Relearning Mathematics the Bourbaki Way

What does it mean to understand mathematics, as opposed to just computing with it? After graduating with a degree in electrical engineering, this question kept surfacing. The standard courses taught how to solve integrals and diagonalize matrices, but never explained why groups matter or what a topology actually is. The computations worked, but the big picture was missing.

The Bourbaki school offers a way to rebuild that picture: structures, axioms, and mappings first, computation second. The core insight is surprisingly simple. Every mathematical object is a set equipped with additional structure. That single idea reorganizes everything. These are the notes from trying to apply it.

Sets as the Foundation

Why do nearly all mathematical definitions start with “let $S$ be a set”? Because in the Bourbaki worldview, a set is not just a container. It is the substrate on which all structure is built. A vector space is a set with addition and scalar multiplication. A group is a set with a binary operation. A topological space is a set with a family of open sets. Once this pattern becomes visible, it clicks.

A set is a collection of objects. If $x$ belongs to $A$, we write $x \in A$. Sets ignore order and repetition: $\{1,2,3\} = \{3,2,1\}$ and $\{1,1,2\} = \{1,2\}$. A subset $B \subseteq A$ means every element of $B$ is also in $A$, formalized as:

$A \subseteq B \iff (\forall x,\ x \in A \Rightarrow x \in B)$

This pattern, defining things through universal quantifiers over elements, shows up everywhere in mathematics. Worth noting: the empty set $\varnothing$ is a subset of every set, since the statement “every element of $\varnothing$ belongs to $A$” cannot be violated. There are no elements to violate it.

What Is a Mapping, Really?

The common understanding of a function is something like $f(x) = x^2 + 1$, an expression you plug numbers into. But this misses the point. Consider $g(x) = 1/x$. Is it a function from $\mathbb{R}$ to $\mathbb{R}$? No, because $x = 0$ has no image. Change the domain to $\mathbb{R} \setminus \{0\}$ and it works. A mapping is not a formula. It is a correspondence between two sets, and specifying where it is defined is part of the definition.

Formally, a mapping $f: A \to B$ assigns to every $x \in A$ a unique $f(x) \in B$. Three properties turn out to be fundamental. Injectivity: no two inputs land on the same output ($f(1) = f(-1)$ kills injectivity for $f(x) = x^2$). Surjectivity: every element of the codomain actually gets hit. Bijectivity: both at once. A bijection is what makes “two sets have the same size” precise.

Equivalence: The Machine for Classification

Here is a surprisingly deep question: when should two things be considered “the same”? Not equal, but the same for the purpose at hand. On $\mathbb{Z}$, if only parity matters, then 2 and 4 are “the same” but 2 and 3 are not. The relation $a \sim b \iff a - b$ is even captures this.

An equivalence relation needs three properties: reflexivity ($a \sim a$), symmetry ($a \sim b \Rightarrow b \sim a$), and transitivity ($a \sim b$ and $b \sim c \Rightarrow a \sim c$). These seem almost trivially obvious, but the definition is load-bearing. The same structure appears behind modular arithmetic, quotient groups, quotient topologies, and many other constructions. Recognizing this single pattern clarifies a surprising amount of mathematics across different branches.

When Is an Operation Not an Operation?

Subtraction on the natural numbers $\mathbb{N}$ seems perfectly natural. But $2 - 5 = -3$, and $-3 \notin \mathbb{N}$. So subtraction is not actually an operation on $\mathbb{N}$, because the result escapes the set.

This is the structural perspective: a binary operation on $A$ is a mapping $*: A \times A \to A$. The crucial word is closure, the result must stay inside $A$. Simple reframing, but it changes how you think. Arithmetic says “subtraction exists.” Structural mathematics asks “on which set?”

From here, the next question is: what properties can an operation satisfy? Associativity: $(a * b) * c = a * (b * c)$. Commutativity: $a * b = b * a$. An identity element $e$ with $e * a = a * e = a$. Inverses: for each $a$, some $b$ with $a * b = b * a = e$. Integer addition satisfies all of these. Subtraction fails associativity: $(5 - 3) - 1 = 1$ but $5 - (3 - 1) = 3$.

Naming Structures by Their Properties

Rather than studying each operation in isolation, the Bourbaki approach names structures by which properties they satisfy. A semigroup is a set with an associative operation. A monoid adds an identity element. A group adds inverses. If the operation is also commutative, it is an abelian group.

Suddenly, familiar objects get clean classifications. $(\mathbb{N}, +)$ is a commutative monoid but not a group (no additive inverse for $3$ in $\mathbb{N}$). $(\mathbb{Z}, +)$ is an abelian group. $(\mathbb{Z}, \times)$ is a commutative monoid but not a group ($2$ has no multiplicative inverse in $\mathbb{Z}$). The beauty is that these are not new facts. They are old facts organized by a single principle.

Why Do Number Systems Keep Expanding?

The progression $\mathbb{N} \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{R}$ looks, in a standard education, like each number system just “appears.” But from the structural viewpoint, each extension is forced. In $\mathbb{N}$, the equation $x + 3 = 1$ has no solution. To make addition invertible, extend to $\mathbb{Z}$. In $\mathbb{Z}$, $2x = 1$ has no solution. To make multiplication invertible for nonzero elements, extend to $\mathbb{Q}$.

