First and foremost, I am aware that a similar question has been asked here and has been touched upon elsewhere. I have found these discussions very compelling but a bit light on external reference, and so wanted to approach the topic on different terms and in one place.

As such, I was wondering if anyone might have recommendations for material on the history of constructions of 'mathematical spaces.' By this, I do not mean a 'mathematical space' in perhaps the most restrictive formal sense, i.e., a set with additional structure. I have a more general interest in the history which pertains to the placement of objects—real numbers, shapes, etc.—in relation to one another. I realize, to some of you this may sound the same; I simply mean to say that I wish to approach this topic rather permissively. In this vein, if anyone sees it fit to bring up material on the history of coordinate systems, geometry, rings, groups, lattices or whatever else, I would more than welcome it.

I've seen this question approached as a topological one—which I suppose it must be to some extent. However, to be clear, if the question were approached as such, I would not be so much interested in the history of topology as the history of the various spaces which topology has studied.

In particular, I'm very interested to learn of alternative spaces which may have developed in antiquity, so material which deals with the more distant past is very much appreciated.

As an aside: I am, at least in part, inspired by the Euler Spiral, Ulam Spiral and other kinds of 'spiral space' which have received some, shall I say, more recent attention. That said, I am curious to learn of even more 'out there' ways of constructing mathematical space.

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    $\begingroup$ Topological spaces are not really related to coordinate systems, the former are from general topology and the latter from algebraic/differential geometry. I can not tell if you interested in spaces (like Ulam spiral) or ways to coordinatize "nice" ones (like Euler spiral, which is identical to the straight line topologically). Topoi is the plural of topos, they are a recent innovation of Grothendieck's, and are probably a red herring given your older time range. $\endgroup$ – Conifold Apr 20 '19 at 0:41
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    $\begingroup$ On the early history of coordinates see When do we see for the first time the use of the Cartesian coordinates? I am still not sure what "spaces" mean, but see Famous Curves Index, Struik's Outline of a History of Differential Geometry, and Origins of Differential Geometry and the Notion of Manifold on Math SE. You may also have in mind differential topology. $\endgroup$ – Conifold Apr 20 '19 at 3:01
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    $\begingroup$ Euclidean space is actually physical space in a thin disguise. So you might be interested in various physical inventions such as twistor space, Rindler coordinates etc which could be retrospectively illuminating. Abstract treatment of coordinates is difficult to find before the medieaval period, e.g. in Nicolas Oresme. $\endgroup$ – sand1 Apr 20 '19 at 8:06
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    $\begingroup$ Historically physical spaces have always come before their mathematical descriptions and/or counterparts. Leafing through Max Jammer's Concepts of space might offer some insights, e.g. that for Aristotle space is a limited collection of places, which obviously is not euclidean space Before Descartes the accepted view was that geometry is about space and algebra is about numbers, but powers above 3 were dificult to conceive, e.g. x^6 was explained as a cubo-cube (3+3) or a cubed plane (3x2). $\endgroup$ – sand1 Apr 20 '19 at 21:42
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    $\begingroup$ (ctd) Introducing the concepts of "synthetic" (pure) geometry and "analytic" (coordinate) geometry attempted to save the distinction. Today we understand that Descartes' brillant idea is what we call a "cartesian product". $\endgroup$ – sand1 Apr 20 '19 at 21:42

This article is concerned to formulate some open problems in the philosophy of space and time that require methods characteristic of mathematical traditions in the foundations of geometry for their solution. In formulating the problems an effort has been made to fuse the separate traditions of the foundations of physics on the one hand and the foundations of geometry on the other. The first part of the paper deals with two classical problems in the geometry of space, that of giving operationalism an exact foundation in the case of the measurement of spatial relations, and that of providing an adequate theory of approximation and error in a geometrical setting. [...]

Suppes, P. (1973). Some open problems in the philosophy of space and time. In Space, Time and Geometry (pp. 383-401). Springer, Dordrecht.

The development of the concept of space was an evolutionary process in early Greek history. Four major periods have been identified (historically and philosophically) by shifts in the way that the Greek Philosophers as a whole looked at the role of space in physical reality. Within each of these periods, the Mythopoeic, the Material, the Non-Definite and the Definite, those philosophers who made contributions to the overall development of the concept of space are discussed. The major themes discussed throughout this essay deal with defining and refining the concept of space and differentiating space from the concepts of time and matter, while keeping track of the influence of the concept of mathematical as opposed to physical spaces.

Beichler, J. E. The Ancient Greek Concept of Space.

Although the concept of space is of fundamental importance in both physics and philosophy, until the publication of this book, the idea of space had never been treated in terms of its historical development. It remained for Dr. Jammer, noted scholar and historian of science, to trace the evolution of the idea of space in this comprehensive, thought-provoking study. The focus of the book is on physical, rather than metaphysical, ideas of space; however, philosophical or theological speculations are discussed when relevant. The author has also given special attention to the cultural settings in which the theories developed. Following a Foreword by Albert Einstein and an introductory chapter on the concept of space in antiquity, subsequent chapters consider Judaeo-Christian ideas about space, the emancipation of the space concept from Aristotelianism, Newton's concept of absolute space and the concept of space from the eighteenth century to the present. [...] An abundance of meticulously documented quotations from original sources and numerous bibliographic references make this an exceptionally well-documented book.

Jammer, M. (1954). Concepts of Space: The History of Theories of Space in Physics. Courier Corporation.

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I am going to keep a running list of suggested material here:

"Discovery and Development of Coordinate Systems", Stack Exchange

History of Analytic Geometry, Carl Boyer

"When do we see for the first time the use of the Cartesian coordinates?", Stack Exchange

"Abscissas and Ordinates", David Pierce

"Oresme and the Invention of Graphing", Mumford

"Outline of a History of Differential Geometry: I", D.J. Struik

"Origins of Differential Geometry and the Notion of Manifold", Stack Exchange

Concepts of Space, Max Jammer

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