Truth and Meaning in Mathematics

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Speakers:

Sorin Bangu (University of Bergen)

Hanoch Ben-Yami (Central European University)
Ásgeir Berg Matthíasson (University of Iceland) 

Silvia De Toffoli (Scuola Universitaria Superiore IUSS Pavia)
Jordi Fairhurst Chilton (Universitat de les Illes Balears)

Andrea Guardo (University of Milan)

Casper Storm Hansen

José Antonio Perez-Escobar (UNED) 

Matteo Plebani (University of Turin) 

Andrei Rodin (University of Lorraine, Archives Henri-Poincaré and Loria)
Deniz Sarikaya (Universität zu Lübeck & Vrije Universiteit Brussels)

Titles and Abstracts:

Sorin Bangu (University of Bergen):  Can there be mathematical knowledge without proof?
This talk answers the question in its title in the negative. The argument exploits the idea that a proof of p removes the possibility that there are counterexamples to p — and this is, we take it, what it means to say that p is necessarily true, a key feature of a mathematical proposition. The paper then turns to discussing the reliability of the methods of proof in mathematics.

Ásgeir Berg Matthíasson (University of Iceland): TBA

Hanoch Ben-Yami (Central European University) False Paradise: On the Mathematical Infinite

The view of the different infinities as differing in size is unjustified, a result of transferring comparisons from the finite domain, where they hold, to the infinite one, where they make no sense. We better think of the so-called ‘larger’ infinities as being more unruly, of infinities as increasing not in size but in unruliness. 

Silvia De Toffoli (Scuola Universitaria Superiore IUSS Pavia)Articulate Intuition (joint work with Elijah Chudnoff)

A common thought about intuition and inference is that they have contrasting epistemologies: intuition, like perception, purports to immediately justify belief, while inference is a way of basing a belief on supporting considerations. This alignment of intuition with perception in opposition to inference encourages the idea that intuitions cannot be fruitfully shaped by rational reflection, and this in turn fuels various skeptical challenges to reliance on intuition in mathematics (and philosophy). In this talk, we argue that some intuitions, which we call articulate intuitions, share key epistemic benefits associated with inference while continuing to provide immediate justification like perception.  In particular, we show that intuitions have a proper place in mathematical practice.

We show that the opposition between intuition and inference is a false opposition.  We agree that intuition, unlike inference, provides immediate justification. However, we explore other ways in which intuition is, at least in some cases, epistemically unlike perception and more like inference. To a first approximation, our thesis is that some intuitions share key epistemic benefits with some inferences because each derives from an appropriate relation to an argument. The inferences are made by following an argument. We’ll call these explicit inferences. The intuitions are enabled by thinking through an argument. We’ll call these articulate intuitions. In order to characterize articulate intuitions, we take into consideration a case study from knot theory.

Jordi Fairhurst Chilton (Universitat de les Illes Balears): TBA

Andrea Guardo (University of Milan) Kripkenstein, Game Theory, and Dispositionalism

I discuss the relationship between Kripkenstein's rule-following paradox, the game-theoretical approach to signaling, and dispositional theories of meaning. I argue (1) that, contrary appearances notwithstanding, the account of meaning suggested by the game-theoretical approach to signaling is perfectly consistent with Kripkenstein's conclusion and (2) that this account of meaning is in fact a form of dispositionalism - although, of course, not of the variety usually put forward with the intent of showing that Kripkenstein's conclusion can be resisted. I then proceed to discuss the consequences of this view of meaning for the idea that mathematical symbols refer to mathematical entities and for the truth of mathematical statements.

Casper Storm Hansen: A Solution to the Liar

I will argue that the liar paradox can be solved by recognizing that certain collective intentions must fail to be satisfied.

It is clear that there is a sense in which it is the intention of the English language community that the negation of a sentence should be true if and only if the negated sentence is not true. In the same sense, it is also the intention of the English language community that a sentence declaring another sentence true should be true if and only if that other sentence is true. Furthermore, there are certain intentions regarding how names are supposed to work in English; these are too complex for me to attempt to spell out in general, but they of course imply, if satisfied, that "the liar" denotes the sentence "The liar is not true".

If all of these intentions were satisfied, then the paradoxical reasoning would go through, i.e., there is a sentence that is both true and not true. This is impossible. Ergo, one of the intentions is not satisfied.

