Spinoza and the Logical Limits of Mental Representation

This paper examines Spinoza’s view on the consistency of mental representation. First, I argue that he departs from Scholastic tradition by arguing that all mental states—whether desires, intentions, beliefs, perceptions, entertainings, etc.—must be logically consistent. Second, I argue that his endorsement of this view is motivated by key Spinozistic doctrines, most importantly the doctrine that all acts of thought represent what could follow from God’s nature. Finally, I argue that Spinoza’s view that all mental representation is consistent pushes him to a linguistic account of contradiction

The Nozick Game

In this article I introduce a simple classroom exercise intended to help students better understand Robert Nozick’s famous Wilt Chamberlain thought experiment. I outline the setup and rules of the Basic Version of the Game and explain its primary pedagogical benefits. I then offer several more sophisticated versions of the Game which can help to illustrate the difference between Nozick’s libertarianism and luck egalitarianism

Spinoza and the problem of other substances

ABSTRACTMost of Spinoza’s arguments for God’s existence do not rely on any special feature of God, but instead on merely general features of substance. This raises the following worry: those arguments prove the existence of non-divine substances just as much as they prove God’s existence, and yet there is not enough room in Spinoza’s system for all these substances. I argue that Spinoza attempts to solve this problem by using a principle of plenitude to rule out the existence of other substances and that the principle cannot be derived from the PSR, as many claim.Abbreviation: PSR: Principle of Sufficient Reason

Rational Tilings of the Unit Square

A rational n-tiling of the unit square is a collection of n triangles with rational side length whose union is the unit square and whose intersections are at most their boundary edges. It is known that there are no rational 2-tilings or 3-tilings of the unit square, and that there are rational 4- and 5-tilings. The nature of those tilings is the subject of current research. In this project we give a combinatorial basis for rational n-tilings and explore rational 6-tilings of the unit square

The Poincar\'e-extended ab-index

Motivated by a conjecture concerning Igusa local zeta functions for intersection posets of hyperplane arrangements, we introduce and study the Poincar\'e-extended ab-index, which generalizes both the ab-index and the Poincar\'e polynomial. For posets admitting R-labelings, we give a combinatorial description of the coefficients of the extended ab-index, proving their nonnegativity. In the case of intersection posets of hyperplane arrangements, we prove the above conjecture of the second author and Voll as well as another conjecture of the second author and K\"uhne. We also define the pullback ab-index generalizing the cd-index of face posets for oriented matroids. Our results recover, generalize and unify results from Billera-Ehrenborg-Readdy, Bergeron-Mykytiuk-Sottile-van Willigenburg, Saliola-Thomas, and Ehrenborg. This connection allows us to translate our results into the language of quasisymmetric functions, and-in the special case of symmetric functions-make a conjecture about Schur positivity.Comment: Expanded implications for matroids (Remark 2.15, Examples 2.16-17), connections to Zeta functions (Remark 2.26), and a new section about quasisymmetric functions (Section 3

Enumerating Parking Completions Using Join and Split

Given a strictly increasing sequence t with entries from [n] := {1, . . . , n}, a parking completion is a sequence c with |t| + |c| = n and |{t ∈ t | t 6 i}| + |{c ∈ c | c 6 i}| > i for all i in [n]. We can think of t as a list of spots already taken in a street with n parking spots and c as a list of parking preferences where the i-th car attempts to park in the ci-th spot and if not available then proceeds up the street to find the next available spot, if any. A parking completion corresponds to a set of preferences c where all cars park. We relate parking completions to enumerating restricted lattice paths and give formulas for both the ordered and unordered variations of the problem by use of a pair of operations termed Join and Split. Our results give a new volume formula for most Pitman-Stanley polytopes, and enumerate the signature parking functions of Ceballos and Gonz´alez D’Le´on