# Mathematics: Objectivity by Representation

## ANR-DFG-Workshop

Coordination between Professor Dr. Gerhard Heinzmann, Laboratoire d’Histoire des Sciences et de Philosophie – Archives Henri-Poncaire, Université de Lorraine/CNRS, Nancy, Professor Dr. Marco Panza, Institut d'Histoire et de Philosophie des Sciences et des Techniques, CNRS/Université Paris1-ENS, and Professor DDr. Hannes Leitgeb, Munich Center of Mathematical Philosophy MCMP, LMU.

## Location & Date

- Dienstag, 11. November 2014

### Directions to the venue

- from the LMU's main building: Take the U3/U6 to "Sendlinger Tor" and the Tram 16 from there (stop "Romanplatz")
- from the conference hotel: Take the Tram 12 from "Rotkreuzplatz" to "Romanplatz"

## Program

Time | Event |
---|---|

09:00 | Opening |

09:15 - 10:45 | Lecture 1: "Three ways in which logic might be normative" (Florian Steinberger, MCMP/LMU) (Watch the Lecture @ LMUcast) |

10:45 - 11:00 | Coffee Break |

11:00 - 12:30 | Lecture 2: "What are the challenges of Benacerrafs Dilemma? A Reinterpretation" (Marco Panza, Paris I) (Watch the Lecture @ LMUcast) |

12:30 - 14:00 | Lunch break. (Lunch served at venue.) |

14:00 - 14:45 | (Short) Lecture 3: "Discernibility from the perspective of a countable language" (Kate Hodesdon, Nancy) (Watch the Lecture @ LMUcast) |

14:45 - 15:30 | (Short) Lecture 4: "Neuropsychology of numbers" (Hourya Benis-Sinaceur, Paris I) (Watch the Lecture @ LMUcast) |

15:30 - 16:00 | Coffee Break |

16:00 - 17:30 | Lecture 5: "IF epistemic logic and mathematical knowledge" (Manuel Rebuschi, Poincaré Archives, University of Lorraine, Nancy) (Watch the Lecture @ LMUcast) |

17:30 - 19:00 | Lecture 6: "On mathematical structuralism. A theory of unlabeled graphs as ante rem structures" (Hannes Leitgeb, MCMP/LMU) (Watch the Lecture @ LMUcast) |

20:00 | Dinner (Schlosswirtschaft Schwaige) |

## Abstracts

### "Neuropsychology of numbers" (Hourya Benis-Sinaceur, Paris I)

How do we extract numbers from our perceiving the surrounding world? Neurosciences and cognitive sciences provide us with a myriad of empirical findings that shed light on hypothesized primitive numerical processes in the brain and in the mind. Yet, the hypotheses based on which the experiments are conducted, hence the results, depend strongly on sophisticated arithmetical models. These sophisticated arithmetical models are used to describe and explain neural data or cognitive representations that supposedly are the roots of primary arithmetical activity. I will give some examples of this petitio principii, which is involved in neuropsychologist arguments, most time without any justification.top

### "Discernibility from the perspective of a countable language" (Kate Hodesdon, Nancy)

In this talk I discuss formal methods for discerning between uncountably many objects with a countable language, building on recent work of James Ladyman, Øystein Linnebo and Richard Pettigrew. In particular, I show how stability theory provides the resources to characterize theories in which this is possible, and discuss the limitations of the stability theoretic approach.top

### "On mathematical structuralism. A theory of unlabeled graphs as ante rem structures" (Hannes Leitgeb, MCMP/LMU)

There are different versions of structuralism in present-day philosophy of mathematics which all take as their starting point the structural turn that mathematics took in the last two centuries. In this talk, I will make one variant of structuralism—ante rem structuralism—precise in terms of an axiomatic theory of unlabeled graphs as ante rem structures. I will then use that axiomatic theory in order to address some of the standard objections to ante rem structuralism that one can find in the literature. Along the way, I will discuss also other versions of mathematical structuralism, and I will say something on how the emerging theory of ante rem structures relates to modern set theory.top

### "What are the challenges of Benacerrafs Dilemma? A Reinterpretation" (Marco Panza, Paris I)

Despite its enormous influence, Benacerraf's dilemma admits no standard, unanimously accepted, version. This mainly depends on Benacerraf's having originally presented it in a quite colloquial way, by avoiding any compact, somehow codified, but purportedly comprehensive formulation. But it also depends on Benacerraf's appealing, while expounding the dilemma, to so many conceptual ingredients so as to spontaneously generate the feeling that most of them are in fact inessential for stating it. It is almost unanimously admitted that the dilemma is, as such, independent of the adoption of a causal conception of knowledge, though Benacerraf appealed to it. This apart, there have not been, however, and still there is no agreement about which of these ingredients have to be conserved so as to get a sort of minimal version of the dilemma, and which others can, rather, be left aside (or should be so, in agreement with an Okkamist policy).

My purpose is to come back to the discussion on this matter, with a particular attention to Field's reformulation of the problem, so as to identify two converging and quite basic challenges, addressed by Benacerraf's dilemma to a platonist and to a combinatorialist (in Benacerraf's own sense) philosophy of mathematics, respectively. What I mean by dubbing these challenges 'converging' is both that they share a common kernel, which encompasses a challenge for any plausible philosophy of mathematics, and that they suggest (at least to me) a way-out along similar lines. Roughing these lines out is the purpose of the two last part of the talk.

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### "IF epistemic logic and mathematical knowledge" (Manuel Rebuschi, Poincaré Archives, University of Lorraine, Nancy)

Can epistemic logicstate anything interesting about the epistemology of mathematics? That's one of Jaakko Hintikka’s claims. Hintikka was not only the founder of modal epistemic logic (1962), since he also worked on the foundations of mathematics (1996). Using what he calls "second generation" epistemic logic (2003), i.e. independence-friendly (IF) epistemic logic, Hintikka revisits the epistemology of mathematics, and in particular the debate between classical and intuitionistic mathematics (2001). The aim of the talk is to show that Hintikka is right regarding IF epistemic logic, for such a logic enables us to account for interesting features of mathematical knowledge. However, the path is not as easy as that Hintikka suggests. I will show that the well-known issue of logical omniscience directly threatens the understanding of intuitionism offered by IF epistemic logic.top

### "Three ways in which logic might be normative" (Florian Steinberger, MCMP/LMU)

Logic, the tradition has it, is, in some sense, normative for reasoning. Famously, the tradition was challenged by Gilbert Harman who argued that there is no interesting general normative link between facts about logical consequence and norms of belief formation and revision. A number of authors (e.g. John MacFarlane and Hartry Field) have sought to rehabilitate the traditional view of the normative status of logic against Harman. In this paper, I argue that the debate as a whole is marred by a failure of the disputing parties to distinguish three different types of normative assessment, and hence three distinct ways in which logic might be said to be normative. I show that as a result of their failure to appreciate this three-fold distinction, authors have been talking past one another. Finally, time-permitting, I show how each of the three types of normative assessments relate to broader epistemological commitments, specifically commitments within the internalism/externalism debate.

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