Physics is the knowledge of the universe works, Mathematics is the knowledge of how all possible universe would work, but how do we know that this is a discovered and not one we are making up? To take a step back further, we need to go to first principles and state some basic truths, these are called Axioms.
Mathematical Axioms:
- The reflexive property of equality and law of identity: a = a
- Symmetric axiom: if a=b, then b=a
- Transitive axiom: if a=b and b=c, then a=c
- Addition axiom: if a=b, then a+c=b+c
- Multiplication axiom: if a=b, then ac=bc
- Distributive axiom: a(b+c) = ab + ac
- Distributive property of multiplication over addition: a x (b + c) = a x b + a x c
- Associative property of addition: (a + b) + c = a + (b + c)
- Commutative axiom: a+b = b+a
- Commutative property of multiplication: a x b = b x a
- Inverse property of addition: a + (-a) = 0
- Identity property of multiplication: a x 1 = a
- Inverse property of multiplication: a x (1/a) = 1
- The law of trichotomy: for any two real numbers a and b, exactly one of the following is true: a < b, a = b, or a > b.
- The law of non-contradiction: ~(p ∧ ~p)
- The law of excluded middle: p ∨ ~p
- The reflexive property of congruence: AB ≅ AB
- The symmetric property of congruence: if AB ≅ CD, then CD ≅ AB
- The transitive property of congruence: if AB ≅ CD and CD ≅ EF, then AB ≅ EF
- The distributive property of addition over multiplication: a x (b + c) = a x b + a x c
- The associative property of multiplication: (a x b) x c = a x (b x c)
- The commutative property of addition: a + b = b + a
- The identity property of addition: a + 0 = a.
- Non-equality axiom: if a is not equal to b, then there exists at least one property that applies to a and does not apply to b, and vice versa
- Infinitude axiom: there are an infinite number of prime numbers
- Fermat’s last theorem: there are no three positive integers a, b, and c such that a^n + b^n = c^n for any integer value of n greater than 2
Geometry Axioms:
- Euclid’s first postulate: Two points determine a unique straight line.
- Euclid’s second postulate: Any finite straight line can be extended infinitely in either direction.
- Euclid’s third postulate: A circle can be drawn with any given point as its center and any given distance as its radius.
- Euclid’s fourth postulate: All right angles are equal to each other.
- Euclid’s fifth postulate: If a straight line intersects two other straight lines in such a way that the sum of the inner angles on one side is less than two right angles, then those two straight lines, when extended indefinitely, will intersect on that side.
- Parallel postulate: Given a straight line and a point not on it, there is exactly one straight line through the point that does not intersect the original line.
- Angle postulate: Any angle can be divided into two equal parts.
- Congruence axiom: If two objects have the same size and shape, they are congruent.
- Similarity axiom: If two objects have the same shape but different sizes, they are similar.
- Continuity axiom: If a line segment is divided into two parts, then there is a point on the segment such that the ratio of the smaller part to the larger part is the same as the ratio of the larger part to the whole segment.
- Triangle axiom: the sum of the angles in a triangle is 180 degrees
- Pythagorean axiom: in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sid
