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The Hidden Side of Visualization

Agustin A. Araya
San Jose State

In the arc of history that extends from early human communities to present day societies, the history of seeing occupies a prominent place. Intimately related to thinking and language, and crucial in most of our comportments, seeing and its history co-determines the essential history of human beings, that is, the history of their essence. Writing, a fateful development in human history, in a sense is an attempt at seeing spoken language. As we move towards our present day, milestones in the history of seeing become increasingly visible, such as printing, perspective drawing, musical notation, tele-scopy, micro-scopy, photo-graphy, cinema, and tele-vision. Computersupported visualization, that is, the use of computers to visualize 'things', is but the currently last element in this series, series whose end is nowhere in sight.

Although a relatively young discipline, computer visualization has already expanded in multiple directions. It can help us to visualize that which is too small and complex for us to see, as in the visualization of the structure of complex molecules, or that which is too big for us to grasp, such as the planet Earth itself. It can also support the visualization of that which only exists in our imagination, without a counterpart in the 'real' world, as in certain applications of Virtual Reality. Visualization can also be used to visualize phenomena that are not visible or sensible in themselves, as in the visualization of mathematical constructs or of abstract relationships between pieces of information. 'Visualization' has even been called the second computer revolution because it goes beyond conventional uses of computers, decidedly stepping into the 'cognitive' domain.¹

A variety of reasons account for the current emergence of computersupported visualization. From the practitioner's perspective, as the raw power of computing and communications technologies grows—accompanied by a significant growth in the amount of data, information, and 'knowledge' available through that technology—the human 'cognitive' capabilities that allow us to deal with them are 'stretched to the limit'. It is this increasing 'mismatch' 'threatening' human-computer interaction that visualization addresses. As the need for innovative interactive technologies grows and, simultaneously, enabling computing technologies such as three-dimensional graphics mature, the terrain is fertile for the rapid dissemination of computer supported visualization.

Standing, then, at the edge of a possible era in which interacting with visualizations and using them in a variety of activities would be as common place, if not even more so, as it is today to write with the help of a computer, we ask the following questions. What is a computer-supported visualization? What is it that we encounter when we interact with a visualization? If visualizations and the machines that generate them turn out to be new kinds of entities, exhibiting novel ontological traits, could this technology transform us in subtle but fundamental ways? And what kinds of transformations would they be?

We start by examining the motivations and principles underlying the technology of computer-supported visualization. After characterizing the approach we will follow to determine what visualizations are and introducing the notions of ontological operations and biases, we enter into a detailed analysis of what we encounter when interacting with visualizations. Because of the close relationships between visualizations on one hand, and geometry as well as applications of geometry in science, on the other, we perform an in-depth exploration of the ontological operations embedded in geometry and its applications. We proceed by developing an ontological reinterpretation of Edmund Husserl's reconstruction of the origin of ancient geometry, of Galilean science, and of Cartesian geometry.

After a critical appraisal of Husserl's reconstruction and of our own reinterpretation of it, we apply the ontological approach to determine both the ontological operations embedded in visualization machines and the ontological traits of the interactive visual worlds supported by these machines. We are then in a position to determine, from an ontological perspective, what is that which we encounter when we become involved with interactive visual worlds. Finally, we explore potential transformations of fundamental ontological traits characterizing us, humans—in particular, transformations of our 'being in the world'—that could emerge from our pervasive interaction with visual worlds. For a related analysis of another kind of computing technology, namely, Ubiquitous Computing see (Araya, 2000).

Computer-Supported Visualization

Before embarking on an ontological analysis of the technology of visualization, we need to gain an understanding of its potential scope and principles. We now consider both the ostensible motives behind the emergence of visualization technology as well as its guiding principles, as they are perceived from within the discipline itself. Preceded by various developments in computer science and other disciplines, computer-supported visualization made its formal appearance in the decade of the nineteen eighties, in terms of a report of the National Science Foundation addressing several 'problems' mainly arising in the context of the scientific community.² A central problem was the so-called 'informationwithout- interpretation dilemma'. As the number and power of sources of data—such as satellites, medical scanners, and supercomputers—increases, the amount of data they make available far surpasses the capabilities of scientists and professionals to process them. Exacerbating the situation, the phenomena under study, such as large molecules, the human brain and body, earth climate, or the large-scale structure of the 'universe', are increasingly complex entities themselves, whose study requires the collection of large amounts of data.

Two additional problems stressed by the report were the means of communicating results among scientists and the ways of providing for close interaction between scientists and the computational analysis of data as it unfolds, to be able to 'steer' the computation in promising directions. As a solution to these problems the report proposed the development of visualization technology:

Scientists need an alternative to numbers. A technical reality today and a cognitive imperative tomorrow is the use of images. The ability of scientists to visualize complex computations and simulations is absolutely essential to insure the integrity of analyses, to provoke insights and to communicate those insights with others (Bruce McCormick? , Thomas DeFanti? , and Maxine Brown 1987, p. 7).

Finally, the report proposed the "implementation of a federally funded ViSC? (Visualization in Scientific Computing) initiative." In addition to the relevance of the problems it addressed, a significant strength of this initiative was that it did not appear in a vacuum, but rather it attempted to build upon a number of existing scientific and technological areas that had developed to a large extent independently of each other, such as Computer Graphics, Image Processing, Computer Vision, Computer-Aided Design, Signal Processing, and Human-Computer Interaction.

In the National Science Foundation initiative the focus was on the interpretation and communication of 'physical' data, that is, data originating from physical sources such as the earth and the human body. But there are other sources of data, more properly referred to as 'information'—such as office, business, and financial information—which are increasingly significant and in which similar problems arise. An important difference between visualizing physical data and visualizing information, understood in the previous sense, is that while in the first case the characteristics of the source of data provide 'natural' visualizations of it, this is not generally the case with information. In consequence, a second area of computer-supported visualization emerged, which has become known as information visualization.³

Visualization technology is now being utilized to support a variety of technological and scientific activities and is a very active area both in terms of research and applications.

Visualization Principles

But, what are the principles underlying computer-supported visualization? What are the fundamental tenets that sustain this discipline, make possible the use of visualizations in the analysis and interpretation of data and information, and orient its future development? Let us identify these principles and examine the way in which they have been understood from within the technical discipline itself.

Scientific and Information Visualization have been characterized as "the use of computer-supported, interactive, visual representations of data [and information] to amplify cognition." (Stuart Card, Jock Mackinlay, and Ben Shneiderman 1999, p. 6, our brackets) A visualization can be a figure (e.g., a map adequately enhanced with the use of colors and patterns, or an image obtained by combining photographic images with computer-generated images), a diagram (e.g., a three-dimensional graph represented in twodimensional space which can be rotated and expanded), or any other kind of visual representation. A crucial characteristic of these visualizations is that it is possible to 'interact' with them with the help of specialized devices. Such interactions allow us to 'manipulate' visualizations as we manipulate things in the 'real' world, and to perform other kinds of manipulations that are not usually possible.

Visualizations contribute to 'amplify cognition'. Those human capabilities and potentialities that come into play when performing a task of analysis and interpretation of data are regarded as cognitive capabilities or 'resources', and conceived as information processing operations. Once cognition is regarded as a human resource for processing information, amplifying or augmenting cognition means increasing, extending, or improving aspects of this resource. Visualization would amplify cognition in a variety of ways:

Visualizations can expand processing capability by using the resources of the visual system directly. Or they can work indirectly by offloading work from cognition or reducing working memory requirements for a task by allowing the working memory to be external and visual...Visualizations allow some inferences to be done very easily that are not so easy otherwise (Card, Mackinlay, and Shneiderman 1999, p. 16).

Interacting with visualizations is regarded as important for furthering our understanding of complex things or systems. In the case of computergenerated visualizations of four-dimensional objects such as hypercubes, translating, rotating, or expanding them can make understandable what at first appears as a confusing collection of interconnected lines.

To render these various notions in a more condensed way, Card et al. characterize the purpose of visualization as "using vision to think," where thinking is understood as a central element of cognition. What emerges from this complex of notions is what we will call the thinking with visualizations principle. This principle is oriented to 'enhance' a thinking that is assisted, mediated, and carried out by interacting with visualizations in a computersupported 'space'. But there is another notion that, although related to the 'thinking with visualizations' principle, brings into sharper relief what computer-supported visualization is about. In his foreword to Richard Friedhoff and William Benzon's Visualization, The Second Computer Revolution, Richard Gregory states that:

The central point that this book makes is that the newly discovered preconscious processes of human vision can be tapped and used to powerful effect by computer images—especially by computer graphics to suggest ideas. Perhaps its most powerful form is interactive graphics, where the hand can control and change the image, much as though it is a solid object lying in the familiar space of the object world. (Friedhoff and Benzon 1989, p. 8, Gregory's italics).

In the same vein, Friedhoff and Benzon consider as crucial the notion that "images have a special ability to trigger, in a controlled way, the exceedingly refined mechanisms of human visual perception." What does this mean?

That beyond the raw power of computing machines to calculate and to store massive amounts of data there appear to lie certain possibilities by which, through an 'increasingly tight coupling of humans and machines', unknown or little known potentialities of the 'brain and mind' could be uncovered and put to use. By means of images with special properties and, possibly, by allowing forms of interaction which go beyond everyday kinds of manipulations of things, it could be possible to trigger preconscious visual mechanisms and put them to work, thus amplifying our cognitive capabilities and thinking. To a limited extent, existing visualization techniques already achieve this. As Friedhoff and Benzon indicate,

Computer graphics provides a seamless fusion between the massive processing power of the visual system and the power of the digital computer...computer graphics, because it bonds mind and machine in a unique partnership, creates an entirely new way of thinking (p. 82, our italics).

Under this characterization of computer graphics, which is one of the most advanced technologies used to generate visualizations, we find two additional and powerful principles that underlie the whole enterprise of visualization. First, what we will call the fusion principle. That is, the notion that cognitive capabilities can be amplified and augmented by integrating, merging, even more, 'fusing' humans and computers, via the utilization of interactive visualizations.⁴ And, second, the 'principle of the possible transformation of thinking by the use of visualizations', which we will call the transformation of thinking principle. That is, the possibility that by using interactive visualizations having properties that could 'trigger' preconscious visual processes in new ways, new kinds of thinking could arise.

Finally, we will mention two other principles at play in the context of visualization. Because the visual system is especially adapted to perform certain kinds of tasks, computer-supported visualizations should be oriented towards those tasks. This leads to what Friedhoff and Benzon call 'objectification': "a phenomenon, whether it is inherently visual or not, should be represented as something that has form, color, texture, motion, and other qualities of objects." (p. 169) We will refer to this basic notion as the objectification principle, which is oriented to make visible that which is not. Closely related to this principle is the principle of naturalism. As stated by Friedhoff and Benzon, "the central issue in computer graphics today is naturalism. The goal is to dispatch forever the angular, harshly colored images of the past and to move towards images that are so realistic as to be indistinguishable from photographs." (p. 85). This principle can be regarded as complementary to objectification in that, once something has been objectified, it strives to give to it a high degree of 'realism'.

