These are the FSB research seminars that have been given in recent years.

Date | Speaker | Title/Subject |
---|---|---|

28 June 2016 | Leon Horsten | The prime in the street |

21 June 2016 | Luca Oliva | Kantian Intuitionism |

17 May 2016 | Alex Jones | Semantic Theories and Plural Conceptions of Truth |

3 May 2016 | Stuart Presnell, James Ladyman | The Philosophy of Homotopy Type Theory: Identity, Intensionality and Univalence (Part III) |

26 April 2016 | Alf Coles | A relational view of the concept of number; pedagogical, neuroacientific and philosophical considerations |

15 March 2016 | Snezana Lawrence | Mathematicians and their Gods |

1 March 2016 | Sam Roberts | A strong reflection principle |

23 February 2016 | Stuart Presnell, James Ladyman | The Philosophy of Homotopy Type Theory: Identity, Intensionality and Univalence (Part II) |

12 January 2016 | Monika Gruber | Is there pragmatic encroachment on epistemic notions? |

In a few articles and a book in the early 1980s, Kit Fine proposed a theory for reasoning with arbitrary objects. The aim of my project is to develop a theory of arbitrary objects that has more and deeper structure than Fine's and has wider and more obvious applications. In the proposed seminar, I want to try to take the first few steps in this direction.

According to the ‘Hintikka-Parsons argument’, Kantian Intuitions (KI) behave like variables, for instance, in first-order quantificational logic. I will defend this thesis by showing 1) the mereological nature of KI, 2) their epistemic role in synthetic and analytic claims, which I will reduce to the basic class relations of inclusion and exclusion, and 3) the shared features of KI and mathematical induction, like the logical operations of generalization (EG and UG, respectively) and proving processes based on variables.

I will be presenting my thoughts in process on how the behaviour of semantic theories of truth for arithmetic supports a plural conception of the nature of truth. I will run through theories of plural conceptions of truth, and their philosophical features, and then look at how theories of truth for arithmetic might support this, at least for arithmetic.

In this seminar, I will move from pedagogical to neuroscientific to philosophical considerations of the concept of 'number'. Two sources of evidence converge to bring into question the orthodoxy that learning mathematics entails a movement from the concrete to abstract. The first source comes from reflection on commonalities across the pedagogy of three of the great mathematics educators of the twentieth century (Caleb Gattegno, Vasily Davydov and Bob Davis). The second source is recent neuroscientific research into number sense and the previously unacknowledged role of ordinality (cf the work of Ian Lyons). Drawing these sources together suggests the potential, for children's learning, of approaching number as a relation, which then leads to philosophical implications for what numbers are.

This talk will look at instances where mathematicians, either as individuals, or in groups, looked at the religious or theological arguments to make sense of their work, and of mathematics. Snezana will present a few such instances, and discuss in particular the case of the history of the fourth dimension.

In this talk I will introduce a new reflection principle and argue that it improves substantially on previously formulated reflection principles.

In this talk we will explain the basic features of HoTT which is a proposed new foundation for logic and mathematics. We will explain the interpretation of types, the unusual features of identity types that give the theory much of its power and interest, and the correct understanding of the content, role and justification of the univalence axiom. We will sketch some connections with structuralism in the philosophy of mathematics and applications to the foundations of physics.

Pragmatic encroachment is the idea that epistemic notions, like belief, knowledge, judgement, or desire, are not “purely epistemic”: they depend upon other, non-epistemic factors, for example, on their relevance to action. Philosophers propounding pragmatic encroachment hold that epistemic notions by themselves have a pragmatic dimension, and hence are in a way “practical”.

In my talk, I will argue that there is, in fact, no such thing as pragmatic encroachment on epistemic notions and that the arguments its proponents hold are very off the track.

