It is only recently that researchers have emerged with sufficiently sophisticated training in more than one area, so that they can participate in the interdisciplinary work necessary for providing a fuller picture of the relevant cognitive capacities. With this background in mind, we intend the LLCC to promote precisely those interdisciplinary initiatives which will lead to groundbreaking results.

The LLCC provides the infrastructure and services necessary for creating an atmosphere which promote the core areas of research and encourage interdisciplinary cooperation among members, fellows and students affiliated with the Center. The LLCC is responsible for a program for outstanding students. The aim of the program is to train extremely promising students in the core areas of interest, thus enabling the students to engage in groundbreaking interdisciplinary research.

The program of study includes advanced courses in syntax, semantics and philosophical logic. We envision the courses offered by the LLCC to be open to research students in programs throughout the university. Skip to main content. So, at some level, you discuss a concrete object, right there on the papyrus.

But then again, because of the ambiguity between icon and symbol, it never gets clarified quite what this means. Are you talking about some ideal triangle, for which the written diagrams serve as a symbol? Or is it about a look-alike for the figure on the page?

This translates into a deeper ontological ambiguity, concerning the nature—physical, or purely geometrical—of the objects studied by Greek geometry. GB: In relation to this, you argue that ancient Greek mathematicians were not so much interested in the ontological status of logic, but rather only in the stabilization and internal coherence of a formal mathematical language.

You write that the diagram acted for them as a substitute for ontology; that is, as a means through which they could seal mathematics off from philosophy.

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## Formal Languages in Logic: A Philosophical and Cognitive Analysis

Yet, to make-believe here, is not just performative; it relies on a complex articulation between what you call the shaping of necessity and what you call the shaping of generality. Could you explain this relation between the relative single-mindedness that seems to have determined the way in which ancient Greek mathematicians were able to retreat from any ontological argument, and the way in which the objective truth procedures that they invented and stabilized also had implications that exceeded mathematics?

RN: As a matter of historical, sociological reality, Greek mathematics was shaped with relatively little debt to specific philosophical ideas.

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Its key practices therefore tend, indeed, to elide specific philosophical questions. How Greek mathematicians ended up this way is a complicated question, but very quickly I can say that they perhaps wished to insulate themselves from a field characterized by radical debate, not because they wished to avoid debate as such but because they wanted to carve out a field of debate of their own professional identity.

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Then, the rich semiotic tools of Greek mathematics—language and diagram—allow a certain room for ambiguity concerning the nature of reference, and this is something Greek mathematicians seem to embrace, and I agree with your formulations there. In general, the attitude is one of make-believe. You ask that a circle may be drawn and, with a free-hand shape in place, you proceed as if it were the circle you wished to have there.

You mentally enter a universe where everything that can be proved to be doable in principle is, indeed, done in practice, once you wish for it. Indeed, this remains the attitude of mathematics even today, and so we may get too accustomed and desensitized to this attitude. But mathematics is essentially a make-believe game, a practice emerging with the Greek geometrical diagrams: taking an imperfect sign as if it were perfect.

This seems to have important consequences for art. Firstly, at least since modernity, art has been very much engaged with overcoming entrenched beliefs and generating novel conventions, moreover formalized procedures have been integral to this turn. Secondly, it could be argued that at least non-representative art is constitutively concerned with what you call de-semantification.

## Universal Logic

It seems that, in this sense, much art could be considered as a kind of operative writing or gesture , as that it is literally a form of thinking that has been externalized onto a particular medium. In the pluralistic conception of human rationality you put forward, how do you view artistic practice within the context of the need to counter doxastic conservativeness?

CDN: Although I have never thought much about the connections between my thoughts on de-semantification and art, I suppose one general connection is that, to some extent, the idea of breaking away from established patterns of thinking by means of cognitive artifacts such as notations is a search for creativity and innovation, in the sense of attaining novel ideas and beliefs. Yet, of course, my emphasis on mechanized reasoning might in fact suggest the exact opposite! Insofar as art is also tightly connected with novelty and creativity, then there might be interesting connections here worth exploring.

Instead, perhaps in this case the user of formalisms might be compared to the skilled artisan, who can execute beautiful objects by deploying the techniques they are good at, without necessarily seeking innovation. But again, this is not something I have given much thought to until now, so these are just some incipient remarks.

