"Under the vastness of the sky I am not at all humbled. The stars may be large, but they do not think nor love; and these qualities touch my heart more than size."
— Frank Ramsay
Figure 1 Frank Plumpton Ramsey (1903-1930)
Frank Plumpton Ramsey (February 22, 1903 – January 19, 1930) was born in Cambridge, England, the son of Arthur S. Ramsey (1867-1954) He is the dean of Magdalene College, Cambridge University, and his mother is Mary (Mary Agnes Ramsey, 1875-1927). At home, Ramsay has a younger brother and two younger sisters.
Ramsay attended secondary school at the prestigious Winchester College. Founded in 1382, the school is the first school in the UK to train priests and public officials, and it is also the first university preparatory school in the UK to accept poor students for free. It created the history of public education in the UK. Today's Winchester College has evolved into a school for the children of nobles, and has trained many dignitaries and celebrities in modern times. After graduating from secondary school, Ramsay entered Trinity College, Cambridge University (Trinity College), majoring in mathematics. He graduated in 1923 and was named Senior Wrangler, the best mathematics undergraduate graduate of Cambridge University. In 1924, he was elected a Fellow of King's College, Cambridge. In the same year he married Lettice C. Baker (1898-1985), who was studying moral science at Cambridge University, with whom he later had two daughters. From 1926, Ramsay worked as a lecturer in mathematics at the University of Cambridge and director of the Mathematics Research Unit at King's College.
In 1930, Ramsay underwent abdominal surgery to treat chronic liver disease. But the operation was unsuccessful and jaundice syndrome occurred at the same time. He died on January 19 at Guy's Hospital in London after treatment failed and was buried in the Ascension Parish Cemetery in Cambridge.
Ramsay never stopped thinking in his short 27-year life. He dabbled in many fields of research, especially made many seminal contributions to mathematics, philosophy and economics.
【1】Mathematics
Let's start with a fun little game.
Suppose there are 6 people, A, B, C, D, E, F, who may or may not know each other. One day, they came together and met each other, as shown by the black border in Figure 1(a). Ramsey once predicted that at least three of these six people knew each other or did not know each other.
You close your eyes and think about it first, is Ramsay right?
Let's take a look at a situation: Suppose A knows all the people, as shown by the red border in Figure 1(b). Then B will not be able to know the rest of the people, as shown in Figure 1(c) with blue edges, otherwise a red triangle will be formed, and Ramsey is right. Okay, now it's C's turn. But from Figure 1(d), it can be seen that whether C knows or not knows the rest of the others, a red or blue triangle will be formed—Ramsey is right.
You may wish to try other possible situations, such as starting from A who only knows one or two people. You'll find out in the end: Ramsey was always right.
Figure 2 Ramsey's theorem: an example of 6 people
In fact, Ramsay published his paper "On a problem in formal logic" in the Proceedings of the London Mathematical Society in 1930, proving that A well-known theorem in logic and graph theory, later known as "Ramsey's theorem": Suppose there are k people attending a party, among whom m people know each other or n people don't know each other. If this situation is expressed as R(m,n)=k, then the simplest case is R(3,3)=6, which is the above example.
Ramsey's theorem is a general statement of logic, and the above game is just a simple special case of it. The reason why this theorem is famous is that it can help us find some ordered substructure that must appear in a large and disordered (such as chaos) structure: "A system must be at least as large as possible to ensure that it has certain properties ?” For example, the above game is to ask: How big must the crowd be at least to ensure that there must be three people in the group who know each other or three people who don’t know each other? Ramsey's answer: At least six people.
Ramsey's theorem is closely related to combinatorics, graph theory, especially graph coloring, and many theories, methods, and problems in discrete mathematics. There are even "Ramsey" classes in infinite-dimensional graph theory and group theory. '", the so-called "Ramsey–Dvoretzky–Milman phenomenon".