Date | Speaker | Title/Subject |
---|---|---|

15 December 2015 | James Ladyman, Stuart Presnell | The Philosophy of Homotopy Type Theory: Identity, Intensionality and Univalence |

8 December 2015 | Aaron Guthrie | Can Accuracy Accounts of Probabilism Explain Truthlikeness? |

1 December 2015 | Can Baskent | Game Theoretical Semantics for Some Paraconsistent Logics |

24 November 2015 | Sam Pollock | The problem of unambiguously communicating/understanding non-algrebraic mathematical theories: Just a special case of Kripkenstein? |

10 November 2015 | Alex Jones | An Introduction to Formal Truth |

27 October 2015 | Michael Shanks | Brouwer, Intuitionism and Society |

20 October 2015 | Leon Horsten | The Logic of Truth |

13 October 2015 | Oliver Tatton-Brown | Frege and the Neo-Fregeans |

29 September 2015 | Monika Gruber | Ramsey on truth, belief and success |

In this talk we will explain the basic features of HoTT which is a proposed new foundation for logic and mathematics. We will explain the interpretation of types, the unusual features of identity types that give the theory much of its power and interest, and the correct understanding of the content, role and justification of the univalence axiom. We will sketch some connections with structuralism in the philosophy of mathematics and applications to the foundations of physics.

Some false beliefs are better than others; false beliefs can be more or less truthlike. Accuracy accounts of probabilism seem in a good position to explain truthlikeness, as they give orderings on beliefs at worlds (or at least we can interpret them as doing such). Joyce has pointed out, however, that there are problems with doing such, and suggested a solution. I will discuss his solution and the problems it encounters.

In this talk, I will give a Hintikka-style game theoretical semantics for a variety of paraconsistent and non-classical logics. I will discuss Priest’s Logic of Paradox, Dunn’s First-Degree Entailment, Routleys’ Relevant Logics, McCall’s Connexive Logic and Belnap’s Four-Valued Logic. I will underline how non-classical logics require different verification games and how different logics require different game theoretical conditions in such games.

Philosophers have made efforts in recent years either to secure or to explain unambiguous communication and/or unambiguous "intuiting" of certain mathematical theories. Whilst some, like Shapiro, give an answer in terms of categoricity to the purely explanatory question, others have looked also at the sceptical question and have given answers in terms of open-ended schemas, eschewing model theory. It looks as though we can interpret these sceptical and explanatory problems for non-algebraic theories as a special case of the famous Kripkenstein problem from philosophy of language, concerning private languages and rule-following. I will give an overview of the Kripkenstein argument and show how we can frame a handful of arguments about the uniqueness of the natural numbers/sets as instances of Kripkenstein. Depending on time, I'd like to explore what consequences this interpretation has if one finds Kripkenstein convincing -- what does it mean for Shapiro's, Parsons' and Martin's arguments for our ability to uniquely communicate/intuit those mathematical subject matters?

This talk will be pretty sketchy, and won't assume any prior knowledge. It will *not* go into matters of how best to interpret Wittgenstein and Kripke (that is a whole other can of worms for another day!).

This talk will be an introduction to the technical background of my PhD proposal. I will introduce first order Peano Arithmetic and then look at some theories of truth which have been defined over the theory and key related theorems. I aim for this to be approachable and informal and will intersperse philosophical questions that I am interested in exploring throughout.

This talk will present a broad introduction to LEJ Brouwer and Intuitionism in general. I will first give an overview of his life and some of the distinguishing features of Intuitionist mathematics before offering some remarks about the nature of Intuisionist mathematics as an activity. This talk is largely based on material from my Undergraduate final year project covering the same topic.

I want to discuss recent results by Graham Leigh (concerning disquotational truth and reflection) and by Carlo Nicolai (concerning the inner logic of KF). The question I will be addressing is whether these results suggest that certain compositional theories of truth are in some sense "complete".

I will discuss the merits of Neo-Fregeanism (of the variety espoused by Wright and Hale) as an interpretation of Frege. This will hopefully serve as an introduction to Frege’s position in the Grundlagen, and the Neo-Fregean point of view, for those not familiar with these. If there’s time I may also discuss other aspects of the Neo-Fregean point of view, such as their assumption of mathematical realism and the Julius Caesar problem.

