Mathematical Inference and Logical Inference

Abstract

The deviation of mathematical proof—proof in mathematical practice—from the ideal of formal proof—proof in formal logic—has led many philosophers of mathematics to reconsider the commonly accepted view according to which the notion of formal proof provides an accurate descriptive account of mathematical proof. This, in turn, has motivated a search for alternative accounts of mathematical proof purporting to be more faithful to the reality of mathematical practice. Yet, in order to develop and evaluate such alternative accounts, it appears as a necessary prerequisite to first possess a clear picture of what the deviation of mathematical proof from formal proof consists in. The present work aims to contribute building such a picture by investigating the relation between the elementary steps of deduction constituting the two types of proofs—mathematical inference and logical inference. Many claims have been made in the literature regarding the relation between mathematical inference and logical inference, most of them stating that the former is lacking properties that are constitutive of the latter. Such differentiating claims are, however, usually put forward without a clear conception of the properties occurring in them, and are generally considered to be immediately justified by our direct acquaintance, or phenomenological experience, with the two types of inferences. The present study purports to advance our understanding of the relation between mathematical inference and logical inference by developing a detailed philosophical analysis of the differentiating claims, that is, an analysis of the meaning of the differentiating claims—through the properties that occur in them—as well as the reasons that support them. To this end, we provide at the outset a representative list of the different properties of logical inference that have occurred in the differentiating claims, and we notice that they all boil down to the three properties of formality, generality, and mechanicality. For each one of these properties, our analysis proceeds in two steps: we first provide precise conceptual characterizations of the different ways logical inference has been said to be formal, general, and mechanical, in the philosophical and logical literature on formal proof; we then examine why mathematical inference does not appear to be formal, general, and mechanical, for the different variations of these notions identified. Our study results in a precise conceptual apparatus for expressing and discussing the properties differentiating mathematical inference from logical inference, and provides a first inventory of the various reasons supporting the observations of those differences. The differentiating claims constitute thus a set of data that any philosophical account of mathematical inference and proof purporting to be more faithful to mathematical practice ought to be able to accommodate and explain.

ProofvsTruth in Mathematics

“…They differ by the properties of those steps. According to Hamami [10] one can distinguish here three types of differences: formality, generality and mechanicality. Informal inferences are meaning dependent, matter dependent and content dependent wheras formal inferences are meaning, matter and content independent.…”

“…Hamimi [10] explains that the claim that logical inference is general means in particular that "it is governed by rules of inference that are generally applicable, i.e., that are applicable to propositions -premisses and conclusions -belonging to any and every topic, subject matter, or domain" [10, pp. 684-685].…”

Plans and Planning in Mathematical Proofs

“…On deductive inferences in ordinary mathematical practice, seeHamami (2018Hamami ( , 2019.3 We assume here that the agent under consideration is indeed able to carry out the deductive inferences corresponding to the deductive steps in the considered mathematical proof, that is, the agent belongs to the intended audience of the written mathematical proof. For an in-depth discussion of the notion of proof as audience-relative, seeCorcoran (1989).…”

Natural Formalization: Deriving the Cantor-Bernstein Theorem in Zf

“…This topic is discussed in greater detail in §5 and Appendix B. 6 In Paulson's early work that uses set theory as a frame for verification, "natural deduction rules" are introduced for all the operations in a rather ad-hoc fashion, not through a uniform mechanism connecting the rules to defining axioms of definitional extensions.-In his (Hamami, 2018), Hamami attempts to make a principled distinction between mathematical and logical inferences. We think, however, that the "mathematical inferences" he discusses, for example, in §3.3 of his article, can all be recast as applications of our uniform lemmas-as-rules mechanism.…”