive method when trying to reach a new conclusion? The quick answer is: no. Mathematics has developed a variety of techniques over the centuries in the continuous pursuit of finding the clearest or most elegant proof for a problem. All these methods show a highly creative thinking process that exploits experience, intuition, and even a sense of adventure. 

By relating the presentation of different types of proofs to historical examples and anecdotes from the lives of famous Mathematicians, from Euclid to Andrew Wiles, the students will embark on a rich investigative journey that will show them why mathematicians can be considered to be part detectives and part artists, thus developing their analytical skills and a broader overview of what a proof is. 

Programme details

Courses start: 19 Apr 2024

Week 0: Course Orientation

Week 1: Introduction to Logic and Sets

Week 2: Proof Techniques: Direct Proof

Week 3: Proof Techniques: Mathematical Induction

Week 4: Relations and Functions

Week 5: Proof Techniques: Proof by Cases and Counterexamples

Week 6: Axiomatic Systems

Week 7: Quantifiers and Logic

Week 8: Proof Techniques: Proof by Contrapositive and Proof by Exhaustion

Week 9: Divisibility Proofs

Week 10: Proof Techniques: Proof by Contradiction

Digital Certification

To complete the course and receive a certificate, you will be required to attend and participate in at least 80% of the live sessions on the course and pass your final assignment. Upon successful completion, you will receive a link to download a University of Oxford digital certificate. Information on how to access this digital certificate will be emailed to you after the end of the course. The certificate will show your name, the course title and the dates of the course you attended. You will be able to download your certificate or share it on social media if you choose to do so.


Description Costs
Course Fee £257.00
Take this course for CATS points £10.00


If you are in receipt of a UK state benefit, you are a full-time student in the UK or a student on a low income, you may be eligible for a reduction of 50% of tuition fees. Please see the below link for full details:

Concessionary fees for short courses


Dr Niccolò Salvatori

Dr Niccolo Salvatori is a Guest Teacher at the London School of Economics and a former Honorary Research Associate of King's College London. In the past, he has lectured Calculus for the University of California, Berkeley for study abroad programmes and has recently joined the Department for Continuing Education, Oxford. Niccolo also teaches at Secondary School and Sixth Form level, including Further Mathematics, and has been creating and delivering projects and enrichment courses for A-Level students since 2017. His interests are in Analysis and Geometry, but he is passionate about Logic and has great admiration for the work of Alan Turing and its far-reaching consequences.

Course aims

To introduce the basic tools of correct thinking and the logical techniques applied in mathematics to create proofs.

Course objectives:

  • To develop a solid understanding of the foundations of mathematical proofs, including logic, sets, and proof techniques.
  • To cultivate critical thinking skills and the ability to construct rigorous mathematical arguments.
  • To enhance problem-solving skills through engaging in-class discussions, problem-solving activities, and practical applications of proof techniques.
  • To understand how proofs have evolved through the centuries thanks to the work and dedication of famous mathematicians.

Teaching methods

Students will have access to a pre-recorded lecture to be watched in advance of the weekly online session.

Learning outcomes

By the end of the course, students will be expected to:

  • have a solid understanding of the basic principles of mathematical proofs, including logic, sets, and proof techniques;
  • have improved their critical thinking skills and ability to construct rigorous arguments through class discussions and problem-solving activities, appreciating how this can be applied in everyday life;
  • have knowledge of the historical development of proofs thanks to famous mathematicians' work, providing them with a broader perspective on the subject. 

Assessment methods

Short exercises and a short report.

Students must submit a completed Declaration of Authorship form at the end of term when submitting your final piece of work. CATS points cannot be awarded without the aforementioned form - Declaration of Authorship form


We will close for enrolments 7 days prior to the start date to allow us to complete the course set up. We will email you at that time (7 days before the course begins) with further information and joining instructions. As always, students will want to check spam and junk folders during this period to ensure that these emails are received.

To earn credit (CATS points) for your course you will need to register and pay an additional £10 fee per course. You can do this by ticking the relevant box at the bottom of the enrolment form or when enrolling online.

Please use the 'Book' or 'Apply' button on this page. Alternatively, please complete an enrolment form (Word) or enrolment form (Pdf).

Level and demands

GCSE Mathematics 

Essential knowledge:

• GCSE level algebra.

• GCSE level number facts and procedures (e.g.: basic operations, factorization, sequences of numbers, etc.).

Desirable knowledge

• Definition of a function and composition of functions.

Students who register for CATS points will receive a Record of CATS points on successful completion of their course assessment.

To earn credit (CATS points) you will need to register and pay an additional £10 fee per course. You can do this by ticking the relevant box at the bottom of the enrolment form or when enrolling online.

Coursework is an integral part of all weekly classes and everyone enrolled will be expected to do coursework in order to benefit fully from the course. Only those who have registered for credit will be awarded CATS points for completing work at the required standard.

Students who do not register for CATS points during the enrolment process can either register for CATS points prior to the start of their course or retrospectively from the January 1st after the current full academic year has been completed. If you are enrolled on the Certificate of Higher Education you need to indicate this on the enrolment form but there is no additional registration fee.

Most of the Department's weekly classes have 10 or 20 CATS points assigned to them. 10 CATS points at FHEQ Level 4 usually consist of ten 2-hour sessions. 20 CATS points at FHEQ Level 4 usually consist of twenty 2-hour sessions. It is expected that, for every 2 hours of tuition you are given, you will engage in eight hours of private study.

Credit Accumulation and Transfer Scheme (CATS)