Piano, Italian mathematician. Born on August 27th, 1858, Spinetta village near Cuneo in Piedmont; 1932 died in Turin on April 20th. Piano devoted himself to developing the symbolic logic system initiated by Boolean. 1889, he published the book Logical Expression of Geometric Principles, in which he used symbolic logic as the basis of mathematics. More than twenty years later, Whitehead continued this work. Piano established an axiomatic system from the undefined concepts of "zero", "number" and "successor number".
Name: Piano.
Peano, Giuseppe
Nationality: Italy
Place of birth: Piedmont
Date of birth:1August 27th, 858
Date of death:1April 20, 932
Occupation: Mathematician
Graduate School: University of Turin
Main achievements: piano axiom
The founder of symbolic logic
Create an international language
Masterpieces: A Course of Differential Analysis, Encyclopedia of Mathematics, and New Methods of Arithmetic Principles.
all one's life
Piano's parents Bartolomeo and Roussillat have four boys and 1 girl, and piano is the second child. Their family lives by farming. Although in a rural area full of illiteracy, piano's parents are knowledgeable, cheerful and educated. His family lives three miles from the provincial capital Cuneo. Every day, piano and his brother Michelle walk to the provincial capital to study. In order to make it easier for their children to go to school, his parents moved their family to the city and didn't move back to the farm until his youngest sister graduated from primary school. His uncle, M. Cavaroz, is a priest and lawyer, and lives in Turin. Because of piano's curiosity and excellent grades, my uncle took him to study in Turin. At the beginning, he received private education (including his uncle's education) and self-study, which enabled him to pass the junior high school entrance examination of Gafur School on 1873. 1876 graduated from high school, won a scholarship because of his excellent performance, and entered the University of Turin. He studied engineering first, and after studying physics and mathematics for two years, he decided to specialize in pure mathematics. After five years in school, he studied a wide range of subjects. 1July, 880, got a college diploma with high marks, stayed in Ovidio as a teaching assistant, and became an analyst A one year later. Professor Ginochi's teaching assistant. 1882 In the spring, Gnozzi broke his kneecap, so piano went to teach the analysis class instead of him. 1884 Lecturer in Calculus, University of Turin. 1890 12 After formal competition, piano became an interim professor at the University of Turin and a full professor at 1895. He taught at the University of Turin until his death.
1887 piano married the painter's daughter, Carola Rosija. They have no children.
Piano is a member of many scientific societies and the Royal Italian Society. He has made great achievements in analytical research. He is the founder of symbolic logic and international language. Piano died of angina pectoris on the evening of April 20th. 1932. According to his wishes, the funeral was simple and buried in Turin cemetery. 1963, his body was moved to the family cemetery in his hometown of Spinetta.
contribution
1903, piano stepped out of the field of mathematics and devoted himself to inventing an international language (at least used by people who speak western European languages). The language form he adopted can be said to be a mixed language, adding Latin stems (but not inflections) to German or English words, as long as it seems feasible. The result is an "international language", which is not difficult for Latin speakers, and it is too difficult for Germanic speakers to be fully familiar with Latin. Nowadays, some scientific magazines take a measure, that is, to publish the abstracts of articles in an internationally common language, so that as many people as possible can see them through the least translation.
He is famous as a pioneer of symbolic logic and an advocate of axiomatic method. 189 1 year, piano founded the Journal of Mathematics, in which he wrote this set of axioms of natural numbers with mathematical logic symbols, and proved their independence. Piano's Compilation of Mathematical Formulas is published in 1895- 1908 in five volumes. The fifth volume alone contains 4200 formulas and theorems, many of which have been proved. The book is rich in history and literature, and some people call it "infinite mathematical mineral resources". Piano introduced and popularized the concept of "small section". Piano thinks that his most important job is analysis. In 1883, he gave a new definition of definite integral, taking the common value when the minimum upper bound of riemann sum is equal to the maximum lower bound. This is an effort to get the definition of integral out of the concept of limit. In 1886, he first proved that the only condition that the first-order differential equation y'=f(x, y) is solvable is the continuity of f, and gave a slightly less strict proof. 1893, piano published the course of infinitesimal analysis, which was listed as "the most important 19 calculus textbook since the time of Euler and Cauchy" by the German Mathematical Encyclopedia. There are many remarkable places in the mathematical encyclopedia written by the piano. For example, the generalization of differential mean value theorem; The judgment theorem of uniform continuity of multivariate functions: the existence theorem of implicit functions and the proof of their differentiability theorems: examples of partially differentiable but globally nondifferentiable functions; Conditions for Taylor expansion of multivariate functions: a counterexample of the popular minimax theory at that time.
