Foundations of AI: Mathematics — Beginner-Friendly Notes

1. Why Maths Is the Engine of AI

Philosophy asks what to do, maths tells AI how to do it precisely — with logic, numbers, and optimisation.

2. Five Maths Pillars (and where you will meet them)

1. Logic (Module 3)

• Propositional & first-order logic give a formal language to represent facts and prove new ones.
• Example: If it rains → ground is wet, it rains ⊢ ground is wet (deduction).
• AI uses resolution, forward/backward chaining to reason.

2. Probability & Statistics (Module 4)

• Real world is uncertain — maths quantifies uncertainty.
• Bayes' theorem is the core of learning and reasoning under uncertainty.
• $P(H|E) = \frac{P(E|H) \cdot P(H)}{P(E)}$ — single line, supported: \frac and literal symbols.
• Example: spam filter learns $P(spam|words)$ from data.

3. Computation & Algorithms (Module 2)

• Theory of computation (Turing, 1936) defines what is computable and at what cost.
• Complexity tells us if search is feasible: BFS is $O(b^d)$ — exponential, needs heuristics.
• Algorithms = search (BFS, DFS, A*), optimisation (hill climbing, simulated annealing).

4. Optimisation (Modules 2, 4, 5)

• Most AI is minimise error / maximise reward.
• Gradient ideas, heuristic functions, loss functions all come from calculus & optimisation.

5. Discrete Maths, Graphs & Linear Algebra (Modules 1, 5)

• Graphs model states and knowledge (state-space, semantic networks, Bayes nets).
• Linear algebra powers ML/neural networks (vectors, matrices).
• Set theory & relations underpin knowledge representation.

3. Tiny Examples That Click

Idea	Maths	AI Use
Is this email spam?	$P(spam|words)$	Naive Bayes
Best route to goal?	graph search + heuristic $h(n)$	A* algorithm
Learn from mistakes?	minimise loss $L = \frac{1}{n}\Sigma(error)^2$	regression, neural nets

4. What You Don't Need to Fear

• You will use Bayes and logic; you don't need to prove Gödel's theorem.
• Focus on intuition: probability = degree of belief, logic = guaranteed truth when premises hold, optimisation = find best among options.

5. One-Line Summary to Write in Exam

Mathematics provides AI with logic for reasoning, probability for handling uncertainty, and optimisation/complexity for search and learning — together turning philosophical ideas into computable algorithms.

6. Quick Self-Check

1. Which maths pillar handles uncertainty?
2. Name one algorithm from computation theory you will study in Module 2.
3. Why is complexity ($O(b^d)$) important for search?