Philosophers of mathematics have discussed various aspects of mathematical diagrams, including their role in proofs and as tools for discovery. In this talk I explore a different function of diagrams, namely their role as unifying objects. More specifically, I consider the mathematical object, known as a lotus, recently introduced by García Barroso, González Pérez, and Popescu-Pampu as a tool for studying plane curve singularities. An important property is that a lotus allows to compute several basic combinatorial invariants of its corresponding reduced plane curve singularity. Moreover, the so-called universal lotus provides a systematic visual presentation of other mathematical objects, such as all 2×2 matrices with natural-number entries and determinant one.

The talk will begin with a brief introduction to the lotus—focusing on its visual representations—and will illustrate how some invariants can be computed from it. I further show how the lotus can be employed to systematically present both the aforementioned 2×2 matrices and the Stern–Brocot tree.

In the second and main part of the talk, I analyse the case in terms of the signs employed to represent the lotus: their appearance, interpretation and use, with the aim of understanding why it unifies. As indicated by the title Lotuses as computational architectures of a recent article by García Barroso, González Peréz and Popescu-Pampu, the architecture, that is, the construction of the lotus plays a key role.

Much effort in philosophy and empirical research has gone into understanding and combating science skepticism. In this paper, I will call attention to a related but much less discussed phenomenon that I will refer to as 'science selectivism.' Broadly, science selectivism may be characterized as an epistemically misaligned approach to science. Central examples of science selectivism occur when, for example, a politician, S, treats a hypothesis, H, as possessing scientific justification that is very different from the scientific justification that S believes that H possesses. For example, S may believe that H is strongly scientifically justified but treat it as if it is not. Conversely, S may believe that the scientific justification for H is weak but nevertheless treat it as if the scientific justification for it is very strong. Science selectivism has severe ramifications for science-informed policy and for public discourse given that it misrepresents the values and beliefs of politicians. I will refer to such purposefully misleading representations of beliefs and values as 'fake views.' Unfortunately, the ramifications of science selectivism are not well understood due to its subtle and elusive nature. Consequently, I will characterize the phenomenon, discuss its ramifications, and articulate mitigation strategies.

It's been suggested (by certain externalist epistemologists) that a belief must be sensitive in order to count as knowledge, and it's also been suggested (by certain philosophers interested in the epistemology of nonempirical areas of discourse) that an agent must take a belief to be sensitive if that belief is to remain justified in the face of a Benacerraf–Field-style reliability challenge. In both contexts, though, it's plausible that what's fundamentally in question is not whether the relevant belief is sensitive but whether its truth is merely an accident or coincidence (i.e., whether the belief is veritically lucky): sensitivity is a modal proxy for a belief's responsiveness to its corresponding fact, which in turn is often taken (at least implicitly) to be necessary for that belief to be nonaccidentally true. In previous work I showed that some standard problem cases for sensitivity-based accounts are best understood as demonstrations that sensitivity and responsiveness come apart in certain circumstances, and I used this diagnosis to construct a new sensitivity-based condition that a belief satisfies if and only if it's indeed responsive to its corresponding fact. In this talk I argue that some other standard problem cases – these involving lottery beliefs and inductive beliefs – are best understood as demonstrations that, in addition, responsiveness is, in certain (very specific) circumstances, not necessary for nonaccidental truth. I then use this diagnosis to modify my previously constructed condition in order to show that a sensitivity-based account can handle lottery beliefs and inductive beliefs after all.

AI systems permeate our epistemic lives. Humans regularly draw on AI systems seeking for advice, explanations, guidance. Very often, AI systems are treated by humans as if they were experts or epistemic authorities and are thereby trusted for what they deliver (Hauswald, 2025). However, is this in general a wise thing to do? Regardless of how AI systems are treated, do they truly deserve the status of experts or epistemic authorities? The answer to these questions depends, at least in part, on whether AI systems are sufficiently reliable; i.e., on whether the output they deliver is correct in the vast majority of cases. However, the literature on expertise (and epistemic authority) shows us very clearly that this is just part of the story. There is more to expertise than mere reliability. The hallmark of expertise is taken to be understanding (Croce, 2019). So, the questions to address in wondering whether AI systems can be experts become: do AI systems understand? Can we genuinely ascribe understanding to AI systems? The performance of AI systems, no doubt, is sometimes outstanding. AI systems have reached remarkable capacities of cognitive and verbal behavior. And yet, many remain hesitant to ascribe genuine understanding to AI. After all, understanding appears to be a uniquely human trait – a deeply personal, internal experience that seems beyond the reach of machines, regardless of their sophistication. 