We have identified five principles, that is, the principles of thinking with visualizations, human-computer fusion, transformation of thinking, objectification, and naturalism, which characterize the area of computersupported visualization as perceived from within the field itself. These principles can be articulated as follows. While the thinking with visualizations principle establishes the 'fundamental purpose' of this technology, namely, the amplification of cognition, the fusion principle, advocating greatly intensified human-computer interaction, specifies the means by which to achieve such amplification. On their part, the objectification and naturalism principles establish particular ways in which human-computer fusion can be achieved. Finally, the principle of the transformation of thinking points towards the possibility that thinking not only be amplified but also transformed as a consequence of the play of the four other principles.

But, what are these principles? What gives them their authority, that they can ground a technological discipline as a whole? We will return to these principles and questions later in the work, once we have gained a deeper understanding of what visualizations themselves are.

Before we conclude this section we need to consider two interrelated questions. First, will this new technology ever be able to transcend the boundaries of research laboratories to become part of the world of work? Second, even if indeed the technology were finally transplanted to the world of science, engineering, medicine, and other disciplines, could this technology make the leap into everyday life, as computers themselves are increasingly doing? Let us hear what practitioners tell us in this regard:

Information visualization is a body of techniques that eventually will become part of the mainstream of computing applications ... At certain points, the development of technology crosses barriers of performance and cost that allow new sets of techniques to become widely used. This, in turn, has effects on the activities to which these techniques are applied. We believe this is about to happen with visualization technology and information visualization techniques. Information visualization is a new upward step in the old game of using the resources of the external world to increase our ability to think (Card, Mackinlay, Shneiderman 1999, p. 34).

Because interactive visualizations rest upon two fundamental and intrinsically related human capabilities, namely, the capabilities 'to see' and 'to manipulate' things, visualization technology has the potential for being used in any activity whatsoever, not just in those specialized activities of the world of work. In addition, other developments of a technological and 'social' character—such as global computer networks and ubiquitous computing—are powerfully contributing to the already 'overflowing river of data, information, and knowledge' that has become accessible in everyday life, creating a 'problem' that visualization technology may contribute to 'solve', thus opening the door for the penetration of everyday life with this technology.

Ontological Approach

Given our aim to understand what visualizations are and, subsequently, to identify ways in which, in the context of highly technologized communities, the pervasive use of visualization technology could invite essential transformations on the way we—humans—are, how should we approach the analysis of this technology? We could start by attempting to determine what the devices enabled by this technology are. If we were to ask a designer of a technological device 'what such a device is', we may be able to learn about the uses that can be made of it, how such uses are supported by the various components of the device, the decisions that were made in its design, and possible justifications for these decisions. If we were to put such question to researchers engaged in the technologies that make a device possible, we could learn about the principles underlying the technologies, as we did in the previous section, as well as about its potentialities and limitations. We could also gain an understanding about what a device is by asking the users of the device, who may be able to identify the reasons why the device is useful in certain situations, and how is best used.

But what would all of this amount to? It would give us an understanding of the technology and related devices from the perspective of 'the present', that is, from a point of view which is almost entirely subsumed within the confines of how things appear to us today. Because the essential ways in which technologies unfold in the course of their long gestation period are for the most part invisible to us, that which is immediately accessible of them constitutes the 'given', the 'historically transmitted'—the obvious—which although familiar to us in its immediacy, becomes largely incomprehensible as we start probing beyond the immediate.

A possible way to free ourselves from this 'tyranny of the present' is to attempt to develop a 'reconstruction' of essential moments in the gestation of a technology, moments that may remain hidden in it but continue to be determinant of its power and of our encounters with the technology. We have to take seriously and assign to it all the weight that it deserves, the notion that human communities in the long span of history they have traversed so far, have not only created myriad artifacts of all kinds but, most important, have been able to create new kinds of beings and entities exhibiting novel ontological traits with respect to what preceded them. Similarly, human communities have developed a variety of practices of all kinds, sometimes giving rise to practices of a new kind relative to those previously known.

Because these new kinds of entities and practices typically take long to develop and to take hold of a community, it is difficult for us to perceive them as novel and to determine in what their novelty consists of. We call ontological operations, or refer to them as operations having ontological import, to certain kinds of human practices or specific actions that take place in the context of practices, which give rise to new kinds of entities and, possibly, new kinds of practices exhibiting novel ontological traits. Similarly, we call ontological biases to certain tendencies in our encounters with things that make us take them to be in certain ways that are essentially different from what they have been to us in the past. Thus, an ontological bias may imply an ontological transformation in the making, but not yet completed.

How do these operations obtain their ontological transformative power? In most cases isolated operations will not give rise to ontologically novel entities. It is only when they are grouped together with related operations in larger practices that they may achieve that capability. In addition, they need to spread themselves among other practices and to acquire a weight within the larger community to the point they begin to supersede other competing practices. Finally, what entitles us to say that certain entities created in the context of human practices have novel ontological traits? That they possess essential traits that they do not share with other known kinds of entities.

If we were able to reconstruct the ontological operations responsible for the emergence of a particular kind of technology, in particular, computersupported visualization, this would give us a good starting point to consider the possibility that in our encounters with devices enabled by such technology certain essential traits that characterize us may be transformed in subtle ways. Our aim in the remainder of this work, then, is to examine the technology of computer-supported visualization from this historic-ontological perspective.

What Do We Encounter In A Visualization Situation?

When we encounter a visualization in the course of an activity, what is that which we become involved with? In encountering a thing, in this case a 'technological thing', what emerges in the encounter is determined, on one hand, by what we bring to the encounter—that is, a particular comportment—and, on the other, by what the thing itself brings. In this section we will concentrate primarily on the technological thing itself, and will ask the following question: What is a visualization?

A Visualization Situation

As a point of reference we will consider the visualization of large molecules, whose characteristics are typical of a large class of visualizations. In comparison with other cases, such as the visualization of four-dimensional objects or of complex mathematical functions, this is a rather conservative domain. Choosing it over the other cases has the advantage that it is relatively simple to understand, and that whatever we may learn from it most likely will also be valid of more radical cases, while the converse may not be necessarily true.

Ball-and-stick models of molecules, made out of wood or plastic material, are commonly used to visualize their structure. Because it is impractical to develop these kinds of models for large molecules, and because they are static and, thus, unable to visualize changes in the molecule's structure, computersupported versions of ball-and-stick models have been developed. These visualizations have many uses, including determining whether a large molecule such as a drug could attach itself to other molecules found in cells of organisms. Such visualizations show detailed, colored, ball-and-stick models on the computer screen, models that can be manipulated to modify their configuration.

In a visualization situation there are entities or systems under consideration and the purpose of the activities taking place in it is to perform certain tasks involving such entities. That which is under consideration is 'represented' in the computer in certain way, for example, in terms of a programmed 'model' of a molecule specifying the kinds of atoms that it contains, their properties, their relative positions, and the bonds between them. To facilitate the performance of tasks, the entity or system is visualized in terms of a computer-supported visualization which is under the control of a second program that produces a visual 'presentation' of the molecule, as represented by the programmed model. A variety of elements enter into play in such presentation.

Elements involved in a computer-supported visualization

A visualization is shown on a flat, two-dimensional surface, the computer screen, which is composed of point-like elements called 'pixels'. The computer screen contains 'windows', that is, rectangular entities that may overlap with each other and can be reshaped and moved within the boundaries of the screen. Windows are under the control of computational processes which result from the execution of computer programs, processes that display visualizations on the windows. A visualization is painted on a window by appropriately coloring selected pixels in the screen region occupied by the window, so as to represent lines and surfaces. The visualization may be organized as a two- or three-dimensional entity. Visualizations can be 'manipulated' by means of point-and-click devices, allowing for a variety of operations including opening, moving, and reshaping.

Visualization Machines and Dimensional Spaces

A computer-supported visualization is a presentation of something on a window under the control of a computer program. We will refer to the complex constituted by computer, screen, windows, and visualization programs as a visualization machine. A visualization, then, is a presentation generated by a visualization machine. But now, what is a visualization machine? We can approach this question by noting that visualization machines are strongly related to the so-called 'Cartesian spaces', which have become pervasive in scientific and technological activities. In a visualization machine the screen and windows function as 'spaces' in which visualization programs display shapes, typically, geometric, which are characteristic of Cartesian spaces. In what follows, we explore in detail the notion of Cartesian space, reserving for later a more precise characterization of the relationships between visualization machines and Cartesian spaces.

Cartesian spaces are constituted by an organized, infinite collection of points—the pixels in the case of the computer screen—each of which can be uniquely identified with respect to a set of 'axes', typically two or three, which intersect on a single point, the 'origin'. Axes are measuring tools that, emanating from the origin, extend to infinity. Points are nothing in themselves but 'measured space'. Because no region of this space can, in principle, escape measurement, that is, escape being uniquely identified in terms of measures, Cartesian spaces turn out to be infinite measuring devices, such that whatever there is of 'space' in them is subordinated to measuring.

Underscoring the centrality of the notion of measure to Cartesian spaces is another common way in which we refer to them, namely, as 'dimensional' spaces. A dimension is a measure of space; as a verb, it refers to an 'act of measuring.' Etymologically, it derives from the Latin dimetiri, dis + metiri, where the particle dis in one of its senses indicates an intensification of the action it modifies, thus signifying 'to measure carefully'. In what follows, we use the terms dimensional and Cartesian interchangeably to refer to these spaces.

What kinds of entities can inhabit such spaces? Entities composed of particular collections of measured points, usually constituting geometrical shapes and surfaces. If, in principle, it is possible to describe any such collection of points by individually specifying each of them in an appropriate order, this would still leave us at the mercy of an unconquered infinity. It then becomes crucial to augment the notion of measure from being a specification of a point on a scale to being a 'formula', which specifies how the points in the collection are to be obtained by means of mathematical operations. Simple incarnations of such formulas describe geometrical shapes. In a broader sense, a Cartesian space is constituted not only by an infinite collection of measured points but also by the complex kinds of measurements we have called formulas. Because of the wealth of significations and practices that obtain in these spaces they have the character of a world, constituting what we will call 'Cartesian' or dimensional worlds. But we will also refer to them interchangeably as Cartesian spaces.

A dimensional or Cartesian space, then, is an infinite measuring device that can be applied as a tool to measure anything measurable—as well as to attempt, unsuccessfully, to measure that which is essentially immeasurable.

