How can we understand reasoning in general and mathematical proofs in particular? It is argued that a high-level understanding of proofs is needed to complement the low-level understanding provided by Logic A role for computation is proposed to provide this high-level understanding, namely by the association of proof plans with proofs Criteria are given for assessing the association of a proof plan with a proof

To those brought up in a logic-based tradition there seems to be a simple and clear definition of proof. But this is largely a twentieth century invention; many earlier proofs had a different nature. We will look particularly at the faulty proof of Euler's Theorem and Lakatos' rational reconstruction of the history of this proof. We will ask: how is it possible for the errors in a faulty proof to remain undetected for several years-even when counter-examples to it are known? How is it possible to have a proof about concepts that are only partially defined? And can we give a logic-based account of such phenomena? We introduce the concept of schematic proofs and argue that they offer a possible cognitive model for the human construction of proofs in mathematics. In particular, we show how they can account for persistent errors in proofs.

A Science of Reasoning.

What is a proof

and concrete syntax Fine details of syntax are of no fundamental importance. Some mathematics is typed, some is handwritten, and people make various essentially arbitrary choices that do not change anything about the structural way symbols are used together. When mechanizing logic on the computer, we will, for simplicity, restrict ourselves to the usual stock of Latin characters and punctuation, and instead of the Greek letters and special symbols that many logicians use, we will use other letters or words, e.g. ‘forall’ instead of ‘∀’. We will, however, continue to employ the usual symbols in theoretical discussions. This continual translation may even be helpful to the reader who hasn’t seen or understood the symbols before. Regardless of how the symbolic expressions are read or written, it’s more convenient to manipulate them in a form better reflecting their structure. Consider the expression ‘x+y×z−w’ in ordinary algebra. This linear form obscures the meaningful structure. To understand which operators have been applied to which subexpressions, or even what constitutes a subexpression, we need to know rules of precedence and associativity, e.g. that ‘×’ “binds tighter” than ‘+’. For instance, despite their apparent similarity in the linear form, ‘y × z’ is a subexpression while ‘x + y’ is not. Even if we make the structure explicit by fully bracketing it as ‘(x+ (y × z))− w’, basic useful operations on expressions like finding subexpressions, or evaluating the expression for particular values of the variables, become tiresome to describe precisely; one needs to shuffle back and forth over the formula matching up brackets. A “tree” structure is much better: just as a family tree makes relations among family members clearly apparent, a tree representation of an expression displays its structure and makes most important manipulations straightforward. As in genealogy, it’s customary to draw trees growing downwards on the printed page, so the same expression might be represented as follows:

Introduction to Logic and Automated Theorem Proving

A proof-centric approach to mathematical assistants

The aim of the MATHsAiD project is to build a tool for automated theorem-discovery; to design and build a tool to automatically conjecture and prove theorems (lemmas, corollaries, etc.) from a set of user-supplied axioms and definitions. No other input is required. This tool would, for instance, allow a mathematician to try several versions of a particular definition, and in a relatively small amount of time, be able to see some of the consequences, in terms of the resulting theorems, of each version. Moreover, the automatically discovered theorems could perhaps help the users to discover and prove further theorems for themselves. The tool could also easily be used by educators (to generate exercise sets, for instance) and by students as well. In a similar fashion, it might also prove useful in enabling automated theorem provers to dispatch many of the more difficult proof obligations arising in software verification, by automatically generating lemmas which are needed by the prover, in order to finish these proofs.

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MATHsAiD: Automated Mathematical Theory Exploration

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A Science of Reasoning.

Citations

What is a proof

Introduction to Logic and Automated Theorem Proving

A proof-centric approach to mathematical assistants

MATHsAiD: Automated Mathematical Theory Exploration

English summaries of mathematical proofs

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