Definitely something to think about in the future. He has this paper 7 where he uses concepts from literary theory like poetry to analyze mathematical proofs, understood as written discourse that is relevantly similar to poems so as to allow for a literary analysis of mathematical proofs also in their aesthetic components. Classical poetry and mathematical proofs have in common the fact that they are forms of discourse constrained by fairly rigid rules, and beauty occurs when creativity and novelty emerge despite, or perhaps because of, these constraints.

But maybe Reviel should talk about this, not me. RN: I would have indeed at least one quick observation to make here. In the paper you mention, I may have in fact over-emphasized the idea of formalism. Mathematicians, in historical reality, are almost always engaged with theories that they understand semantically. Because after all this is how you understand and you must operate through your understanding. And the same is true in art.

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Artists, in historical practice, are people, they tell stories. There are those few aberrant moments in high modernism where this seems to have been avoided but even tonal music is not very far from narrative forms and is in fact embedded, historically, in various forms of song and opera. GB: The idea that logic has a prescriptive, or normative, import for reasoning—that is, a logically valid argument should compel an agent to act in accordance with the moral law that can be deduced from it—has been heavily criticized within philosophy, and is mostly vehemently rejected within the humanities.

Catarina, you agree that the notion of necessary truth preservation that is embedded in many logical systems, in particular the classical framework, is not descriptively valid for reasoning in most everyday situations, and further concede that there are good arguments against considering logic as prescriptive for thought. However, you argue that a historically informed reconceptualization or rational reconstruction of the deductive method according to a multi-agent dialogical framework shows that the normativity of logic may be upheld but with significant modifications to the classical image.

The normative import of logic for reasoning that results seem to imply a potentially different take on the problem of social relativism. Could you expand on this?

The same question could also be asked to you Reviel, notably in relation to the difference you seem to make in the introduction of your book with what Simon Schaeffer and Steven Shappin have argued in their seminal book, The Leviathan and the Air-pump CDN: In a weak sense, I am a social constructivist in that I am interested in how deductive practices emerged and developed in specific social contexts. But I do not think that this entails rampant relativism of the kind that is often correctly or not associated with the strong program of social constructivism.

Prima facie, he seems to be saying that any set of principles that satisfy very minimal conditions maybe consistency, or in any case non-triviality counts as a legitimate logical system. But he also emphasizes that, ultimately, what will decide on the value of a logical system is its applicability.

The pragmatic focus is made clearer in his concept of explication, which I wrote a paper on with Erich Reck. But it is not the case that any logical system is as good as any other for specific applications. As it so happens, deduction is a rather successful way of arguing in certain circumstances, and indeed the Euclidean model of mathematical proofs remained extremely pervasive for millennia.

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It was only in the modern period that other, more algebraic modes of arguing in mathematics were developed. You can still ask yourself what makes deduction so successful and popular for certain applications, despite or perhaps because being a cognitive oddity. A lot of my work has focused precisely on addressing this question.

But my strategy is usually to look for facts about human practices, and how humans deal with the world and with each other, to address these questions, rather than a top-down, Kantian transcendental approach that seeks to ground the normativity of logic outside of human practices. By the way, just to be clear, I am equally interested in the biologically determined cognitive endowment of humans, presumably shared by all members of the species, and the cultural variations arising in different situations. It is in the essence of human cognition to be extremely plastic, and so within constraints, there is a lot of room for variation.

I find what they write compelling and I think they do not disagree too much with me. The relations between mathematics and its languages In Macbeth , D. Let us examine each of them in turn. Written mathematical symbols are constitutive of mathematical rea- soning Macbeth attributes this position to Kant3 and, as a more recent source, to Rotman But the most outspoken recent defenders of the idea that external devices such as inscriptions of any kind — writing, diagrams, nota- tions etc.

Quoting from the abstract: This paper draws on the extended mind thesis to suggest that math- ematical symbols enable us to delegate some mathematical opera- tions to the external environment. I use these terms merely for the sake of brevity; this should not be interpreted as an endorsement of an overly homogeneous view of mathematics as a discipline, or of the idea that there is a core essence to mathematical reasoning — there may well be one, but arguing for such a view is not the aim of the present analysis.

Mathemat- ical symbols are epistemic actions, because they enable us to rep- resent concepts that are literally unthinkable with our bare brains. But they disagree on whether mathematical reasoning is inherently linguistic or not. In contrast, according to the third position, mathematical reasoning is not in any way language- dependent, or in any case not dependent on any public language such as speech or writing.