Measurement Practices

But now, what is an infinite measuring device? Let us approach this question by way of examining what takes place in practices in which something is being measured, which we will call measuring or measurement practices. Measuring, which is an extremely broad and pervasive kind of practice taking place in everyday life as well as in technological and scientific activities, is a particular way of encountering things. In measuring something, say the temperature of a person or the length of a piece of furniture, we come to the encounter with a 'purpose in mind' and with a measuring device or instrument in hand. We 'apply' the device to the thing in question, obtain a measurement, and terminate the encounter. At the heart of such encounter, when we apply the instrument to measure the thing, we pay attention primarily to that which the instrument can measure. No matter what the thing is, while taking its measure it recedes into the background and the property or properties being measured come sharply to the fore.

For its part, the instrument plays a crucial role in the encounter, bringing with it a way of measuring as well as the particular kind of measures, that is, the 'units' in terms of which the property will be measured. Thus, the specificity of the thing is superseded by the specificity of the instrument. In applying the instrument we have to 'follow the instructions' associated with the measuring procedure and, in so doing, we have to 'adjust' ourselves to the device and its procedure, we have to 'attune' ourselves to them. In the end, what counts most in the encounter is the measurement itself, to the point that once it is obtained, the encounter comes to an end. A measurement is the 'result' of the encounter, what comes out of it, and what remains. It is the specific purpose of the encounter.

In a measurement practice, then, we have multiple encounters, including encounters with that which is being measured, with the measuring device, and with the measurement itself. But we also encounter a purpose—in that we come 'to have in mind' a measuring-related purpose—and, most important, we encounter or engage in a particular measuring procedure: We carry out an act of measuring thus becoming, for the duration of the activity, 'a being that measures'. Depending on the specific characteristics of the measurement practice and on the purposes behind it, these different kinds of encounters may be emphasized differently. In taking the temperature, say, of an animal, we may do so in the context of trying to cure the animal or in the context of an animal experiment. In the second case, as it is the measurement that counts most, we will have a different kind of encounter with the animal than in the first, in which it is the animal that matters.

Thus, a measuring comportment defines a space of possibilities in which various kinds of encounters take place, encounters which can be emphasized or accentuated differently, depending on the characteristics of the specific encounter. But it should be possible to distill certain biases and tendencies underlying such diversity, which may be characterized as follows. First, a measuring encounter unfolds in ways that are governed by the measuring procedure that, in turn, is primarily determined by characteristics of the measuring device. This is an indication of a certain primacy or preeminence of the measuring device and its associated procedure over the thing being measured. Second, during the measuring activity the property that is to be measured of the thing comes sharply to the fore, while the thing recedes into the background. This suggests the existence of 'operations' to 'put aside' the thing to attend to one of its properties. Third, the purpose of the encounter is to obtain a measurement of the thing, which is what we take with us, leaving the thing behind. This is an indication of a certain primacy or preeminence of the measurement over the thing being measured.

Because through these biases, tendencies, and operations what is being encountered in a measuring practice may be significantly altered, we will say that they have a potential ontological import. These are biases and operations which, by underlying measuring encounters, have the possibility of changing the character of what we encounter in things, ultimately, the possibility of giving birth to new kinds of beings, with peculiar ontological characteristics. What we have here are subtle biases and operations, themselves subject to change, which could transform the way we encounter things thus giving rise to new kinds of encounters, that is, new kinds of practices in which we can engage. We note that with 'advances' in technology, through which measuring devices become increasingly 'powerful' and measurements increasingly more 'complex and systematic', most likely these inner trends characterizing measuring practices are exacerbated.

Ontological Operations and Biases Embedded in Dimensional Spaces

Having gained a basic understanding of what measuring is and the kind of possibilities that are opened in it, we now return to our previous question, that is, to the question of what dimensional spaces, as infinite measuring devices, are. Because a Cartesian space is a measuring device, it should reflect in itself central characteristics of measuring practices, that is, their tendencies and operations and, given that it is an infinite measuring device, it should reflect them in an intensified way. Thus, to understand what a Cartesian space is we need to identify the specific tendencies and operations that are embedded in such a device. To this effect, we must go beyond the consideration of such a device as it appears to us in the present, and consider its 'genesis' or 'origin'. Behind the device in question hide operations and biases that generated it in the first place.

In fact, a Cartesian space is a very elaborate historical creation, resulting from the accumulation of operation upon operation, of bias upon bias during the course of centuries, literally, of millennia. Successive waves of schools of mathematics and natural philosophy—nested in and nurtured by successive historical ages—have contributed to its creation. A Cartesian space, carrying the name of one who made decisive contributions to it, is an extraordinary historical achievement, whose power in terms of infinity and universality continues to be expanded, and 'for which new and significant horizons have been opened up with the advent of the digital computer'. Computer-supported visualization is but the current last step in a long—very long—chain of developments.

What are the ontological operations and biases that have contributed to the genesis of this device? We are fortunate to have available a powerful and imaginative work which will be very helpful in approaching this question. In The Crisis of the European Sciences and Transcendental Phenomenology (Husserl 1970a), Edmund Husserl attempted to grasp, from what turned out to be a 'historico-intentional-praxical' perspective, essential moments that could account for the emergence of modern science and modern geometry with Galileo and Descartes, respectively.⁵ Central to Husserl's analysis is a characterization of the 'origin of geometry', which is a fundamental element in the notion of Cartesian space, as we are considering it here. Because of the relevance of Husserl's analysis to what concerns us, we will now consider it in detail.⁶

Husserl asks for what was 'given' to Galileo, as transmitted by the tradition, that he took it for granted and that served as a basis for his own contributions to modern science. Unless we are engaged in critical reflection, that which is given is so familiar to us that we barely have an awareness of its being there. Or, if we do have some awareness it is of something that is so close to us, something with which we have such an intimate connection that it is difficult to separate it from ourselves. It constitutes us; it is us in some sense. Because of this intimacy and the attendant difficulty in establishing a distance from which we could confront the given, this is something of which we don't typically talk about.

How does Husserl proceed to gain access to that which is given at a particular historical moment, say, the 'Galilean moment'? He does not rely primarily on Galileo's works, although certainly he assumes a close familiarity with them. Rather, from his general knowledge of a historical moment, in particular, from what was available in everyday life and in the practices that obtain in it, Husserl identifies significant elements, which on one hand it is plausible to assume as pervasive, and on the other, it makes sense to assume their relevance to, in this case, Galileo's own practices. Husserl's analysis, then, focuses on practices and transformations of practices as they take place in everyday life. In addition, these transformations are not regarded as being primarily triggered from the 'outside', but are understood as emerging from the inner development of those same practices.

Among the important givens for Galileo, geometry certainly occupies a significant place. At first, Husserl focuses on the differences between modern geometry and mathematics and their Greek counterparts, in an attempt to understand what is peculiar to the modern developments. But soon the focus changes towards the 'origin of geometry'. It is possible that after the analysis of the differences between modern and ancient mathematics, Husserl may have concluded that although they are significant in several respects, ancient mathematics and geometry had already taken steps so decisive that in order to gain a fundamental understanding of geometry it was necessary to go beyond the moderns toward the ancients, and even beyond the ancients themselves, in an effort to understand the origin of geometry.

In The Origin of Geometry,⁷ a work closely related to the Crisis in which a more radical understanding of the insights gained in the Crisis is attempted, Husserl indicates that:

...our interest shall be the inquiry back into the most original sense in which geometry once arose, was present as the tradition of millennia, is still present for us, and is still being worked on in a lively forward development; we inquire into that sense in which it appeared in history for the first time -- in which it had to appear, even though we know nothing of the first creators and are not even asking after them. Starting from what we know, from our geometry, or rather from the older handed-down forms (such as Euclidean geometry), there is an inquiry back into the submerged original beginnings of geometry as they necessarily must have been in their "primally establishing" function. This regressive inquiry unavoidably remains within the sphere of generalities, but, as we shall soon see, these are generalities which can be richly explicated, with prescribed possibilities of arriving at particular questions and self-evident claims as answers (Husserl 1970b, p. 354).

Husserl sees here the necessity of a bold kind of 'historical' inquiry, which attempts to go beyond historical facts and aims at a 're-construction' of the foundational notions of geometry. In Husserl's perspective such reconstruction attempts to grasp the 'original meanings' of geometric notions. Because we lack historical sources the attempt takes the form of a 'regressive inquiry' that takes as its point of departure, say, Euclidean geometry as it was handed-down to us, and goes back towards the origins. But what could those origins be and how could we know about them given the lack of sources? Husserl suggests that, ultimately, geometry must have emerged from the 'prescientific' world:

...even if we know almost nothing about the historical surrounding world of the first geometers, this much is certain as an invariant, essential structure: that it was a world of "things" (including the human beings themselves as subjects of this world); that all things necessarily had to have a bodily character .... What is also clear, and can be secured at least in its essential nucleus through careful a priori explication, is that these pure bodies had spatiotemporal shapes and "material" qualities (color, warmth, weight, hardness, etc.) related to them. Further, it is clear that in the life of practical needs certain particularizations of shape stood out and that a technical praxis always (aimed at) the production of particular preferred shapes and the improvement of them according to certain directions of gradualness (p. 375).

In the 'prescientific' world there are already practices oriented to the production of smooth surfaces and edges, which require estimates of sizes and, consequently, measuring techniques of varying degrees of precision. It is the gap between these origins on one hand, and ancient and modern geometry, on the other, that is necessary to bridge by reconstructing possible intermediate steps. This insight, namely, that geometry must have emerged from measuring practices already prevalent in everyday activities, provides the starting point for the approach Husserl will follow to explore possible ways in which it originated. In the Crisis, and in a summary way in the Origin, Husserl identifies several moments, which we will now consider. In the presentation below, although we follow the general thrust of Husserl's analysis, taking into account the question we have raised about the ontological operations and biases embedded in Cartesian spaces, we are freely reinterpreting, and at times extending, the analysis from an ontological perspective. In a subsequent section we will perform a critical appraisal of Husserl's approach and of our own reinterpretation of it.

Ancient Geometry

Let us start by examining basic geometric notions. If we consider what a geometric shape such as a line or a circle is, as opposed to the shape of a thing, we come to see that it is a 'limit' case of a shape. That is, it is a shape that, although for the most part may not be immediately available in our encounters with everyday things, can nonetheless be obtained by performing certain operations upon 'naturally' occurring shapes. As Husserl suggests (1970a, p. 26), in many everyday practices there are activities oriented to develop smooth surfaces and edges, such that the notion of a straight edge appears in those activities as a limit, specifically, as a 'perfect' limit to which the activity tends to. We add that the notion of something 'perfectly straight' may appear in a variety of activities, for instance, in the act of walking. To approach something we typically walk towards it maintaining a constant direction. If we 'disregard' the 'irregularities' of walking and concentrate purely on its constant direction, we are left with a straight trajectory of movement. It is at the limit, when all irregularities have been discarded, that we find a perfectly straight trajectory. Or again, if we use a string to measure the length of something, as we stretch the string it goes through a variety of shapes until it reaches one that no further stretching can change. In the limit we have a straight string.

