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**1 - 2**of**2**### Formal Proof: Reconciling Correctness and Understanding

, 2009

"... Hilbert’s concept of formal proof is an ideal of rigour for mathematics which has important applications in mathematical logic, but seems irrelevant for the practice of mathematics. The advent, in the last twenty years, of proof assistants was followed by an impressive record of deep mathematical th ..."

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Hilbert’s concept of formal proof is an ideal of rigour for mathematics which has important applications in mathematical logic, but seems irrelevant for the practice of mathematics. The advent, in the last twenty years, of proof assistants was followed by an impressive record of deep mathematical theorems formally proved. Formal proof is practically achievable. With formal proof, correctness reaches a standard that no pen-and-paper proof can match, but an essential component of mathematics — the insight and understanding — seems to be in short supply. So, what makes a proof understandable? To answer this question we first suggest a list of symptoms of understanding. We then propose a vision of an environment in which users can write and check formal proofs as well as query them with reference to the symptoms of understanding. In this way, the environment reconciles the main features of proof: correctness and understanding.

### What Do We Mean by Mathematical Proof? 1

"... Mathematical proof lies at the foundations of mathematics, but there are several notions of what mathematical proof is, or might be. In fact, the idea of mathe-matical proof continues to evolve. In this article, I review the body of literature that argues that there are at least two widely held mean ..."

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Mathematical proof lies at the foundations of mathematics, but there are several notions of what mathematical proof is, or might be. In fact, the idea of mathe-matical proof continues to evolve. In this article, I review the body of literature that argues that there are at least two widely held meanings of proof, and that the standards of proof are negotiated and agreed upon by the members of math-ematical communities. The formal view of proof is contrasted with the view of proofs as arguments intended to convince a reader. These views are examined in the context of the various roles of proof. The conceptions of proof held by students, and communities of students, are discussed, as well as the pedagogy of introductory proof-writing classes. What is a mathematical proof? This question, and variations on it, have been debated for some time, and many answers have been proposed. One variation of this question is the title of this article: “What do we mean by mathematical proof? ” Here we may stand for the international community of mathematicians, a classroom of students, the human race as a whole, or any number of other mathematical communities. When the question is phrased this way, it becomes clear that any answer to this question must, in one way or another, take into account the fact that mathematics and mathematical proof are endeavors undertaken by people, either individually or communally.2 This article will discuss two answers to this question that are held by mathematicians and mathematics educators, and how those answers affect 1Portions of this article previously appeared in the author’s doctoral dissertation, [7]. 2By this statement, I do not mean to take sides in the debate over whether mathematics is “discovered ” or “created”; in either case, it is people who discover or create mathematics.