This paper analyzes some examples of diagrammatic proofs in elementary mathematics. It suggests that the cognitive features that allow us to understand such proofs are extensions of the cognitive features that allow us to navigate the physical world.

Convinced that the classically undecidable problems of mathematics possess determinate truth values, Gödel issued a programmatic call to search for new axioms for their solution. The platonism underlying his belief in the determinateness of those questions in combination with his conception of intuition as a kind of perception have struck many of his readers as highly problematic. Following Gödel's own suggestion, this article explores ideas from phenomenology to specify a meaning for his mathematical realism that allows for a defensible epistemology.

Skepticism about the epistemic value of intuition in theoretical and philosophical inquiry has recently been bolstered by empirical research suggesting that people’s concrete-case intuitions are vulnerable to irrational biases (e.g., the order effect). What is more, skeptics argue that we have no way to ‘‘calibrate” our intuitions against these biases and no way of anticipating intuitional instability. This paper challenges the skeptical position, introducing data from two studies that suggest not only that people’s concrete-case intuitions are often stable, but also (...) that people have introspective awareness of this stability, providing a promising means by which to assess the epistemic value of our intuitions. (shrink)

At the most general level, the concept of finitism is typically characterized by saying that finitistic mathematics is that part of mathematics which does not appeal to completed infinite totalities and is endowed with some epistemological property that makes it secure or privileged. This paper argues that this characterization can in fact be sharpened in various ways, giving rise to different conceptions of finitism. The paper investigates these conceptions and shows that they sanction different portions of mathematics as finitistic.

Gödel initiated the program of finding and justifying axioms that effect a significant reduction in incompleteness and he drew a fundamental distinction between intrinsic and extrinsic justifications. Reflection principles are the most promising candidates for new axioms that are intrinsically justified. Taking as our starting point Tait’s work on general reflection principles, we prove a series of limitative results concerning this approach. These results collectively show that general reflection principles are either weak ) or inconsistent. The philosophical significance of these (...) results is discussed. (shrink)

Gilbert’s four modes of communication include the logical, the emotional, the visceral and the kisceral, which last has not received much attention at all. This mode covers the forms of argument that rely on intuition and undefended basal assumptions. These forms range from the scientific and mathematical to the religious and mystical. In this paper these forms will be examined, and suggestions made for ways in which intuitive frameworks can be compared and valued.

The view that mathematics deals with ideal objects to which we have epistemic access by a kind of perception has troubled many thinkers. Using ideas from Husserl’s phenomenology, I will take a different look at these matters. The upshot of this approach is that there are non-material objects and that they can be recognized in a process very closely related to sense perception. In fact, the perception of physical objects may be regarded as a special case of this more universal (...) way of recognizing objects of any kind. (shrink)

What counts as an intuitively plausible set theoretic content (notion, axiom or theorem) has been a matter of much debate in contemporary philosophy of mathematics. In this paper I develop a critical appraisal of the issue. I analyze first R. B. Jensen's positions on the epistemic status of the axiom of constructibility. I then formulate and discuss a view of intuitiveness in set theory that assumes it to hinge basically on mathematical success. At the same time, I present accounts of (...) set theoretic axioms and theorems formulated in non-strictly mathematical terms, e.g., by appealing to the iterative concept of set and/or to overall methodological principles, like unify and maximize, and investigate the relation of the latter to success in mathematics. (shrink)

Many philosophers and psychologists hold that intuitions are, or reduce to, beliefs. The argument from intuition without beliefs threatens to undercut any such doxastic account: since there are clear cases of intuition without belief, intuitions cannot be beliefs. Advocates of the intellectual seeming account conclude that intuitions belong to the basic mental kind of intellectual seeming. I argue that rightly understood, apparent cases of intuition without belief are cases of someone having the inclination to believe that p whilst believing that (...) not-p. These can be accommodated by a disjunctive doxastic account holding that to have an intuition is to either have a belief, or to have an inclination to believe. I conclude that intuitions reduce to beliefs, and that there is no need to acknowledge intellectual seemings as basic states in our mental taxonomy. (shrink)

The paper focuses on the Epistemic Challenge for mind-independent accounts of modality. The challenge can be formulated as an inconsistency problem among three premises and, therefore, any strategy to meet the challenge will require the negation of (at least) one of its premises. The aim of the paper is not to offer a positive solution to the challenge, but rather to argue for the claim that to follow a hybrid strategy is probably the best way to meet it. With some (...) qualifications, reasons are given as to why empiricism should be the way to meet the challenge as far as de re modality is concerned, whereas rationalism might be the correct way of addressing it for the case of de dicto modality. (shrink)

The question of how we can know anything about ideal entities to which we do not have access through our senses has been a major concern in the philosophical tradition since Plato's Phaedo. This article focuses on the paradigmatic case of mathematical knowledge. Following a suggestion by Gödel, we employ concepts and ideas from Husserlian phenomenology to argue that mathematical objects – and ideal entities in general – are recognized in a process very closely related to ordinary perception. Our analysis (...) combines Husserl's insights into the nature of perception and the phenomena of evidence and justification with case studies of the role of intuition in axiomatic set theory. (shrink)