In this paper, we want to assess the validity of this intuition. We will start by delving into the notion of understanding. Elaborating on previous work (Malfatti, 2019), we will show that understanding comes in different kinds: it is one thing to understand an epistemic mediator (a theory, a model, a representational system, …), and quite another thing to understand phenomena, or reality, on the basis of it. An agent who understands an epistemic mediator has what we suggest calling symbolic understanding; an agent who understands phenomena on the basis of an epistemic mediator has what we suggest calling noetic understanding. Symbolic understanding is mainly a matter of having certain abilities: to explain, infer, predict on the basis of the relevant epistemic mediators. Noetic understanding certainly requires a set of abilities of this kind; we cannot understand a phenomenon without being in the position to make competent use of the mediator or mediators that account(s) for it (De Regt, 2017). However, there is more to noetic understanding than just this. Understanding and being capable to cognitively deploy an epistemic mediator that does not answer to the facts, e.g., does not lead one to understand reality on its basis. Moreover, noetic understanding requires a noetic profile of the right kind. An agent with noetic understanding is reasonably committed to the mediator(s) that she deploys (Khalifa, 2017). I.e., the noetic understander has rationally internalized one of the most plausible candidates for being the accurate epistemic mediator of the phenomenon under consideration. We want to argue that while AI systems can display (some measure of) symbolic understanding, they do not, at least currently, exhibit noetic understanding. This is because they are (still) incapable of reasonable commitment. So, even if an AI system were to behave indistinguishably from a human agent, and were exceptional in its capacity to deploy an epistemic mediator of the right kind, a crucial element of noetic understanding would still be absent. 

Where does this leave us for our question, namely for the question whether AI systems deserve the status of experts? Our analysis shows that the answer to this question crucially depends on the kind of understanding that expertise requires: if it’s noetic, then AI systems (probably) cannot be experts; if it’s symbolic, even AI systems incapable of reasonable commitment, which are missing the right kind of noetic profile, could qualify as experts. Drawing on examples from real scientific practice, we tentatively show that the latter possibility is more promising.

According to Stalnaker's well-known common ground model of conversational context, a conversation's common ground is the set of propositions that are mutually believed/accepted by the conversational participants for the purpose of the conversation itself.  Stalnaker was well aware that participants can be mistaken about the propositions in this set — their respective representations of common ground can diverge from one another — but he was not much concerned to address such "defective" cases.  In this paper I suggest that we would do well to do so, and that we can do so by supplementing Stalnaker's common ground with what I will call the "expected common ground": the set of propositions which the conversation's participants are (conversationally) entitled to regard as common ground (whether or not they actually do so).  The aim of the paper is to motivate the notion of the expected common ground, and to suggest why such a notion is useful for philosophy of language.  Since my argument is meant to appeal to claims and principles that Stalnaker does or should accept, it is meant to develop his account in ways that make it more explanatorily robust.

Arithmetical knowledge is standardly understood as knowledge of numbers.  The axioms of an axiomatization of arithmetic, accordingly, express the basic truths of these objects in terms of the basic properties that hold of them and the basic relations that hold between them.  In my talk I present an axiomatization of arithmetic—termed CA for counting arithmetic—developed from an alternate conception of arithmetical knowledge whereby it is practical knowledge.   The alternate conception of arithmetical knowledge from which CA is developed results in an alternate formal picture of it.  Arithmetical equations are analyzed as equivalences (not identities) and arithmetical generality is secured by the pigeonhole principle (not mathematical induction).

