Axioms of Arithmetic 2

Here is a list of the axioms of Arithmetic as stated by Gottlob Frege.


 * 1) Background Information
 * 2) All Real numbers form an algebraic field
 * 3) Fundamental Operations
 * 4) Addition – resulting in a “sum”
 * 5) Multiplication – resulting in a “product”
 * 6) Axioms of Addition
 * 7) Commutative Law of addition: a + b = b + a
 * 8) Associative Law of Addition : a + (b + c) = (a + b) + c
 * 9) Closure Property: a + b is a Real number.
 * 10) Axioms Derived from Axioms of Addition
 * 11) Additive Identity Property: a + 0 = a
 * 12) Additive Inverse Property: For every “a” there is a “-a” such that “a + (-a) = 0”
 * 13) Axioms of Multiplication
 * 14) Commutative Law of Multiplication: a X b = b X a
 * 15) Associative Law of Multiplication: (a X b) X c = a X (b X c)
 * 16) Closure Property: a X b is a Real number
 * 17) Axioms Derived from Axioms of Multiplication
 * 18) Multiplicative Identity Property: a X 1 = a
 * 19) Multiplicative Inverse Property: For every “a” there is a “1/a” such that “a X 1/a = 1”
 * 20) Distributive Property combining Addition & Multiplication
 * 21) (a + b) X c = a X c + b X c
 * 22) Four Additional Axioms – Real numbers are Ordered
 * 23) Given a & b only one of the conditions is true: “a < b” or “a = b” or “a > b”
 * 24) If a > b & b > c, then a > c
 * 25) Monotonic Property of Addition: If a > b then a + c > b + c
 * 26) Monotonic Property of Multiplication: If a > b & c#0 then a X c > b X c

Imagined Mathematical Realities

In his book Sapiens, social scientist Yuval Harari proposes a theory for the ascendance of humans on this Earth. He primarily attributes it to the human ability to imagine "worlds" whose ideas help humans to cooperate and collaborate.

According to him our ideas of religion, nation, patriotism, human rights etc are examples of such worlds.

We can think of "axioms" as "imagined worlds" created by mathematicians which have the rules which control the development of math.

We cannot "prove" the correctness of these axioms but nevertheless they seem intuitively correct. In this sense mathematics is purely a creation of the human mind.