Trivium & Quadrivium

 Three quotes from the article that made me stop and that surprised me in some way.

(1) “Logistic was practical and utilitarian, a study for children and slaves; logic was a liberal art, a study for free men” (Dorothy V. Schrader , 2018, Page 266).

 This tells us something about the social classes of medieval Europe. The slaves (I guess peasants and other workers) worked to produce food, did the construction and maintenance of buildings, etc. These workers studied logistics. The free men studied logic and arithmetic not for the purpose of practical works but philosophical.

Those who prayed (clergy), those who fought (knights), those who worked (peasants)

According to Wikipedia, the period between 1000AD and 1347 AD was an expansion of the population. By estimate, the population grew from 35 million to 80 million. Around 90% of the population was rural peasants, many of them settled into villages called manors. Peasants paid noble overlords rent for places and for services. So one wonders who went to the schools or universities to study logic and arithmetic to know some properties of numbers, proofs and some formal demonstrations, etc (Dorothy V. Schrader , 2018, Page 266)? Meanwhile, good nutritious food production by peasants might be a contributing factor in population expansion.

 (2) In the first arithmetical period (5^th to 10^th century approximately) , the emphasis was on the art of computation especially on the method of establishing the date of Easter (Dorothy V. Schrader , 2018, Page 267).

 Free men went to study logic in the lower level, and then arithmetic in the upper level. Almost all they did, in terms of applications, was to learn how to correctly calculate the date of Easter.  The date of Easter was one very important matter during that period. Many important days of the Christian Church were dependent on the date of Easter. We can imagine the difficulties these free men faced partly due to the Roman numeral system they were using. 

 Only in the second arithmetical period (end of 10^th to end of 12^th century) all four arithmetic operations on abacus were possible due to improvement made by Gerbert (Dorothy V. Schrader , 2018, Page 269). Compared to the scribes doing math on base 60 positional system in Babylonian period more than 2500 years earlier (~ Hammurabi in old Babylon, 1700 BCE), it seems that development of mathematics went through several darknesses.

 (3) The third arithmetic period (end of 12^th century to the end of middle ages ~ late 15^th century) was one of great activity and great change, an almost intellectual revolution in Europe. The Hindu Arabic number (positional decimal number system used today) was introduced and zero was added to the previously zero-less world. It was possible by translation into Latin from Arabic and Syriac (old Syrian language) (Dorothy V. Schrader , 2018, page 267).

We have learned that, by evidence, positional system with base 60 was used in Babylon of Mesopotamia (current Iraq) in 1700-300 BCE (Victor J. Katz, 2009, page 10-12).  Greeks used this base 60 positional system for astronomy but it is not clear if they used this system in other situations. Even though earlier ancient Chinese used base 10 system, the clear evidence of positional system including zero in China came from 12^th century (Victor J. Katz, 2009, page 198). In India, Syria and Cambodia earliest evidence of the use of decimal place value system came from 7^th century.  Meanwhile, until the third arithmetic period, European were using the cumbersome Roman numeral which requires a great number of symbols to write a large number. Calculations with Roman numerals are difficult.

 It took more than 2500 years for human being to finally adopt the positional numeral system after first  discovering it in Babylon. If the positional numeral system was a bird species, it was an amazingly difficult journey dotted with deaths and rebirths for that species to spread its seeds around the world for humans to make good use of.