Epistemic Justification and the Possibility of Computer Proof (original) (raw)
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Correctness, Artificial Intelligence, and the Epistemic Value of Mathematical Proof
We argue that it is neither necessary nor sufficient for a mathematical proof to have epistemic value that it be “correct”, in the sense of formalizable in a formal proof system. We then present a view on the relationship between mathematics and logic that clarifies the role of formal correctness in mathematics. Finally, we discuss the significance of these arguments for recent discussions about automated theorem provers and applications of AI to mathematics.
WHAT IS A MATHEMATICAL PROOF IN THE AGE OF MODERN THEOREM PROVERS
A mathematical proof is notable for its clear language which satisfies the logical rules of inference and which convinces us by its intrinsic explanation relatively easy to reproduce. Nevertheless, since the publication of the computer aided proof of the four-colour theorem in 1976, the status of proof has been widely discussed, and that discussion was recently reopened following the verification of the Feit–Thompson theorem by a modern theorem prover. Modern theorem provers enable us to verify mathematical theorems, construct formal axiomatic derivations of remarkable complexity and, potentially, increase confidence in mathematical statements. Proof assistants, therefore, are the result of the efforts by logicians, computer scientists, and mathematician to obtain complete mathematical confidence through computers. In this paper, I will discuss how classical mathematical theorems strongly contrast with the " trivial " use of modern theorem provers. I will specifically address the Feit–Thompson theorem, ⎯ proof of which was recently verified by the interactive theorem prover Coq ⎯ in order to assess the proof's status in the era of modern theorem provers, and, more clearly, whether a machine proof may still be considered a calculus of reasoning.
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The idea that science can be automated is so deeply related to the view that the method of mathematics is the axiomatic method, that confuting the claim that mathematical knowledge can be extended by means of the axiomatic method is almost equivalent to confuting the claim that science can be automated. I argue that the axiomatic view is inadequate as a view of the method of mathematics and that the analytic view is to be preferred. But, if the method of mathematics and natural sciences is the analytic method, then the advancement of knowledge cannot be mechanized, since non-deductive reasoning plays a crucial role in the analytic method, and non-deductive reasoning cannot be fully mechanized.
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Cognitive Structures in Scientific Inquiry, 2005
Most philosophers still tend to believe that mathematics is basically about producing formal proofs. A consequence of this view is that some aspects of mathematical practice are lost out of view entirely. As I will defend, it is precisely in those aspects that similarities can be found between practices in the exact sciences and in mathematics. Hence, if we are looking for a (more) unified treatment of science and mathematics, it is necessary to incorporate these elements into our view of what mathematics is about. As a helpful tool I will introduce the notion of a mathematical argument as a more liberalized version of the notion of mathematical proof.
Tennessee Technological University Mathematics Department Technical Report, No. 1999-1., 1999
Mathematics departments rarely require students to study very much logic before working with proofs. Normally, the most they will offer is contained in a small portion of a "bridge" course designed to help students move from more procedurally-based lower-division courses (e.g., calculus and differential equations) to more proof-based upper division courses (e.g., abstract algebra and real analysis). What accounts for this seeming neglect of an essential ingredient of deductive reasoning? We will suggest a partial answer by comparing the contents of traditional logic courses with the kinds of reasoning used in proof validation, our name for the process by which proofs are read and checked. First, we will discuss the style in which mathematical proofs are traditionally written and its apparentutility for reducing validation errors. We will then examine the relationship between the need for logic invalidating proofs and the contents of traditional logic courses. Some topics emphasized in logic courses donot seem to be called upon very often during proof validation, whereas other kinds of reasoning, not often emphasized in such courses, are frequently used. In addition, the rather automatic way in which logic, such as modus ponens, needs to be used during proof validation does not appear to be improved by traditional teaching, which often emphasizes truth tables, valid arguments, and decontextualized exercises. Finally, we will illustrate these ideas with a proof validation, in which we explicitly point out the uses of logic. We will not discuss proof construction, a much more complex process than validation. However, constructing a proof includes validating it, and hence, during the validation phase, calls on the same kinds of reasoning. Throughout this paper we will refer to a number of ideas from both cognitive psychology and mathematics education research. We will Þnd it useful to discuss short-term, long-term, and working memory, cognitive load, internalized speech and vision, and schemas, as well as reßection, unpacking the meaning of statements, and the distinction between procedural and conceptual knowledge.
1999
Mathematics departments rarely require students to study very much logic before working with proofs. Normally, the most they will offer is contained in a small portion of a "bridge" course designed to help students move from more procedurally-based lower-division courses (e.g., calculus and differential equations) to more proof-based upper division courses (e.g., abstract algebra and real analysis). What accounts for this seeming neglect of an essential ingredient of deductive reasoning? 1 We will suggest a partial answer by comparing the contents of traditional logic courses with the kinds of reasoning used in proof validation, our name for the process by which proofs are read and checked.