Mathematics
What mathematical patterns recur across all domains?
These aren't abstract — they're the hidden structure of reality. Recognizing them changes what you see.
The Patterns
Power Laws (Pareto)
~80% of effects come from ~20% of causes. Distribution is unequal by nature.
| Domain | Manifestation |
|---|---|
| Wealth | 20% own 80% |
| Customers | 20% generate 80% of revenue |
| Bugs | 20% of code causes 80% of errors |
| Earthquakes | Frequency drops 10x per magnitude unit |
| Implication | Application |
|---|---|
| Focus | Find the vital few |
| Expectations | Outliers dominate; averages mislead |
| Strategy | Be in the 20%, or serve them |
The question: Where's the 20% that matters?
Compound Effects
Small consistent gains accumulate exponentially. Time is the multiplier.
| Formula | Meaning |
|---|---|
| A = P(1 + r)^t | Principal × rate × time = exponential growth |
| Domain | Application |
|---|---|
| Finance | Compound interest |
| Skills | Daily practice compounds |
| Relationships | Small investments accumulate |
| Debt | Interest works against you too |
The question: What's compounding? For you or against you?
Regression to Mean
Extreme results tend to be followed by more average ones.
| Mechanism | Implication |
|---|---|
| Random variance | Hot streaks end |
| Selection effects | Winners regress |
| Noise vs signal | Extreme = more noise |
| Application | How It Works |
|---|---|
| Investing | Past performance ≠ future results |
| Hiring | Star performers regress |
| Sports | Rookie of the year curse |
The question: Is this signal or noise? Will it persist?
Network Effects
Value grows non-linearly with connections.
| Law | Formula | Meaning |
|---|---|---|
| Sarnoff | V = n | Broadcast: value = users |
| Metcalfe | V = n² | Network: value = connections |
| Reed | V = 2^n | Groups: value = possible groups |
| Implication | Application |
|---|---|
| Winner-take-most | First mover advantage |
| Critical mass | Networks need minimum scale |
| Lock-in | Switching costs increase with size |
The question: What network effects apply? Where's the tipping point?
Probability & Base Rates
Most predictions fail because they ignore base rates.
| Concept | Meaning |
|---|---|
| Base rate | How often something happens in general |
| Bayes' theorem | Update beliefs with new evidence |
| Expected value | Probability × outcome |
| Mistake | Correction |
|---|---|
| Ignoring base rates | Start with "how often does this happen?" |
| Overweighting recent | Weight evidence by reliability |
| Ignoring low-probability high-impact | Expected value includes tail risks |
The question: What's the base rate? How does this evidence change it?
Scale Invariance (Fractals)
Patterns repeat at different scales. What works small often works large.
| Domain | Manifestation |
|---|---|
| Coastlines | Same jaggedness at any zoom |
| Organizations | Team dynamics mirror company dynamics |
| Markets | Daily charts look like yearly charts |
The question: What pattern at one scale appears at another?
Diminishing Returns
Each additional unit yields less benefit than the previous one.
| Domain | Application |
|---|---|
| Economics | First hire > tenth hire |
| Learning | First 80% faster than last 20% |
| Optimization | Easy gains first; hard gains later |
The question: Where are you on the curve?
The Meta-Principle
Reality is not normally distributed. Extremes matter more than averages. Small effects compound. Time transforms everything.
This is why:
- Consistency beats intensity
- Position beats effort
- Patience beats timing
- Focus beats breadth
Context
- Predictions — Applying probability
- Business Leverage — Mathematical advantage
- Compounding — The snowball effect
- Portfolio Management — Risk mathematics