A Note on the Relation Between Formal and Informal Proof

Sjögren, Jörgen

doi:10.1007/s12136-009-0084-y

Cited by 10 publications

(8 citation statements)

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“…proved that no formal theory extending Robinson arithmetic admits a provability predicate for which the reflection schema and Nec (if ' φ then ' Pð⌜φ⌝Þ) hold. So, at least prima facie, if we wanted to axiomatise our intuitions about 1 See (Antonutti Marfori, 2010;Rav, 1999;Sjögren, 2010) for a detailed discussion of these issues.…”

Section: Motivationsmentioning

confidence: 99%

“…(In Chapter 5 we will consider further the procedure for embedding mathematics in set theory.) However, many philosophers argue that these two notions of a proof are substantially different (Antonutti Marfori, 2010;Horsten, 1996;Leitgeb, 2009;Myhill, 1960;Rav, 1999;Rav, 2007;Sjögren, 2010;Tanswell, 2015). The standard view, they insist, does not fully explain why informal proofs are quite good at convincing mathematicians, whereas formal ones are not.…”

Section: Motivationsmentioning

confidence: 99%

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Informal provability and dialetheism

Pawlowski

Urbaniak

2023

Theoria

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According to the dialetheist argument from the inconsistency of informal mathematics, the informal version of the Gödelian argument leads us to a true contradiction. On one hand, the dialetheist argues, we can prove that there is a mathematical claim that is neither provable nor refutable in informal mathematics. On the other, the proof of its unprovability is given in informal mathematics and proves that very sentence. We argue that the argument fails, because it relies on the unjustified and unlikely assumption that the informal Gödel sentence is informally provable.

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Section: Motivationsmentioning

confidence: 99%

Section: Motivationsmentioning

confidence: 99%

Informal provability and dialetheism

Pawlowski

Urbaniak

2023

Theoria

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“…Let me stress that one should not expect the mid-level step of the sharpening of the clarified explicandum to occur in every explication whatsoever, but only in certain complex cases where many different formal explicata are meant to replace a single explicandum. Apart from the CTT case, other examples of concepts that may exhibit different sharpenings of the same clarified explicandum and for which the threestep explication seems an appropriate tool of analysis are formal theories of truth (Horsten 2011;Halbach 2014), different conceptions of set (Incurvati 2020), theories of informal proofs (Leitgeb 2009;Sjögren 2011), notions of logical consequence (Etchemendy 1990), and mathematical conceptions of infinity (Mancosu 2009).…”

Section: Explication As a Three-step Procedure: The Semi-formal Sharpening Of The Clarified Explicandummentioning

confidence: 99%

Explication as a Three-Step Procedure: the case of the Church-Turing Thesis

Benedetto

2021

Euro Jnl Phil Sci

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In recent years two different axiomatic characterizations of the intuitive concept of effective calculability have been proposed, one by Sieg and the other by Dershowitz and Gurevich. Analyzing them from the perspective of Carnapian explication, I argue that these two characterizations explicate the intuitive notion of effective calculability in two different ways. I will trace back these two ways to Turing’s and Kolmogorov’s informal analyses of the intuitive notion of calculability and to their respective outputs: the notion of computorability and the notion of algorithmability. I will then argue that, in order to adequately capture the conceptual differences between these two notions, the classical two-step picture of explication is not enough. I will present a more fine-grained three-step version of Carnapian explication, showing how with its help the difference between these two notions can be better understood and explained.

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“…5 A representative list of the 1 We shall not discuss in this paper whether, and in which sense, formal proof constitutes the mathematical ideal of proof, nor shall we discuss whether there could be a plurality of mathematical ideals of proof. 2 We refer the reader to the following list of references: (Myhill, 1960), (Kreisel, 1967(Kreisel, , 1970(Kreisel, , 1981, (Corcoran, 1973), (Feferman, 1979(Feferman, , 2012, (Robinson, 1991(Robinson, , 1997(Robinson, , 2000(Robinson, , 2004, (Detlefsen, 1992a(Detlefsen, ,b, 2009, (Thurston, 1994), (Rav, 1999(Rav, , 2007, (Azzouni, 2004(Azzouni, , 2009(Azzouni, , 2013, (Bundy, Atiyah, Macintyre, & MacKenzie, 2005), (Bundy, Jamnik, & Fugard, 2005), (Suppes, 2005), (Aberdein, 2006), (Avigad, 2006(Avigad, , 2008, (Cellucci, 2008), (Leitgeb, 2009), (Goethe & Friend, 2010), (Sjögren, 2010), (Larvor, 2012), (Tanswell, 2015), (Burgess, 2015), and (Weir, 2016). 3 See, for instance, Kitcher: "I do not intend to deny that much mathematical knowledge is gained by constructing or following the sequences of statements contained in mathematics books and labelled "proofs."…”

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confidence: 99%

Mathematical Inference and Logical Inference

Hamami

2018

The Review of Symbolic Logic

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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.

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A Note on the Relation Between Formal and Informal Proof

Abstract: Using Carnap's concept explication, we propose a theory of concept formation in mathematics. This theory is then applied to the problem of how to understand the relation between the concepts formal proof (deduction) and informal, mathematical proof.

Cited by 10 publications

References 32 publications

Informal provability and dialetheism

Informal provability and dialetheism

Explication as a Three-Step Procedure: the case of the Church-Turing Thesis

Mathematical Inference and Logical Inference

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