What these scenarios suggest is that there are practices in which the notion of limit-shape arises, either as that toward which the activity explicitly tends to, or as extreme cases that emerge in the course of the activity. These practices can be regarded as including operations oriented towards a limit in which irregularities in shape are eliminated, thus constituting 'smoothing' operations. But, we need to add, there is more to these operations than that because in the limit—a limit that although never reached is still possible—a new kind of being or entity emerges, characterized by an absolutely non-irregular shape. What is novel in it, is that it is a limit, something possible but never attainable. For this reason, because through these operations new kinds of entities arise, we will refer to them as ontological smoothing operations. This last phrase can be understood in at least two senses, both of which are intended. First, it refers to 'smoothing operations' that have an ontological character because they give rise to new kinds of entities. But, second, it refers to 'ontological smoothing', that is, to operations leading to simpler, smoother kinds of beings.

Yet the question may still arise: Are these perfectly regular shapes of things 'really' new kinds of beings? Why not just say they are shapes with the very special characteristic of being perfectly regular, in a particular sense. To us, their ontological novelty resides in that they are 'limit' shapes, unsurpassable from the point of view of their regularity and unachievable in terms of concrete practices. Underlying limit shapes, there is a new kind of encounter with things which does not remain with the thing as it emerges, but attempts to surpass it absolutely by positing a 'counter-thing' that is perfectly regular in some sense.

As Husserl noted (p. 25), these transformations of 'empirical' shapes towards a limit still leave us with empirical limit-shapes. A straight edge, in its perfected shape remains a straight edge, something that is accessible to us empirically. But these perfected empirical shapes are not yet, strictly speaking, geometrical notions. Even if, with our imagination, we 'eliminate' the irregularities still present in the edge of a piece of furniture, that imagined perfect shape is not yet a 'straight line', in a geometric sense. To arrive at this last notion an additional and quite powerful operation is necessary, namely, to discard the edge of the piece of furniture, the trajectory of the movement, or the measuring string itself, in order to be left with the 'pure' straight line.

We have, then, a second kind of operation by which the 'body' of a perfected thing is erased out of existence. It is the body that is irregular, the body that contains the impurities affecting the not-yet-perfect empirical shape. How do we attain to the limit? To put it figuratively, we reach the limit by means of ontological 'surgical' operations through which we, first, separate the body of things from their perfected, totally visible 'skin' and, second, discard the former and hold onto the latter. We will refer to this kind of operations as ontological excising and lifting operations, by which a world of geometric 'idealities' is lifted from the world of everyday practices. Again, this last phrase can be understood in at least two senses. It refers to excising and lifting operations that create new kinds of beings, hence operations that have an ontological character. In addition, these operations act by ontological excising and lifting, that is, by performing a radical incision splitting the realm of experience into a realm of 'empirical' shapes and a realm of 'ideal' shapes, giving rise to excised and lifted beings.

Although a geometric shape may appear to be just a contour or line, because it is a limit-shape that has been obtained by operations of ontological smoothing, excising, and lifting it possesses unique, exquisite properties. It is a being that is not just 'pure skin', but is endowed with properties that distinguish it from any other geometrical shape. When elementary shapes are combined with each other in multiple ways, they give rise to constructions of considerable complexity, exhibiting complex properties. As a result, we have not just a collection of geometrical shapes but an entire 'world' of idealities populated by ideal geometric beings.

Once geometrical shapes have been attained something else becomes possible. By getting rid of the body of things we also get rid of what is 'unknown', invisible in them, while keeping only perfected, visible shapes. In doing so, we have created entities that are absolutely regular and absolutely visible. These two distinctive characteristics of geometric shapes make simple shapes such as lines, circles, and polygons immediately understandable, thus suggesting the possibility that all the properties and relationships that can be conceived of them could become 'fully known' to us. Hence, a new kind of practice arises which, by positing initial, self-evident assertions about geometric relationships, is able to determine the kinds of relationships that should follow from them by taking a series of steps, each of them justified by prior steps. This practice emerges by lifting operations from practices that occur in everyday life (p. 26). While in the 'empirical' world we can determine properties and relationships involving things by measuring them—achieving in this way what may be called 'empirical truths'—in the ideal geometric world we gain knowledge of properties and relationships by means of logicodeductive practices, achieving a lifted form of empirical truths, namely, 'universal truths'.

Husserl can then say:

So it is understandable how, as a consequence of the awakened striving for "philosophical" knowledge, knowledge which determines the "true," the objective being of the world, the empirical art of measuring and its empirically, practically objectivizing function, through a change from the practical to the theoretical interest, was idealized and thus turned into the purely geometrical way of thinking (p. 28).

We can come to appreciate, then, that these ontological excising and lifting operations not only give birth to new kinds of beings but, also, to new practices concordant with them, thus inaugurating a new world, even more, inaugurating a world of a new kind.

As if to confirm its character of being a world, we observe that this new kingdom of idealities has its own foundational work or 'Bible'. Euclid's Elements, which because of its influence is probably the most important mathematical text ever written, establishes the foundations of a new world.⁸ It is not so much the specific results it submits that is significant, but it is the manner of proceeding, by first laying the ground in terms of 'definitions' of the entities with which it deals—definitions by which these entities 'come to life'—followed by 'postulates' which identify basic geometric practices and properties of mathematical entities, and by 'common notions' identifying what today we could call logical axioms of equality.⁹ All of this crowned with the introduction and utilization of a new kind of practice appropriate to this new kind of world, namely, that of establishing the 'truth' of geometric properties and relationships by a rigorous logico-deductive method.

What ontological biases can be identified in these new kinds of practices? As indicated before, ontological smoothing, excising, and lifting operations have given birth to absolutely regular and absolutely visible beings. Through these practices an ontological bias is introduced, namely, a tendency towards assuming that things, be them 'ideal' or 'empirical', are fully knowable, and that they are fully knowable for us. We call this bias ontological rather than epistemological because it concerns 'essential' characteristics of things as well as our own essential traits, insofar as we deal with things, specifically, by getting to 'know' them. On the basis of this ontological tendency certain epistemological consequences should follow. We will refer to this kind of bias as the ontological bias towards full knowability of things, or ontological bias towards transparency.

We should note two related and important additional biases introduced by the previously mentioned operations. First, because through excising and lifting operations the body of things is discarded together with all its accompanying phenomena while retaining only smooth shapes, an ontological bias towards visibility is instituted. In the richly endowed world of geometry only shaperelated properties obtain while every other kind of property has been eradicated. Second, by smoothing, excising, and lifting operations, the characteristics of entities in the 'empirical' world that make them unique, concrete, distinguishing them from all others have been obliterated. Thus, geometrical entities are no longer 'individuals' but 'types'. Only in this way whatever properties they exhibit have a 'universal' character defining that particular type of shape. We will refer to this bias as an ontological bias towards the obliteration of the concrete. Through these biases—towards transparency, visibility, and the obliteration of the concrete—in our encounters with things we may be inclined to take them to be fully knowable and primarily visible, and may tend to pass over what distinguishes them uniquely from any other. By their own characteristics and by the kinds of entities they give rise to, ontological operations favor the emergence of certain biases in our encounters with things.

So far we have presented an interpretation of the main moments that Husserl identifies in relation with the basic notions underlying ancient geometry. Husserl then proceeds to examine the moments underlying the development of modern science and modern geometry, with Galileo and Descartes.

The Galilean Moment

Although our primary interest resides in understanding ontological operations and biases inherent in Cartesian spaces, a consideration of what we are broadly referring to here as the 'Galilean moment'¹⁰ is important, in particular, for the 'application' of geometry—and more generally, of mathematics—to the 'study of nature' that took place in it. Certain kinds of operations that originate in this application are also relevant to 'visualizations', as will be shown later.

To the question of 'What is given to Galileo that may have contributed to a new notion of natural science?' Husserl replies: Geometry and mathematics, the 'art of measuring', and 'applied' geometry and mathematics (p. 28). If, as indicated earlier, geometry emerged from the art of measuring, conversely, once geometry was established it itself was applied to measuring practices, helping to perfect them. In spite of their ideality, insofar as ideal geometric entities are about shapes and visible forms, they retain a strong reference to the empirical world. For this reason, they have been applied in the context of practical measuring activities. In technical activities such as engineering design, the application of geometric notions is, for the same reason, very 'natural' to make. Although these measuring practices are very diverse, covering a variety of activities, they are not yet directed to the 'study' of nature.

It is at the Galilean moment that geometry¹¹ is decisively applied to the understanding of nature. In the context of natural philosophy, ideal shapes are made to descend into the world of bodily things and empirical phenomena. While in the movement of ascent toward ideal shapes the body of things is set aside by means of ontological excising and lifting operations, in the movement of descent or application, there is again a setting aside of their bodies, this time accompanied by an additional operation where the now bodiless things or phenomena are 'assigned' an ideal shape. Thus, in the study of the movement of planets or of balls rolling on inclined planes, the empirical trajectory of the movements is set aside and assigned an elliptical or linear shape, while the 'objects', such as the sun and the planets, are represented by geometric points, circles, or spheres.¹² We should note that the application of geometry to celestial objects is very 'natural' to make, given that the large distances involved and the limitations of human visual capabilities produces a 'smoothing' of such objects.

What is the character of this setting aside of the body and the accompanying assignment of an ideal shape? From an ontological perspective, in this movement of descent of idealities towards the empirical world we recognize the emergence of yet another kind of world and its corresponding entities—the so-called 'physical world' and 'physical entities'—in which, by means of ontological reconstitution operations, the empirical world is reconstituted on a new basis. Among these operations we find what we will call ontological shape-regularizing operations, corresponding to the setting aside and assignment operations mentioned above. Through these regularizing operations, objects and phenomena are reconstituted with respect to their 'shapes' in terms of ideal geometric shapes.

We are not suggesting at this point that through these operations the actual trajectories of the planets, for whoever applies the operations, have been regularized by assuming them to be ellipses. Rather, we indicate that a new kind of world is being created, the 'physical world', in which these trajectories are regularized. It could be argued that this operation produces only an 'interpretation' of the empirical world, but no new world is created. But what is such interpretation if not something constituted by new kinds of entities created by such operations?

To better understand what takes place in the context of these reconstitution operations we need to attend carefully to additional considerations Husserl makes. Although the primary ingredients of geometry are the ideal geometric objects, equally important are the properties characterizing these objects and the myriad relationships that obtain in complex configurations of objects, as well as the apodictic method of determining them. In the reconstitution of the empirical world by geometrical means, how is this complex geometric machinery brought into play?

