# Mathematical Indispensability in Philosophy (16-17 February 2023)

## Idea & Motivation

The application of formal methods has been a hallmark of analytic philosophy since its inception. It then flourished broadly during the post-war period, not just in the traditionally more technical disciplines of philosophy of logic and language (Quine, Kreisel, Kripke, Montague, Lewis), but also in philosophy of science (Carnap, Hempel, Putnam), epistemology (von Neumann, de Finetti, Jeffrey, Hintikka), ethics (von Wright, Harsanyi), and social choice (Arrow, Sen). A notable consequence is that these subjects are now developed using techniques and results not just from logic but from core mathematics.

The integration of mathematical methods into philosophical methodology is widely regarded as having been fruitful, giving rise to sub-developments such as formal semantics, formal epistemology, axiomatic theories of truth, and logical philosophy of science. But at the same time, the reception of mathematical results within philosophy has been largely unsystematic and non-critical — e.g. in regard to the framework by which philosophical principles and mathematical theorems are combined, the logical principles required for the derivation of the latter, and their appropriateness within different philosophical contexts. Contemporary formal philosophy has thus developed in relative isolation from the concerns of both mathematical logic and philosophy of mathematics — e.g. in regard to axiomatically calibrating the strength of core mathematical theorems required to sustain philosophical arguments or assessing the acceptability of axioms relative to foundational considerations.

The general goal of this workshop will be to highlight mathematical results which find applications in philosophy. More specific questions to be addressed include the following: Is it possible to provide an axiomatic characterization of “philosophically applicable” mathematics? How does this compare with the characterization of “empirically applicable” mathematics? Are there instances in which mathematical results are explicitly required as premises in philosophical arguments? If so, does this suggest that some portion of mathematics is indispensable for the development of certain philosophical positions or theories? To what extent can the methods and results of Reverse Mathematics and computability theory be used to clarify these questions?

## Confirmed speakers

- Marianna Antonutti Marfori (Paris 1 Panthéon-Sorbonne, IHPST)

- Mark Colyvan (University of Sydney, Department of Philosophy)

- Marta Fiori Carones (Novosibirsk, Sobolev Institute of Mathematics)

- Walter Dean (University of Warwick, Department of Philosophy)

- Damir Dzhafarov (University of Connecticut, Department of Mathematics)

- Benedict Eastaugh (University of Warwick, Department of Philosophy)

- Martin Fischer (LMU Munich, MCMP)

- Sam Sanders (Ruhr University Bochum, Institute for Philosophy II)

- Sean Walsh (UCLA, Department of Philosophy

## Program

Thursday 16 February 2023 | |

09:30 | Welcome and Coffee |

09:45 - 11:15 | Walter Dean: From the method of arithmetization to reverse philosophy via representation theorems |

11:30 - 13:00 | Marta Fiori Carones: Different base theories for reverse mathematics |

13:00 - 14:30 | Lunch |

14:30 - 16:00 | Sam Sanders: Indispensability arguments and Reverse Mathematics |

16:15 - 17:45 | Damir Dzhafarov: Reverse Sorites |

Friday 17 February 2023 | |

10:00 - 11:30 | Mark Colyvan: Mathematical Assumptions in Representation Theorems |

11:30 - 13:00 | Benedict Eastaugh: Social choice theory and non-computable sets |

13:00 - 14:30 | Lunch |

14:30 - 16:00 | Marianna Antonutti Marfori: The role of mathematical premises in philosophical arguments |

16:15 - 17:45 | Martin Fischer: The role of truth in categoricity arguments |

18:00 - 19:30 | Sean Walsh: Algorithmic Randomness and Lévy's Upward Theorem |

**Abstracts**

**Marianna Antonutti Marfori: The role of mathematical premises in philosophical arguments**The Quine-Putnam argument for the indispensability of mathematics to science argues that scientific realists should extend their commitment to our best scientific theories to mathematics on the grounds that mathematics is indispensable to our best science. Recently, an enhanced version of this argument based on the indispensable role of mathematics in our best scientific explanations brought new life to mathematical platonism; a major advantage of this version of the argument is that it does not rely on confirmational holism. There seem to be several instances of philosophical arguments whose premises make ineliminable appeal to mathematical results, so the question arises whether the indispensability of mathematics to philosophical arguments grants similar support to mathematical platonism, and if so, whether this is due to the mathematical premises' role in philosophical explanations. This talk will highlight some important points of contact as well as some important asymmetries between the case of the indispensability of mathematics to science and to philosophy, and will assess the prospects for the latter to provide support for mathematical platonism.

**Mark Colyvan: Mathematical Assumptions in Representation Theorems**

One, arguably indispensable, application of mathematics in philosophy is in the proofs of various representation theorems (e.g. in decision and game theory). Here I’ll tell a cautionary tale about some of the pitfalls of such applications. I’ll consider one such representation theorem: Cox’s Theorem. By failing to properly understand the role of the mathematical assumptions underpinning this theorem, it is very easy to slide from a correct statement of the theorem to stronger, unsupported philosophical theses in epistemology. I’ll conclude by showing how the lessons from Cox’s Theorem generalise to other representation theorems.

**Marta Fiori Carones: Different base theories for reverse mathematics**

Commonly proofs in reverse mathematics are carried out over the base theory $RCA_0$, Recursive Comprehension Axiom. This means that during a proof in $RCA_0$ one can define objects computable in the objects that are assumed at the outset, having but a limited amount of induction available.

Despite this, the choice of the base theory is certainly not univocal and its very choice can raise interesting philosophical questions about its legitimacy, as about the meaning of (possibly) different results obtained over different base theories. In this talk I present two alternatives and some results which can be obtained over these base theories.

