Non-Euclidean Geometry

Non-Euclidean geometry began as an inquiry into a possible weakness in Euclid’s Elements and became the source of the ideas that there are geometries of spaces other than the one imagined in elementary geometry and that many mathematical theories, not only in geometry but in algebra and analysis, can be fully and profitably axiomatized.

Space in the Seventeenth Century

During the seventeenth century, a profound shift occurred: whereas space and its structure were not major subjects of philosophical analysis in the early part of the century, by 1800 they had become central to many debates. This shift was due to the influence of Leibniz and Newton: despite their famous disagreements, they agreed on the limitations of Cartesian natural philosophy and on the importance of analyzing space and its structure.

Geometry and Visual Space from Antiquity to the Early Moderns

This chapter examines the development of a geometrical framework for understanding and explaining spatial aspects of visual perception, including perception of the sizes, shapes, and positions of things in the field of view. The structure of this framework is built on the fact that vision typically occurs in straight lines (rectilinearly). Within this framework, the chapter selectively focuses on size perception. This focus allows for a comparative examination of how a single problem was treated geometrically by various theorists, ancient, medieval, and modern. The theorists examined are Euclid and Ptolemy, who were extramissionists, and Ibn al-Haytham, Kepler, Descartes, and Berkeley, each of whom adopted, in one way or another, an intromissionist scheme. In comparing Descartes and Berkeley, notice is taken of Berkeley’s interpretive bent in treating Descartes’s account of distance perception in a way that requires mental calculation, where Descartes sometimes offered psychophysiological mechanisms (avoiding mental calculation).

Space, Vision, and Faith

Linear perspective served as a useful tool in the pursuit of verisimilitude in early modern art. But the technique’s enduring appeal among visual artists from this period did not stem from its ability to transcribe the world’s appearance alone. Rather, as this Reflection highlights, it was the perspectival image’s capacity to build a direct, one-to-one rapport with a viewer that most excited art practitioners. Painters like Piero della Francesca and Giovanni Bellini took full advantage of linear perspective’s potential for simulating vision. The spaces they constructed in altarpieces and other devotional imagery often showed supernatural realms that clearly defy human optical function. This essay presents several examples of early modern works that evoke spiritual vision through linear perspective constructions.

Space in Ancient Times

This chapter tells the story of the way in which, in ancient Greek thought, space first came to be established as an independent and unified dimension. The story begins with prephilosophical as well as philosophical understandings of space, in which spatial notions are often not clearly distinguished from time and matter. This leads to difficulties accounting for motion and change. While Plato’s Timaeus conceives of time and space for the first time as two independent magnitudes, this chapter shows that they are assumed to be different to such a degree that it is unclear how they could be related to each in an account of motion and change. The task of distinguishing time and space in a way that they can, nevertheless, be intelligibly related, is finally accomplished in Aristotle’s Physics. There, time and space are both conceived as (distinct) continua, which can be combined.

Space in Kantian Idealism

Kant’s conception of space can be understood only against the background of the famous debate between Leibniz and Newton in the early eighteenth century. Throughout his life, Kant sought a middle path between the philosophical extremes represented by Leibniz and Newton, but he eventually concluded that only transcendental idealism is able to explain the knowledge of nature expressed in Newton’s theory of universal gravitation, whose postulation of action at a distance among material bodies Kant embraced. Kant’s argument for transcendental idealism involves both his conception of geometry as providing synthetic a priori knowledge, but also his contention that the pure concepts of the understanding, such as substance and community, must be “schematized” or infused with spatiotemporal content if Newton’s science of nature is to be explained.

Ants in Space

Patterns in space can be used to infer ecological and evolutionary processes. This is a foundational practice in the study of natural history. To make good inferences, patterns must be viewed at the appropriate scale. Humans may perceive and study phenomena on scales that accord with their own experiences rather than the experiences of the organism under consideration. Using the global invasive Argentine ant, Linepithema humile, as a case study, this reflection shows how a bias toward studying phenomena at the anthropocentric scale, relevant to the imagination and experience of human researchers, can lead to bizarre ideas about the lives of other organisms. To avoid such scaling problems in the venture to know something about another being it is essential that research stays with the embodied experience of that being.

A Mathematical Sculptor’s Perspective on Space

In this Reflection, a mathematician discusses four sculptures he created to express important aspects of various kinds of spaces, including ordinary Euclidean, hyperbolic space. The sculptures represent the transcription into physical objects of conceptual ideas concerning figures and the spaces they inhabit. Different sculptures exhibit various aspects of different spaces: e.g. whereas a handheld sculpture may lack orientation, thereby exhibiting an aspect of classic Euclidean space, a large sculpture fixed to the ground may have an orientation, thereby exhibiting an aspect of physical space.

Space for Thought

This Reflection concerns how the brain represents space and how such spatial representations may relate to our cognitive abilities. Space is central to how the brain encodes information, whether it concerns what we see, hear, or feel or how we move through our environment. Two different kinds of spatial signals have been observed in the brain: maps, in which different neurons are responsive to different locations of external stimuli, and meters, in which neurons are sensitive to a broad range of locations but can signal the position of a stimulus via an overall level of activity. These spatial codes may be recruited in the brain not only for processing the immediate spatial environment but also for thought and language. Evidence for this view comes from patterns of spatial sensory and motor metaphors in language and from brain-imaging studies suggesting a relationship between the neural substrates for language and those deployed for sensory and motor processing. Such parallels in functionality may have emerged in an evolutionary process of duplicating the brain’s primary sensory and motor areas and repurposing them for new tasks, i.e. our cognitive abilities.

Imagine a Place

Proclus conceived of complex kinematic constructions in geometry as involving motions of points and lines that produced figures like conic sections within the phantasia (imagination) of the reasoner. He contended that in such constructions, the geometer projects various figures in three-dimensional space and not on a two-dimensional screen, as many commentators believe. Hence Proclus believes in what one might call the three-dimensionality of imagination. He also presents the remarkable argument that space should be understood as a kind of intelligible matter, where the latter is understood to be three-dimensional (not merely two-dimensional). Finally, the chapter discusses the extent to which Proclus foresees various early modern conceptions of space.