As Husserl indicates, already in the world of everyday practices the notion that different kinds of phenomena are related to each other in certain ways is familiar to us. Everyday things and phenomena not only appear in close proximity to other things and phenomena, but they affect each other in typical ways. In consequence, the world appears as permeated by a connectedness and relatedness by which things and phenomena depend on each other. In many cases even a clear understanding of the existence of causal relationships between phenomena develops (pp. 30-31).

What this suggests is that in the movement of descent or application of geometry, just like geometric shapes are assigned to empirical objects, the properties and relationships that obtain in the geometric world are assigned to or put into correspondence with—in an abstract sense—dependencies that obtain in the empirical world. But for this to be achieved, the notion of causality needs to be understood in a way that is amenable to the establishment of that correspondence, that is, it needs to be regularized. This regularization implies a clarification of the notion of causality, as distinct from vague notions of dependency.¹³ In the reconstituted world that emerges in this way mathematically measurable causal relationships among phenomena obtain.

Reinterpreting these notions from an ontological perspective, we realize that among the ontological reconstitution operations introduced above we need to include what we will refer to as ontological link-regularizing operations by which phenomena are understood as being essentially linked to each other by means of regularized causal relationships. It is on the basis of these operations and the application of geometry that these linkages between phenomena are regarded as causes, and understood as essentially measurable. Are we justified in referring to these operations as ontological? In what way do they contribute to the emergence of 'entities' with unique ontological traits? By transforming the way in which the relatedness and connectedness of phenomena appear to us they play a major part in the emergence of what we have called the 'physical world'. We will return to this point below.

With respect to the logico-deductive method of geometry used to establish geometric truths, at the moment of application of geometry to the empirical world that method will be transformed into the more complex method of natural science. This last method involves not only logico-deductive practices but many others, such as the development of mathematical 'measures', models, and theories, of specialized measuring instruments, and of experimental methods, thus incorporating in a transformed way elements of measurement practices.

In the application of ontological shape-regularizing operations described above, objects and phenomena are attributed geometric shapes. But shapes are only one among a multiplicity of sensible qualities obtaining in phenomena, some of which are somewhat related to shape as in the case of colors, while others appear to be totally unrelated to it, as in the case of sound, taste, and heat. How could geometry be applied to measure these other kinds of sensible qualities? We cannot do justice here to the complexity of Husserl's analysis of what he called the 'indirect mathematization of the plena' (p. 34) and will pursue it only as it is related to the issue of visualization.

Because of what we referred to earlier as ontological link-regularizing operations, in the reconstituted 'physical world' it is taken for granted that there are causal relationships between different kinds of phenomena. In particular, this assumption leads to the question as to whether for sensible qualities other than shape, which Husserl called 'specific sense-qualities', it would be possible to establish causal dependencies with shape-related qualities. If this were the case, the mathematization of specific sense-qualities could be done, first, by measuring their causal dependency with shape-related qualities, and, second, by establishing and measuring dependencies between these shaperelated qualities and other shape-related qualities of interest. After acknowledging Galileo's affirmative answer to this question,¹⁴ Husserl invites us to appreciate in its full force "the strangeness of [Galileo's] basic conception in the situation of his time."

In the reconstitution of the empirical world as a 'physical world' by means of the application of geometry, requiring that all sense-specific qualities be regarded as dependent on shape-related qualities, we identify additional ontological operations, namely ontological shape-reductive operations. These operations are based on prior ontological link-regularizing operations but involve an additional step through which all sense qualities are linked with shape-related qualities, in such a way that some kind of primacy of the latter over the former is, implicitly, established. Such operations may invite what we will call an ontological bias towards visibility in our understanding of the 'physical world'. In a sense, this bias extends to the physical world a previously identified, similar bias that obtains in the ideal geometric world.

Before concluding this examination of reconstitution operations, we need to consider an additional operation not explicitly examined by Husserl. Any concrete situation in the empirical world, even those that appear to be quite simple, involve a multiplicity of relationships. Moving 'objects' are restrained in their movement by the surrounding air or by the unevenness of the plane over which they slide; their movement is disrupted or impeded as they collide with other objects. Even after shape- and link-regularizing operations, the application of geometry to concrete situations turns out to be difficult to realize. It becomes crucial to drastically simplify the situations.

How can this be achieved? All the 'impediments' to movement are to be eradicated. Any remaining object, which is not absolutely necessary from the particular point of view from which the situation is being considered, must be discarded. Undesirable characteristics of the media surrounding the objects are to be straightened out.¹⁵ In brief, the situation as a whole must be made smooth. We will refer to these operations as ontological situation-smoothing operations. While they complement the other reconstitution operations, in a sense these smoothing operations are more powerful than them because they transform a situation as a whole. In the limit, these operations create the 'vacuum'—the smoothest of media—that is then populated by 'physical' entities, from which all that hinders the application of geometry has been removed. Such a physical world is still different from a geometric world because—even if faintly—it refers to empirical situations from which it originated.

Universalization at the Galilean Moment

What is involved in this 'strange' step by which all sense-specific qualities are taken to be relatable to and measurable in terms of shapes? This question leads us into the final step taken in Husserl's analysis of the Galilean moment: the notions of 'universalization' and 'infinitization'. Husserl is struck by a decisive movement toward universalization he senses at the dawn of modern science and modern mathematics, a multi-pronged movement that impulses natural science, geometry, and mathematics as a whole beyond the boundaries known to the ancients. In the case of modern mathematics, Husserl finds an "immense change of meaning" with respect to the geometry and mathematics inherited from the, primarily Greek, tradition. As Husserl indicates, "universal tasks were set, primarily for mathematics (as geometry and as formal-abstract theory of numbers and magnitudes)—tasks of a style which was new in principle, unknown to the ancients" (p. 21, Husserl's emphasis).

At the Galilean moment, the notion of universalization arises in at least two different but related senses. First, in the sense of a universal causal regulation obtaining among diverse phenomena. Second, in terms of "what was taken for granted by Galileo, i.e., the universal applicability of pure mathematics" (p. 38, our emphasis). With respect to the first, we already mentioned the linkregularizing operations by which phenomena are regarded as dependent on other phenomena by means of particular dependency relations understood as causal relationships. But Husserl suggests that more is required regarding causality in the quest to achieve "a scientific knowledge of the world," namely, that the world be understood, in advance, as an infinitude of causalities (p. 32). While link-regularizing operations regularize dependency relationships in terms of causality, the notion of a world conceived in advance as an infinitude of causalities has a far broader scope and announces a transformation at the level of the empirical world as a whole, as will be discussed below.

Returning now to the second sense in which the notion of universalization arises, we need to consider the universal applicability of mathematics, taken for granted by Galileo. We can approach this particular sense of universalization by considering the ontological bias toward full knowability identified in the context of ancient geometry. The ideal geometric world is populated by entities that, because of their complete regularity and visibility are, in principle, fully knowable, entirely open to us, transparent. Consistent with this characteristic of geometric objects, the logico-deductive practices employed in finding new geometric truths already had a universal character, in the sense that they could be applied to consider any geometric relationship whatsoever.

But while with the 'ancients' transparency is contained within the boundaries of the ideal geometric world, at the Galilean moment and through a series of ontological operations previously identified, this ontological bias is extended to the whole of the empirical world. It is not just particular kinds of phenomena or particular regions of beings to which geometry is applied. If at first the old distinction between sublunar—including terrestrial—and celestial phenomena, may have kept the impulse toward the application of mathematics within the boundaries of the 'world' known to humans, astronomical observations of the planets by means of the telescope—most preciously of the rings of Saturn and of the moons of Jupiter—brought these boundaries down.¹⁶

Universalization of causality and universal application of mathematics are the two faces of a sweeping movement out of which the 'physical world' was born. Nothing can escape this movement. Hence the 'strange' emergence of shapereductive operations mentioned earlier, by which non-shape related phenomena are linked to and measured in terms of shape-related phenomena. Universalization, then, can be understood as a movement by which all boundaries demarcating what is possible to know from what is beyond human possibilities are, in principle, erased, thus opening the whole of the empirical world to human perusal and understanding. This does not means, though, that humans are understood as having in practice the actual capacity to understand everything.¹⁷

Because of the transformative power of universalization in the two senses indicated above, we regard them as ontological operations but with a far greater scope than those mentioned earlier, because they operate on the empirical world as a whole. Thus, we introduce the ontological, world-scope operation of causalization that refers to the universalization of causality. Through this operation, not only causal relationships are supposed to obtain between phenomena of all kinds, but rather the new kind of world thus constituted, the 'physical world', is regarded as nothing else but an infinite network of causal relationships. While these relationships become 'entities' of the first rank, empirical phenomena no longer occupy center stage.

Similarly, we introduce the ontological, world-scope metricizing operation, which refers to the universal application of geo-metry, and mathematics to measure the empirical world as a whole. More properly, metricizing is a fundamental 'inclination' towards declaring the empirical world to be what is measurable of it, where measuring is understood as an attempt at reconstituting the empirical world as a whole in terms of the ideal, mathematical world. Because this ideal world has emerged itself as a particular distillation of the empirical world—via ontological smoothing, excising, and lifting operations—metricizing comes to be the inclination to reconstitute the empirical world on an ontologically purified basis, where full knowability obtains. As indicated, because of their scope the two operations of causalization and metricizing need to be clearly distinguished from the more focused operations previously identified.

Regarding infinitization, a notion closely related to universalization, Husserl identifies what may be called a meta-operation—in the sense that it affects all operations and practices previously mentioned—namely, the infinite perfectibility of the application of mathematics in the creation of the 'physical world' and of the corresponding scientific measuring practices. All operations and practices are understood as having in them the possibility of being carried out more exactly, more pervasively, more perfectly, thus becoming stronger. Even more, the specialized operations by which perfectibility is carried out are themselves perfectible, thus creating what may be regarded as second order, 'accelerated', enhancing effects.

We now realize that the ontological reconstitution operations and biases identified earlier, such as shape- and link-regularizing, shape-reductive, and situation-smoothing operations, owe much of their ontological character precisely to the movements of universalization and perfectibility just described. Each operation taken by itself does not seem to amount to much, but when they are applied coordinately, in a mutually-reinforcing manner, and when they are regarded as having a universal scope and thus being applicable to the whole of the empirical world, they acquire an unsuspected transformative power, the power to inaugurate a world of a new kind. Because of their universal applicability they not only engender new kinds of beings but, because all beings in the empirical world can be operated upon in this way, they contribute to transform that empirical world as a whole, thus leading to the creation of the physical world.

Let us conclude by stating in a summary way the relationships between the various ontological operations identified as characterizing the Galilean moment. Link-regularizing operations, by which dependencies between phenomena are understood in terms of measurable causal relationships, are universalized by a world-scope causalization operation positing an infinite network of causalities and attributing to them first rank of existence. Ontological reconstitution operations—among which we included ontological shape- and link-regularizing, shape-reductive, and situation-smoothing operations—which reconstitute empirical phenomena on a new basis, are made possible by a world-scope metricizing operation. Because of their world scope, causalization and metricizing operations bring the physical world into existence and grant the other operations their transformative power, which is also strengthened by their infinite perfectibility.

