Plans and Planning in Mathematical Proofs

Abstract

In practice, mathematical proofs are most often the result of careful planning by the agents who produced them. As a consequence, each mathematical proof inherits a plan in virtue of the way it is produced, a plan which underlies its “architecture” or “unity.” This paper provides an account of plans and planning in the context of mathematical proofs. The approach adopted here consists in looking for these notions not in mathematical proofs themselves, but in the agents who produced them. The starting point is to recognize that to each mathematical proof corresponds a proof activity which consists of a sequence of deductive inferences—i.e., a sequence of epistemic actions—and that any written mathematical proof is only a report of its corresponding proof activity. The main idea to be developed is that the plan of a mathematical proof is to be conceived and analyzed as the plan of the agent(s) who carried out the corresponding proof activity. The core of the paper is thus devoted to the development of an account of plans and planning in the context of proof activities. The account is based on the theory of planning agency developed by Michael Bratman in the philosophy of action. It is fleshed out by providing an analysis of the notions of intention—the elementary components of plans—and practical reasoning—the process by which plans are constructed—in the context of proof activities. These two notions are then used to offer a precise characterization of the desired notion of plan for proof activities. A fruitful connection can then be established between the resulting framework and the recent theme of modularity in mathematics introduced by Jeremy Avigad. This connection is exploited to yield the concept of modular presentations of mathematical proofs which has direct implications for how to write and present mathematical proofs so as to deliver various epistemic benefits. The account is finally compared to the technique of proof planning developed by Alan Bundy and colleagues in the field of automated theorem proving. The paper concludes with some remarks on how the framework can be used to provide an analysis of understanding and explanation in the context of mathematical proofs.

Reliability of mathematical inference

“…They and other authors have therefore begun to explore the broader epistemic benefits of informal proof [4,5,42,55,62]. It is even reasonable to maintain that philosophy of mathematics has suffered from excessive fixation on proof itself, and a number of philosophers of mathematics are turning their attention to aspects of mathematical understanding that extend well beyond justification and correctness [3,20,21,22,23,27,36,37,38,46,49,50,57,58,59,65].…”

“…Morris has argued that the kinds of proofs we value tend to be motivated, which is to say, they convey information as to how individual steps are suggested by previous ones and how they constitute progress towards the final goal [49]. Drawing on philosophical theories of planning and agency by Bratman [17], Hamami and Morris have argued that the comprehensibility and effectiveness of an informal proof requires us to interpret them as instantiations of a rational plan [36]. With these insights, we can start to be more specific about the features of informal proof that convey understanding and support higher-level inference.…”

On the Epistemological Relevance of Social Power and Justice in Mathematics

“…Epistemologies of mathematics currently on offer in the philosophy of mathematics literature largely focus on questions of validity, rigour, and correctness of proofs, that is on stage 3 (e.g., Manders 2008). Even explicitly agent-focussed accounts, such as Hamami and Morris (2020), concentrate on these stage 3 questions. The exploration of the relevance of stages 1 and 2 to the epistemology of mathematics remains an under-researched topic in the field.…”

Reading Mathematical Proofs as Narratives

“…Hence, a written proof is a telling of how something happened. A written proof is a telling of a sequence of actions performed on mathematical objects by an agent with the aim of proving a given proposition (in line with Hamami and Morris 2020). This implies that a proof as written is a narrative in at least a minimal sense.…”