Each step is driven by the same structural pressure: the need to close off an operation by giving every element an inverse. Negative numbers and fractions are not arbitrary inventions. They are forced into existence.

$\mathbb{Z}$ also illustrates what happens when two operations coexist. Addition makes it an abelian group. Multiplication makes it a commutative monoid. The distributive law $a(b + c) = ab + ac$ binds them together. This combination is essentially what defines a ring.

What Mappings Are Worth Studying?

Not all mappings between groups are interesting. Consider $f: (\mathbb{Z}, +) \to (\mathbb{Z}, +)$ defined by $f(n) = 2n$. Check: $f(m + n) = 2(m + n) = 2m + 2n = f(m) + f(n)$. The operation is preserved across the mapping. Now try $g(n) = n + 1$: $g(m + n) = m + n + 1$, but $g(m) + g(n) = m + n + 2$. It breaks. Looking simple doesn’t mean it preserves structure.

A homomorphism between groups $(G, *)$ and $(H, \circ)$ is a mapping satisfying $f(a * b) = f(a) \circ f(b)$. Do the operation first then map, or map first then do the operation, the result is the same. This condition alone forces many consequences: the identity must map to the identity, and inverses must map to inverses.

Perhaps the most beautiful example: the exponential map $\varphi(x) = e^x$ from $(\mathbb{R}, +)$ to $(\mathbb{R}_{>0}, \times)$. Since $e^{x+y} = e^x e^y$, it turns addition into multiplication. Two seemingly different worlds connected by a single structure-preserving map.

Image, Kernel, and a Surprise About Parity

Every homomorphism $f: G \to H$ produces two natural objects. The image $\operatorname{Im}(f)$ is the subset of $H$ that actually gets hit. The kernel $\ker(f) = \{x \in G \mid f(x) = e_H\}$ is the set of elements that collapse to the identity.

Here is a small surprise. Define $\varepsilon: (\mathbb{Z}, +) \to (\{1, -1\}, \times)$ by $\varepsilon(n) = (-1)^n$. This is a homomorphism (check: $(-1)^{m+n} = (-1)^m(-1)^n$). Its kernel is $2\mathbb{Z}$, the even integers. So “parity,” which looks like a trivial arithmetic property, is actually the kernel of a homomorphism. This is what structural mathematics does: it finds structure behind facts you thought were trivial.

A clean theorem ties this together: a homomorphism is injective if and only if its kernel is $\{e_G\}$. The kernel controls whether information is lost in the mapping.

When Are Two Structures “the Same”?

A homomorphism that is also a bijection is called an isomorphism, written $G \cong H$. This means the two groups are structurally identical, even if they look completely different.

$(\mathbb{Z}, +) \cong (2\mathbb{Z}, +)$ via $f(n) = 2n$. These are different sets ($1 \in \mathbb{Z}$ but $1 \notin 2\mathbb{Z}$), yet structurally the same. The exponential map gives $(\mathbb{R}, +) \cong (\mathbb{R}_{>0}, \times)$: the additive world and the multiplicative world are the same group in disguise. On the other hand, $(\mathbb{Z}, +) \not\cong (\mathbb{N}, +)$, because one is a group and the other is not, and isomorphisms preserve all structural properties.

Isomorphism is itself an equivalence relation (reflexive via the identity map, symmetric via inverses, transitive via composition). So mathematical objects can be classified up to isomorphism. This is exactly what modern algebra does: not studying individual objects, but studying isomorphism classes.

Quotients: Systematically Forgetting

The last foundational idea is the quotient. When only certain distinctions matter, everything else can be compressed away. If only parity matters, $\mathbb{Z}$ compresses to $\{[0], [1]\}$: two equivalence classes, one for even, one for odd. Mod 3 gives three classes. Mod 12 gives a clock.

Given an equivalence relation $\sim$ on $A$, the equivalence class of $x$ is $[x] = \{y \in A \mid y \sim x\}$. The quotient set $A/{\sim}$ is the collection of all such classes. Note that this is a new set, not a subset. Its elements are classes, not original elements. This distinction matters: a subset selects from the original, a quotient compresses the original into something new.

The natural projection $\pi: A \to A/{\sim}$ sends $\pi(x) = [x]$, forgetting within-class differences. Any mapping $f: A \to B$ induces an equivalence relation ($x \sim y \iff f(x) = f(y)$), whose classes are the fibers of $f$. For a group homomorphism, this connects directly to the kernel: $a \sim b \iff a - b \in \ker(f)$. The kernel tells you exactly what gets identified. This is the motivation for quotient groups, and once you see it, quotient constructions start appearing everywhere.

The Skeleton

One pattern runs through all of this:

$\text{Set} \longrightarrow \text{Structure} \longrightarrow \text{Homomorphism} \longrightarrow \text{Isomorphism} \longrightarrow \text{Quotient} \longrightarrow \text{Classification}$

Start with a set. Equip it with structure. Study mappings that preserve that structure. Identify when two structures are the same. Compress by ignoring inessential differences. Classify what remains. Groups, rings, vector spaces, topological spaces: the objects change, the methodology does not.

That is the power of the Bourbaki viewpoint. Not any single definition, but the unifying language it provides across all of mathematics. The intuition and computation were already there. What was missing was the bones.