Regarding truth, Ramsey is best known for a redundancy theory he laid down in "Facts and Propositions” (1927), and which has been ever since falsely attributed to him. This theory became an inspiration for modern deflationary truth theories. Moreover, on Ramsey's account it is possible to bring into harmony the deflationary attitudes confined within his ideas with his correspondence theoretical intuitions, thus constructing a new theory of truth. However, a definition of truth was not what Ramsey was after. His goal was to present a logical analysis of the terms “judgement” or “belief" which did not presuppose the concept of truth, for as he said “if we have analysed judgement we have solved the problem of truth”.

I will discuss Ramsey’s attitude towards truth and the role played here by the notion of “belief”. I will also consider Ramsey’s Principle and its role in his truth theory as well as its impact on decision theory and success semantics.

Date | Speaker | Title/Subject |
---|---|---|

27 January 2015 | Mona Simion, Sam Roberts, Markus Eronen, Sam Pollock, Stefan Buijsman | Bristol-Leuven Workshop |

3 February 2015 | Peter Koepke | Natural Formalism |

10 February 2015 | Alexander Jones | Satisfaction Classes and Deflationary Truth |

17 February 2015 | Oliver Tatton-Brown | Counting without numbers |

17 March 2015 | Andrew Brooke-Taylor | Categoricity in AECs from large cardinals via category theory |

24 March 2015 | Can Baskent | Epistemic Game Theory and Paraconsistency |

21 April 2015 | Ali Enayat | Tarskian Satisfaction Predicates, Revisited |

26 May 2015 | Michael Sheard |

The language of mathematics as found in textbooks and research articles is a variant of natural language which admits symbolic terms and formulas within argumentative natural language contexts. As it is written to be unambiguously understandable by experts it is susceptible to definite translations into (first-order) formal statements by standard (computer-)linguistic methods. Conversely those methods determine a class of accepted statement and hence implement a controlled natural mathematical language that can be seen as an enriched formal language.

Formal mathematics, which aims at complete formalisations of mathematical statements and proofs, has recently seen considerable advances using powerful software like automatic theorem provers for bridging proof steps. There is now a large selection of computer-checked proofs, including major results like the Four-Colour Theorem or the Feit-Thompson Theorem. Computer proof systems can be viewed as implementing strong formal calculi which allow sophisticated deductions known from ordinary mathematical proofs.

The combination of controlled natural language and argumention will lead to rich natural logics, which are completely formal extensions of first-order logic or some variant, but which accept proofs in the familiar language of mathematics.

In my talk I want to discuss some aspects of that development:

- relevant systems in formal mathematics and linguistics;

- associated work in the foundations of mathematics;

- implications for the practice of mathematics;

- implications for mathematical formalism.

Satisfaction classes are an axiomatic definition of truth for nonstandard sentences of arithmetic. I will run through the definition of a satisfaction class and argue that we should be interested in the truth of these sentences and that a satisfaction class is the minimum such structure we should consider. I will then discuss the implication that this has for deflationist conceptions of truth and the questions that it raises.

I will discuss a way of phrasing statements of finite cardinality (including additions and multiplications of cardinality) that does not require any objects, numbers, to do the counting (actually I'll discuss two).

Abstract elementary classes are a framework proposed by Shelah for generalising classical model theory to non-first order cases. A key test question for whether this is a good generalisation was whether the classical categoricity results generalise; it turns out that under suitable large cardinal hypotheses they (more or less) do. In joint work with Jiri Rosicky, we have shown how to reduce the large cardinal assumption from the one originally used, and notably our proof goes by way of category theory.

A two-person Russell’s Paradox can be given within the context of epistemic game theory. In this talk, I will discuss the well-known paradox, called the Brandenburger - Keisler paradox, and suggest an inconsistency friendly approach, which can allow us to reason with inconsistencies in a non-trivial fashion. At the end, I will construct a counter-model that can satisfy the “paradoxical” sentence.

This talk reports on joint work with Albert Visser. Suppose we start with a foundational 'base theory' B formulated in a language L (such as B = Peano arithmetic, or B = Zermelo-Fraenkel theory), and we extend B to a new theory B+ := B ∪ Σ, where Σ is a set of sentences formulated in the language L+ := L ∪ {S}, such that Σ describes certain prominent features of a Tarskian satisfaction predicate. I will give an overview of the status of our current knowledge of the relationship between B and B+ in connection with the following four questions (for various canonical choices of B and Σ) :

1. Is B+ semantically conservative over B? In other words, does every model of B expand to a model of B+?

2. Is B+ syntactically conservative over B? In other words, if some L-sentence φ is provabe from B+, then is φ also provable from B?