Another important significance of Ramsey's theorem is that it provides the first concrete example for Kurt F. Gödel's (1906-1978) theory of the incompleteness of logical systems, that is, in any first-order form A logical system must contain the assertions allowed in a certain system, and it cannot use the logic of the system itself to prove whether it is right or wrong. Ramsey's theory was a failed attempt to solve such a problem in classical logic, but it provided a useful result. He introduced and analyzed the first-order logic function that was later called Bernays-Schönfinkel-Ramsey type, proved a series of important assertions, and made substantial contributions to logic.
[2] Philosophy
While at Cambridge, Ramsey read and translated Ludwig J. J. Wittgenstein's (1889-1951) German work Tractatus Logico Philosophicus, published in 1921. The author Wittgenstein was an Austrian-British man who was later considered to be the most influential philosopher of the 20th century. His research fields are mainly in logic, philosophy of language, philosophy of mind and philosophy of mathematics. Ramsey was deeply influenced by Wittgenstein's book. In early 1923, Ramsey published a review article "Mind of Wittgenstein's Tractatus" (Mind of Wittgenstein's Tractatus), which made some criticisms of Wittgenstein's work. In September of the same year, Ramsey made a special trip to Austria to visit the philosopher who was fourteen years his senior. The two had a two-week conversation devoted to this work. Ramsey's pragmatism brought Wittgenstein's pure rational thinking from the pure logic in early philosophy back to real life, allowing him to clarify the relationship between logic and the world, rather than "just some logical operation symbols".
The complete philosophical thought formed by Wittgenstein's later masterpiece "Philosophical Investigations" benefited from the important influence of Ramsey at that time. Wittgenstein said in the preface of the book: "Since I started to focus on philosophy again sixteen years ago, I have to admit my major mistakes in the first book. The criticism and help from Ramsey made me Aware of these mistakes (in a way, I hardly realize it myself). In the last two years of Ramsey’s life, I have had countless conversations with him about these mistakes of mine.” For this reason, later generations even regard Ramsey as a better philosopher than Wittgenstein.
Ramsey's philosophical contribution and its historical significance lie in leading the "analytic philosophy" established by Bertrand AW Russell (1872-1970) and Wittgenstein to a new stage. Ramsay even made a radical criticism of Alfred N. Whitehead (1861-1947) and Russell's famous book "Principia Mathematica" (Principia Mathematica) in the early days. He contributed to Russell's reduction of mathematics to logic in his final theoretical framework.
In fact, Ramsey has raised some ideas of Russell and Wittgenstein to a new level, and is also an inheritor of their philosophical theories. A typical example is Ramsey's first distinction between Logical Paradox and Semantic Paradox. The well-known Russell's paradox (does the barber who "cuts everyone who doesn't shave himself" shave himself?) falls into the former category, while the liar's paradox (does the person who says "I'm lying" lie?) belongs to the latter. Ramsey's "simple types" theory (Ramsey theory of the simple types) states that logical contradictions involve the mathematical or logical terms "type" and "number", thus indicating a logical problem. Semantic contradictions (i.e., epistemological contradictions) involve concepts such as "thought", "language", and "symbols" in addition to purely logical terms, which are empirical rather than formal terms. Today, in philosophy and logic, Ramsey's distinction is used as a standard method of classifying paradoxes.
Figure 3 Ramsay is reading leisurely in the suburbs
Ramsey's 1926 paper "Truth and probability" laid the foundation for some aspects of subjective probability and decision theory. He used it to measure the degree of "partial belief", thus providing a certain measure for the so-called "subjective" or "personal" probability analysis. He developed the first quantitative theory of how people make decisions, illustrating how such decisions depend on the strength of individual beliefs and desires. He uses the equivalence between "believing a proposition" and "believing that the proposition is true" to define truth in terms of the quantified beliefs described above. He proposed two theories of natural laws: on the one hand, the laws are generalizations of the axioms and theorems in the simplest real theory; Forecast and judge accordingly.