Scientific contribution
initiator
Piano is famous as a pioneer of symbolic logic and a popularizer of axiomatic methods. His works were completed independently of Dai Dejin. Although Dai Dejin once published an article about natural numbers, his views were basically the same as those of the piano, but his expression was not as clear as that of the piano, which did not attract people's attention. Piano's research on mathematical logic and mathematics based on concise symbols and axioms has created a new situation. His first article on logic appeared in his book Geometric Calculus Based on grassmann, published in 1888. This article is an independent chapter of ***20 pages, which is about "Operationsofdeductivelogic". Piano disagreed with Russell, but integrated and developed the work of Bull, Schroeder, Pierce and Maccoll. 1889, piano's masterpiece Arithmetic e Sprincipia (Novamedoexpostia) was published. In this booklet, he completed the axiomatic processing of integers and made many innovations in logical symbols, thus making reasoning more concise. In the book, he gave the world-famous axiom of natural numbers and became a classic. 189 1 year, piano founded the Journal of Mathematics, in which he wrote down the axioms of logical symbols of natural numbers RivistadiMatematica and proved their independence. Piano defined natural numbers with two undefined concepts "0" and "successor" and five axioms.
The set n of natural numbers refers to a set that meets the following conditions: ① There is an element in n, which is recorded as 0. ② Every element in n can find an element in n as its successor. ③0 is not the successor of any element. ④ Different elements have different successors. ⑤ (inductive axiom) Any subset m of n, if 0∈M, and as long as X is in M, it can be deduced that the successor of X is also in M, then M = N. ..
Continue to learn logic
65438+90 years continued to study logic, which contributed to the first international congress of mathematicians. 1900, piano and his collaborators Blary Folthy, Padoa (A.Padoa) and Pieri (M.Pieri) presided over the discussion. Russell later wrote: "This meeting was a turning point in my academic career, because I met piano at this meeting." Piano played a great role in the development of logic in the mid-20th century and made outstanding contributions to mathematics.
Piano published his and his followers' achievements on the basis of logic and mathematics in the Journal of Mathematics. He also published his huge formula plan on it, and spent 26 years on this work. He expects to establish the whole mathematical system according to some basic axioms of his mathematical logic symbols. He profoundly changed the views of mathematicians and had a great influence on the Bourbaki school.
Compilation of mathematical formulas
Piano's formula Mathematico has five volumes published in 1895- 1908. The fifth volume alone contains 4200 formulas and theorems, many of which have been proved. The book is rich in historical and documentary information, which some people call "endless mathematical mineral deposits". He doesn't aim at logic. He only pays attention to the development of logic in mathematics and calls his own system the logic of mathematics.
Other fields
Piano also used axiomatic methods in other fields, especially in geometry. Starting from 1889, he adopted the axiomatic method of elementary geometry and gave several axiomatic systems. 1894, he extended this method and simplified the undefined terms in geometry into three (point, line segment and motion) on the basis of M. Pasch's work. Later, in 1899, Pieri simplified the undefined terms in geometry into two (point and motion).
Many of his papers give clearer and stricter descriptions and applications to the existing definitions and theorems. For example, in 1882, H.A.Schwarz introduced the concept of surface area, but it was not clear. A year later, piano independently defined the concept of surface area.
Introduce and popularize "measures"
Piano introduced and popularized the concept of "small section". From 65438 to 0888, he extended grassmann's vector method to geometry, and his expression Bigrat Mann was much clearer, which greatly promoted the research of Italian vector analysis.
