Wittgenstein and the formal sciences 2

Wittgenstein and the Liar Paradox
by Hanoch Ben-Yami

Abstract: What you’re now reading isn’t true! But it isn’t just in case it is… How should we remove this paradox? – Should we remove it at all? What harm can it do, this singularity in our language? – “The real harm comes in when there’s an application, in which case a bridge may fall down or something of that sort.” – But it won’t come in if we can quarantine these ‘ungrounded’ statements. We should better say, there is the liar sentences phenomenon, not paradox, a fact about our language, which is not a problem and needs no resolution.

Remarks on Tractarian Mathematics
by Jean Paul van Bendegem 

Abstract: For many philosophers, Wittgenstein in his Tractatus Logico-Philosophicus has little to say about mathematics, apart from arithmetic, that is at least spelled out in some detail (and further refined by, e.g., Pasquale Frascolla). The mere fact that mathematical theories as theories are rejected poses the problem how mathematics can go beyond arithmetic. Although the shared opinion is that this task is to be considered hopeless, I will defend the opposite thesis. In addition, the conception of mathematics that I propose, stresses the continuity with Wittgenstein’s later thoughts, as expressed in the Remarks on the Foundations of Mathematics.  

Mathematical normativity: dealing with counterexamples in the scientific practice

by José Antonio Pérez Escobar

Abstract: This talk will explore how the later Wittgenstein’s view of mathematics applies to mathematical models in practice. First, I will argue for the exegesis that Wittgenstein viewed mathematics as having two aspects: an empirical and a normative one. Which aspects trumps the other depends on practical nuances. I will show this via a rational reconstruction of Euler’s conjecture for polyhedra. Second, I will present a case study from neuroscience showing how mathematical models can sometimes be challenged by empirical exemplars but may also dictate how measurement is to be carried out so that exemplars conform to the mathematics.

The Computational Effectiveness of Diagrams

by John Mumma 

Mathematical proofs are often presented with notation designed to express information about concepts the proofs are about.  How is such mathematical notation to be understood, philosophically, in relation to the proofs they are used in? 

A useful distinction in approaching the question is that between informational and computational equivalence, formulated by Larkin and Simon in Simon and Larkin's 'A Diagram is (Sometimes) Worth 10,000 Words.'.  Informational equivalence holds between two representations if and only if ``all of the information in the one is also inferable from the other, and vice versa.''  Computational equivalence holds between two representations if and only if  ``they are informationally equivalent and, in addition, any inference that can be drawn easily and quickly from the information given explicitly in the one can also be drawn easily and quickly from the information given explicitly in the other, and vice versa.''   Applying this to the case of mathematical proofs, we can take informational equivalence to be determined by the content of proofs as revealed by a logical analysis, and computational equivalence to be determined by the notations used to express the content of proofs.  

In what is perhaps the standard view, we need only consider mathematical proofs modulo informational equivalence in developing a philosophical account of them.  The capacity of a notation to present a proof more clearly and effectively is a pragmatic matter, something perhaps for psychologists to consider, but not philosophers.  An alternate view, in line with philosophical work attempting to illuminate mathematical practice, regards distinctions of computational nonequivalence as philosophically significant.  On such a view, the structure of mathematical methods and theories is inextricably linked with the way the mind takes in and holds mathematical information.

In my talk, I aim to make some small steps in elaborating the second view by applying some observations made by Wittgenstein in Remarks on the Foundations of Mathematics to the diagrammatic proofs of elementary geometry.  A central challenge in elaborating the view is being precise about what, exactly, the effectiveness of effective mathematical notations amounts to.  The notion of informational equivalence has the well developed philosophical resources of logic to support it.  No such philosophical resources exist for the notion of computational equivalence.  The only option available for investigating it at present would seem to be a bottom-up, case study approach.  Accordingly, one looks at cases that dramatically illustrate computationally nonequivalent notations  and aims to articulate the features that account for the nonequivalence.  Wittgenstein can be understood to be doing exactly this in section III of RFM.  The cases that figure in his discussion are proofs of arithmetic identities like 27 +16 =43.  He in particular contrasts calculations of such identities using Arabic numerals with derivations of them using a purely logical notation, and identifies features that recommend the former over the latter from the perspective of computational effectiveness.  After presenting Wittgenstein's observations, I contrast diagrammatic and purely sentential proofs in elementary geometry and argue that analogous observations distinguish the former as computationally effective.