On the one hand, one can further restrict induction, so to obtain a theory known as $RCA_0^∗$. Simpson and Smith defined this theory in 1986, and recently some authors, including Kołodziejczyk, Kowalik, Wong, Yokoyama pursued this line of research. On the other hand, one may restrict the allowed basic operations to the primitive recursive ones. In 2000 Kohlenbach introduced the theory $PRA^2$, Primitive Recursive Arithmetic in the second order, which allows to study the implications between theorems having only the primitive recursive procedures as the basic ones.

**Walter Dean: From the method arithmetization to reverse philosophy via representation theorems**

I will present the method of "reverse philosophy" -- that of using the methods of reverse mathematics to study the reliance of philosophical arguments on mathematical principles -- as a natural extension of the method of arithmetization developed within the Hilbert program -- that of axiomatizing the subject matter of diverse theories (e.g. of geometry, physics, or psychology), demonstrating their consistency by the construction of arithmetical models, and then using proof- and computability-theoretic techniques to study their properties. A step in this process is often the isolation of representation theorems -- results which show that any structure satisfying a given set of axioms can be mapped isomorphically into a prescribed sort of model, typically with the real numbers as domain. I will illustrate how such results are illuminated by the method of reverse mathematics by considering the representation theorems of Hölder (1901) and Debreu (1954).

**Damir Dzhafarov: Reverse Sorites**

Sorites is an ancient piece of paradoxical reasoning pertaining to sets with the following properties: (Supervenience) elements of the set are mapped into some set of “attributes”; (Tolerance) if an element has a given attribute then so are the elements in some vicinity of this element; and (Connectedness) such vicinities can be arranged into pairwise overlapping finite chains connecting two elements with different attributes. Obviously, if Supervenience is assumed, then (1) Tolerance implies lack of Connectedness, and (2) Connectedness implies lack of Tolerance. Using a very general but precise definition of “vicinity”, Dzhafarov and Dzhafarov (2010) offered two formalizations of these mutual contrapositions. Mathematically, the formalizations are equally valid, but in this paper, we offer a different basis by which to compare them. Namely, we show that the formalizations have different proof-theoretic strengths when measured in the framework of reverse mathematics: the formalization of (1) is provable in $RCA_0$, while the formalization of (2) is equivalent to $ACA_0$ over $RCA_0$. Thus, in a certain precise sense, the approach of (1) is more constructive than that of (2).

**Benedict Eastaugh: Social choice theory and non-computable sets**

Arrow’s impossibility theorem appears to place substantial limits on the existence of methods for social decision-making that are fair, rational, and uniform. The theorem shows that there is no way of aggregating individual preferences into an overall social preference or general will, known as a social welfare function, assuming only that any such social welfare function must satisfy conditions of unanimity, independence, and non-dictatoriality. Arrow’s theorem has therefore been of major interest in economics and political philosophy as well as many other fields, including philosophy of science. It has also been the subject of many different formalisations, using modal logic, first-order logic, dependence logic, and intuitionistic logic. This talk presents a new approach to formalising Arrow’s theorem and related results in social choice theory, with a focus on infinitary counterexamples to Arrow’s theorem, since although the theorem holds for finite societies it fails in infinite ones. I will show that such infinitary counterexamples necessarily involve appeal to non-computable objects, namely non-principal ultrafilters, and that therefore it is difficult to regard them as able to act as ideal social planners or as genuinely expressing the general will of a society. However, these non-computable social welfare functions are also not overly complex (in terms of the computability-theoretic and proof-theoretic hierarchies used to measure this complexity), and can therefore be proved to exist in a system with the same proof-theoretic strength as first-order Peano arithmetic.

**Martin Fischer: The role of truth in categoricity arguments (joint work with Matteo Zicchetti)**

Categoricity theorems have played an important role in arguments for the determinateness of arithmetical statements. In recent years the attention has shifted towards internal categoricity theorems. Parsons (2008) made a case for first-order versions of such internal categoricity arguments. In this talk I reconsider such arguments by employing a primitive notion of truth and argue that it can play an important role. The notion of truth can strengthen the first-order case and help to express a natural version of 'intolerance'.

**Sam Sanders: Indispensability arguments and Reverse Mathematics**

We survey recent results in reverse mathematics (jww Dag Normann) on a considerable extension of the Big Five and the limit thereof. Of particular import/interest is the observation that the usual hierarchy of function classes from real analysis looks markedly different in weak(-ish) systems. We use this observation to critically examine (and reject) indispensability arguments from analysis.

**Sean Walsh: Algorithmic Randomness and Lévy's Upward Theorem**

Much recent work in algorithmic randomness has concerned characterizations of randomness notions in terms of the almost-everywhere behavior of suitably effectivized versions of functions from analysis or probability. In this work, we examine the relationship between algorithmic randomness and Lévy's Upward Martingale Convergence Theorem, in the setting of arbitrary computable Polish spaces. We show that Schnorr randoms are precisely the points at which the conditional expectations of L^1-computable functions converge to their true value. This result has natural applications to formal epistemology and the philosophical interpretation of probability: for, the natural Bayesian interpretation of this result is that belief, in the form of an agent's best estimates of the true value of a random variable, aligns with truth in the limit, under appropriate effectiveness and randomness assumptions. We also consider other randomness notions such as Martin-Löf Randomness and density randomness. This is joint work with Simon M. Huttegger (UC Irvine) and Francesca Zaffora Blando (CMU).

## Organizers

## Venue

Ludwigstr. 31, Room 021, 80539 Munich. The conference will be a hybrid event.

## Registration

Please register here via our online tool.

## Acknowledgement

The conference is generously funded by the Alexander von Humboldt Foundation and the MCMP.