Methodology and free will

The existence question asks whether actual human agents have free will. By contrast, the compatibility question asks whether free will is compatible with determinism. Most philosophers hope that answering the compatibility question will subsequently help to answer the existence question. However, standard approaches to free will tend to result in empirical and dialectical stalemates about how to analyze the concept of free will. At the very least, these approaches leave us unable to definitively establish the application or reference conditions of the concept, and so they make it difficult to answer not only the compatibility question, but also the existence question. The main goal of Chapter 1 is to survey a number of problems with standard approaches to free will, which rely heavily on intuitions. The aim in doing so is to motivate an alternative approach, which does not rely on intuitions.

Naturally free action

Adopting the alternative approach motivated in Chapter 1, this chapter argues that free will is a natural kind, by relying on the influential idea that natural kinds are homeostatic property clusters (HPCs). The resulting HPC natural-kind view about free will answers the existence question positively: free will exists and we act freely. Moreover, it does so without directly addressing the compatibility question, although the view favors compatibilism over libertarianism. The chapter also rebuts a prominent objection to natural-kind views about free will, including the HPC view. Finally, the HPC view builds on Andrew Sims’s recent view that agents are a natural kind and it yields an appealing alternative to standard approaches as well as to recent revisionist approaches to free will and moral responsibility.

Contradiction Medium and the Existence Question

The process of human understanding of the world starts from viewing complex chaos, proceeding to monism followed by the contradiction theory, and finally returning to complexity theory, but without giving up the pursuit of monism. Lao-Tzu and Heraclitus put forward their own theories of unity of opposites at almost the same time. The thought of unity of opposites has long been contained in the theory of yin and yang and the Book of Changes. In the ontology of information evolution, existence and nonexistence (you and wu in Chinese) can also be roughly interpreted as a contradictory relationship. Existence and nonexistence are two opposing worlds. Our understanding of existence needs medium. We can only indirectly grasp the current meaning of existence after the transmission of multilayer mediums and the distortion and loss of information. Aristotle mentioned the notion of medium, but the real world cannot be explained by his ideas. All transformational processes of existence rely on medium. The transformation process of existence and nonexistence is different from the transformation process in the domain of existence. There is no need to rely on medium.

Contradiction Medium and the Existence Question

The process of human understanding of the world starts from viewing complex chaos, proceeding to monism followed by the contradiction theory, and finally returning to complexity theory, but without giving up the pursuit of monism. Lao-Tzu and Heraclitus put forward their own theories of unity of opposites at almost the same time. The thought of unity of opposites has long been contained in the theory of yin and yang and the Book of Changes. In the ontology of information evolution, existence and nonexistence (you and wu in Chinese) can also be roughly interpreted as a contradictory relationship. Existence and nonexistence are two opposing worlds. Our understanding of existence needs medium. We can only indirectly grasp the current meaning of existence after the transmission of multilayer mediums and the distortion and loss of information. Aristotle mentioned the notion of medium, but the real world cannot be explained by his ideas. All transformational processes of existence rely on medium. The transformation process of existence and nonexistence is different from the transformation process in the domain of existence. There is no need to rely on medium.

On dimensions supporting a rational projective plane

A rational projective plane ([Formula: see text]) is a simply connected, smooth, closed manifold [Formula: see text] such that [Formula: see text]. An open problem is to classify the dimensions at which such a manifold exists. The Barge–Sullivan rational surgery realization theorem provides necessary and sufficient conditions that include the Hattori–Stong integrality conditions on the Pontryagin numbers. In this paper, we simplify these conditions and combine them with the signature equation to give a single quadratic residue equation that determines whether a given dimension supports a [Formula: see text]. We then confirm the existence of a [Formula: see text] in two new dimensions and prove several non-existence results using factorization of the numerators of the divided Bernoulli numbers. We also resolve the existence question in the Spin case, and we discuss existence results for the more general class of rational projective spaces.

Abstract Objects

An abstract object is a non-physical, non-mental object that exists outside of space and time and is wholly unextended. For example, one might think that numbers are abstract objects; e.g., it is plausible to think that if the number 3 exists, then it is not a physical or mental object, and it does not exist in space and time. Likewise, one might think that properties and relations are abstract objects; e.g., it is plausible to think that if redness exists, over and above the various red balls and red houses and so on, then it is an abstract object—i.e., it is non-physical, non-mental, non-spatiotemporal, and so on. Other kinds of objects that are often taken by philosophers to be abstract objects are propositions, sentence types, possible worlds, logical objects, and fictional objects. The view the that there are abstract objects—known as platonism—is of course extremely controversial. Many philosophers think there are just no such things as abstract objects. Philosophers who endorse this antiplatonist view have to endorse some other view of objects of the above kinds—i.e., numbers, properties, propositions, etc.; in particular, in connection with each of these kinds of objects, they have to say either that these objects are physical or mental objects or that there are just no such things. There is a vast literature on the existence and nature of abstract objects. This article focuses mostly (but not entirely) on the existence question—that is, the question of whether there are any such things as abstract objects. In addition, it focuses to some extent (though, again, not entirely) on the specific version of this question that is concerned with the existence of abstract mathematical objects.

Symmetric Union Diagrams and Refined Spin Models

An open question akin to the slice-ribbon conjecture asks whether every ribbon knot can be represented as a symmetric union. Next to this basic existence question sits the question of uniqueness of such representations. Eisermann and Lamm investigated the latter question by introducing a notion of symmetric equivalence among symmetric union diagrams and showing that non-equivalent diagrams can be detected using a refined version of the Jones polynomial. We prove that every topological spin model gives rise to many effective invariants of symmetric equivalence, which can be used to distinguish infinitely many Reidemeister equivalent but symmetrically non-equivalent symmetric union diagrams. We also show that such invariants are not equivalent to the refined Jones polynomial and we use them to provide a partial answer to a question left open by Eisermann and Lamm.

A FAMILY OF TWO-DIMENSIONAL LAGUERRE PLANES OF KLEINEWILLINGHÖFER TYPE II.A.2

Kleinewillinghöfer classified Laguerre planes with respect to linearly transitive groups of central automorphisms. Polster and Steinke investigated two-dimensional Laguerre planes and their so-called Kleinewillinghöfer types. For some of the feasible types the existence question remained open. We provide examples of such planes of type II.A.2, which are based on certain two-dimensional Laguerre planes of translation type. With these models only one type, I.A.2, is left for which no two-dimensional Laguerre planes are known yet.

A New Approach for Examining $q$-Steiner Systems

One of the most intriguing problems in $q$-analogs of designs and codes is the existence question of an infinite family of $q$-analog of Steiner systems (spreads not included) in general, and the existence question for the $q$-analog of the Fano plane in particular.We exhibit a completely new method to attack this problem. In the process we define a new family of designs whose existence is implied by the existence of $q$-Steiner systems, but could exist even if the related $q$-Steiner systems do not exist.The method is based on a possible system obtained by puncturing all the subspaces of the $q$-Steiner system several times. We define the punctured system as a new type of design and enumerate the number of subspaces of various types that it might have. It will be evident that its existence does not imply the existence of the related $q$-Steiner system. On the other hand, this type of design demonstrates how close can we get to the related $q$-Steiner system.Necessary conditions for the existence of such designs are presented. These necessary conditions will be also necessary conditions for the existence of the related $q$-Steiner system. Trivial and nontrivial direct constructions and a nontrivial recursive construction for such designs are given. Some of the designs have a symmetric structure, which is uniform in the dimensions of the existing subspaces in the system. Most constructions are based on this uniform structure of the design or its punctured designs.