We will reserve for a later section a discussion of Husserl's characterization of the Galilean moment as a whole, in which he tries to understand how the different elements he identified are articulated.

The Cartesian Moment

So far we have considered Husserl's characterization of the 'origin' of ancient geometry and its application to the empirical world, as it took place at the Galilean moment. We now consider the 'Cartesian moment', which gave rise to what we referred to earlier as Cartesian or dimensional spaces, moment which Husserl characterizes as the "arithmetization of geometry" (p. 44), and which in the context of the historiography of mathematics is understood as the 'application of algebraic methods to geometry leading to analytical geometry'. With these considerations we will conclude the analysis of Cartesian spaces in terms of the ontological operations that gave rise to them, and will be ready to return to our consideration of visualization machines.

What is the arithmetization of geometry? How can this development be characterized ontologically? In ancient geometry, the logico-deductive method made possible to acquire knowledge of geometric shapes, in particular of their properties and relationships. Regarding the circle, for instance, it was possible to know about its properties, and even it was possible to establish the relationship between the radius and its corresponding circumference, but the circular shape itself could not be mathematically specified. Although that shape could be described in a constructive manner, say, as the shape generated by the radius while it rotates around a fixed point, still it could not be fully measured, if by measuring we understand putting something into correspondence with 'numbers'. Arithmetization of geometry refers precisely to the notion of measuring shapes in that sense. Because Husserl is interested in providing an 'essential' characterization of the Cartesian moment, he deliberately uses arithmetization rather than 'algebraization', which would be more appropriate from a historiographical point of view, but would obscure the essential connection with numbers.

Summarily, algebra, as an extension of arithmetic, is a particular kind of measuring practice that deals with known and unknown quantities expressed in terms of numbers. In this practice, measurements are represented in terms of equations that establish relationships between what is unknown and what is known to us. As such, an equation is a particular way of extending our knowledge on the basis of prior knowledge. Because an algebraic equation consists of quantities combined by means of certain operations, where quantities represent nothing but a measurement of whatever they are about, an equation is a very abstract expression of knowledge about something. Attending to its 'essential' character, the operation of coming up with an equation we will call measuring by 'counting'. The result of such an operation, a specific algebraic equation, is a particular kind of measurement, which we will call a 'count', and which provides an account of something.

In the application of algebra to geometry we can identify several steps that can be characterized preliminarily as follows. First, there is a positing of 'axes' emanating from an origin, axes which are the simplest possible kind of lines, that is, straight lines, each of which acts as a measuring device. Second, there is a placing of geometric shapes in the context of these axes. Third, there is the introduction of equations as measuring 'tools'. Fourth, for a particular shape, there is the establishment of a relationship between the measurements (coordinates) of points belonging to the shape along both axes, relationship that is expressed in terms of an equation. Fifth, there is an analysis of the effect of variations in the algebraic equation on the geometric shapes that correspond to it.

Let us try to characterize this application and its steps from an ontological perspective. In the first step what takes place is the emergence of a new kind of 'entity', namely, a dimensional space. We can distinguish the following moments in this emergence: i) An emptying of the ideal geometric world by which geometric shapes are set aside, giving rise to ideal geometric 'space', ii) An instauration of an 'origin' in this space, origin which stands for us, humans, as measuring beings, signaling a sub-ordination of geometric space to us. Hence, geometric space now appears as 'emanating' from this origin, us, in all directions, infinitely, iii) An introduction of dimensional axes, again emanating from the origin, and embracing geometric space as a whole, in the sense that they uniquely identify each and every point of this infinite space. These three moments conjugate the act of creation of dimensional spaces, and we will refer to this step as a whole as constituting ontological dimensional spatialization operations.

In the second step, there is a populating of dimensional spaces with ideal geometric shapes by which the latter are referred to an origin with its corresponding axes. Next, we have the introduction in geometric space of a particular kind of 'measuring tool', namely, algebraic equations which, according to our prior characterization of them as particular kinds of measurement by 'counting', make possible shape-counting operations. In the fourth step there is the actual carrying out of shape counting operations for particular shapes, giving rise to shape-counts, namely, equations representing shapes, which we had called earlier 'formulas'. With these three steps, in which dimensional spaces are populated with shapes, shape-counting operations, and shape-counts, we have the emergence of Cartesian worlds, which we mentioned earlier.

Finally, in the fifth step, equations are no longer regarded as simply measurements of geometric shapes. Rather, they appear as 'generators' of shapes, such that given a shape-count we can generate its corresponding shape at any desired level of precision. Because of their ontological smoothness, geometric shapes can be generated typically by using relatively simple shape counts. What we have here is a powerful new kind of practice that contributes to the emergence of Cartesian worlds in their most proper sense. It is not only that Cartesian space emanates from the origin and that every point and every shape can be counted. More than that, using shape counts under the guise of algebraic equations, we can generate an infinite class of geometric shapes. In consequence, because they give rise to Cartesian worlds proper, we refer to these operations as ontological shape-generating operations. As we will consider later, such operations turn out to be crucial in the context of computer-supported visualization because, going beyond knowing as measuring, they give us the power of creation.

Overall, we interpret Husserl's notion of the arithmetization of geometry as the creation of Cartesian, dimensional worlds by means of ontological dimensional spatialization, shape-counting, and ontological shape-generating operations. This development is related to the ontological metricizing operations identified at the Galilean moment, except that instead of being directed at the empirical world as a whole they are now directed at geometry itself, that is, at metricizing geometry. In turn, metricized geometry can be applied to the empirical world, thus intensifying its metricizing.

Critical Appraisal of Husserl's Analysis

We now examine critically both Husserl's analysis and our reinterpretation of it. In the sections of the Crisis we have examined, Husserl attempted to uncover 'essential moments' leading to modern science and modern geometry, going in this way beyond historiography, that is, beyond the way in which a 'history of science' would have typically approached the subject. In addition, contrary to what could have been expected given his phenomenological perspective, in these sections rather than starting from an analysis of intentionality as it unfolds in geometry and natural science, Husserl focuses on practices, more specifically, on what certain practices take for granted and on their transformation. A guiding assumption orienting his work is that, ultimately, these practices must arise from everyday life practices and from the world in which they take place, the 'lifeworld'.

By focusing on essential moments, Husserl remains at the appropriate level of analysis without loosing himself in an overly detailed examination of practices, making possible for him to cover ample territory and to gain a sense of the overall development. At the same time, Husserl was well aware of the risks involved in this manner of proceeding (p. 57ff), in which most of the time he did not support his analysis with explicit references to concrete practices or to relevant works. For Husserl, the main purpose of the analysis was "to bring "original intuition" to the fore" by breaking through what has become too obvious to us: "we who all think we know so well what mathematics and natural science "are" and do." (p. 58).

But, after all, it appears that Husserl could not leave entirely behind the notion of intentionality that was central in his phenomenological approach, and he conceived of at least some of the moments he identified as acts of "pure thinking." As Husserl has it in the Origin, referring to the construction of geometric space, "this new sort of construction will be a product arising out of an idealizing, spiritual act, one of "pure" thinking, which has its materials in the designated general pregivens of this factual humanity and human surrounding world and creates "ideal objects" out of them" (1970b, p. 377). This conception forces Husserl to deal at great length with the problem of "how does geometrical ideality (just like that of all sciences) proceeds from its primary intrapersonal origin, where it is a structure within the conscious space of the first inventor's soul, to its ideal objectivity?" (p. 357).

It is this same general orientation that makes Husserl, for the most part, to fail to consider the role of technical instruments, tools in general, and their associated practices in the origin of geometry and natural science. Most striking, Husserl does not take into account the possible significance of the use of the telescope by Galileo on the birth of Galilean science. As Don Ihde has it, "Husserl's Galileo is a Galileo without the telescope."¹⁸ In the context of our reinterpretation of Husserl, we need to consider additional practices involving the use of instruments as crucially important for the emergence of what we called the 'physical world'. In the case of the telescope, although far from allowing us to land on the surface of the moon, it allows us to examine it as if it were far closer than it indeed is, transforming in a significant way our relationship with what is beyond the confines of the Earth.

But the telescope is only the most conspicuous example of a variety of artifacts which were available at Galileo's time, and that played an important role in science. A simple but very important device used in the study of the motion of falling bodies is the 'inclined plane', which presents the phenomena of interest in 'slow motion', thus effectively slowing 'physical' time down, making the phenomena accessible to us. While some artifacts, such as the telescope, have as primary purpose the augmentation of human capabilities, others, like the inclined plane, constitute an 'intervention in nature', and are related to the important notion of experiment in science. This is another, crucial aspect of natural science that, because of his general orientation, Husserl could not see. Because in this work we are primarily concerned with visualization rather than with science, and given that experimentation does not play a crucial role in a visualization situation, we will not approach the notion of experiment from the ontological perspective, and will leave it unconsidered.

While examining the markedly different ways in which the phenomena of falling bodies and swinging pendulums appear when viewed from Galilean and Aristotelian perspectives, Thomas Kuhn, rightly or wrongly, attributes them to the different 'paradigms' on which they are based. A paradigm is a complex notion, but in one sense that Kuhn came to prefer it is a 'shared example' that gives practitioners a way of seeing situations, in particular, of seeing phenomena under study.¹⁹ In Husserl's work there is no clear equivalent to this notion, possibly because in his suggestions regarding the Galilean moment Husserl was working at a very basic level, trying to identify the essential ingredients of the Galilean perspective. On the other hand, in his discussions of what was taken for granted by Galileo, Husserl refers to the pre-givens in Galileo's time, which could be understood as constituting a shared background for the community or communities to which Galileo belonged. In this sense, there is a possible connection with a second notion of paradigm, less preferred by Kuhn, as the shared commitments of a community of scientists.²⁰

Insofar as what Husserl uncovered at the Galilean and Cartesian moments constitutes a significant transformation in the notion of science and in the conception of 'nature', and considering the 'universal' character of some of these transformations, Husserl is identifying an epochal transformation of the highest significance in the history of the West. In this sense there is an important connection with the notion of the 'history of being' elaborated by Heidegger within a few years after the publication of the Crisis. Heidegger's notion suggests that there are essential transformations on the way the world appears to human communities, transformations that determine the character of historical ages. In the Crisis, Husserl not only seems to anticipate this insight, but in a sense goes beyond it because it attempts to uncover the way a particular transformation 'originated'.

