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In Arabic treatises on algebra, Book II of Euclid’s Elements quickly became a traditional work of reference, especially for justifying quadratic equations. However, in many of these treatises we find a representation of Euclid’s notions that deviates from the “original Euclid.” In this article, I focus on the way in which propositions of Book II were understood and reported by al-Karaǧī (11th c.) in two of his algebraic writings. Inspired by the variety of arithmetical practices of his time, al-Karaǧī transposed these Euclidean propositions from geometrical objects to numbers and applied them to an algebraic context. This allowed him to combine various argumentative strategies deriving from different fields. Building upon al-Karaǧī’s work, al-Zanǧānī (13th c.) no longer needed to mention Euclid and instead conceived of a justification of quadratic equations (the “cause” of the equation) which is completely internal to algebra. These case studies provide evidence for the use of the Elements as a toolbox for the development of algebra. More importantly, they shed further light upon a typical feature of medieval mathematics, namely the existence of a plurality intrinsic in the name “Euclid.”
Avicenna’s distinction between external existence and mental existence is seminal to logic and philosophy in the Islamic tradition. This article examines philosophers who depart from Avicenna’s external-mental existence framework. They view the former as failing to support a general analysis of reality and truth, as mental existence is neither necessary nor sufficient for analyzing propositional truths, i.e., true propositions are true irrespective of “the very existence of minds” and “the perceptual acts of perceivers.” They propose that Avicenna’s semantics for categorical propositions needs revision, as there are true metathetic and hypothetical propositions, i.e., subject terms need not exist – in external reality or in a mind – for such propositions to be true. This counter-Avicennan current of thought articulates a third distinction in the analysis of reality, which focuses on the mind-independent nature of propositional content – particularly propositions with empty, hypothetical, or impossible subject terms – as a way to think generally about reality, in contrast to the Avicennan emphasis on the existential status of terms and essences. Notably, the analysis of mind-independent reality is supported by a novel semantics of “real” (ḥaqīqī) categorical propositions, which avoids external and mental existence conditions.
Nous entendons dans cet article éditer, traduire et analyser un texte datant du xiiie siècle dans lequel figurent des preuves arithmétiques de la proposition selon laquelle la somme des carrés de deux nombres impairs ne peut pas être un carré. Cette proposition avait déjà été démontrée par al-Ḫāzin au xe siècle par le biais des propositions 3 et 5 du livre II et de la proposition 22 du livre IX des Éléments d’Euclide.
Avicenna is well-known for rejecting Aristotle’s dichotomy between perception and the intellect by introducing the so-called estimative power, which connects perception and the intellect. The estimative power is similar to sensory cognition because what is estimated is always mixed with the sensibles. Additionally, the proper object of estimation is the individualised macnā, which seems similar to the object of the intellect as the intelligible macnā. Given the special role of estimation, scholars have recently begun debating whether Avicenna has a conceptualist theory of perception. This article contributes to that debate by focusing on Avicenna’s discussions about the perception of externals in Al-taclīqāt. I argue for a reading that steers between Mohammad Azadpur’s conceptualist reading and Luis Farjeat’s anti-conceptualist reading. For Avicenna, the presence of the sensible form in a sensory power is non-conceptual, but the perceptual judgement exhibits a weak epistemic conceptualism.
This study tries to shed further light on Avicenna’s (d. 1037) philosophical and linguistic innovations as suggested in his various accounts of the problem of individuation. To better contextualize his discussions, a background is given from both Porphyry’s (d. 305) Isagoge and Fārābī’s (d. 950) remarks in his Isāġūǧī. I have also enumerated all the candidates for the principle of individuation in Avicenna’s œuvre. It is argued in this paper that the pre-Avicennian Peripatetic tradition hardly engaged, both epistemologically and ontologically, with individual per se as having its own unique identity. Instead, individual was ontologically treated as instantiation of universals and epistemologically it was inquired about to the extent that it could be only told apart. Introducing the notion of individuation as tašaḫḫuṣ, instead of the traditional individuation as tamayyuz, Avicenna offers a new way of looking at intra-species differences for a more complex understanding of the individual per se. According to this view, individual with its unique šaḫṣiyya must be understood on its own through sense perception. This approach appears to propose that the individual should not be deemed as subordinate to Aristotelian universals whose assemblage, in Peripatetic thought, was vainly expected to lead to the knowledge and definition of the individual.
