The normativity of the formal 

Paul Ernest (U Exeter)

Jordi Fairhurst (UIB)

José Antonio Pérez-Escobar (UNED) 

Sander Pouliart (VUB)

Colin Jakob Rittberg (VUB)

Participants: 

Igor Roberto Amorim
Hakob Barseghyan (Victoria College, University of Toronto)
Ambra Benvenuto (IULM University)
Antonella Bilotta (Scuola Normale Superiore di Pisa, Italy)
Fabio Ceravolo (Sichuan University)
Michalis Christou (Institute of Philosophy and Scientific Method, Johannes Kepler University, Linz)
Alejandro Chmiel (Udelar (Uruguay))
Emillie de Keulenaar (University of Copenhagen)
Atahan Erbas (KU Leuven)
Konsman Jan Pieter (CNRS)
Henry Kalter (University of Amsterdam (ILLC))
Miguel Angel León Untiveros (Universidad Nacional Mayor de San Marcos)
Gareth Norman (Massachusetts Institute of Technology)
Tuomas Pernu (University of Eastern Finland)
Simone Paolo Roca (IUSS Pavia)
Maximino Robles
Andrei Rodin
Ainu Syaja (University of Liverpool)
Fatih Taş (Bartın University)
Natalia Tomashpolskaia
Robert Thomas (St John's College Winnipeg)
Kulesh Vandan
Helena Winiger (ETH Zurich)

And 19 who chose not to be listed

Semiotics and the Normativity of the Formal by Paul Ernest (University of Exeter)

From a semiotic perspective the normativity of the formal can be seen as part of three nested sets of language games.

First, at the macro pragmatics level, the formal has a normative function in that it provides a template for how knowledge should be structured and represented. Euclid’s Elements stood for over two thousand years as the model and formal archetype for how knowledge should be presented. With applications from Newton’s Principia, Spinoza’s Ethics to Aleister Crowley’s Black Magick the formal logical structure of axioms-postulates-methods-lemmas-theorems is the normative paradigm for soundly structured knowledge. The formal is normative in a metaphorical way, organising the rhetoric of knowledge presentation.

Second, at the semantic level, the formal sciences were once regarded as objective and metaphysically necessary truths of mathematics and logic. However, these are now understood by many to rest on norms, customs and rules (Wittgenstein 1953). This is the central and most important meaning of the normativity of the formal. Within the texts and formal proofs of mathematics and the formal sciences (at the semantic level) imperatives and commands are the main verbs whether direct (add, integrate, infer) or indirect (let us …, consider) making the statements injunctions, deontologically based orders,  rather than assertions or declarative statements of truth

Third, at the syntactic level, the normative is present in the formal rules for forming and transforming strings of symbols. Uninterpreted formal representations of theories as symbol strings and rules for operating on them, enable the derivation of strings with certainty and without ambiguity. The fully formal counterpart of Peano’s axioms and theory permits the derivation of the new string 2+2=4. This is a necessary outcome of the formal system and can be interpreted at the semantic level as a truth (based on the system’s assumptions). Ultimately  the nnormativity of the formal at the syntactic level is the basis for the knowledge and results on the semantic and pragmatic levels, that is the philosophical level. 

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Mathematical normativity outside mathematics: descriptions and explanations by José Antonio Pérez-Escobar (UNED)

The later Wittgenstein made the case that mathematics are rules of description of empirical phenomena, and thus have normative power. This aspect of his philosophy of mathematics not only employs toy examples but also, in contrast to the rest of his philosophy and even his philosophy of pure mathematics, remains under-explored. In this talk, I will 1) discuss how Wittgenstein’s view of mathematics as rules of description apply to contemporary applied mathematics in the sciences, and 2) how the normative power that Wittgenstein had in mind also applies to mathematical explanations outside mathematics (the subject of another contemporary debate in the philosophy of science), thus constituting rules of explanation. 

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The Social Life of Mathematical Norms by Sander Pouliart (VUB)

This talk examines mathematical normativity from a sociological standpoint, drawing on the contrasting accounts of David Bloor, Bettina Heintz, and Sal Restivo. Each challenges the idea of mathematics as a purely logical or self-justifying discipline: Bloor treats mathematical objectivity as socially stabilised through communal norms, Heintz views mathematics as an autonomous communication system whose norms enable mutual understanding; and Restivo sees mathematics as a collective representation shaped by cultural and historical forces. Together, these perspectives reveal mathematical necessity as a social achievement, its authority grounded not in transcendence, but in shared human practices that define what counts as truth and proof.

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Normativity beyond the formal: exploring non-idealities in mathematical knowledge-making by Colin Rittberg (VUB)

This talk side-steps the question of the workshop. Rather than asking whether the mathematical standards of normativity are grounded in objective necessities or human practices, I argue that the formal is not the only, not even the always-trumping, normative force in mathematics. To do so, I explore some of the various real-world and messy factors that impact which bit of mathematics "makes it". The result is the beginning of an epistemology of mathematics which, even under strong assumptions about the objectivity of the normative force of mathematical proof, reveals that mathematical knowledge is situated and context-dependent. 

 FWO-project "The Epistemology of Big Data: Mathematics and the Critical Research Agenda on Data Practices" 

This Event is endoresd by the Young Network for Wittgensteinian Philosophy