The philosopher and the mathematician are both on the same epic quest: the search for absolute truth.

The philosopher begins with axioms and first principles—self-evident truths upon which to build an argument.

The mathematician begins with axioms and definitions—the foundational, unproven rules of the game.

The philosopher uses the tools of formal logic (like modus ponens) to construct a valid, step-by-step argument.

The mathematician uses the tools of algebra (like the distributive property) to construct a valid, step-by-step proof.

The philosopher's goal is a sound conclusion, a statement of truth about the nature of reality or knowledge.

The mathematician's goal is a proven theorem, a statement of truth about the nature of numbers and structure.

In both fields, every single step must be justified. There is no room for unsupported leaps or emotional appeals.

Both require a high degree of comfort with abstraction—the ability to work with pure concepts.

Algebra can be seen as a perfect, simplified model of the philosophical search for truth. It is philosophy with unwavering certainty.