1890, piano found a strange curve. As long as the function and a continuous parametric curve defined by are properly selected, when the parameter t is in the interval, the curve will traverse all points in the unit square and get a curve full of space. Later, D. Hilbert and piano discovered some other curves.
Piano thinks that his most important job is analysis. Indeed, his analytical work is very novel, many of which are groundbreaking. In 1883, he gave a new definition of definite integral, and defined Riemann integral as the common value taken by riemann sum when its minimum upper bound is equal to its maximum lower bound. This is an effort to get the definition of integral out of the concept of limit. In 1886, he first proved that the only condition for the solvability of first-order differential equations is the continuity of F, and gave a slightly less strict proof.
In 1890, he extended this result to general differential equations through another proof, and gave a direct and clear description of axiom of choice. This was 14 years earlier than that of E.F.F Zermelo, but piano refused to use axiom of choice because it was beyond the ordinary logic used in mathematical proof. 1887, he discovered the successive approximation method for understanding linear differential equations, but people attributed the credit to E.Picard, who gave this method one year later than him. Piano also gave the error term of integral equation and developed it into the theory of "asymptotic operator", which is a new method to solve mathematical equations. 1901-1906, which contributes to insurance mathematics. As a member of the National Committee, he was asked to estimate the amount of pension. 1895- 1896 He wrote articles on theoretical mechanics, several of which were about the motion of the earth's rotation axis. His work also involves the generalization of special determinant, Taylor formula and integral formula. 1893, piano published the course of infinitesimal analysis, and the clear and rigorous expression in the book is amazing. It and piano's book "Principles of Differential and Integral Calculus" are listed as "the most important 19 calculus textbook since the time of L. Euler A. L. Cauchy" by the German Mathematical Encyclopedia.
Write an encyclopedia of mathematics
Piano's mathematical encyclopedia has many remarkable places. For example, the generalization of differential mean value theorem; The judgment theorem of uniform continuity of multivariate functions: the existence theorem of implicit functions and the proof of their differentiability theorems: examples of partially differentiable but globally nondifferentiable functions; Conditions for Taylor expansion of multivariate functions: a counterexample of the popular minimax theory at that time.
The founder of the international language
From 65438 to 0900, piano became interested in international auxiliary language because of its strong language ability. He wrote various book reviews in English, Italian, German and Polish. 1903 published his views on international languages in the Journal of Mathematics. He wants to build an international language for scholars, especially scientists. He thinks that there are a large number of scientific words from Latin and tries to choose the combination form of each word. He added Latin stems to German or English words so that scholars could identify them quickly. He believes that the best grammar is grammar-free, and advocates canceling complicated inflections. 1908 piano was elected president of the International Language Association until his death. He led the free discussion of this association, and published "Lexical Commune Latin-Italy-France-English-German" in19, which contains 14000 entries. Piano put himself in the later stage. He is regarded as the founder of international languages.
teaching
Piano's teaching work is also excellent, so he was hired as a part-time teacher by military schools and engineering schools. He has a strong interest in education and has made some contributions. He is firmly opposed to putting too much pressure on students. 19 12 published the essay "Opposing Examination" in primary schools. He said: "It is really a crime against humanity to torture poor students with exams and let them master things that ordinary educated adults don't know. The same principle applies to middle schools and universities. " He is very concerned about the rigor of teaching content. He believes that the definition must be accurate and clear, the proof must be correct, and those difficult contents can be omitted. He organized a series of discussions among middle school mathematics teachers, trying to promote the development of mathematics education in a clear, accurate and simplified direction.
Research on the history of mathematics
Piano also paid attention to the study of the history of mathematics, and he made an incisive exposition on the origin of mathematical terms. In mathematics teaching, he often introduces the knowledge of the history of mathematics and excavates the mathematical thoughts of G.W. Leibniz and I. Newton, which has a great influence on contemporary people.
Start a school
Piano also started a school with his mathematical formula collaborators. His knowledge and tolerance for students attracted a group of people with similar interests in mathematics and philosophy, and formed his school, which played an important role in the development of Italian mathematical logic and vector analysis.