Arithmetic as Symbolic Knowledge: notes for a Leibnizian reading of Wittgenstein's Tractatus
by Gisele Secco 

Abstract: Tracing the Leibnizian lineage of Wittgenstein's Tractatus Logico-Philosophicus does not constitute any hermeneutic novelty. Hidé Ishiguro [1], for example, has pointed out various Leibnizian themes in the TLP: the central aspects of the doctrine of logical atomism; the remarks on the nature of propositional representation and its consequences for the place and role of logic in the businesses of human knowledge of the world, or even the idea that the superficial grammar of our language does not reflect its philosophical grammar (or logical form).

But when it comes to the Leibnizian ancestry of Wittgenstein's early remarks about mathematics, experts do not say enough. Even the best readings of the TLP’s relevant passages do nothing but mention Leibniz en passant – while the connection between these passages and the notion of symbolic knowledge, so dear to Leibniz and so critical for a range of subsequent philosophies of logic and mathematics, including Wittgenstein’s, is hardly mentioned (with the exception of Sören Stenlund [2]).

Indeed, as we have shown in [3] when writing about the symbolic constructions of arithmetic, the young Wittgenstein puts in motion (rather than explicitly asserts) the most outstanding features of Leibniz’s concept of Symbolic Knowledge, its computational and expressive (or “ecthetic”) functions. In my talk, after presenting a few conceptual distinctions typical of Leibniz’s characterizations of Symbolic Knowledge, I discuss Pasquale Frascolla’s reading of those same Tractarian passages on arithmetic (mostly in [4] and [5]). My main aim is to show that, when illuminated by the Leibnizian distinctions previously presented, one can not only make a better sense of the Tractarian aphorisms on arithmetic within the general economy of the treatise but also offer a renewed and didactically fruitful way of telling the story of Wittgenstein's reception and reshaping of his predecessors' logicism.

[1] Hidé Ishiguro, Leibniz’s Philosophy of Logic and Language. (London: Duckwork, 1972).

[2] Sören Stenlund, The Origin of Symbolic Mathematics and the End of the Science of Quantity. (Uppsala: Uppsala Universität, 2012).

[3] Gisele Secco & Pedro Noguez, “Operar e Exibir: Aspectos do Conhecimento Simbólico na Filosofia Tractariana da Matemática.” Revista Portuguesa de Filosofia 73, 3-4 (2017), 1463-1492. DOI 10.17990/RPF/2017_73_3_1463

[4] Pasquale Frascolla, Wittgenstein’s Philosophy of Mathematics. (London/New York: Routledge, 1994).

[5] Pasquale Frascolla, “The Tractatus System of Arithmetic”. Synthese 112, n.3 (Sep./1997), 353-378.

The Idea of “the right ethics”. Pursuing Wittgenstein’s Suggested Analogy to Logic and Mathematics.
by Vincent Vincke 

Abstract: Wittgenstein wrote very little on ethics during his lifetime, and the few remarks he made on the matter are notoriously scarce and dense. As a result, the scholarship surrounding his (meta)ethical thought has inevitably been plagued by issues concerning both the assessment of its continuity and its reconstruction. One of the very few glimpses we get of his thoughts on the matter appear in Rush Rhees’s (2015) transcription of a conversation both had on September 12, 1945. In this conversation Wittgenstein suggests an analogy between the “arbitrariness of mathematics and logic”, and the arbitrariness of ethics. The aim of this talk is to explore this suggested analogy.

Wittgenstein, Mathematics and Conceptual Engineering  
by Wei Zeng 

Abstract: Although revising and replaying concepts is not a new method in the history of philosophy, it is only recently that conceptual engineering became a popular independent field in philosophy. In the literature on conceptual engineering, in addition to the general discussion of the conceptual engineering approach (Cappelen 2018), there are also discussions about projects of revising or replacing specific concepts, such as the concept of “woman” (Haslanger 2000), “truth” (Scharp, 2013). However, there is still little literature on whether mathematical concepts can be engineered. Tanswell (2018) argued that mathematical concepts are open-textured and therefore, could be engineered, but she didn’t fully explore the practical issues of how to engineer mathematical concepts. Following Tanswell, I examine the possibility of applying conceptual engineering method to mathematics from a practice-oriented perspective. Moreover, I argue that Wittgenstein’s philosophy of mathematics, in particular, his account of mathematical concept formation and conceptual change, provides an insightful view to this discussion. I propose that although Wittgenstein might share Cappelen’s anxiety that from a semantic externalist’s position, we don’t have sufficient control over the meaning of concepts, Wittgenstein would have a more optimistic view, a view closer to Pinder (2021), instead of aiming at a change of semantic meaning, starting from a change of speaker-meaning should also be considered as a step of conceptual engineering.

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