Computation as a Mathematical Tool
Computation has become a third pillar alongside theory and experiment, extending mathematics rather than replacing rigour.
| Approach | What it Does | Core Method | Example |
|---|---|---|---|
| Numerical | Solves problems without closed-form solutions approximately | Iteration, refinement | Newton's method |
| Symbolic | Manipulates expressions exactly via computer algebra systems | Exact algebra | Factoring, integrals |
| Simulation | Turns abstract structures into visible, testable data | Monte Carlo random sampling | Estimating probabilities |
| Verification | Checks millions of cases or verifies formal logic | Proof assistants | Four-colour theorem |
Programming turns static mathematics into something you can probe, but output is only as trustworthy as its logic.
Pitfalls and Monte Carlo Methods
Common pitfall: treating a program's numerical output as exact mathematical truth. Floating-point results (Lesson 2) are approximate and accumulate rounding error. A simulation estimates rather than proves, and a bug produces confident wrong answers. Computation supports work, but a printed number is not automatically a theorem.
Programs let you experiment with mathematics to test conjectures and visualise systems. A classic example is the Monte Carlo simulation:
Random accent points are scattered in a square with an inscribed quarter-circle. The fraction landing inside times 4 estimates pi, converging as more points are added.