Condensed mathematics, developed by Clausen and Scholze over the last few years, proposes a generalization of topology with better categorical properties. It replaces the concept of a topological space by that of a condensed set, which can be defined as a sheaf for the coherent topology on a certain category of compact Hausdorff spaces. In this case, the sheaf condition has a fairly simple explicit description, which arises from studying the relationship between the coherent, regular, and extensive topologies. In this paper, we establish this relationship under minimal assumptions on the category, going beyond the case of compact Hausdorff spaces. Along the way, we also provide a characterization of sheaves and covering sieves for these categories. All results in this paper have been fully formalized in the Lean proof assistant.

This paper provides a detailed exploration of the sale of offices in China and France in the seventeenth and eighteenth centuries. In France, the sale of offices became deeply integrated into the officialdom, effectively serving as a formal institution. It was a part of the financial framework and contributed to the operation of the bureaucracy. However, this practice led to public dissatisfaction due to concerns about the fairness of the judicial system. Conversely, in China, the sale of offices was less formalized and more directly associated with corruption due to the close connection with informal income and the challenge to the formal system of imperial civil examinations. It was considered an ad hoc, informal, and pragmatic solution to financial emergencies. The sale of offices, in a different context, was seen as having both positive and negative aspects, with its impact varying, depending on the specific function it served.

This Article describes and analyzes the role that various supranational actors from the European level play in the rise of formalization in the European states of the legal framework surrounding their judiciary. It starts from the observation that those supranational actors act as a driving force towards increased formalization via a variety of recommendations, reports, or decisions. It first provides several concrete examples of where the supranational actors act as such a driving force towards more formalization. In a second step, the Article looks more generally at how the various supranational actors think about informality surrounding the judiciary and tries to discern the rationales that underlie their position. In a third step, it assesses the various advantages and disadvantages of the rise in formalization that is propagated by the supranational actors. In doing so, the Article shows how the topic of informal judicial rules and practices cannot be understood fully without having due regard to the supranational level and contributes to the literature on the relationship between formal and informal institutions.

Purity is known as an ideal of proof that restricts a proof to notions belonging to the ‘content’ of the theorem. In this paper, our main interest is to develop a conception of purity for formal (natural deduction) proofs. We develop two new notions of purity: one based on an ontological notion of the content of a theorem, and one based on the notions of surrogate ontological content and structural content. From there, we characterize which (classical) first-order natural deduction proofs of a mathematical theorem are pure. Formal proofs that refer to the ontological content of a theorem will be called ‘fully ontologically pure’. Formal proofs that refer to a surrogate ontological content of a theorem will be called ‘secondarily ontologically pure’, because they preserve the structural content of a theorem. We will use interpretations between theories to develop a proof-theoretic criterion that guarantees secondary ontological purity for formal proofs.

According to Karl Popper, we can tell good theories from poor ones by assessing their empirical content (empirischer Gehalt), which basically reflects how much information they convey concerning the world. “The empirical content of a statement increases with its degree of falsifiability: the more a statement forbids, the more it says about the world of experience.” Two criteria to evaluate the empirical content of a theory are their level of universality (Allgemeinheit) and their degree of precision (Bestimmtheit). The former specifies how many situations it can be applied to. The latter refers to the specificity in prediction, that is, how many subclasses of realizations it allows. We conduct an analysis of the empirical content of theories in Judgment and Decision Making (JDM) and identify the challenges in theory formulation for different classes of models. Elaborating on classic Popperian ideas, we suggest some guidelines for publication of theoretical work.

This article develops an integrated perspective to study whether formalization can significantly reduce precariousness for informal workers. This perspective combines the analysis of employment dualism with that of rural–urban dualism and the analysis of the production sphere with that of the social reproduction sphere. By applying this integrated framework to the case of a state-owned enterprise (SOE) in China, this article finds that formalization does little to reduce precariousness for the migrant agency workers there. Migrant agency workers in China are in a precarious position not only because of their employment status but also because of their incomplete citizenship and the commodification of social reproduction materials. With the compensation gap between formal and agency workers narrowed primarily owing to the deterioration of formal employment, formalization has little effect on increasing the income of agency workers or alleviating the financial pressure upon them in the sphere of social reproduction; neither can formalization raise migrants up to full citizenship or reduce related precariousness.

Scholars have long argued that institutional context significantly influences business strategy and economic performance. Research on the relationship between institutions and business strategy, however, has overwhelmingly focused on the decisions of larger, established corporations, mostly neglecting the strategic thinking of smaller, more entrepreneurial ventures. This article seeks to correct this bias by focusing the analysis directly on the critical decision of small-scale entrepreneurs to move from the informal and largely unregulated sector into operation as formal companies. Using a unique dataset and ranking of provincial governance institutions from Vietnam, the authors show that improvements in institutions make firms more likely to choose the formal sector from the start and, for those who do not, to spend less time in the informal sector. The study also finds that property rights have a more salient impact on formalization than other types of institutions.

This article identifies the problems and malfunctions of the Chinese apartment ownership regime that necessitates its formalization. The benefits of formalizing the apartment ownership, such as resolution of the social housing problem, and establishment of the mortgage-backed real estate market, are presented. Thereafter, we indicate that the success of formalizing apartment ownership in China depends on one hand on the extent to which the unique property market in China can be reconciled with acknowledged ways of dealing with apartment ownership in various other jurisdictions acting as comparative examples. On the other hand, simply grafting borrowed laws or legal institutions into a new context cannot achieve the desired effects. Legal culture matters and it cannot be downplayed in the formalizing process. Our conclusion is that the booming but immature Chinese apartment ownership market will benefit greatly if the casual link between the market economy and well-defined property rights are confirmed through legal formalization.

We propose a criterion to regard a property of a theory (in first or second order logic) as virtuous: the property must have significant mathematical consequences for the theory (or its models). We then rehearse results of Ajtai, Marek, Magidor, H. Friedman and Solovay to argue that for second order logic, ‘categoricity’ has little virtue. For first order logic, categoricity is trivial; but ‘categoricity in power’ has enormous structural consequences for any of the theories satisfying it. The stability hierarchy extends this virtue to other complete theories. The interaction of model theory and traditional mathematics is examined by considering the views of such as Bourbaki, Hrushovski, Kazhdan, and Shelah to flesh out the argument that the main impact of formal methods on mathematics is using formal definability to obtain results in ‘mainstream’ mathematics. Moreover, these methods (e.g., the stability hierarchy) provide an organization for much mathematics which gives specific content to dreams of Bourbaki about the architecture of mathematics.