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This chapter focuses on Native American concepts of time and process. Western culture often perceives time as a linear progression that advances from past to present to future in a straight line. In contrast, many Native American cultures observe that the rhythm of the world is circular, as is the life of all beings and forms. In these cultures, time tends to be experienced as cyclical and rhythmic, rather than linear and progress oriented. Most Native American languages, for instance, do not have past and future tenses; they reflect instead a perennial reality of the present. These differing perceptions of time have contributed to the misunderstandings that characterize so many interactions between Native and non-Native Americans.

This chapter examines Thomas Burnet's The Sacred Theory of the Earth, an influential account of the earth's history based on the text of the Bible, published in the seventeenth century. In The Sacred Theory, Burnet, who considers himself among the first last men, articulates the myth of the last of the race in its most extreme form; yet, his text contains all the major factors that were to bring about the myth's development: the diminishing apocalyptic convictions, the tension between individual and communal attitudes, and the concept of linear time. The Sacred Theory is a manifestation of traditional beliefs and among the first futuristic fictions. Burnet's creation of the Sacred Theory was seen to be taking over God's role as ‘Author’ while also underscoring the scriptural history's fictional aspect.

A central problem in the study of comparative cognition is to identify an animal's psychological representation of information in its environment. A psychophysical approach to studying the representation of time seeks to identify the relation between psychological (that is, subjective) and physical estimates of time. There is a long history of controversy over the psychophysical function for time. Some of the earliest investigations in experimental psychology focused on identifying aspects of the psychophysical relation described above. Later research suggested that psychological time is linearly related to physical time. The hypothesis that subjective estimates of time are linearly related to physical time is referred to as linear timing. There is a growing body of recent evidence which suggests that subjective estimates of time are nonlinearly related to physical time; this will be referred to as the nonlinear timing hypothesis. The observation that time perception across many ranges is nonlinear may provide a basis for the development of a unified theory of timing that encompasses the discrimination of temporal intervals across several orders of magnitude — from milliseconds to days.

This chapter assesses Western ideas about early Indian notions of time and history. At the turn of the eighteenth century, the theory emerged that the Indian sense of time was entirely cyclic, was tied into infinite recurring cycles, and did not therefore recognize historical change; and in the absence of a sense of history there was no differentiation between myth and history. Cyclic time was seen as diametrically opposite to linear time and linear time was associated with dialectical change. Two hundred years later, the received wisdom on the subject remains largely unchanged. The chapter discusses that constructions of time in early India are also linked to the study of astronomy and mathematics. It argues that the cyclic time too has a genesis and a termination and its dichotomy with linear time is not as distinct as it seems. Narratives in this period, therefore, have an underlying sense of time: it is sequential, moving from the earliest to the most recent

This chapter explores the nexus of time, truth, and death as it emerges hermeneutically from the symbolic world of medieval kabbalah. It focuses on the letters alef, mem, and tau, the consonants of the word emet, “truth,” which stand respectively for beginning, middle, and end, the three points of the curvature of the timeline. The correlation of truth and divinity underscores that truth, which embodies in its semiotic constellation the triadic structure of temporality, is the mark of the divine eternally becoming in time. The letters of emet are examined to discern something of the truth of time manifestly concealed in the time of truth, the beginning that cannot begin if it is to be the beginning, the middle that remarks the place of origin and destiny, and the end that is the figuration of the impossible disclosing the impossibility of figuration. Linear time is eternalized in the circular rhythms of the sacred time of liturgy and ritual, a process exemplified especially in the celebration of Sabbath.

Linear systems theory, based on the mathematics of vector spaces, is the backbone of all “classical” DSP and a large part of statistical machine learning. The basic idea -- that linear algebra applied to a signal can of substantial practical value -- has counterparts in many areas of science and technology. In other areas of science and engineering, linear algebra is often justified by the fact that it is often an excellent model for real-world systems. For example, in acoustics the theory of (linear) wave propagation emerges from the concept of linearization of small pressure disturbances about the equilibrium pressure in classical fluid dynamics. Similarly, the theory of electromagnetic waves is also linear. Except when a signal emerges from a justifiably linear system, in DSP and machine learning we do not have any particular correspondence to reality to back up the choice of linearity. However, the mathematics of vector spaces, particularly when applied to systems which are time-invariant and jointly Gaussian, is highly tractable, elegant and immensely useful.

This chapter examines Shakespeare and Donne’s use of the die pun, referring both to mortal death and orgasmic death, in order to argue that the works of both authors utilize the die pun as an escape from the confines and damages of linear time through the pure potentiality of embodied time. Throughout Songs and Sonnets, Donne uses imagery of perpetual, pre-orgasmic sex as a way to imagine escaping the destructive passages of time. Shakespeare uses the same pun to create productive, autoerotic self-replication in the first fifteen sonnets to the Young Man. For Shakespeare and Donne, language, specifically the die pun, is a vehicle for the manipulation of both meaning and time. Puns, as these two poets deploy them, deny both a unifying central meaning and diametrically opposed binary meanings; instead, they create a multiplicity of meanings manifested in embodied time.

This chapter introduces the main themes and arguments of the book, claiming that, in the later eighteenth century, European art music began to take seriously the flow of time from past to future. Until then, music was simply “in time”; “earlier” and “later” mattered little to the way it was experienced and understood. From that point on, music added the experience of linear time, of time's arrow, to its essential subject matter. It could no longer be experienced with understanding unless one recognized the temporal ordering of events. The chapter also argues that this change in the shape of musical time was not a development internal to music alone, but rather, with the onset of modernity, part of a larger transformation in the way educated Europeans began to conceive of time.

