February 27, 2025
Euclid came up with the notion of deducing geometric truths from axioms and postulates.
axiom vs postulate
Descarte. Cartesian coordinates.
Move from geometry to algebra
When you enter a PhD math program the first thing you learn is logic and set theory. George Boole invented propositional calculus to reduce logic to algebra.
Predicate calculus. First order logic.
rules of inference
Gentzen natural deduction, modus polens.
Sequent calculus.
| Language | Logic | Set | Algebra |
|---|---|---|---|
| not | \neg P | A' = \Omega\setminus A | 1 - 1_E |
| and | P\wedge Q | A\cap B | 1_E 1_F |
| nand | \neg(P\wedge Q) | (A\cap B)' | 1 - 1_{EF} |
| or | \vee P | A\cup B | 1_E + 1_F - 1_{EF} |
| implies | P\Rightarrow Q | A'\cup B | 1 - 1_E + 1_{EF} |