Deducing false propositions from true ideas: Nieuwentijt on mathematical reasoning
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Deducing false propositions from true ideas: Nieuwentijt on mathematical reasoning Sylvia Pauw1,2 Received: 22 November 2017 / Accepted: 17 September 2018 © The Author(s) 2018
Abstract This paper argues that, for Bernard Nieuwentijt (1654–1718), mathematical reasoning on the basis of ideas is not the same as logical reasoning on the basis of propositions. Noting that the two types of reasoning differ helps make sense of a peculiar-sounding claim Nieuwentijt makes, namely that it is possible to mathematically deduce false propositions from true abstracted ideas. I propose to interpret Nieuwentijt’s abstracted ideas as incomplete mental copies of existing objects. I argue that, according to Nieuwentijt, a proposition is mathematically deducible from an abstracted idea if it can be demonstrated that that proposition makes a true claim about the object that idea forms. This allows me to explain why Nieuwentijt deems it possible to deduce false propositions from true ideas. It also implies that logic and mathematics are not as closely related for Nieuwentijt as has been suggested in the existing secondary literature. Keywords Nieuwentijt · Mathematical deduction · Logical deduction · Abstraction
1 Introduction In his 1720 work on the nature of pure and mixed mathematics, Grounds of Certainty1 (Gronden van zekerheid), the Dutch philosopher Bernard Nieuwentijt suggests that he regards logic and mathematics as intimately related. In a chapter that is concerned with logic, Nieuwentijt claims to show “[…] that true Logicians […] only differ from the true Mathematicians in the outer Ways [uiterlyke Omstandigheden] of expressing 1 Translation adopted from Ducheyne (2007). Throughout this paper, I use GC as abbreviation for Grounds of Certainty (i.e. Nieuwentijt 1720), and use it to give references to page numbers of this work.
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Sylvia Pauw [email protected]
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Department of Political Science, University of Amsterdam, PO box 15578, 1001 NB Amsterdam, The Netherlands
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Department of Philosophy and Moral Sciences, Ghent University, Blandijnberg 2, 9000 Ghent, Belgium
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Synthese
their proofs, yet not at all in the force of Arguments”2 (GC, p. 218). He also claims to show that “[…] the true mathematicians agree with the logicians, and do not fundamentally differ [in den gront niet verschillen]”3 (GC, p. 204). Nieuwentijt’s remarks have led Ducheyne (2017b, p. 287n.) to claim that Nieuwentijt regards logic as a part of pure mathematics. Petry suggests that mixed mathematics forms, to a certain extent, a “branch of logic” (1979, p. 6) for Nieuwentijt.4 Beth (1954, pp. 451–452) even compares Nieuwentijt’s position to logicism. In this paper, I argue that Nieuwentijt does not identify mathematical reasoning with logical reasoning, and that the existing accounts of his views on the relationship between logic and mathematics cannot be correct. In his analysis of the nature of mixed mathematics, Nieuwentijt claims that it is possible to mathematically deduce false propositions from true abstracted ideas.5 An abstracted i
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