## Final Report Summary - ORCHA (Order/Chaos: Genealogy of Two Concepts in the Culture of European Mathematical Physics)

The ORCHA project is an interdisciplinary research driven by the following question: When and how did order and chaos become objects of scientific inquiry? This question, highly relevant both for history and philosophy of science, is investigated over a considerably long time range spanning from the Enlightenment to the early 1930s and is restricted mainly to mathematical physics. I decided to concentrate on the concepts of order and chaos because, surprisingly enough, they have been conspicuously neglected by both historians and philosophers of science. More importantly, in the literature there is no truly multidisciplinary account able to combine the technical aspects of a scientific treatment of order/chaos and its philosophical implications.

It should also be noted that order and chaos are not ordinary scientific problems. They are, so to say, second-order problems because they concern the global behavior of natural systems. As a consequence, they often appear as by-products or hidden assumptions of other issues rather than independent questions. This is the reason why the project covers such a long span of time. Only by following the traces of order and chaos over a long period it is possible to paint an unitary picture.

The workflow of the project was divided into three parts. The first part was devoted to the question of the stability of the solar system in the eighteenth century. This is the first instance of order treated as an autonomous scientific problem. The second part was concerned with the emergence of statistical mechanics in the mid-nineteenth century and its relations with chaos theory, a mathematical field of research born in the last part of the century. Being the study of the complicated evolution of countless mechanical points in rapid collision, statistical mechanics was the first branch of mathematical physics to make wide use of disorder assumptions. Interestingly, statistical mechanics shares some fundamental mathematical techniques with chaos theory, which deals with deterministic systems with unpredictable behavior. However, the interconnections between these two fields only started to emerge in the work of George David Birkhoff in the early decades of the twentieth century. This is the topic of the third part of the project, which climaxes in the discovery of Birkhoff’s ergodic theorem.

From the methodological point of view, the ORCHA project adopted an innovative approach to the history of scientific concepts. Instead of mapping out the succession of conceptual representations, order and chaos were treated as concepts-in-action. In other words, the research focused mainly on the way in which order and chaos were deployed in different theories and for different problems. By unpacking the complex interplay between concepts and mathematical practices used to “put them to work”, this methodology opened up a novel perspective in which formal techniques acquired a primary epistemological role in shaping scientific concepts.

While the project relies on an overall narrative spanning from the Enlightenment to the twentieth century, the most significant result has been achieved in the first part. The historical analysis of the proof of stability of the solar system has led to a dramatic change of the traditional narrative. According to the common wisdom, the question whether the solar system is stable or not was first raised by Isaac Newton and then answered—within the limits of eighteenth century mathematical rigor—by Pierre Simon Laplace in 1773. By contrast, the research carried out in the ORCHA project has shown that Newton never set the stability issue and that the first real proof of stability was given by Joseph Louis Lagrange in 1781. To achieve this conclusion it was necessary to enlarge the analysis and include also the broader cultural and philosophical context, which had been hitherto neglected. It thus became possible to demonstrate that, until Lagrange, the stability of the solar system was part of the theological discourse and was never considered a genuine scientific problem. What eventually allowed the problem of stability to emerge was the transformation of the notion of natural order from a metaphysical concept to an effective tool to solve mathematical problem. This transformation occurred over a long period of time, beginning with Leonhard Euler in mid-eighteenth century and culminating in Lagrange’s proof in 1781.

These results have been published in a major article on a leading peer-reviewed journal (Historical Studies in the Natural Sciences) and have been presented in a number of conferences and invited lectures. What is perhaps more important, however, is that the proposed methodology has proven extremely effective to analyze the dynamics of scientific concepts in general. By investigating the interplay between concepts and mathematical practices it is possible to illuminate the way in which scientific concepts are disseminated, appropriated, reflected upon, and manipulated. As mathematical practices are historically and culturally situated, they enable the researcher to bring into the discourse also the larger context. This methodological platform will be further developed in the next future to foster a new kind of interdisciplinary cooperation between history and philosophy of science.

It should also be noted that order and chaos are not ordinary scientific problems. They are, so to say, second-order problems because they concern the global behavior of natural systems. As a consequence, they often appear as by-products or hidden assumptions of other issues rather than independent questions. This is the reason why the project covers such a long span of time. Only by following the traces of order and chaos over a long period it is possible to paint an unitary picture.

The workflow of the project was divided into three parts. The first part was devoted to the question of the stability of the solar system in the eighteenth century. This is the first instance of order treated as an autonomous scientific problem. The second part was concerned with the emergence of statistical mechanics in the mid-nineteenth century and its relations with chaos theory, a mathematical field of research born in the last part of the century. Being the study of the complicated evolution of countless mechanical points in rapid collision, statistical mechanics was the first branch of mathematical physics to make wide use of disorder assumptions. Interestingly, statistical mechanics shares some fundamental mathematical techniques with chaos theory, which deals with deterministic systems with unpredictable behavior. However, the interconnections between these two fields only started to emerge in the work of George David Birkhoff in the early decades of the twentieth century. This is the topic of the third part of the project, which climaxes in the discovery of Birkhoff’s ergodic theorem.

From the methodological point of view, the ORCHA project adopted an innovative approach to the history of scientific concepts. Instead of mapping out the succession of conceptual representations, order and chaos were treated as concepts-in-action. In other words, the research focused mainly on the way in which order and chaos were deployed in different theories and for different problems. By unpacking the complex interplay between concepts and mathematical practices used to “put them to work”, this methodology opened up a novel perspective in which formal techniques acquired a primary epistemological role in shaping scientific concepts.

While the project relies on an overall narrative spanning from the Enlightenment to the twentieth century, the most significant result has been achieved in the first part. The historical analysis of the proof of stability of the solar system has led to a dramatic change of the traditional narrative. According to the common wisdom, the question whether the solar system is stable or not was first raised by Isaac Newton and then answered—within the limits of eighteenth century mathematical rigor—by Pierre Simon Laplace in 1773. By contrast, the research carried out in the ORCHA project has shown that Newton never set the stability issue and that the first real proof of stability was given by Joseph Louis Lagrange in 1781. To achieve this conclusion it was necessary to enlarge the analysis and include also the broader cultural and philosophical context, which had been hitherto neglected. It thus became possible to demonstrate that, until Lagrange, the stability of the solar system was part of the theological discourse and was never considered a genuine scientific problem. What eventually allowed the problem of stability to emerge was the transformation of the notion of natural order from a metaphysical concept to an effective tool to solve mathematical problem. This transformation occurred over a long period of time, beginning with Leonhard Euler in mid-eighteenth century and culminating in Lagrange’s proof in 1781.

These results have been published in a major article on a leading peer-reviewed journal (Historical Studies in the Natural Sciences) and have been presented in a number of conferences and invited lectures. What is perhaps more important, however, is that the proposed methodology has proven extremely effective to analyze the dynamics of scientific concepts in general. By investigating the interplay between concepts and mathematical practices it is possible to illuminate the way in which scientific concepts are disseminated, appropriated, reflected upon, and manipulated. As mathematical practices are historically and culturally situated, they enable the researcher to bring into the discourse also the larger context. This methodological platform will be further developed in the next future to foster a new kind of interdisciplinary cooperation between history and philosophy of science.