Epistemology of Mathematics: Knowledge, Proof, and Explanation

I am very grateful to the British Academy for supporting this workshop, as part of the project Mathematics: The Unlikely Engine of Scientific Discovery.

Date: 29/11/2018

Location: University of Leeds, Leeds Humanities Research Institute (LHRI) Seminar Room 1.

Provisional schedule:

8:30am-9.00am: Coffee

9.00am-10.30am: Justin Clarke-Doane (Columbia University)

Title: SET-THEORETIC PLURALISM AND THE BENACERRAF PROBLEM

Abstract: Set-theoretic pluralism is an increasingly influential position in the philosophy of set theory (Balaguer [1998], Linksy and Zalta [1995], Hamkins [2012]). According to the pluralist, “whenever you have a consistent [formulation of set theory], then there are…objects that satisfy that theory under a perfectly standard satisfaction relation…[A]ll the consistent concepts of set… are instantiated side by side [Field 2001, 333].” Of course, the Completeness Theorem ensures that every consistent theory – set-theoretic or otherwise – has a model. What the pluralist adds is that it has an intended model. The intuition is that set theoretic “truth comes cheaply” (given consistency), but not because it depends on us. Set theoretic truth comes cheaply because the set-theoretic universe – or, better, pluriverse – is so rich, and the semantics of set-theoretic discourse so cooperative, that consistent theories are automatically about the entities of which they are true, and there are always such entities.

There is considerable room for debate about how best to formulate set-theoretic pluralism, and even about whether the view is coherent. But there is widespread agreement as to what there is to recommend the view (given that it can be formulated coherently). Unlike Godelian “universalism”, set-theoretic pluralism affords an answer to Benacerraf’s epistemological challenge. In this talk, I seek to determine what Benacerraf’s challenge could be such that this view is warranted. I argue that it could not be any of the challenges with which it has been traditionally identified by its advocates, like of Benacerraf and Field. Not only are none of the challenges easier for the pluralist to meet. None satisfies a key constraint that has been placed on Benacerraf’s challenge, independent of the universalism-pluralism debate. However, I argue that Benacerraf’s challenge could be the challenge to show that our set-theoretic beliefs are safe – i.e., to show that we could not have easily had false ones (using the method that we actually used to form ours). Whether the pluralist is better positioned to show that our set-theoretic beliefs are safe turns on a broadly empirical conjecture which is outstanding. If this conjecture proves to be false, then it is unclear what the epistemological argument for set-theoretic pluralism could be.

10:30am-12.00pm: Jack Woods (University of Leeds)

Title: ABDUCTIVE IMMODESTY IN MORALITY AND MATHEMATICS

It’s increasingly common to recognize that philosophical arguments are typically abductive in character. From Lewis’s point that philosophy is a game of costing views, to the rise of anti-exceptionalism in logic, to the now nigh universal use of reflective equilibrium (and like methods) in ethics and metaethics, philosophers have started to recognize that often the best we can do is search for our best explanation of some phenomena. This is especially true in esoteric areas like logic, aesthetics, mathematics, and morality where the target explanandum often include our trenchant intuitions.

While this methodological shift is largely a welcome change, abductive arguments typically involve significant theoretical resources which can be part of what’s be- ing disputed. This means that we will sometimes find otherwise good arguments where coming to accept their conclusion conflicts—in some way—with some of the supporting materials we used to justify that conclusion. These arguments are interesting for a number of reasons, but perhaps most importantly because this feature—the self-effacing character of these arguments—occurs most often in the context of arguments for modifying our standing views on subject matters like logic, mathematics, aesthetics, and morality; subject matters which play a basic role in how we reason about other cases. Self-effacement, it turns out, gives us the resources to resist at least some of these skeptical arguments. Which arguments and which subject matters is the subject of this paper.

12:00pm-1.00pm: Lunch (Provided by Opposite)

1.00pm-2.30pm: Marcus Giaquinto (University College London)

Title: A PRIORI AND A POSTERIORI IN MATHS: SOME COMMON VIEWS ARE WRONG

Abstract:

Mathematics is an a priori science, in which proofs play a central role. This is largely because thinking through an argument warrants high confidence in its conclusion only if the argument is a proof. If, in thinking through an argument, visual experience is used not merely to grasp the argument, but also to accept it, the argument is not purely a priori but contains an a posteriori element, and for that reason is not a proof.” 

I will cast doubt on all of this, apart from the claim that proofs play a central role. 

2:30pm-4.00pm: Catarina Dutilh Novaes (VU University Amsterdam)

Title: A DIALOGICAL CONCEPTION OF EXPLANATION IN MATHEMATICAL PROOFS

Abstract: The distinction between non-explanatory and explanatory mathematical proofs is often formulated in terms of the difference between proofs that merely establish that the conclusion is the case and proofs that establish why the conclusion is the case; the former merely demonstrate, while the latter explain. The issue has received significant attention in recent decades, both among philosophers and among mathematics educators, and a number of different accounts of the explanatoriness of mathematical proofs have been proposed. I here apply the dialogical account of mathematical proof that I have developed elsewhere to the issue of mathematical explanation. The key idea is to emphasize the observation that a proof is a piece of discourse aimed at an intended audience, with the intent to produce explanatory persuasion. The result is an ‘embodied’, pragmatic and independently motivated dialogical account of the explanatoriness of mathematical proofs, which is able to explain a number of otherwise puzzling features of the practices of mathematical proofs.

7:00pm: Dinner (Eat Your Greens)