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 Calcolo Geometric o Secondo L 'Ausdehungslehre di H. Grassmann published in 1888. This article is an independent chapter of ***20 pages, which is about "the operation of deductive logic". Piano disagreed with Russell, but integrated and developed the work of G. Bull, F.W.K.E E. Schleede, C. S. Pierce and H. Maccoll. 1889, piano's masterpiece "The Principle of Arithmetic" (Nova Methodo Exposita) 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 (Rivista di Matematica), wrote this group of axioms of natural numbers in this magazine with mathematical logic symbols, 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. ③ 0 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 ... 65438+90 years continued to study logic, which contributed to the first international congress of mathematicians. 1900, piano and his collaborators Blary-Folthy, Padoa and 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. 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 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. Schwartz introduced the concept of surface area, but it was not clear. A year later, piano independently defined the concept of surface area. 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 study of Italian vector analysis.

1890, piano found a strange curve. As long as the function and a continuous parameter curve defined by are properly selected, when the parameter t is in the interval of [0, 1], 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 is 14 years earlier than 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 Calculo Differential e Principi i di Calco Integrale are listed as "the most important 19 calculus textbook since the times of L. Euler and A. L. Cauchy" by the German Mathematical Encyclopedia. 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.