Intuition and surveyability in mathematical proof: a Franco-German perspective

Voici quelques résumés :

• Daniel Forster (University of Jena) - "Wittgenstein’s Notion of ‘Perspicuous Representation’ in his Method of Conceptual Clarification and Elucidation: The Example of the Family Tree of Psychological Concepts”

In this talk I try to get a grasp of what Wittgenstein could have meant by ‘perspicuous representation’ (or: ‘surveyable representation’; ‘übersichtliche Darstellung’) in the context of his later philosophical method of conceptual clarification and elucidation.

Although he describes the notion of ‘perspicuous representation’ as “of fundamental significance for us” (PU 122), he rarely tells us explicitly what exactly he means by it. Given their importance, it is surprising how few examples he gave. In fact, the only explicit example in his writings (excluding VoW) is the colour octahedron in a remark from 1930 (Ms 108:89). But this has barely anything to do with what he himself practiced during the development of the Philosophical Investigations and afterwards.

I will first try to outline the characteristic features of what he could have meant by ‘perspicuous representation’ in referring to central remarks in this context. Then I will present the most and only elaborate example Wittgenstein gave of a perspicuous presentation as a tool for conceptual elucidation. Although he never explicitly called it that, it is his attempt to create a family tree of our psychological concepts (see RPP I 722, RPP II 62, 148). But, as I will show, in the course of his examination he came to the conclusion that such large-scale overviews are heuristically futile.

This leaves him with what he already did before such an endeavor: describing the use of our concepts by, for example, pointing out similarities and differences or stating elusive grammatical rules for the purpose of conceptual clarification. On a charitable reading, such descriptions could be called perspicuous representations. Nevertheless, it should be noted that he does not return to this way of speaking about perspicuity and representability afterwards. So it seems more reasonable to treat them as part of the way to achieving those perspicuous representations. A goal which he ultimately abandoned.

It has been claimed that perspicuous representations share some important similarities with proofs as they can be found in mathematics. At the end of my talk, I will make a brief comparison between what has already been presented and Wittgenstein’s notion of the ‘surveyability’ of mathematical proofs in RFM.

• Vincenz Nitschke (University of Jena) - “Hertz’s Principles lacks examples”

In the Introduction to his Principles of Mechanics (1894), Hertz announced to eliminate contradictions in the notion of force and to render the whole body of mechanics more limpid to (trained) physicists. Once the new representation of mechanics has been understood, people will cease to ask the illegitimate question: “What is force?”

What contradictions does Hertz mean? And what difference do his modifications make? In contrast to previous publications, my talk will not follow Hertz’s own entanglement in analytical mechanics, but will answer these questions with regard to a concrete physical example, i.e. the swinging stone.

• Reinhold Schwenzer (University of Jena) - "Did Ludwig Wittgenstein Shape Hans Hahn’s Philosophy of Mathematics?”

Hahn described the Tractatus as “the most important contribution to philosophy since the publication of Russell’s basic writings.” Gödel states that “around 1930 [...] H. Hahn, largely under the influence of L. Wittgenstein, developed a view about the nature of mathematics […].” 

However, Goldfarb (1996) suggests that the overlap between Wittgenstein and the Vienna Circle in the 1930s was largely terminological, masking important differences. Instead, he argues that their views are best understood as the adoption of a broadly logicist agenda. This paper reexamines Hahn’s position around 1930 by emphasizing affinities with Wittgenstein’s philosophy of mathematics (most notably the view that mathematics concerns language and is best understood as a logical method) while reconsidering the differences (most notably their respective understandings of tautologies, Hahn’s claim that mathematics is reducible to logic, and his critique of intuition in mathematics).

• Manon Prost (Université de Strasbourg) - "The Status of the Equation in Wittgenstein’s Tractatus”

In the Tractatus, Wittgenstein draws a strict distinction between meaningful propositions and nonsense. Yet the status of mathematical equations within this framework is unclear. It seems implausible to treat equations such as 2×2 = 4 as nonsensical, but they cannot easily be understood as meaningful propositions either, since a meaningful proposition must represent a possible state of affairs whose negation is conceivable.

This paper examines Wittgenstein’s treatment of equations in connection with his conception of number as the exponent of an operation (TLP 6.02). I argue that equations do not assert the equality of two objects or meanings. Rather, they express formal properties of relations between structures and display the substitutability of expressions within a calculus.

On this view, equations neither describe states of affairs nor function as logical tautologies. Instead, they belong to the domain of formal relations that mathematics reveals. This interpretation clarifies Wittgenstein’s critique of Frege and Russell and highlights the role of equations in exhibiting the formal structure of logic.