Module 01 Interactive
Correlation vs. Causation
Why two things moving together does not show that one causes the other.
Math × Debate
Mathematics as a Tool for Better Reasoning
Numbers can strengthen an argument, but they can also conceal weak assumptions. Math × Debate explores how statistics, probability and mathematical models help us interpret evidence, recognise uncertainty and test the reasoning behind real-world claims: whether evidence supports a conclusion, whether a relationship is causal, and how much a conclusion depends on uncertain assumptions.The aim is not to turn every debate into a calculation, but to use mathematics where it helps evaluate a claim. Each module also states where the mathematics stops: it can clarify evidence and consequences, but it cannot decide which values should take priority. That question belongs to Values & Clashes.
Quantitative reasoning modules
Fifteen modules, each with a mathematical foundation, an intuitive explanation, a debate connection, a real-world application, an interactive visualisation, a practice exercise and its limitations.
- Module 02 Interactive
Confounding Variables
How a third factor can create or hide an apparent relationship.
- Module 03 Interactive
Selection Bias
When the way a sample is chosen distorts what it appears to show.
- Module 04 Interactive
Base Rate Fallacy
Why a highly accurate test can still produce mostly false alarms.
- Module 05 Interactive
Bayes' Theorem
How to update confidence in a hypothesis when new evidence arrives.
- Module 06 Interactive
Absolute vs. Relative Risk
Why “risk doubles” can describe a tiny change or a large one.
- Module 07 Interactive
Simpson's Paradox
How a trend in every group can reverse when the groups are combined.
- Module 08 Interactive
Confidence Intervals
What a range estimate does and does not tell you about uncertainty.
- Module 09 Interactive
Sample Size and Variability
Why small samples swing widely and large samples settle down.
- Module 10 Interactive
Regression to the Mean
Why extreme results tend to be followed by less extreme ones.
- Module 11 Interactive
Expected Value
Weighing outcomes by how likely they are, and knowing when that is not enough.
- Module 12 Interactive
Probability Distributions
Averages hide spread, skew and extreme risks.
- Module 13 Interactive
Game Theory
How rational choices by each player can produce outcomes no one wants.
- Module 14 Interactive
Multi-objective Optimization
When policies trade off goals that cannot all be maximised at once.
- Module 15 Interactive
Sensitivity Analysis
Testing whether a conclusion survives changes to its assumptions.
Mathematical Thinking and Debate Reasoning
Six lessons on how the habits of mathematical reasoning map onto argument, each with an interactive explanation, a concrete motion example, and the point where the analogy breaks down.
Lesson 1
Assumptions and Axioms
Every proof rests on axioms; every argument rests on premises. Change the starting points and the conclusions change.
Open lessonLesson 2
Proof and Burden of Proof
A mathematical proof establishes certainty; a policy case establishes that a claim is more likely than not, or likely enough to act on.
Open lessonLesson 3
Counterexamples and Rebuttals
One counterexample refutes a universal claim. It barely dents a claim about what usually happens.
Open lessonLesson 4
Bayesian Updating and Evidence Assessment
Evidence should move belief in proportion to how much better one hypothesis predicts it than the other.
Open lessonLesson 5
Optimization and Value Weighing
Optimisation finds the best option given objectives and weights. It cannot tell you what the weights should be.
Open lessonLesson 6
Sensitivity Analysis and Argument Robustness
A robust argument survives pessimistic assumptions. A fragile one flips when one uncertain number moves.
Open lesson