Roughly, two problem-solving plans are inferentially equivalent when what you need to know to carry out one plan is the same as what you need to know to carry out the other plan. Using this account of inferentially equivalent plans, I develop an account of trivial notational variants. Two notations are trivial variants when they support the same problem-solving plans. My account applies Gibbard's (2012) account of synonymy and builds on work by Morris and Hamami on understanding proofs in terms of plans. I will also engage with recent and forthcoming work by De Toffoli on these issues.

Our doubt talk is varied. We talk of having doubts that P, about doubting that P, and being in doubt with respect to P, to give a few examples. Some epistemologists have taken this variety in our 'doubt' talk to indicate a variety of doubt states. In this talk, I argue that this is a bad way of accruing ontological commitments in one's epistemological theorising. I'll propose a better one, which has to do with normative conflict between doxastic states. In short, we should be committed to as many distinct doxastic states as are required to explain genuine normative conflicts in one's doxastic profile. Linguistic data suggests that there are, after all, differences in the normative profiles of having doubts, doubting that and being in doubt, suggesting three different doubt states. I'll consider some ways of accounting for these normative differences using extant theories of doubt, and find them wanting. I'll then propose my own theory of the nature of doubt, which accounts for this normative difference, but manages to do so without positing multiple kinds of doubt. Rather, the difference in doubt that underpins these normative differences is argued to be a difference in degrees of one doubt-state, rather than a difference in kind between doubt-states.

In this paper, I plan to tackle the question of what makes a mathematical proof "good". Such a question has been in the centre of the philosophy of mathematics for a very long time. Recently, Granville (2023) investigated the question from the perspective of a working mathematician. In that paper, he distinguishes between "formal" and "culturally appropriate, intuitive" proofs. The first kind of proofs are the gap-less sequences of instances of axioms or application of rules that logicians know and love. According to Granville, these formal proofs are not "good" mathematical proofs, since they are not intuitive, and tend to muddle the reasons behind a mathematical result. On the other hand, we have "good" proofs. These are the intuitive proofs that we can (easily) understand, that convince us of the results they claim to prove, and that can be used as a springboard to prove novel and original theorems. Granville's claims are perfectly plausible and appealing, but he doesn't give any reason (other than some examples) on why formal proofs are not intuitive and culturally appropriate. I argue that adherence to Grice's Cooperation Principle (Grice (1975)) provides those reasons. In other words: a formal proof is not a "good" mathematical proof because it doesn't satisfy Grice's maxims of the Cooperation Principle.

Automated Theorem Proving (ATP) and Automated Theorem Finding (ATF) are well-established research domains in mathematics, characterized by a wealth of methods and results, as well as numerous open questions that attest to their continuing vitality. Among these, three stand out as particularly relevant for current investigations. The first concerns the definition of a criterion to measure the simplicity of a proof; the second, a criterion to assess its readability; and the third, a criterion to evaluate the degree of interest of a theorem. Each of these three problems holds intrinsic value for mathematics as a whole. However, this seminar will focus on their implications within the context of automated theorem proving and theorem finding in geometry. It will also examine how these three questions are conceptually connected, and how a unified perspective on them may contribute to a clearer understanding of both the process and the product of automated reasoning in geometry.

The genealogical contingency of our ideas has long concerned European thinkers, but until relatively recent times philosophers in the analytic tradition used to dismiss debunking genealogies as instances of a genetic fallacy arising from a confusion between two distinct and fundamentally independent contexts – the context of discovery and the context of justification. More recently, the appearance of reliabilist accounts of knowledge and epistemic justification and the formulation of evolutionary debunking arguments (henceforth: EDAs) drawing on scientifically respectable evolutionary psychology rather than speculative socio-historical reconstructions has led many analytic philosophers to reevaluate the status of debunking genealogies. In the talk, I focus on two different types of EDAs – blocking and defeating ones – and bring the distinction between propositional and doxastic justification to bear on the different ways such arguments affect (if successful) the epistemic status of their target beliefs.