Proof and the Art of Mathematics | May-Jul 2025

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Proof and the Art of Mathematics

Joel D. Hamkins

05 May 2025 to 09 Jul 2025
About the Book

From the preface of the book:

[…] I have organized the book around a selection of mathematically rich topics, having interesting theorems with elementary proofs. I have chosen what I find to be especially enjoyable mathematical topics—at times fanciful, yet always amenable to a serious mathematical analysis—-because I feel students will learn best how to write proofs with material that is itself intrinsically interesting. Please enjoy!

My main motivation for this reading is that I am teaching discrete mathematics course this summer and attempting to write a companion text. Hamikins’ book is a joy to read and I am looking forward to the good vibes! No specific pre-requisites required.

You can find this book on Amazon. More pointers on the book site.

Timings and Venue

We meet on Zoom at 9AM IST for about 60-90 minutes every day according to the daily-ish schedule below: the plan is to meet on all weekdays with a break about half-way, and cover an average of two subsections in one session. Here’s a link to a WA group for general discussions and meetup reminders, and announcements of any changes to the schedule. The Zoom details are below:

Meeting ID: 977 9556 2618
Passcode: irrational

Alternatively, you can use this link to join directly. See you there!

Date Lecture Slides Notes Video
05 May, 2025 Chapter 1
A classical beginning.
06 May, 2025 Exercises from Chapter 1
Working through a general characterization of when the k-th root of n is irrational.
07 May, 2025 Sections 2.1, 2.2
$n^{2}-n$ is even; One theorem, seven proofs
08 May, 2025 Sections 2.3, 3.1
Different proofs suggest different generalizations; Prime numbers
09 May, 2025 Sections 3.2, 3.3
The fundamental theorem of arithmetic; Euclidean division algorithm
12 May, 2025 Sections 3.4, 3.5
Fundamental theorem of arithmetic, uniqueness; Infinitely many primes
13 May, 2025 Sections 4.1, 4.2
The least-number principle; Common induction
14 May, 2025 Sections 4.3, 4.4
Several proofs using induction; Proving the induction principle
15 May, 2025 Sections 4.5, 4.6
Strong induction; Buckets of Fish via nested induction
16 May, 2025 Sections 4.7, 5.1
Every number is interesting; More pointed at than pointing
19 May, 2025 Sections 5.2, 5.3
Chocolate bar problem; Tiling problems
20 May, 2025 Sections 5.4, 5.5
Escape!; Representing integers as a sum
21 May, 2025 Sections 5.6, 5.7
Permutations and combinations; The pigeon-hole principle
22 May, 2025 Sections 5.8, 6.1
The zigzag theorem; A geometric sum
23 May, 2025 Sections 6.2, 6.3
Binomial square; Criticism of the 'without words' aspect
26 May, 2025 Sections 6.4, 6.5
Triangular choices; Further identities
27 May, 2025 Sections 6.6, 6.7
Sum of odd numbers; A Fibonacci identity
28 May, 2025 Sections 6.8, 6.9
A sum of cubes; Another infinite series
29 May, 2025 Sections 6.10, 6.11
Area of a circle; Tiling with dominoes
30 May, 2025 Sections 6.12, 7.1
How to lie with pictures; Twenty-One
02 Jun, 2025 Sections 7.2, 7.3
Buckets of Fish; The game of Nim
03 Jun, 2025 Sections 7.4, 7.5
The Gold Coin game; Chomp
04 Jun, 2025 Sections 7.6, 7.7
Games of perfect information; The fundamental theorem of finite games
11 Jun, 2025 Sections 8.1, 8.2
Figures in the integer lattice; Pick's theorem for rectangles
12 Jun, 2025 Sections 8.3, 8.4
Pick's theorem for triangles; Amalgamation
13 Jun, 2025 Sections 8.5, 8.6
Triangulations; Proof of Pick's theorem, general case
16 Jun, 2025 Sections 9.1, 9.2
Regular polygons in the integer lattice; Hexagonal and triangular lattices
17 Jun, 2025 Sections 9.3, 10.1
Generalizing to arbitrary lattices; The polygonal dissection congruence theorem
18 Jun, 2025 Sections 10.2, 10.3
Triangles to parallelograms; Parallelograms to rectangles
19 Jun, 2025 Sections 10.4, 10.5
Rectangles to squares; Combining squares
20 Jun, 2025 Sections 10.6, 10.7
Full proof of the dissection congruence theorem; Scissors congruence
23 Jun, 2025 Sections 11.1, 11.2
Relations; Equivalence relations
24 Jun, 2025 Sections 11.3, 11.4
Equivalence classes and partitions; Closures of a relation
25 Jun, 2025 Sections 11.5, 12.1
Functions; The bridges of Königsberg
26 Jun, 2025 Sections 12.2, 12.3
Circuits and paths in a graph; The five-room puzzle
27 Jun, 2025 Sections 12.4, 13.1
The Euler characteristic; Hilbert's Grand Hotel
30 Jun, 2025 Sections 13.2, 13.3
Countability; Uncountability of the real numbers
01 Jul, 2025 Sections 13.4, 13.5
Transcendental numbers; Equinumerosity
02 Jul, 2025 Sections 13.6, 13.7
The Shröder-Cantor-Bernstein theorem; The real plane and real line are equinumerous
03 Jul, 2025 Sections 14.1, 14.2
Partial orders; Minimal versus least elements
04 Jul, 2025 Sections 14.3, 14.4
Linear orders; Isomorphisms of orders
07 Jul, 2025 Sections 14.5, 14.6
The rational line is universal; The eventual domination order
08 Jul, 2025 Sections 15.1, 15.2
Definition of continuity; Sums and products of continuous functions
09 Jul, 2025 Sections 15.3, 15.4
Continuous at exactly one point; The least-upper-bound principle
10 Jul, 2025 Sections 15.5, 15.6
The intermediate-value theorem; The Heine-Borel theorem
11 Jul, 2025 Sections 15.7, 15.8
The Bolzano-Weierstrass theorem; The principle of continuous induction
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