Why Mathematics Feels Certain
- Mathematics often carries a special aura in TOK discussions because it's seen as the gold standard of certainty.
- This reputation comes from its unique approach to building and verifying knowledge through axioms, theorems, and proofs.
Axiom
A basic assumption taken to be true without proof. Axioms are the foundations of a mathematical system.
Theorem
A statement that has been logically proven based on axioms and previously established theorems.
Proof
A step-by-step logical argument that demonstrates why a theorem must be true.
Axioms are self-evident truths that form the foundation of a mathematical system. Theorems are statements proven to be true based on axioms and previously established theorems. Proofs are logical arguments that demonstrate the truth of a theorem.
Why Mathematics Is Perceived as Certain
- Mathematics relies on logical deduction , where each step follows inevitably from the previous one.
- This creates a sense of certainty that is hard to find in other areas of knowledge.
Pythagorean Theorem: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides ($a^2 + b^2 = c^2$). This theorem is proven using logical steps, making it universally true in Euclidean geometry.
Why Mathematical Knowledge Feels Unique
Internal consistency
The quality of a system where all statements and proofs agree with its starting axioms, with no contradictions.
- The validity of mathematics comes from internal consistency: if the starting axioms are accepted and the reasoning is sound, the conclusion must follow.
- There aren't any instruments to malfunction or documents to misinterpret. just logic working within its own framework.
- This gives mathematics a kind of certainty that feels rare elsewhere.
- Because of this, mathematics often becomes the backbone of other fields.
- Physics formulas, economic models, and computer algorithms all lean on its deductive structure to give their own claims strength.
The Limits of Mathematical Certainty
- While mathematics is internally consistent, it is not infallible.
- Humans do the proving, which means errors creep in.
- Famously, proofs can be published, accepted, and only later revealed to have hidden flaws.
- Andrew Wiles’ proof of Fermat’s Last Theorem was celebrated in 1993, only for a serious error to be found.
- It took Wiles a year to repair the gap.
- The theorem is now secure, but the episode shows that the process of proving is human, even if the logic itself is abstract.
- Another limitation is that mathematics is only as solid as the axioms it rests on.
- Different axioms can produce entirely different mathematical systems.
- This ties to the core concept of frameworks.
- Mathematical certainty exists within a chosen system, not across all systems.
- How does the certainty of mathematics compare to the uncertainty of natural sciences or history?
- If mathematical systems depend on human-defined axioms, can maths ever be called “objective”?
- When a proof contains an error, does it undermine mathematics as a whole, or just the individuals involved?
- Does the existence of multiple mathematical systems strengthen or weaken the claim that maths provides certain knowledge?
