Logistics:
| Credits: | 4 |
| Classes: | Mondays, Tuesdays, Thursdays, Fridays, 2:10pm to 4:10pm |
| Classroom: | AC-04-LR-302 |
| Teaching Staff: | Anwesha Ghosh (teaching assistant) and Rhea (teaching fellow) |
Links:
| Course Outline and Lectures |
Course Description:
“Wonder not then, what God for you informs,
If your objects may be steps to ascend to God.”
— John Milton
There is a particular kind of grief that belongs to those who reach for the infinite and find, at the last moment, that their arms are too short. The mystic knows it. The poet knows it. And, as this course will argue, so does the mathematician.
Once exiled from the Heaven of the Infinite, we have found ourselves in a world that feels chaotic, broken, and opaque. Since that imagined abandonment, human beings have attempted to reconstruct a stairway to the skies: through art, music, poetry, science, and even mathematics. However, we are not made for mathematics. We are creatures of bone and breath and endings, furnished with ten fingers, a little din in our throats, and a brief corridor of years between birth and disappearance. And yet, from within this corridor, we have had the audacity to attempt a description of the infinite.
In the first module of this course, we will trace how different civilisations have constructed systems for counting, comparing, and compressing the world into symbol and language. What we discover, when we look closely, is that every such attempt bears the fingerprints of its makers. What presents itself as the neutral grammar of reality turns out, on closer inspection, to be a series of choices made by particular people in particular moments: what symbols to use, how many to use, what operations to permit, when to divide by zero, and what to use as a yardstick for proof.
We will then turn to what may be the most ambitious intellectual project in human history: the attempt to build a formal language capable of capturing all of mathematics, a tower of symbols tall enough to touch the sky. This is the Babelian dream, and like Babel, it ends in a ruin. A language powerful enough to describe arithmetic is powerful enough to describe, most fatally, itself. It can encode well-formed sentences like “this statement cannot be proved,” which are neither provable nor refutable. With merciless rigour, Kurt Gödel showed that this is not a flaw of a particular construction but an inescapable feature of logic. In the second module of the course, we will sketch the proof of this incompleteness and show that any symbolic language powerful enough to do arithmetic will contain truths it cannot prove. We will see that mathematics is not merely incomplete in the way any unfinished project is incomplete, but rather broken from the inside and beyond repair.
And yet, somehow, most mathematicians wake up every morning and go to work. This is because mathematics works, with an extravagance that borders on the offensive. Structures conjured in abstraction turn out to describe the world with almost indecent precision. The same incomplete, historically contingent mathematics turns out to be unreasonably effective in physics, economics, and the social sciences. In the final module of this course, we will sit with utility as a consolation for incompleteness.
By the end of this course, you will have witnessed the grief of the subject. You will also have witnessed something else: human beings, when confronted with a proof of their own limitations, chose to keep rolling the boulder anyway. This is, in the end, what mathematics is. Not a stairway to the infinite; not a view from above. It is the stubborn, fingerprinted, perpetually incomplete work of creatures like us who refused to stop counting.
Learning Outcomes: This course studies mathematics as a language. The profit of this course is the same as that of Caliban learning language:
- To learn to name: We will familiarise ourselves with mathematical objects such as numbers, primes, functions, sets, logic, and proof.
- To learn to speak: We will become accustomed to operating within a formal grammar of symbols and inference, and to constructing arguments that count as valid inside a system.
- To learn to curse: We will discover how self-reference, paradox, and incompleteness arise from within the system itself — and how the language can be turned against itself.
- To learn to lie: The final presentations of this course will explore how mathematical language, especially statistics, acquires authority — and how that authority can be manipulated.
Grading: A serious student of mathematics learns best by active involvement. This course is structured to incentivise involvement through the following:
- Quizzes (31%): There will be four in-class quizzes. The top two will contribute 20% of the grade, the third highest will contribute 7%, and the lowest will contribute 4%.
- Participation & Engagement (17%): Attendance, in-class exercises, and participation in lecture and discussion sessions.
- Course Project (29%): A final course project.
- Final Exam (23%): A closed-book in-class exam.
Policies: From Ashoka's Academic Integrity Policy (MyAshoka → Information and Documents → Office of Academic Affairs): plagiarism—which is a matter of producing academic work that borrows, without acknowledging, from another person's work—is a serious academic offense. All violations of Academic Integrity Policy (including but not limited to plagiarism and the use of AI or other tools) will result in an F grade for the course. Please familiarize yourself with the policies and sanctions.
Support: Students are encouraged to reach out to University offices such as the Office of Learning Support, Ashoka Center for Well-Being, and Center for Writing and Communication for additional support.