Reference: Axioms

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.  Axioms are unique types of assertions in mathematics: they are foundational, self-evident truths accepted as being true.

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

Axioms and Assertions

Axioms are just one type of mathematical assertion. Being foundational, self-evident truths, they serve as the starting points for logical reasoning in a given mathematical system. For example:

• In Euclidean geometry: “Through any two points, there exists exactly one straight line.”
• In set theory: “There exists a set that contains no elements” (this is the empty set).

Axioms differ from all other types of assertions because they are often assumed to be true without proof. They are the “starting points” from which other mathematical truths (theorems, corollaries, and even definitions) are derived. Other assertions, such as theorems, corollaries, and conjectures, are subject to proof and are not assumed to be true without evidence.

Other Assertions in Mathematics

In addition to axioms, there are other types of assertions in mathematics, with distinctions based on whether these assertions require proof or are simply accepted as true. Here are a few types:

a. Theorems

A theorem is a statement proven based on previously established axioms, definitions, and logical reasoning.

Example: The Pythagorean Theorem, which states that in a right triangle, the square of the hypotenuse’s length is equal to the sum of the squares of the other two sides:

a2 + b2 = c2

The proof of the Pythagorean Theorem relies on axioms of geometry, algebra, and logical reasoning, so it’s not assumed; it’s an assertion that must be proven.

b. Lemmas

A lemma is an intermediate result used to assist in proving a larger theorem. It’s a “helping” assertion that may not be significant on its own but is useful for proving more substantial results.

Example: In proving the Pythagorean Theorem, a lemma might involve the properties of similar triangles.

Lemmas are still proven, like theorems, but serve a different purpose.

c. Corollaries

A corollary is a statement that follows directly from an already-proven theorem, usually requiring little to no additional proof.

Example: Once the Pythagorean Theorem is proven, a corollary might be the specific case where the triangle is a right-angled isosceles triangle (with equal-length legs), leading to a simplified result, such as the hypotenuse being √2 times the length of each leg.

d. Postulates

Postulates are often used interchangeably with axioms, particularly in geometry, but “postulate” generally refers to basic assumptions within a specific context (such as Euclidean geometry).

Example: One of Euclid’s postulates is: “All right angles are equal to one another.”

In modern mathematics, “postulate” and “axiom” are often used synonymously, although “axiom” is more common in set theory and abstract mathematics.

e. Conjectures

A conjecture is an assertion believed to be true based on empirical evidence or observed patterns, but it has not yet been proven.

Example: Fermat’s Last Theorem was a conjecture for over 350 years, stating there are no whole number solutions to the equation:

xn + yn = zn

for any integer n > 2. It was eventually proven by Andrew Wiles in the 1990s.

A conjecture remains unproven, though it is still an assertion believed to be true based on evidence or reasoning.

f. Definitions

Definitions are another type of assertion, but unlike axioms, theorems, or conjectures, they don’t claim truth. Instead, they specify the meaning of terms used in mathematics, providing the basis for making other assertions.

Example: A “prime number” is defined as a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

Definitions clarify the objects of study in mathematics but don’t require proof.