Paper 2 - The Business of Fibonacci Leonardo of Pisa, who went by the nickname Fibonacci, was born in Pisa, Italy in 1170 (O Connor & Robertson, 1998). He contributed much to the field of mathematics, including the sequence that bears his name. It is important to note that these contributions not only advanced the field of mathematics but also business finance. The development of the mathematics of business is a logical and under examined legacy of Fibonacci. It is important to note that it was his father s role as a customs officer and diplomat that brought Fibonacci to north Africa (O'Connor & Robertson, 1998). It is there where he began to learn about historically important advances in mathematics that had been erased in Europe. It is logical that his father s duty to represent merchants from the Republic of Pizza in this part of the Mediterranean influenced his perspective on mathematics. Many of the scenarios that his example problems were based on were not only practical but also based in finance. Some prominent examples of the obvious relationship of Fibonacci s work and business finance are the famous text Liber Abaci (1202) and the lost work on commercial arithmetic Di minor guisa which consisted solely of business and trade related mathematics which he completed after he returned to The Republic of Pizza around 1200 (O Connor & Robertson, 1998) Geotzmann (2004) asserts that Liber Abaci is focused on financial mathematics and it is easier to point out the sections of the book that do not relate to finance than those that are purely theoretical. One of the few examples of work contained in Liber Abaci that does not
relate to business finance is the opening section on basic arithmetic. A second example is the introduction of Fibonacci sequences. Otherwise, most of the material is financial in nature. The connection is so strong that even research focused solely mathematics mention his finical innovations. Debnath (2011, p 341) points out the parallels seen in Fibonacci s work and modern financial mathematics and states In his Liber Abaci, Fibonacci solved the following problem and found 383 apples as his solution. This problem contained an essence of the idea of an annuity. This shows that not only are the writings of Fibonacci financial in nature, but contain the elements needed to develop modern financial principles. The problem is as follows: A man entered an orchard through seven gates and there took a certain number of apples. When he left the orchard he gave the first guard half the apples that he had and one apple more. To the second guard he gave half his remaining apples and one apple more. He did the same to each of the remaining five guards, and left the orchard with one apple. How many apples did he gather in the orchard? Other authors like Goetzmann (2004) have gone even further. He credits Fibonacci with laying the groundwork for modern present value analysis. He points out that the formal calculation did not take form until the early twentieth century, but the birth of present value analysis can be credited to Leonardo of Pizza. Geotzmann (2004) continues the common theme of crediting him with rescuing the most useful and valuable aspects of Indian and Arabic mathematics from antiquity and reintroducing them to the west. This is followed up, as in most cases, with crediting Fibonacci with new applications and innovative additions to the same. At the heart of his attribution of
present value analysis is a problem called On a Soldier Receiving Three Hundred Bezants for his Fief. The problem states (Sigler, 2003, p 392); A certain soldier because of his fief received from a certain king 300 bezants each year, and it is satisfied in IIII payments, and in each payment he takes 75 bezants; this is a payment for three months which by necessity is collected together; he asks for a certain compensation in order to accommodate himself for interest because he accepts the 300 bezants instead of the 75 bezants of each payment, namely from payment to payment, of the capital and profit. Voluntarily acquiescing to this he invests the bezants at a profit of two bezants per hundred in each month. It is sought how many bezants he makes in his investment. The solution Fibonacci suggests is to discount a payment by four periods (Goetzman, 2004). This is the basis of later western ideas about the time value of money and how structuring payments can affect the actual value of a contract. Goetzman believes that these ideas probably arose in Asia Minor before the time of Fibonacci, but there is no doubt that he refined and formalized them. The most striking theory that I have discovered so far during this course revolves around the economic development of Europe compared with that of China. Goetzmann (2004) awards the Fibonacci with the distinction of initializing modern economic sophistication. He makes a very convincing argument.
His theory is based on the logic that during the 15 th century governments in Europe were disorganized and economic systems were primitive. In contrast, the Chinese bureaucracy was highly organized and the Chinese economic systems were sophisticated in comparison. The question one must ask is, how did western economic systems overtake and surpass the Chinese within an extremely short time period? Goetzmann (2004) gives the credit to Fibonacci. This logic is hard to argue with. All of the sources I could find claim Fibonacci as the greatest mathematician of the Middle Ages. He gave Europeans the best tools of the ancient world. Adding to this that a large portion of his brilliant work revolved around practical business mathematics and inventing modern business concepts, the conclusion that Goetzmann comes to is inevitable. The average student of tertiary mathematics in the United States has only a vague knowledge of what Fibonacci accomplished over his lifetime. My personal knowledge of what he accomplished was limited to the sequence of his name sake being mentioned during Calculus I. Those who are more studied on the life of Leonardo may know of the practical work and theoretical mathematics represented in works like Liber quadrotorum. But certainly very few people are aware of his contributions to financial mathematics. As a high school teacher, my students usually ask me about why they should learn mathematics. I inform them that many mathematical concepts were developed for financial gain. The Greeks were interested in mathematics for spiritual reasons but this seems to be the exception, not the rule. Most other cultures were interested in financial gain. Now I have
another specific example in Fibonacci to illustrate the influence of business and trade in mathematics.
References Debnath, L. (2011). A short history of the Fibonacci and golden numbers with their applications. International Journal of Mathematical Education in Science and Technology, 42(3), 337-367. Goetzmann, W. N. (2004). Fibonacci and the financial revolution (No. w10352). National Bureau of Economic Research. O'Connor, J. Robertson, E. (1998) Leonardo Pisano Fibonacci University of St. Andrews. Retrieved from; http://www-groups.dcs.stand.ac.uk/~history/biographies/fibonacci.html Sigler, L. (2003). Fibonacci s Liber abaci: a translation into modern English of Leonardo Pisano s Book of calculation. Springer Science & Business Media.