Proofs and Refutations (Imre Lakatos)
A short novel exploring what mathematics is and how mathematics happens through a historical lens. Presented as a fictional conversation between teachers and students. Found it to be generally gripping, even though I think most of it went over my head. Things that stuck:
Mathematics is a human endeavor
Rigor is never achieved, only pursued. For instance, let us start with the assertion of Euler’s formula that VE+F=2 % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } % magnitude etc \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % cardinality \newcommand{\card}{ \t{cd} } % disjoint untion \newcommand{\dcup}{ \sqcup } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ \newcommand{\apar}{ {-} } % tuples \newcommand{\tup}[1]{ \langle {#1} \rangle } % reverse-order composition %\newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\then}{ {\scriptsize\ \rhd\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } % category names \newcommand{\cat}[1]{{ \bf{#1} }} V - E + F = 2. Since this is unproven, it is a boundary of rigor. We can give a proof of this assertion. This pushes our boundary of rigor out further. However, our proof will assume certain things about what a polyhedron is. Digging into those assumptions will reveal issues with the proof. These assumptions represent a new boudnary of rigor. The book explains this better than me...
Would like to read again, so: #onwards