In addition, Ramsey also made outstanding contributions to the philosophy of science. In a striking paper, "Theories," Ramsey proposes a new method for eliminating excessive reference to theoretical entities in many formal statements of scientific theories. The method consists in substituting an appropriate variable for each constant denoting a theoretical entity in the axioms expressing the formal system of scientific theoretical inquiry, and then quantifying the matrix of propositions thus obtained applying a full name. Ramsey went on to show that universally quantified statements derived from the original axioms would have the same observations as the original axiom system. This quantitative technique is of great interest to philosophers concerned with the ontological implications of scientific theories. In the study of philosophy of science, Ramsey introduced the Ramsey-Lewis method to define the terms in scientific theory. By using this approach, the set of theoretical terms that appear in a theory can be implicitly defined by the assertions of the theory itself.
【3】Economics
During the 20th century, mathematics gradually became the lingua franca of economics. However, only a few mathematicians had an immediate and lasting impact on the world at the time, leading figures including Ramsey, John von Neumann (1903-1957), and John F. Nash Jr., 1926-2015).
Ramsey published three seminal economic papers on subjects such as subjective probability and returns (1926), optimal taxation (1927), and optimal single-sector growth (1928). These papers became the main reference works of public finance theorists and monetary economists. In addition, another of his papers on savings became a touchstone for various theories of economic growth.
Ramsey left behind a famous Ramsey-Cass-Koopmans model of economic growth (often shortened to "the Ramsey growth model") for optimizing economic returns. Figure 4 is a dynamic phase diagram used to explain the Ramsey growth model, in which the blue line represents the dynamic adjustment path of the economy, which satisfies all the constraints of the model, and is a stable evolution path of the dynamic system; the red line represents the unstable dynamic Evolution path; the black line represents the boundary between stable and unstable states, described by a certain revenue function g(.).
Fig.4 Kinetic phase diagram of Ramsey growth model
Today, there is also a "Ramsey problem" in economics, which is the mathematical problem of how policymakers should set prices to regulate market monopolies in order to ensure the public interest. This problem was first clearly stated in the book "The Economics of Imperfect Competition" (The Economics of Imperfect Competition) by British female economist Joan V. Robinson (1903-1983) in 1933 to describe and analyze the wage gap between equally productive male and female workers. She pointed out that monopoly applies to buyers of labor, and that employers have pricing power, can exploit, and pay wages below workers' marginal productivity. But later I learned that in 1927 Ramsey had discovered this pricing law. Therefore, later generations named it the Ramsey problem.
【Four】
After Ramsey's death, the academic circle has published many articles, essays and books commemorating him, especially the collection of Ramsey's essays "Foundations: Essays in Philosophy, Logic, Mathematics and Economics" (Foundations: Essays in Philosophy, Logic, Mathematics and Economics), comprehensively reflecting his main academic contributions and their far-reaching influence.
Figure 5 Some anthologies and books commemorating Ramsey
In memory of Ramsey, Cambridge University established the honorary title of "Frank P. Ramsey Professor Emeritus of Economics"; Harvard University also established the honorary titles of "Professor Frank P. Ramsey of Managerial Economics" and "Professor Frank P. Ramsey of Political Economy". "Two Chair Professorships.
About Huayuan Computing
Huayuan Computing Technology (Shanghai) Co., Ltd. ("Huayuan Computing" for short), established in 2002, focuses on algorithm research and innovative applications, focusing on the research, application and development of cognitive intelligence technology. Based on the development of mathematics application and computing technology, the company focuses on cognitive intelligence technology and innovates self-developed underlying algorithms; based on the scenario application of the cognitive intelligence engine platform, it provides AI+ industry solutions for digital governance, intelligent manufacturing, digital cultural tourism, retail finance and other industries Solutions, to achieve comprehensive empowerment, so as to promote the transformation and upgrading of industry intelligence, and make the world smarter.