Bernoulli's Theorem and Its Reception in Türkiye
Zekeriya DuruBernoulli’s Theorem has a very special position in probability and is the first crucial theoretical achievement in the history of probability. Bernoulli’s Theorem, which forms the basis of indispensable topics in mathematics and statistics, such as the law of large numbers and the central limit theorem, was introduced to Turkey by Salih Zeki in the first probability work published in Turkey, Hulâsa-i Hesâb-ı İhtimâlî (1898). Although there is a large amount of research literature on Ottoman-period mathematical works, there is still a long way to go in terms of evaluating the mathematical content of these works. In order to contribute to this situation, Bernoulli’s theorem and how this theorem is discussed in Hulâsa-i Hesâb-ı İhtimâlî are emphasized. The entire work, written in the old Turkish script, was read, the relevant parts were written in modern Turkish, and Bernoulli’s theorem in the work was compared and analyzed with foreign sources of the period. Our research confirms that Salih Zeki is a pioneer in that he introduced a work on probability to Turkish science for the first time. However, it was determined that the theorem he called Bernoulli’s Theorem in his book corresponded to a different concept, Littlewood’s Law, as it is known today.
Bernoulli Teoremi ve Türkiye'ye Girişi
Zekeriya DuruBernoulli Teoremi, olasılıkta çok özel bir konuma sahiptir ve olasılık tarihinin ilk önemli teorik başarısıdır. Büyük sayılar yasası, merkezi limit teoremi gibi matematik ve istatistiğin vazgeçilmez konularının temelini oluşturan Bernoulli Teoremi, elimizdeki verilere göre Salih Zeki tarafından yayınlanan ilk olasılık eseri Hulâsa-i Hesâb-ı İhtimâlî’sinde (1898) Türkiye’ye tanıtılmıştır. Osmanlı dönemi matematik eserleri ile ilgili geniş bir araştırma literatürü oluşmasına rağmen bu eserlerin matematiksel içeriğinin değerlendirmesi açısından daha alınacak uzun bir yol vardır. Bu duruma bir katkı olması için Bernoulli Teoremi ve bu teoremin Hulâsa-i Hesâb-ı İhtimâlî’de nasıl ele alındığı üzerinde durulmuştur. Eski harfli Türkçe ile yazılmış eserin tamamı okunmuş, ilgili kısımları günümüz Türkçesi ile yazılmış ve eserde yer alan Bernoulli Teoremi dönemin yabancı kaynakları ile karşılaştırılmış ve analiz edilmiştir. Araştırmamız, Salih Zeki’nin bir olasılık eserini ilk kez Türk bilimine kazandırması yönüyle bir öncü olduğunu teyit etmektedir. Ancak kitabında Bernoulli Teoremi olarak adlandırdığı teoremin farklı bir kavramı, günümüzdeki adıyla Littlewood Yasasını karşıladığı belirlenmiştir.
Salih Zeki (1864-1921) returned to Istanbul after attending L’École Supérieure de Télégraphie in Paris between 1883 and 1887. He started to teach mathematical physics (hikmet-i riyaziyye) at the Mühendishâne-i Berrî-i Hümâyun, the Ottoman military engineering school. There are indications that he had covered probability topics as part of this course. In 1898 he published Hulâsa-i Hesâb-ı İhtimâlî, a booklet on probability which has the distinction of being the first in its field in Turkey. The small treatise has two chapters totaling fifty-eight pages. The first 40-page chapter of the book presents the theoretical concepts of probability. The second 18-page chapter includes examples on empirical probability.
Bernoulli’s theorem was first presented with its proof by Jacques Bernoulli (1655-1705) in Ars Conjectandi (1713), his posthumous work in Latin. The theorem can be simply stated as follows;
With the probability approaching 1 or certainty as near as we please, we may expect that the relative frequency of an event E in a series of independent trials with constant probability p will differ from that probability by less than any given number , provided the number of trials is taken sufficiently large. (J. V. Uspensky, Introduction to Mathematical Probability, 1937, 96.)
Unlike this statement, Salih Zeki wrote the theorem as follows and solved its examples in this context: “If the number of experiments for an event is increased to equal the simple probability of the event, the probability of that event will eventually be brought closer to the level of mathematical certainty.” We will now briefly represent Salih Zeki’s three examples.
Example 1: The probability of getting a 6 when a dice is rolled is . If the dice were rolled twice, the probability of at least one of them getting a 6 would be . Similarly, if a dice is rolled three times, the probability of getting a 6 for at least one of them is . The probability of getting a 6 at least once when rolled four times is and other situations can be calculated by continuing in this way. The resulting probabilities are getting larger and closer to 1 for each case where the number of experiments is increased.
Example 2: Let the probability of an event occurring be very small. If the number of experiments for the same event is gradually increased, the probability of the event approaches the level of certainty. For example, while the probability of drawing a white ball from a box containing 40 black and one white ball is , this probability can be approached to the degree of mathematical certainty in 100 consecutive draws. Indeed, when drawn 100 times, the probability of being white is 0,91526.
Example 3: Consider the following sixteen situations that occur when balls a, b are drawn in quaternary:
aaaa aaab aaba abaa baaa aabb abab baab baba abba bbaa bbab abbb babb bbba bbbb
If two classes are accepted according to whether the quaternary order of balls is mixed or uniform, the probabilities of these two classes of events will be 14/16, 2/16 respectively. Accordingly, the ratio of mixed-type events to all events is 14/16. When the number of experiments is increased, this ratio also increases. Indeed, when the number of experiments is five, 30/32 ratio is obtained, and when six experiments are performed, 62/64 ratio is obtained. Thus, it is possible to bring the probability of mixed-type events closer to the level of mathematical certainty by increasing the number of experiments as much as possible.
Bernoulli’s theorem has a privileged place in calculus of probability. Salih Zeki allocated a subchapter to this theorem in a small and concise textbook, Hulâsa-i Hesâb-ı İhtimâlî (1898), thus emphasizing its importance in a sense. The text of the theorem in the work and three related examples were examined. Here, it is stated that in experiments conducted for an event whose probability is not zero, if the number of experiments is increased as much as possible, this event will occur at the level of mathematical certainty. However, in the Bernoulli theorem texts that we see in the 18th and 19th-century probability works that Salih Zeki was aware of, it is emphasized that as the number of experiments increases, the ratio of the results obtained approaches a probability value p. Since Salih Zeki did not take into account approaching a certain probability value in his examples, the definition he gave does not belong to Bernoulli’s theorem but to another rule called Littlewood’s law today.
