Mathematical Philosophy Seminar

I shall describe a simple historical thought experiment showing how our attitude toward the continuum hypothesis might easily have been very different than it is. If our mathematical history had been just a little different, I claim, if certain mathematical discoveries had been made in a slightly different order, then we would naturally have come to view the continuum hypothesis as a fundamental axiom of set theory, necessary for mathematics, indispensable even for the core ideas of calculus.

According to representationalism about physical quantities, there is nothing inherently numerical about (say) the property five kilograms mass. Rather, mass properties bear certain higher-order relations to each other which license a representation theorem of the form: “There is a unique function that assigns pairs of properties to real numbers and respects these higher-order relations.” This function can be thought of as a ratio function, and we call a property “five kilograms mass” when, if we feed it and the property we called “one kilogram” into the function, the number 5 comes out. 

Representationalism has many adherents, and works well for scalar quantities. We know when a family of properties licenses a representation theorem. But there are also vector quantities in science, and representationalism has had precious little to say about them. In this paper I provide an axiomatization of a system of gizmos and show that the elements of any model of the system are well-represented by a vector space over a real field. We thus extend the representationalist project a more general class of physical quantities.

In this presentation, we will consider the categoricity problems, or Carnap’s problems, of some non-classical logics. We will first look into the basics of three non-classical logics, K3, LP and FDE and introduce the categoricity problems. Then we will review the categoricity results of LP and K3 proved by Tabakci (2024) and my correction to one of the results. After that we will prove some categoricity results for FDE and a theorem connecting those of FDE with those of LP and K3.

This paper offers a historical and philosophical analysis of the exchange between Christiaan Huygens and René de Sluse from 1669 to 1672 concerning their solutions to “Alhazen’s problem,” the determination of the point of reflection on a spherical mirror given the positions of the eye and the object. Huygens first provided a general solution using analytic geometry, to which Sluse responded, through Henry Oldenburg, with an alternative general solution that he considered superior. This initiated a polite but sustained debate in which both mathematicians produced additional solutions to the problem, based on their different choices of coordinates and the geometric constructions derived from them. While the technical aspects have received scholarly attention, the motivations for repeatedly providing solutions to an already-solved problem remain unexplored. I argue that the repeated formulation of different general solutions, beyond showcasing their mathematical expertise, also reflects their competing views on the proper application of Cartesian mathematics to concrete problems, as well as their interpretations of epistemic criteria such as elegance and simplicity in mathematics. Their correspondence thus serves as an arena where such criteria within the emerging discipline of analytic geometry were articulated and negotiated through actual problem-solving practices.

For the past 15 years, Woodin’s HOD conjecture has been a driving force in the study of large cardinal hypotheses. Although it is perfectly clear what it would mean to prove the conjecture, it is more subtle to say what it would take to refute it. That is not to say that the HOD conjecture is ill-defined, or that it fails to be a mathematical problem. In this talk, which is based on joint work with Gabriel Goldberg, I seek to clarify the problem and its current status.

In a series of works, Szabó (1995, 2001), Bana and Durt (1997), and Rédei (2010, 2012) developed the Kolmogorovian censorship hypothesis, according to which all the probabilities associated with a quantum state can be recovered as Kolmogorovian probabilities provided they are modeled as conditional probabilities of outcomes conditional on choices of measurement. However, these authors encountered a roadblock in trying to extend their construction to an uncountable number of possible measurements. Such an extension is necessary to recover all the probabilities associated with the quantum state. After examining ways to overcome the roadblock, Rédei (2012) declared that there is “no hope” of completing the construction, and theorists have since abandoned the censorship project. If the project is indeed doomed, this also calls into question a number of positions in the philosophy of quantum mechanics according to which classical logic and probability theory can be recovered via an appropriate solution to the measurement problem. In this talk, based on joint work with Benjamin Feintzeig (University of Washington), we argue that, pace Rédei, such an extension is in fact possible, using the Kolmogorovian theory of regular conditional distributions (or rcds). Thus, it is possible to recover quantum probabilities as Kolmogorovian probabilities. Whether this is ultimately the physically correct way of viewing these probabilities is, of course, another question, which I will touch on but not aim to settle.