Public spaces, as places of consumption, are windows onto unequal economic structures. In this chapter, I discuss different aspects of real and perceived inequalities in Tehran. I demonstrate that massive structural changes, such as the expansion of infrastructure and public transportation, have facilitated access to different parts of Tehran and a more equal experience of the city, yet different forms of inequality persist and are reproduced. Many public spaces offer a variety of opportunities for using space, ranging from walking in a public park to eating in high-end restaurants, all in very close proximity. Depending on what can be consumed and where it happens, public spaces bring inequalities to the fore as different groups often segregate within the same public space, following patterns that usually correlate with their ability to pay for products and services. Thus, in Tehran, as much as urban development may appear to work as an equalizer – bringing different socioeconomic groups together in newly shared public spaces – it highlights economic and social inequalities and makes disparities even more visible.
In Chapter 6, I offer a narrative of how Tehran, as both a physical reality and a conceptual entity, captures the imagination of its residents. The chapter is organized around two emerging cities. The first is a material city that is sometimes admired as “modern,” “developed,” or “comparable to other modern capitals,” and sometimes criticized as “a betrayal of Tehran’s history,” “superficial,” “fake,” “a parody of other cities, with no authenticity.” I explore a second emerging city, a perceptual Tehran, through the narratives that engage with the city as a symbolic entity. Through these expressions, I lay out how Tehran is perceived by its residents, showing that identifying with the city is common and that place identities are more influenced by a sense of belonging to the city than to specific neighborhoods. Furthermore, Tehran has become a new source of inspiration for an unprecedented number of artworks and literature in recent years. Accordingly, while the chapter explores perceptions of the city through narratives of its residents, it also draws on examples of works of art and literature to examine how the city is reproduced and, thus, remembered and celebrated.
In this chapter, I argue that geographical location and spatial orientation influence how residents of Tehran think about who they are and how they define and negotiate boundaries. In Tehran, the spatial locations of self and others in the hierarchical structure of the city remain signifiers of social status, yet the use of public spaces in different parts of the city and easier access to these spaces have complicated established social relations. I discuss how social, symbolic, and spatial boundaries are negotiated in a changing urban environment and how such processes create a sense of belonging or alienation – of being included or excluded – in different spaces. Furthermore, I show that while dividing lines among social groups in Tehran are conceptually powerful, they are not entirely class-based; rather, they are defined by a complex set of values and relations that are constantly questioned and renegotiated in public spaces.
The Introduction situates the book within the context of urban sociology, highlights the importance of the study, and outlines the arguments and contributions. I discuss my approach to the study of public spaces as multilayered sociological entities, rather than mere physical containers of events, people, and the built environment of cities. Studying how public spaces function at the city level, I argue that the meanings and values assigned to places are closely tied to where they are located and how they are used. Approaching public spaces as places where economic, political, gender, and social hierarchies are both reinforced and undermined, I show the complexity of social relations and coexistence in a rapidly changing urban environment. Key themes from urban sociology, sociology of culture, and inequality will be used to lay out the book’s arguments and contributions. I will also discuss my methods and provide an overview of the rest of the chapters.
Economic inequality is not the only form of inequality in urban contexts. In this chapter, I discuss other forms of marginalization in public spaces. Although my main focus is on social relations among citizens, the state’s control of public spaces is consequential in creating and sustaining structural inequalities that directly or indirectly impact social relations in public spaces. Whether controlling appearance and behavior (particularly for women) or suppressing certain belief systems and lifestyles, these state-imposed restrictions create inequalities that extend well beyond economic inequality in use of space. I argue that discriminatory laws or conventions (especially against women and those whose lifestyles or beliefs are not aligned with the ideals promoted by the state) are translated into unequal power relations in public spaces. This chapter examines how these inequalities impact perceptions of class and culture as social groups interact in public spaces and how public spaces are used to create spaces of being and belonging for marginalized groups.
In the conclusion, I bring the components of the book together, arguing that the findings in each chapter relate to a broad framework that explains the social functions and meanings of public spaces. I discuss how perceptions of self and others, in both the economic and cultural senses, act as essential components of urban experience. Through these discussions, this concluding chapter lays out the opportunities and limits of studying public spaces as a means of understanding social relations in changing urban contexts, and it suggests potential paths for future research.