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:

  1. The reflexive property of equality and law of identity: a = a
  2. Symmetric axiom: if a=b, then b=a
  3. Transitive axiom: if a=b and b=c, then a=c
  4. Addition axiom: if a=b, then a+c=b+c
  5. Multiplication axiom: if a=b, then ac=bc
  6. Distributive axiom: a(b+c) = ab + ac
  7. Distributive property of multiplication over addition: a x (b + c) = a x b + a x c
  8. Associative property of addition: (a + b) + c = a + (b + c)
  9. Commutative axiom: a+b = b+a
  10. Commutative property of multiplication: a x b = b x a
  11. Inverse property of addition: a + (-a) = 0
  12. Identity property of multiplication: a x 1 = a
  13. Inverse property of multiplication: a x (1/a) = 1
  14. 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.
  15. The law of non-contradiction: ~(p ∧ ~p)
  16. The law of excluded middle: p ∨ ~p
  17. The reflexive property of congruence: AB ≅ AB
  18. The symmetric property of congruence: if AB ≅ CD, then CD ≅ AB
  19. The transitive property of congruence: if AB ≅ CD and CD ≅ EF, then AB ≅ EF
  20. The distributive property of addition over multiplication: a x (b + c) = a x b + a x c
  21. The associative property of multiplication: (a x b) x c = a x (b x c)
  22. The commutative property of addition: a + b = b + a
  23. The identity property of addition: a + 0 = a.
  24. 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
  25. Infinitude axiom: there are an infinite number of prime numbers
  26. 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:

  1. Euclid’s first postulate: Two points determine a unique straight line.
  2. Euclid’s second postulate: Any finite straight line can be extended infinitely in either direction.
  3. Euclid’s third postulate: A circle can be drawn with any given point as its center and any given distance as its radius.
  4. Euclid’s fourth postulate: All right angles are equal to each other.
  5. 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.
  6. 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.
  7. Angle postulate: Any angle can be divided into two equal parts.
  8. Congruence axiom: If two objects have the same size and shape, they are congruent.
  9. Similarity axiom: If two objects have the same shape but different sizes, they are similar.
  10. 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.
  11. Triangle axiom: the sum of the angles in a triangle is 180 degrees
  12. Pythagorean axiom: in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sid

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