Looking retrospectively at his own Galilean Studies, Alexandre Koyré characterized "the revolution of the seventeenth century" as bringing about two fundamental changes. First, the "infinitization of the universe," which in that work he had expressed in terms of "the replacement of the idea or concept of the Cosmos—a closed whole with a hierarchical order—by that of the Universe—an open ensemble interconnected by the unity of its laws." Second, the geometrization of space, that is, the replacement of the Aristotelian conception of space by that of Euclidean geometry.²¹ Although Husserl's notion of the universalization of causality bears a close relationship to Koyré's infinitization, the latter goes beyond it and points to a breaking of limits in such a way that now nothing escapes human perusal. Husserl's 'mathematization of nature' is more comprehensive than Koyré's geometrization of space.²²

Working from the perspective of the philosophy of science, Patrick Heelan (1987) has assessed critically Husserl's approach to natural science. Heelan suggests that Husserl's notion of modern science, referred to as 'Galilean science', is strongly influenced by Husserl's familiarity with 'Göttingen science', that is, a view of science that understands it as 'theory making'. Husserl's Galilean science corresponds to 'the philosophical core of Göttingen science', whose most prominent figures were influential mathematicians such as Hilbert, Klein, and Minkowski. According to Heelan, from Hilbert's point of view, "physics needs the help of mathematics to construct the ideal physics, and the ideal physics has the form of theory, and all theory ideally has the form of an axiomatic system." (p. 371). The particular aims of Göttingen science may have led Husserl to the extreme view that Galilean science, ultimately, attempted the 'mathematization of nature', a notion that we will examine in a later section.

Among the many possible ways to approach Galileo's works, there is the epistemological perspective. Joseph Pitt elaborates such an approach (Pitt 1992) in the context of the philosophy of science. In contrast to Husserl, Pitt works primarily from Galileo's text, and attempts to clarify the methodological principles underlying Galilean science, which Pitt regards as Galileo's most important contribution. Although emerging from different perspectives, what from this particular epistemological approach appear as principles oriented to secure a 'pragmatic' path toward the acquisition of knowledge about nature, from the ontological reinterpretation appear as operations leading to the emergence of a new kind of world, the physical world. The principles distilled by Pitt—of quantification, abstraction, universality, and evidential homogeneity—bear complex relationships to the ontological operations identified above.

In the proposed ontological reinterpretation of Husserl's analysis we introduced the notion of ontological operations and biases, identified specific instances of them, and established certain kinds of relationships among them. Now, from this analysis what it is possible to retain and what must be put under question marks? By ontological operations we mean practices or specific actions within practices, which either by their overall orientation or because of their characteristics, invite the emergence of new kinds of entities as well as new kinds of practices with novel ontological traits. In a summary way, a practice is an unfolding of human comportments carried out with the help of other humans, things, and tools, in a particular situation. While Husserl emphasizes intentional phenomena—for instance, the 'thinking' of the original geometers—a practice involves many different kinds of comportments, among them 'thinking', which may precede, accompany, or succeed other comportments.

Practices interact in complex ways with other practices, in particular, by being part of them, and they are grounded in specific 'ways of being human' characteristic of particular communities. Going even further, and considering Heidegger's notion of the history of being, given that a practice takes place in the context of a particular historical age, as such it responds to what may be called the predominant 'mode of revealing or unconcealing' characteristic of that age.²³ Although most practices keep this kind of correspondence with the mode of revealing, there are practices that either survive from prior ages—thus keeping alive in their own ways older modes of revealing—or that go against the grain and contain seeds of future ages.

What this tells us is that practices, in addition to having a complex structure of their own, maintain complex relationships with other practices, with particular ways of being human as embodied in particular communities, and with modes of revealing. Thus, attributing an ontological import to specific kinds of practices or actions within practices as we have done, entails an oversimplification. Rather, we should think of these practices as the most visible embodiments of transformations that are taking place in more or less pervasive ways in the context of human communities, transformations that are difficult to disentangle from each other.

Consider what we called 'ontological smoothing operations'. These are actions that take place in the context of practices by which shapes of things are made increasingly smooth, from which the notion of limit-shape arises. How do these operations themselves emerge? They could arise simply as intensifications of similar, pre-existing operations or they could be the echo, in this particular kind of practice, of trends emerging in other practices. Or consider two operations we identified in the context of the Galilean moment, namely, ontological link-regularizing operations and ontological, world-scope causalization operation. In the first kind of operation, dependency relationships between phenomena are regularized in terms of measurable causal relations; in the second, the notion of causality is universalized. How to understand the relationships between these two kinds of operations? Do linkregularizing operations emerge because of causalization, or vice versa? In the context of the present discussion this question doesn't make sense. We are in the presence of a very complex phenomenon of which these two operations are the most visible embodiments. At most, what makes sense is to say that because of the different scopes of these operations, of which the second has a 'world scope', the transformative power of the first owes much to the scope of the second.

Regarding the specific operations and biases identified in the previous analysis, the following needs to be said. Husserl's reconstruction of the essential moments of the 'origin' of geometry takes as its point of departure Euclidian geometry and, assuming that the basic geometric notions must have emerged from what was available in the 'lifeworld', it attempts to fill this immense gap by identifying certain practices that could have led to such notions. It is clear that, at best, what can be obtained in this way is one among many possible reconstructions. Because we also find the emergence of 'ideal' objects in language and in writing, which are far more pervasive achievements of human communities than geometry, it is conceivable that the emergence of geometric idealities could have followed an entirely different route than that suggested by Husserl in the Crisis. Some of these issues are explored in Husserl's Origin.

In his analysis of the emergence of Galilean science and modern geometry, Husserl was relying on his own mathematical background, on general knowledge of science, and on historical sources, such as Galileo's works. Although the emergence of Galilean science is far closer to us than that of ancient geometry, the intricacy and complexity of this development, and the fact that we are still caught under its spell, forces us again to be cautious. We already noted Husserl's failure to consider the possible relevance of technical artifacts in the emergence of natural science. More than that, it is possible that techniques, tools, and associated practices by the time of Galileo had already achieved a level of pervasiveness such that the universal character of causalization and metricizing operations was latent in them, insofar as certain forms of causality and measurement are intrinsic to such techniques and practices.

Because Husserl attributed the 'crisis of the European sciences' to a large extent to a 'loss of meaning', for Husserl the main purpose of the reconstruction was to recover the 'forgotten meaning' of geometry and natural science, that is, to regain the lost 'experiences' that were at the source of their emergence from the lifeworld. And he attempted this recovery by building explicit bridges between lifeworld practices and experiences, and those characterizing geometry and natural science. Any plausible reconstruction along these lines would effectively give us a 'sense' of what geometry and natural science are in an essential way. As a consequence, whether the specific transformations of practices identified correspond to what a precise historiographic account would produce, is not decisive for Husserl's analysis.

On the other hand, our reinterpretation of this analysis has a different orientation and is guided by a particular notion of practice. In this notion, in general, it is not us, particular human beings, that carry out practices. Rather, it is practices that carry us out. Although it is the case that we imprint upon them our own style, more important, practices imprint upon us their own seal. If this were the case, transformations of practices, in particular transformations with an ontological import, carry with them the possibility of transforming us in significant ways. In consequence, the felicity of our analysis depends to an extent on the appropriateness of the transformations that were identified.

To conclude, we can say—with the previously indicated caveats—that the notions of ontological operations and biases, as visible markers of more or less pervasive transformations with ontological import that are taking place in human communities, make sense. On the other hand, particular operations and biases we identified may be more problematic, and must be regarded only as plausible suggestions.

Ontological Operations Embedded in Visualization Machines

After examining the notion of Cartesian world and, previously, the application of geometry to the study of nature, we now return to consider the relationships between visualization machines and dimensional spaces. A visualization machine, as indicated earlier, is constituted by a computer, screen, windows displayed on the screen, and a variety of programs. Among these programs there are some which contain algorithmic representations of equations describing geometric shapes, while others, visualization programs, are in charge of displaying the geometric shapes on windows. Finally, these programs can be executed by a computer, thus effecting the display of Cartesian spaces populated by geometric shapes.

A visualization, on the other hand, is a presentation or exhibition of something by means of a visualization machine. This requires the application of dimensional spaces to a particular phenomenon under consideration—say, the structure of molecules—application that can be performed in a manner similar to the application of geometry we examined at the Galilean moment. Measures of the phenomenon or of causal relationships related to it are developed in terms of formulas, with a collection of formulas constituting a model of the given phenomenon. A model is specifically tailored to the case at hand by incorporating in it 'data' obtained through appropriate measurements. Then, geometric shapes are associated with elements of the model, thus giving rise to a visualization of it.²⁴

But, what is a visualization machine, when considered from an ontological perspective? It is a new kind of entity, in particular, a new kind of machine that emerged in the second half of the twentieth century, as a result of certain kinds of practices that we will now attempt to ascertain. As a whole, these practices can be characterized with the obvious rubric of 'computerization of geometry' which, coming after the 'arithmetization of geometry' and made possible by it, gives rise to visualization machines. Focusing on 'essential' moments and ignoring historiographical considerations, visualization machines emerged by certain kinds of practices involving Cartesian worlds. In ancient geometry, the ideal geometric world emerged from smoothing, excising, and lifting operations. At the Cartesian moment, dimensional worlds emerged from dimensional spatialization, shape-counting, and shapegenerating operations. With Galilean science, geometry was applied in the context of natural philosophy by means of reconstitution practices including shape- and link-regularizing, shape-reductive, and situation-smoothing operations. We have here a movement of ascent toward idealities, followed by a counter-movement in which they are made to descend to the empirical world giving rise to the so-called physical world.

At the 'computerization moment', dimensional worlds undergo yet another movement of descent that differs from, but is related to, that which took place with geometry at the Galilean moment. To some extent, this new descent can be understood as a retracing of the steps that led to the ideal geometrical world in the first place. By an overall movement of real-inaction, those smoothing, excising, and lifting operations are in some sense reversed by descending, embodying, and roughing operations to give rise to what we will call 'interactive visual worlds'. Real-ization reverts idealization but does not lead us back to the empirical world. Because realization takes place primarily by 'embodying' operations we will also refer to it as embodization.

Two different but related sets of practices need to be considered in the movement of realization or embodization of Cartesian worlds, that is, practices that led to the emergence of visualization machines, and practices that apply visualization machines to particular situations. Because a detailed consideration of these practices would require a careful examination of the ontological characteristics of computer machines, which is beyond the scope of the present work, we will proceed by focusing only on 'essential' ingredients of this movement, emphasizing its visualization-related aspects. We start with a preliminary analysis to be followed by a second, ontologically oriented consideration.

What takes place when geometric complexes come alive on a computer screen? Consider an 'animated visualization' of a large molecule, say a protein, as it folds upon itself under the effects of its own characteristics. Let us call this entity a v-molecule. Bright-colored, multiple, initially exhibiting an elongated configuration the v-molecule slowly curbs upon itself until it comes to rest, and remains in this state. Throughout this silent maneuver within the confines of the screen, the v-molecule retains its identify and structure. If we so desire we can return the v-molecule to its initial configuration and repeat the entire maneuver. Or, we can introduce a change in its composition and observe whether it has any effect on its folding and final configuration. In the course of its existence, the v-molecule takes residence in the generally flat computer screen. Now, a computer screen is a particular kind of embodiment of a dimensional space, uniquely distinguished from any other kind of screen by its being 'attached' to a computer, thus giving rise to what we called visualization machines.

