Three Hundred Years of Making Conjectures

Features

  • Author: Leila Battison
  • Date: 17 Jun 2013
  • Copyright: Image of Jacob Bernoulli appear courtesy of Wikipedia.

Whether you’re waging your last few pennies on a game of blackjack, or flying high in the casinos of Vegas, chances are the probability of you winning is somewhere in the back of your mind. Yet today, probability penetrates our everyday life, from the decisions in the stock market that affect the value of our investments, to the personal quandary of whether or not it will rain today. Probability surrounds us, and many of its central themes have a three hundred-year heritage, having been developed and expanded over the centuries into the particular brand of mathematics that we recognise today.

This year, mathematicians are celebrating the 300th anniversary of the publication of one of the most influential texts on probability, Ars Conjectandi.

thumbnail image: Three Hundred Years of Making Conjectures

Celebration of the Real World

Building on the game-based works of Pascal, Fermat and Huygens in the mid- to late-seventeenth century, Jacob Bernoulli dedicated his life to the study of mathematics that would later form the basis of modern probability. He published numerous preparatory works, and his research culminated in the preparation of a four-part thesis, which was left partially unfinished after his death and was published posthumously by his nephew Niklaus Bernoulli in 1713. This year marks the three hundredth anniversary of the appearance of Ars Conjectandi, or the ‘Art of Conjectures’, and despite significant advances in the field, it is still celebrated as a seminal founding work in probability and stochastic studies.

Despite significant advances in the field, it is still celebrated as a seminal founding work in probability and stochastic studies.

Bernoulli’s principal aim was to demonstrate the applicability of mathematical probability to everyday life. Beginning with an exposition on previous work by Huygens, he solved mathematical problems posed in those studies, before expanding his own theories on enumerative combinatorics, permutations and combinations that would become components of the ’12-fold way’, and frequency-based reasoning that is known today as the Law of Large Numbers. In the final two sections, Bernoulli applies his techniques first to common games of chance involving cards and dice, and later to ‘real-world’ personal, judicial, and financial decisions. Ars Conjectandi was the source of some of his eponymous theories, including the Bernoulli distribution, Bernoulli trials, and Bernoulli numbers, and it was the first to refer to this particular brand of mathematics as ‘Probability’.

Logic before Philosophy

Following its publication, Ars Conjectandi had an uncertain reception. Jacob Bernoulli apparently didn’t add to his theories in the final fifteen years of his life, and a further eight years passed before its ultimate publication. In this time, the mathematical community had moved the concept of probability forward, and theories had departed somewhat from Bernoulli’s thinking. Contemporary French mathematician Pierre Raymond de Montmort even questioned the central theme of the work: that probability could be applied to everyday or legal problems, rather than simply games of chance.

Front cover of Ars Conjectandi (Copyright - Wikipedia).

Over the following centuries, probability theory surged onward, with the continuing works of De Moivre and Laplace, and ultimately the rise of frequentism and twentieth century Bayesian approaches. Yet each great advance, rather than building on the relatively philosophical structure of Bernoulli’s work, tended to instead construct and expand a more logical framework, in which the ‘real-world’ applications that were a motivating factor for Ars Conjectandi, are relegated to external applications of an internally consistent theory. But certain aspects of Bernoulli’s theories stuck, in particular the concept of representing probability as a portion of certainty, which spring boarded the later development of probabilistic logic.

In a recent essay on the significance of Ars Conjectandi today, Glenn Shafer of Rutgers University highlights its importance in considering the philosophy of probability, rather than the mathematical execution of it. In the late seventeenth century, Jacob Bernoulli was limited by a lack of practical examples to apply his theoretical framework to, yet we now benefit from three centuries of additional experience. Shafer explains that: “the central philosophical problem of probability is
still to understand its limits of application. When is probability applicable and when is it not?” With modern-day pressure to apply probabilistic arguments to economics and other specialist fields, this ‘Philosophy of Probability’, first posited in Ars Conjectandi, is a source of much discussion and study.

Three Hundred Years and Counting

For many, the three hundredth anniversary of this involved statistical framework may mean little, but for those that work directly in the field of probability and stochastics, it is a chance to pause and reflect on the advances that were sparked and guided by Ars Conjectandi.

In May, a symposium organised by the Swiss Statistical Society and based jointly in Freiburg and Basel, brought together leading stochastic scientists from around the globe to discuss the historical development of Bernoulli’s work alongside current and future goals and perspectives. Covering topics as diverse as Brownian motion and evolution, to the application of stochastics to interest rates and pure mathematics, the conference was testament to Jacob Bernoulli’s guiding principle: that non-deterministic probability resounded through all of life.

According to Shafer, the future looks rosy for probability. He foresees some resolution between the differing perspectives of frequentist and Bayesian mathematics, whose specific definitions of probability have precluded any broad-scale synthesis. Shafer claims that “the simple-minded empiricism on which both the frequentist and the Bayesian pictures are founded is out of fashion”, and that future statisticians are no longer merely satisfied with the simple modelling of a problem, but are more focused on the subtle differences between actual and theorised situations.

Thus it would seem that after three hundred years, we are finally able combine the logical tools of probability with the experience of the centuries, in a philosophical perspective more in line with Jacob Bernoulli’s original thinking. We look forward to the powerful results of this modern philosophy, and the real-world benefits it will bring, in the future Bernoulli probably hoped for.

 

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