Wittgenstein and the formal sciences 3

Wittgensteins’s Radical Conventionalism in the Lectures on the Foundations of Mathematics.

Ásgeir Berg

Dummett’s reading of Wittgenstein as a radical conventionalist is well known. However, Dummett has been understood as claiming that for Wittgenstein, each mathematical theorem is a deliberate choice.
In this talk, I will argue that (a) this is a superficial reading of Dummett and (b) that the real definition of radical conventionalism is best seen through comparing it with more moderate forms where we lay down axioms and derive theorems via inference rules. The radical version, in comparison, sees each mathematical truth as a result of a convention, and not as consequences of a prior choice.
I give a number of examples from the Lectures on the Foundations of mathematics that support this reading.

Logic and Conventions
Kai Michael Buttner and Hans-Johann Glock

Wittgenstein and the logical positivists attempted to explain logical necessity in terms of linguistic conventions. It is often thought, however, that their respective accounts have been conclusively refuted by objections from Quine, Dummett and others. In this paper, we shall argue that this verdict is premature. Several of the most popular anti-conventionalist arguments fail, partly because they misconstrue the positivists’ notion of truth by convention. Correctly understood, the claim that certain sentences are true by convention is difficult to deny, and the corresponding conventionalism about logical necessity remains a viable position.

Against a Global Conception of Mathematical Hinges: A Holistic Reading of Wittgenstein and Mathematics in Practice 

 Jordi Fairhurst, José A. Perez-Escobar, and Deniz Sarikaya 

Rule following paradox and domain expansion in mathematics
Mikkel Willum Johannsen

 In his book Wittgenstein on Rules and Private Language Saul Kripke gives his own account of Wittgenstein’s rule following paradox. Kripke’s account and suggested solution to the paradox has attracted much debate and criticism.  

In his account of the rule following paradox Kripke draws on relatively basic mathematical examples, primarily examples concerning addition of pairs of natural numbers. In such examples it may seem counterintuitive that the rule does not directly apply to new instances. There are however other examples where the application of the rule to new examples is less clear. These examples typically involve what can be called domain expansion for a mathematical operation. For addition the move from adding a finite to adding an infinite number of addends is a case of domain expansion where it is less clear how the rules of addition are to be applied to the new instances and where mathematicians historically have disagreed about how to proceed. In this paper I will analyze and discuss if and in what way there is a qualitative difference between the simple examples used by Kripke and examples involving domain expansion. Additionally, I will explore how domain expansion scenarios impact our understanding of the rule following paradox.

Wittgenstein and Set Theory 

Felix Mühlhölzer

Wittgenstein was hostile towards set theory; see his remark in §22 of RFM II on set theory’s extensional point of view: “I believe and hope that a future generation will laugh at this hocus pocus.” At the same time, he says in a manuscript that what philosophy owes set theory is ‘tremendous’, and that this is something deep. I want to clarify these two statements and to reconcile them. The hocus pocus remark is mainly directed at the temptation to talk about an own world of sets, focussing on extensions and thereby forgetting the actual mathematical maneuvers leading to them, comparable, as Wittgenstein says, to a magician who performs his tricks in front of a mirror such that he sees them like his audience. Wittgenstein does not shy away from using set theoretical symbols and expressions, but he regularly reinterprets them, depriving them of their genuine set theoretical character. So, he rather often uses the symbol “ℵ0” but understands it in the everyday sense of an unlimited technique. He even uses “2א0” and the formula that 2א0 is greater than ℵ0, but in this case he refers to Cantor’s diagonal method which he interprets as a rather elementary technique, and he interprets the concept of uncountability not simply as non-countability but as encompassing the diagonal method. According to Wittgenstein, uncountability in the sense of non-countability is not, as understood in set theory, the general concept which is then applied via the diagonal method to the real numbers, and so on, the generality involved rather resides in the different applications that the concept of uncountabiliy has within mathematics. This is Wittgenstein’s view of generality in mathematics through and through; as he says: “the general calculus is ‘general’ only by referring to the special calculi”. When speaking of the ‘depth’ we encounter with regard to set theory, what he means is the depth explained in PI §111: it arises “through a misinterpretation of our forms of language”, and this is in particular so in set theory.

POSTPONED TO 2025: Wittgenstein and Mathematical Understanding

Wei Zeng

The nature of understanding is one of the central topics of Wittgenstein’s Philosophical Investigation. In contrast to conceiving understanding as a mental process, Wittgenstein urges us to shift our focus from deciphering the mental state of the 'Aha moment' to observing the circumstances under which we are convinced that we know how to go on. Avigad (2008) extends this perspective of Wittgenstein in PI to his discussion of mathematical understanding, and proposes that mathematical understanding should be understood in terms of a group of abilities. However, in Remarks on the Foundation of Mathematics, Wittgenstein presents a more nuanced view of mathematical understanding. In this talk, I argue that while the central idea of Wittgenstein’s conception of mathematical understanding is that in order to claim one understands a mathematical proposition or proof one must have a clear view of its application, his account of mathematical understanding has a layer of complexity not present in his general account of understanding. Moreover, I posit that Wittgenstein’s account challenges the view that understanding is merely the consequence of explanation.