When the computer machine executes the programs corresponding to a visualization a 'computational process' comes into being. Extraordinarily fastpaced, pulsating invisible and silent in the hard body of the machine, the process manifests itself in the v-molecule that emerges on the screen. It is the computational process that is at the heart of the embodization of Cartesian worlds. Visualization machines are fundamentally characterized by being programmable. As suggested above, at least two kinds of programs can be found in such machines, that is, programs that 'embody' formulas and visualization programs that 'display' them. Briefly, a program is an 'algorithmic' embodization of a practice or aspect of a practice, which can be 'executed' in a computer, thus giving rise to a computational process. More specifically, a program is constituted by a sequence of 'instructions' expressed in a particular programming 'language'. Computers are programmable machines because they can carry out these instructions. To be able to embody a practice in terms of a program, the practice must be understood in its minute details so that it can be thoroughly specified by the program.

Finally, then, a computational process is a process that takes place in a computer machine, and that comes about by the enactment by such a machine of a practice constituted by operations of various kinds, practice which has been embodied in a computer program. In the case of a visualization machine, in which computational processes manifest themselves on a computer screen with 'interactive capabilities', these processes give rise to 'interactive visual worlds' constituted by a variety of practices with which we can become engaged. Thus, we can set a v-molecule in motion and see how fittingly or unfittingly attaches itself to an element in the wall of a v-cell. And we can 'browse' the space of possible v-molecules of a certain class to determine which ones could give a better fit, if necessary.

Let us now reconsider these preliminary observations from an ontological perspective, in an attempt to identify practices and operations that have contributed to the emergence of visualization machines. We are struck by a most powerful movement of embodization by which geometrical, mathematical, and 'physical objects'—the latter understood in the sense previously discussed in the Galilean moment—as well as related practices, possessing traits that characterize them as ideal objects and practices, are made to descend into the empirical world and 'brought to life' in visualization machines. First, we identify ontological digital embodization operations by which digital bodies—known as computer machines—and digital places, usually referred to as interactive computer screens, are born. A digital body, 'imprinted' upon a 'material substrate', is a body whose capabilities are 'triggered' numerically.²⁵ But now, what is embodied in digital bodies? Nothing less than 'agents' that can carry out practices, which in the case of visualization machines are geometric and mathematical practices. Digital places, on the other hand, are digital embodizations of dimensional or Cartesian spaces proper, such that whatever takes place in them is also activated numerically.

Second, and complementing these operations, we identify ontological practice- proceduralizing operations, by which practices, in particular of the geometric and mathematical variety, are embodied into procedures or programs. To be executed by a computer, a program must be expressed in terms of statements in a programming 'language', where a language contains a fixed set of kinds of statements. Whatever of the practice that cannot be expressed in the language must be either twisted, so that it fits the language or extirpated, if twisting is not sufficient. Consequently, we identify here two particular ontological operations needed for proceduralization, namely, practice-regularizing and practice-smoothing operations. It is through these operations that, for instance, equations describing shapes, which we called above 'shape counts' and which are important elements of geometrical practices, are embodied into programs.

Third, we have ontological practice-enactment operations by which programs embodying practices are enacted by digital bodies giving rise to computational processes and manifested in digital places. Through practice-enactment operations, digital bodies become 'agents' carrying out practices in the context of more encompassing practices in which human 'agents' can also participate. It appears that all the embodization operations we have noted are ultimately geared towards making possible these practice-enactment operations, by which regularized practices are actually carried out.

We referred to all the operations mentioned above as ontological because, as a whole, they give rise to a new kind of machine, namely, visualization machines which, in turn, enact new kinds of worlds, that is, interactive visual worlds. Visualization machines, then, emerged by ontological embodization operations of Cartesian worlds. To some extent, this movement is made possible precisely because geometric and mathematical entities emerged themselves from the empirical world by certain kinds of operations, so that, in spite of their ideality, these entities and practices are pregnant with ingredients typical of the empirical.

Let us now return to the case of visualization of molecules. Given this talk of 'embodization' operations—in particular, of geometric shapes which are used to visualize molecules—the following questions arise: Where is the 'body' of a v-molecule? If it has one, what kind of body it is, what are its characteristics? A v-molecule is the 'manifestation' on a digital place of a computational process taking place in a digital body. While it emerges and continuous to be, the computational-digital process constitutes what we will call a d-molecule. In turn, the d-molecule comes about from the enactment of a program, which itself constitutes a p-molecule. Finally, the p-molecule results from proceduralizing 'models' of molecules, the m-molecules, which have emerged from scientific practices.

From the above, it appears that the v-molecule has a fragmented body that gives to it a peculiar mode of existence. While the d-molecule, as that which comes to be in a digital process, could be regarded as the 'living body' of the vmolecule, it itself has an ephemeral mode of existence: It comes to be from the enactment of a program, the p-molecule, and ceases to be at the end of the enactment. As a program, the p-molecule enjoys a more permanent mode of existence but, by itself, is 'non-emergent', to the point that the v-molecule could not come to be directly from the program. Now, the v-molecule, although visible and possessing interactive capabilities, is nothing but an echo of the computational process. How about the 'hard' body of the computer itself, on top of which the digital body has been imprinted: Could we find the body of the molecule in this? As a machine with certain 'universal' characteristics, which are particularized by the specific program being enacted, the computer lends its body to whatever was embodied in the program, in this case, m-molecules. But the way it does so is by supporting a computational process, which takes us back to d-molecules.

What is the import of this fragmentation? What is made possible by it? Because of its peculiar fragmentation, an important characteristic of the body of a v-molecule is its 'replicability'. Programs can be replicated at will, without any loss of properties. Also, digital bodies and places are, to a very high degree, replicas of other digital bodies and places. As a consequence, these fragmented bodies have an existence which is to a large extent 'time' and place independent, giving rise to the replicable, place-independent enactment of practices. This tells us that even after the embodization operations to which they are subjected, the ideality of geometric, mathematical, and physical entities is in some sense preserved. Although the fragmented body is a real-ization, it retains much of the ideality of ideal objects.

But there is an additional characteristic of 'digital' embodization operations, which, although implicitly included in the above considerations, needs to be addressed more forcefully. In this context, 'digital' has two connotations that play upon each other. On one hand, as already noted digital refers to the 'numerical' aspect of the embodization, thus making explicit the ideal mathematical character of that which is embodied. On the other, digital also refers to the specific character of the way in which the embodization is carried out, namely, by means of 'digital electronics'. Briefly, digital bodies and digital places are constituted by what to us, today, are some of the nimblest and lightest phenomena we can encounter, and to which we will refer as a whole simply as 'light'. In consequence, we will say that an 'essential' characteristic of digital embodizations is their lightness, in all the proximal senses in which this word can vibrate here, including the fact that digital bodies are swiftly 'propelled' by electrons, that digital places are expressions of light, and that visualizations are not subject to gravity.

In brief, visualization machines and the interactive visual worlds they bring to life are born from ontological embodization operations of Cartesian and physical worlds. Through specific operations such as digital embodization, practice proceduralization—which include practice-regularizing and practicesmoothing operations—and practice enactment, interactive visual worlds come to life. Because much of the character of Cartesian and physical worlds is preserved through these operations, interactive visual worlds inherit and have embedded in them ontological operations that gave birth to those kinds of worlds in the first place. With interactive visual worlds a full movement of ascent and descent toward and from idealities has been completed, constituting as a whole a peculiar movement of migration towards new kinds of worlds.

Ontological Traits of Visualization Elements

After all the operations previously identified, concluding with a movement of descent towards 'visualized' idealities, what do we end up with? What are the ontological traits of that which comes about from this proliferation of practices of great transformative power, accumulating one upon the other? To address these questions let us examine the elements out of which visualizations are made.

Computer-based visualizations are 'painted' on computer windows, that is, selected pixels in the screen region in which a window is located are appropriately colored to display the visualization. Let us introduce the notion of a 'surf', which we write as a contraction of the word 'surface' to suggest how it differs from the surface of an empirical object. While pixels are individual points on a computer screen, a surf is a collection of pixels on a computer window whose purpose is to real-ize a geometrical surface representing something. We can say that surfs are the building blocks from which visualizations are constructed. In the visualization of molecules, a surf can visualize an atom or a relationship between atoms, while a collection of surfs constitutes a v-molecule, which is an embodization of an m-molecule, that is, a molecule defined by a physico-geometrical model.²⁶

Let us now attempt to understand what a surf is, beyond this preliminary characterization. As a digital real-ization of a geometric or physical object, what kind of real-ity a surf has? We could at first consider a surf to be a 'body' of some kind, but if counting on it we tried to grab or just to touch a surf, we would either end up empty-handed or touching the screen. Perhaps we could say that if surfs don't have bodies, at least they have 'skins', but not even this would be appropriate. How, then, something that doesn't have some kind of body can be visible?

In this regard, as well as others, surfs are ontological cousins of the 'shadows', those surprising 'entities' that light engenders as it plays upon things. Like surfs, shadows 'happen' on a surface in a place, can move or displace, and are untouchable. On the other hand, like shadows, surfs are also the play of light, although this time not upon things, but from screen and pixels. A shadow involves at least three elements, i.e., the projecting light, the thing whose shadow the light projects or creates, and the thing upon which the shadow is projected. In the case of surfs, the emitted light that brings pixels into existence corresponds to the projecting light, while the screen or windows correspond to the thing where the shadow is projected, but what is the 'thing' whose surf is projected upon the screen? As indicated earlier when examining the fragmented body of a v-molecule, there are several elements involved in it such as a computational process and a program but, ultimately, it is the model whose 'shadow' or surf is projected upon the screen.

With respect to their mode of persisting, surfs also share the persistence of shadows. Just like that proximally-remote power, the Sun, recreates everyday the shadows of trees upon the walls, a proceduralized practice, when—at our command—is enacted by a digital body, can re-create surfs upon screens. Unlike 'material things', whose mode of persisting is continuous in that they cannot cease to be and 'come back to life again', surfs have an 'intermittent' mode of duration. Even something as delicate as a particular instantiation of a sign—say, any of the letters on a written page—has what might be called a 'material existence': it is a 'physical' mark on the page. As the page ages, the mark may become lighter and lighter in color or the whole document to which the page belongs may be completely destroyed, and with it the instantiation of the sign. But from the moment the mark was made to the moment of destruction, there was a continuity to it that the surf, for its part, lacks.

This intermittence of surfs is intimately related to the replica