I believe that this, in all its simplicity, is the solution to the paradox. This simplicity will probably seem suspicious given that the paradox is widely considered unsolved after more than two millennia's worth of solution attempts. However, the reason that the paradox has seemed so extremely difficult is, I will suggest, because the discussion about it has tended to conflate metaphysics and linguistics.

Let me exemplify with the first of the three intentions mentioned above. On the one hand, we have the metaphysical principle that any state of affairs either obtains or fails to obtain, and never both; on the other, the linguistic principle that in any pair consisting of an indicative sentence and its negation exactly one is true. They may be conflated because the latter is intended to be valid as a reflection of the former, and one might naively assume that when it comes to making such linguistic principles valid, the intention to make it so by the language community is all that is required. However, they are distinct, and the former is inviolable, and as consequence of that the latter may fail. 

José Antonio Perez-Escobar (UNED) Analogies in mathematics and the disagreement between Turing and Wittgenstein

The discussions between Wittgenstein and Turing has been used to clarify their views of mathematics. The structural analogies employed in their exchanges remain underexplored, despite playing a key role: they illustrate Turing’s arguments for his view of mathematics as structures that mimic empirical structures, and Wittgenstein’s rejection of such a view. They also regulate each one’s tolerance of contradictions in mathematics, both in pure and applied contexts. Both Turing and Wittgenstein attempt to convince each other of their views via these analogies, often to no success. I clarify Turing’s and Wittgenstein’s attitudes towards structural intuitions in mathematics by analyzing structural analogies in the Lectures on the Foundations of Mathematics. This 1) clears up misconceptions about their views, 2) shows how analogies regulate mathematical practice, and 3) helps understand the applicability of contradictory mathematics in physics and engineering and assess the potential risks. 

Matteo Plebani (University of Turin)  Semantic paradoxes as collective tragedies

Abstract: What does it mean to solve a paradox? A common assumption is that to solve a paradox we need to find the wrong step in a certain piece of reasoning. In this talk, I will argue while in the case of some paradoxes such an assumption might be correct, in the case of paradoxes such as the liar and Curry’s paradox it can be questioned. 

Andrei Rodin (University of Lorraine, Archives Henri-Poincaré and Loria)  Truth and Meaning in the Automated Proof Verification

Automated proof verification is an emerging mathematical practice that amounts to representing mathematical proofs in form of executable computer programs. A successful execution of such a program is supposed to serve as an evidence that the corresponding proof is correct and the corresponding mathematical statement is true. Assuming that a judgement to the effect that a given mathematical proposition is true ultimately belongs to a human, one may wonder how exactly a computer code is interpreted into a meaningful mathematical argument and what exactly constitutes a truth value of a given proposition in such contexts.  I show that in order to achieve its expected epistemic goals the automated proof verification needs to be supported with appropriate foundations of mathematics, and argue that the received set-theoretic foundations are hardly appropriate for the job. Finally, I shall briefly outline the idea of Univalent foundations (UF) as a candidate foundation that supports automated proof verification and enhances the human understanding of mathematical reasoning. 

Deniz Sarikaya (Universität zu Lübeck & Vrije Universiteit Brussel) Wittgenstein, Big Strokes and the mathematical landscape
Is mathematics as a whole driven forward by only a few geniuses—those who prove the most important theorems and generate groundbreaking ideas? Or is progress the result of many mathematicians, each contributing incrementally? Perhaps it is only in hindsight that we can distinguish which mathematicians made significant contributions and which ultimately pursued fruitless endeavors?
In this talk, we will present work in progress. Heuer and Sarikaya (2023) argued that the daily work of mathematicians can, rather pessimistically, be modeled as random walks driven by local variations of previous theorems. We now propose that the next, more complex step involves codifying local patterns and theorems into broader ‘metatheorems’—the large, defining strokes that shape our picture of the mathematical landscape. To illustrate this, we sketch a few possible case studies from combinatorics—one from infinite combinatorics and one from probabilistic methods.
We then extend this idea to a Wittgensteinian model of seeing the mathematical landscape as something and discuss the role of different perspectives and paradigmatic examples.
Literature: Heuer, Karl and Deniz Sarikaya (2023). Paving the cowpath in research within pure mathematics – a medium level model based on text driven variations. Studies in History and Philosophy of Science.

Organized by:

Ásgeir Berg Matthíasson (University of Iceland)
Deniz Sarikaya (Universität zu Lübeck & Vrije Universiteit Brussels)