This chapter tells the story of the transition from the premodern Christian moral-political outlook to the modern post-Christian worldview, providing a larger context for the musical developments, a context for understanding how people got, in a mere sixty years, from a work like the St. Matthew Passion to a work like Don Giovanni. It presents the perspective from which the music is seen here, recounting the emergence of human autonomy and a consequent transformation in the fundamental experience and understanding of how time is shaped. This is the story of how people exchanged time's cycle for time's arrow, and why. In their thinking about both the moral and the natural realms, the moderns shifted the balance of their esteem decisively from eternity, rest, and immutability toward time, motion, and change. Modernity, scientific as well as moral-political, is at bottom an attempt to emancipate linear time.

This chapter returns to the Passion's opening chorus to ask why it is here, before the story of the Passion even gets under way, that Bach wants to abolish time. It shows that in the Passion, time is nested within the structurally and ontologically more primordial timeless eternity embodied, as it happens, in what is musically the work's most substantial layer. The story of humanity that is the implied context of Jesus's story possesses the same complex temporality, the same embedding of the linear flow of time within the framework of eternity which people find in Bach's setting. The linear time of human earthly history is not infinite; it had a beginning and will come to an end. “Before” and “after” there is God's infinite time, eternity. It is this fundamental structure of irreversible time embedded in eternity that Bach replicates in the Passion.

This chapter draws on texts by Walter Benjamin and Hannah Arendt to establish the framework of the book as a whole. For both Arendt and Benjamin, the narrative form is the predominant expression of historical memory, and this chapter emphasizes how both thinkers articulate its relation to the finitude of human experience by virtue of its having a beginning, middle, and an end. While Arendt argues that this structure has been lost in the open-endedness of modern conceptions of progress and Benjamin suggests that its loss has contributed to the demise of storytelling as individual remembrance, the latter nonetheless suggests that something of human finitude has been retained in modernity, even within its commitment to never-ending progress. In the course of the exploration, the chapter argues that this structure of open-endedness can provide a resource for theorizing historical narratives in terms of their withheld endings, or as an experience of reading about the past that is charged with a future-oriented suspense. It pursues this jarring experience of reading the past as one which has the potential to suspend or interrupt a straight-forward conception of linear time.

This chapter analyzes the television series, Supernatural, which breaks traditional linear narration in numerous episodes. It argues that by departing from a chronological structure and also deconstructing seemingly fixed temporal markers such as death, Supernatural self-reflexively draws attention to the constructed nature of (television) narratives while also highlighting the cultural construction that is the concept of linear time. This program is thus indicative of a larger trend in our contemporary society in which the differentiation between objective and subjective time has evolved into conceiving of temporality as discontinuous and fragmented.

This chapter shows the extent to which the self-confidence that culminated in The Magic Flute was undermined as the Revolution failed to achieve its democratic goals, arguing that the traditional image of Beethoven as the tone poet of the heroic Revolutionary and Napoleonic history, while not false, is one-sided. The Beethovenian abstraction out of time is the obverse of the Beethovenian heroic quest and its temporal teleology. Similarly to Rousseau's writing, Beethoven's composition is torn between the ideal of engagement in the historical social world and the wish to disengage from it, to escape into the private refuge beyond or within. In Beethoven's Ninth Symphony, the ideal of engagement in the historical social world of an emancipating autonomous humanity is nostalgically commemorated as an infinitely postponed utopia, thus transcending in a complex way the simple opposition of cyclical and linear time.

Chapter 3 offers a close reading of the book’s framing poems. In these two texts, the narrator of Qohelet outlines the understanding of time, upon which he continues to build throughout his book. The framing poems reflect explicitly upon the cosmic, temporal structure, describing it as a reality of continuous repetition. They consider how the human experience of life in time fits within this temporal framework, focusing in particular on the tensions between the cosmic, temporal reality and the life-experience of the individual. The framing poems argue that there is no real human continuity in time to match that of the world. Rather, the individual’s finite existence, linearly shaped, is on a collision course with the cosmic temporal order. In particular, structured world time hinders human cognition: human beings are not able to understand the temporal reality within which they live. An excursus discusses the main leitmotif of the book, hebel.

This chapter focuses on the computation of the saddle-point equilibrium of a zero-sum continuous time dynamic game in a state-feedback policy. It begins by considering the solution for two-player zero sum dynamic games in continuous time, assuming a finite horizon integral cost that Player 1 wants to minimize and Player 2 wants to maximize, and taking into account a state feedback information structure. Continuous time dynamic programming can also be used to construct saddle-point equilibria in state-feedback policies. The discussion then turns to continuous time linear quadratic dynamic games and the use of dynamic programming to construct a saddle-point equilibrium in a state-feedback policy for a two-player zero sum differential game with variable termination time. The chapter also describes pursuit-evasion games before concluding with a practice exercise and the corresponding solution.

This introductory chapter provides an overview of time-delay systems. Time-delay systems, also called systems with after-effect or dead-time, hereditary systems, equations with deviating argument, or differential-difference equations, are ubiquitous in practice. Some representative examples are found in chemical industry, electrical and mechanical engineering, biomedical engineering, and management and traffic science. The most common forms of time delay in dynamic phenomena that arise in engineering practice are actuator and sensor delays. Due to the time it takes to receive the information needed for decision-making, to compute control decisions, and to execute these decisions, feedback systems often operate in the presence of delays. The chapter then illustrates the possible methods in control of time-delay systems. This book develops adaptive and robust predictor feedback laws for the compensation of the five uncertainties for general linear time-invariant (LTI) systems with input delays.