3. Is B+ interpretable in B?

4. What type of speed-up (if any) does B+ have over B?

Date | Speaker | Title/Subject |
---|---|---|

21 October 2014 | Leon Horsten | Revising the concept of truth |

28 October 2014 | Hazel Brickhill | Vagueness and Continuity: how gnomes can help us understand the Sorites paradox |

4 November 2014 | Pavel Janda | Accuracy-Difficulty of a Single-Number Credence Representation in Belnap’s Four-Valued Logic |

11 November 2014 | Sam Roberts | Reinhardt's view of the universe of sets |

25 November 2014 | Kate Hodesdon | Discernibility Relations |

9 December 2014 | Kentaro Fujimoto | Second-order arithmetic and set theory, and theories of truth over arithmetic and set theory |

16 December 2014 | Philip Welch | Informal Categoricity |

I will introduce the basic ideas of the revision theory of truth by Gupta, Belnap, Herzberger. Then I will hint at a possible new revision idea, where the revision process is given not as a linear ordering but as a tree.

When looking at the phenomena of vagueness there is so much to think about- language, meaning, epistemology, philosophy of mind etc. So I construct a simple toy world, with gnomes as agents, including things I thought key to the phenomena of vagueness (continuity in the underlying properties, restrictions on epistemic access, communication, transience). As we build up this model, something very like vagueness in human languages emerges. The insights gained here can then be used to look more closely at the sorities paradox, in particular the gnomes shed light on the notion of being "indistinguishable". The result of this (if you are convinced!) is a solution to the sorites, but perhaps surprisingly we don't come much closer to understanding the logic of vague predicates.

There is nothing very complicated in this talk, I will introduce the sorites paradox and the maths involved is just intervals on the real line. It should be of fairly broad interest.

View the handout (PDF, 131kB) for this talk.

An alternative philosophical approach to the representation of uncertain doxastic states is suggested, by considering how to model an agent who is concerned about the accuracy of her credences in Belnap’s four-valued logic B4. The paper has three parts. The first part motivates the whole project by showing that the single-number representation of credences is not appropriate for measuring an agent’s accuracy in B4. In the second part, we will introduce ordered pairs to represent an agent’s uncertain doxastic states, and we will show how they can solve the problem faced by the single-number representation. In the third part, we will show how Joyce’s idea of non-pragmatic vindication of probabilism works for ordered pairs in the classical two-valued and non-classical logical systems.

I will introduce a view in the philosophy of set theory due to William Reinhardt. It is based on the idea that the universe of sets is inherently potential, and can be used to justify very strong set-theoretic large cardinal axioms. I will discuss the significance of this view for some debates in the philosophy of set theory.

I will 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 will show how stability theory in model theory provides the resources to characterize theories in which this is possible, and discuss the limitations of the stability theoretic approach.

There is a certain strong disanalogy between second-order theories of numbers and those of sets. A parallel disanalogy exists between theories of truth over arithmetic and set theory. I'll explain some of the differences. If time permits, I'll touch upon some issues related to deflationism of truth.

We outline an extension of Martin’s view of a conceptual realism, to a Cantorian realm of absolute infinities, focussing on, essentially, the Zermelian categoricity arguments.

Date | Speaker | Title/Subject |
---|---|---|

8 October 2013 | Benedict Eastaugh | A tutorial on countable coded omega-models |

15 October 2013 | Benedict Eastaugh | Computable Entailment |

22 October 2013 | Aadil Kurji | Formality |

29 October 2013 | Leon Horsten | Description terms for languages with intensional operators |

5 November 2013 | Aaron Guthrie | |

12 November 2013 | Benedict Eastaugh | |

26 November 2013 | Samantha Pollock | Meadows (2013), 'What can a categoricity theorem tell us?' |

10 December 2013 | Richard Pettigrew | |

17 December 2013 | Ben Eva |