The earliest uses of mathematics were confined to:

• Agriculture - calculation of areas of fields
• Business - keeping track of supplies, inventory and taxes (!)
• Time - ritual calendars

Mesopotamia - region surrounding the Tigris and Euphrates rivers, now in the region of Iraq. This region is one of the oldest, continously inhabited regions in the world.

• Sumerian (- 2350 B.C.E.)
• Akkadian (2350 B.C.E. - 2150 B.C.E.)
• Assyrian and Babylonian (2150 B.C.E. - 539 B.C.E.)
• Persian (539 B.C.E. - 331 B.C.E.)
• Greek (331 B.C.E. - 138 B.C.E.)
• Parthians (138 B.C.E. - 224 A.D.)
• Sasanian (224 A.D. - 634 A.D.)
• Arab (634 A.D. -present

A table of Babylonian Numerals . Alternate link [PDF file]

Example of Babylonian algebra.

Another (humorous) example of Babylonian tablets containing algebra problems for students.

Babylonian geometry.

Example modeled on Babylonian astronomy.

Notes:

• The number system is base 60 (not 10). No one knows the reason for this choice.
• There is no symbol for zero.
• Some fractions, such as 1/2, 1/3 and 2/3, had unique symbols.
• There is no well-defined concept of proof, or rigorous justification.
• Methods are used because they worked!
• Tables of fractions, square roots, cube roots were created.
• Used 3 as an approximation to pi.

Egypt - Civilizations existed along the Nile from before 4000 B.C.E. to the present time.

A table of Egyptian Numerals . Another reference (with color illustrations).

Overview of Egyptian Mathematics.

Example of Egyptian algebra.

Example of Egyptian geometry.

Egyptian astronomy.

Notes:

• Limited positional notation
• No concept of zero.(for use in numerical position)
• According to Herodotus (Greek historian), geometry was developed in order to re-determine boundaries of land after periodic flooding of the Nile.
• Used (16/9)² ~ 3.16049 as an approximation to pi.

Summary:

1. Mathematics was empirically devloped (from practical problems)
2. Methods were justified by whether they worked (approximately) or not.
3. No symbolic variables.
4. No abstraction or generalization or results.
5. No concept of rigorous proof, or even plausibility arguments.
6. Mathematics is a tool, not a discipline.