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Proofs and Refutations

 
Proofs and Refutations

 
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Proofs and Refutations
 
 
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Proofs and Refutations is a book by the philosopher Imre Lakatos
Imre Lakatos

Imre Lakatos was a philosopher of Philosophy of mathematics and Philosophy of science, most famous today worldwide for his thesis of the fallibility of mathematics and its 'methodology of proofs and refutations', and also for introducing the concept of the 'research programme' in his methodology of scientific research programmes....
 expounding his view of the progress of mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
. The book is written as a series of Socratic dialogue
Socratic dialogue

Socratic dialogue is a genre of prose literary works developed in Ancient Greece at the turn of the fourth century BC, preserved today in the dialogues of Plato and the Socratic works of Xenophon - either dramatic or narrative - in which characters discuss moral and philosophical problems, illustrating the Socratic method....
s involving a group of students who debate the proof of the Euler characteristic
Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent....
 defined for the polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
. A central theme is that definition
Definition

A definition is a statement of the Meaning of a word or phrase. The term to be defined is known as the definiendum . The words which define it are known as the definiens ....
s are not carved in stone, but often have to be patched up in the light of later insights, in particular failed proof
Mathematical proof

In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive reasoning or empirical arguments....
s. This gives mathematics a somewhat experimental flavour.

The pupils in the book are named after letters of the Greek alphabet.

Many important logical ideas are explained in the book.






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Proofs and Refutations is a book by the philosopher Imre Lakatos
Imre Lakatos

Imre Lakatos was a philosopher of Philosophy of mathematics and Philosophy of science, most famous today worldwide for his thesis of the fallibility of mathematics and its 'methodology of proofs and refutations', and also for introducing the concept of the 'research programme' in his methodology of scientific research programmes....
 expounding his view of the progress of mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
. The book is written as a series of Socratic dialogue
Socratic dialogue

Socratic dialogue is a genre of prose literary works developed in Ancient Greece at the turn of the fourth century BC, preserved today in the dialogues of Plato and the Socratic works of Xenophon - either dramatic or narrative - in which characters discuss moral and philosophical problems, illustrating the Socratic method....
s involving a group of students who debate the proof of the Euler characteristic
Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent....
 defined for the polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
. A central theme is that definition
Definition

A definition is a statement of the Meaning of a word or phrase. The term to be defined is known as the definiendum . The words which define it are known as the definiens ....
s are not carved in stone, but often have to be patched up in the light of later insights, in particular failed proof
Mathematical proof

In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive reasoning or empirical arguments....
s. This gives mathematics a somewhat experimental flavour.

The pupils in the book are named after letters of the Greek alphabet.

Many important logical ideas are explained in the book. For example the difference between a counterexample
Counterexample

In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule, i.e., a specific instance of the falsity of a universal quantification ....
 to a proof (local counterexample) and a counterexample to a conjecture (global counterexample) are discussed.

The 1976 book has been translated into more than 15 languages worldwide, including Chinese, Korean and Serbo-Croat, and went into its second Chinese edition in 2007.