1. numbers (numerals: basic arithmetic operations)

1-4-2-1. Speaking of evolution of mathematics chronologically, necessitates to bring up the first arithmetic operations because they were conceived while or ahead of the time that people invented and developed numerals: addition, subtraction and multiplication.

Addition is one of the undefined concepts (we know that a word can’t be defined with its synonyms). We’re simply adding a notch to a in order to write b (It shows that how intertwined adding and counting are), however it might have taken time that people came to awareness about what they were doing, therefore they called the process addition.

Subtraction on the other hand is taking an amount away from another (can we state that subtraction is an undefined concept too?). Both addition and subtraction were conceived when people found the change in quantity of things around. For example they wanted to know how many sacks of rice are left when they cooked or sold out two sacks.

To count eeeeee they needed to add five to itself six times. That’s why they came up with multiplication to simplify addition.

So we see that the concepts of basic arithmetic operations were conceived by people who used the tally marks, and they were one of the reasons for evolution of numerals, apart from writing or reading them.

P.S. I found out that the tally marks font can't be shown on some systems which don't support it, so in the second paragraph if, instead of one notch and two notches, you had a and b, they'd mean one and two respectively; and in the forth paragraph if you had six e's, it'd mean six fives. 

1. numbers (numerals: Tally marks)

1-4-2. Carving numbers on wood or stone made them more abstract, instead of using objects to represent numbers they used symbols; so the easiest symbol to be carved is a straight line such as a, b, c, or d. These numerals were the first symbols invented and they are called tally marks. They notched the quantities as they counted objects; however for large quantities they needed to recount the notches, which was not easy. Imagine that you’re asked to count 1000 banknotes. You’d count them one by one, but what would you do if you doubted the current number let’s say if it’s 500 or 501. Then you should start over counting which is tedious and tardy. Even if you used tally marks, you’d need to count them.

The easy solution for this problem is bundling the banknotes, normally we count up to ten, and then we fold the tenth banknote around the bundle. That’s exactly what they did with the fifth notch, they carved it across the four notches (d) to create e.

Why 5? Because we have 5 fingers attached to a hand.