scholarly journals A type-free formalization of mathematics where proofs are objects

1998 ◽  
pp. 88-111
Author(s):  
Gilles Dowek
Keyword(s):  
Formalization Of Mathematics

The formalization of mathematics

1954 ◽  
Vol 19 (4) ◽  
pp. 241-266 ◽  
Author(s):  
Hao Wang
Keyword(s):  
Large Body ◽  
Axiom System ◽  
Original Sin ◽  
Axiomatic Method ◽  
Formalization Of Mathematics ◽  
Rigid System ◽  
Mathematical Experience ◽  
Mental Operations ◽  
The Mind

Zest for both system and objectivity is the formal logician's original sin. He pays for it by constant frustrations and by living ofttimes the life of an intellectual outcaste. The task of squeezing a large body of stubborn facts into a more or less rigid system can be a painful one, especially since the facts of mathematics are among the most stubborn of all facts. Moreover, the more general and abstract we get, the farther removed we are from the raw mathematical experience. As intuition ceases to operate effectively, we fall into many unexpected traps. The formal logician gets little sympathy for his frustrations. He is regarded as too rigid by his philosophical colleagues and too speculative by his mathematical friends. The life of an intellectual outcaste may be a result partly of temperament and partly of the youthfulness of the logic profession. The unfortunate lack of wide appeal of logic may, however, be prolonged partly on account of the fact that very little of the well-established techniques of mathematics seems applicable to the treatment of serious problems of logic.The axiomatic method is well suited to provide results which are both exact and systematic. How attractive would it be if we could get an axiom system in which all the axioms and deductions were intuitively clear and all theorems of mathematics were provable? Such a system would undoubtedly satisfy Descartes who admits solely intuition and deduction, which are, for him, the only “mental operations by which we are able, wholly without fear of illusion, to arrive at the knowledge of things.” Indeed, according to Descartes, intuition and deduction “are the most certain routes to knowledge, and the mind should admit no others. All the rest should be rejected as suspect of error and dangerous.”

scholarly journals An experimental library of formalized Mathematics based on the univalent foundations

2015 ◽  
Vol 25 (5) ◽  
pp. 1278-1294 ◽  
Author(s):  
VLADIMIR VOEVODSKY
Keyword(s):  
Type Theory ◽  
Proof Assistant ◽  
Formalization Of Mathematics ◽  
Constructive Type Theory ◽  
Short Overview ◽  
Formalized Mathematics ◽  
The Coq Proof Assistant ◽  
Constructive Type ◽  
Coq Proof Assistant ◽  
Univalent Foundations

This is a short overview of an experimental library of Mathematics formalized in the Coq proof assistant using the univalent interpretation of the underlying type theory of Coq. I started to work on this library in February 2010 in order to gain experience with formalization of Mathematics in a constructive type theory based on the intuition gained from the univalent models (see Kapulkin et al. 2012).

scholarly journals N.G. de Bruijn’s contribution to the formalization of mathematics

2013 ◽  
Vol 24 (4) ◽  
pp. 1034-1049
Author(s):  
Herman Geuvers ◽  
Rob Nederpelt
Keyword(s):  
Formalization Of Mathematics

scholarly journals Exploring the Educational Value of Mathematical Beauty and Effective Ways of Discovering Mathematical Beauty

2021 ◽  
Vol 5 (10) ◽  
pp. 82-87
Author(s):  
Cunrong Wang
Keyword(s):  
Problem Solving ◽  
Mathematics Teaching ◽  
Mathematical Reasoning ◽  
Knowledge Systems ◽  
Mathematical Language ◽  
Mathematical Culture ◽  
Educational Value ◽  
Formalization Of Mathematics ◽  
Mathematical Beauty ◽  
Opening Up

The precision of mathematical reasoning, the abstractness of mathematical language, the profundity of mathematical thought and method, as well as the excessive formalization of mathematics teaching have formed an impassable gap, hindering students in approaching mathematics. This has concealed the beauty of mathematics and the light of mathematical culture. However, if students are able to cross this gap, they would find that mathematics is a vast world full of vitality, imagination, wisdom, poetry, and beauty. The pursuit of mathematical beauty is one of the motivations for scientists to research this field. Experiencing mathematical beauty is of great significance to students’ learning and growth. In teaching, the value of mathematical beauty is explored, such as stimulating emotions, opening up to the truth, and cultivating goodness. Several effective ways are suggested in this article to guide students to discover the mathematical beauty in life while finding it in problem-solving methods and exploring it in knowledge systems.

scholarly journals Formalization of the Equivalence among Completeness Theorems of Real Number in Coq

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 38
Author(s):  
Yaoshun Fu ◽  
Wensheng Yu
Keyword(s):  
Real Number ◽  
Fundamental Theorem ◽  
Completeness Theorem ◽  
Accumulation Point ◽  
Formal Proof ◽  
Theorem Prover ◽  
Monotone Convergence ◽  
Formalization Of Mathematics ◽  
Sequential Compactness ◽  
Completeness Theorems

The formalization of mathematics based on theorem prover becomes increasingly important in mathematics and computer science, and, particularly, formalizing fundamental mathematical theories becomes especially essential. In this paper, we describe the formalization in Coq of eight very representative completeness theorems of real numbers. These theorems include the Dedekind fundamental theorem, Supremum theorem, Monotone convergence theorem, Nested interval theorem, Finite cover theorem, Accumulation point theorem, Sequential compactness theorem, and Cauchy completeness theorem. We formalize the real number theory strictly following Landau’s Foundations of Analysis where the Dedekind fundamental theorem can be proved. We extend this system and complete the related notions and properties for finiteness and sequence. We prove these theorems in turn from Dedekind fundamental theorem, and finally prove the Dedekind fundamental theorem by the Cauchy completeness theorem. The full details of formal proof are checked by the proof assistant Coq, which embodies the characteristics of reliability and interactivity. This work can lay the foundation for many applications, especially in calculus and topology.

scholarly journals Formalization of mathematics through proof assistants

2019 ◽  
Author(s):  
Henrique Yuji Rossetti Inonhe ◽  
Walter Alexandre Carnielli
Keyword(s):  
State Of The Art ◽  
Learning Curves ◽  
The State ◽  
Mathematical Notation ◽  
Proof Assistants ◽  
Formalization Of Mathematics ◽  
Formalized Mathematics ◽  
Automatic Theorem Provers ◽  
Wide Gap ◽  
Automatic Theorem

The formalization of mathematics in practice relies heavily on proof assistants and automatic theorem provers, therefore we studied what are the state of the art proof assitants and their limitations to understand what are the main challenges in making formalized mathematics common practice among mathematicians. We found out that curretly the two major dificulties in formalizing mathematics with proof assistants are due to steep learning curves in how to use these tools and due to a wide gap between the notation employed in these proof assistants and the currently used mathematical notation. We also developed a C++ library to develop proof assistants with great notational flexibility.

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