11: The deeper meaning of space

Blank leaf

What is space? The answer to this seemingly esoteric question has, in fact, tremendous practical significance for our everyday lives. Our conventional thinking about space can alienate us from others and ourselves by creating the illusion that we are all separated by an unbridgable distance. How can we rethink space to bring ourselves closer to our fellow human beings and to ourselves?

What’s in a container?

Let’s begin by taking a look at the conventional concept of space. Normally, we tend to think of space as a kind of three-dimensional container of things - an undifferentiated, infinite and infinitely divisible emptiness. Indeed, Wikipedia defines space as “… the boundless three-dimensional extent in which objects and events have relative position and direction.” But if space is “boundless”, how is it that finite creatures like us can “make space”? How do we accomplish this seemingly cosmic feat whenever we clear a table or tidy up the house?

Clearly, there is something left out of the conventional definition of space as a 3D container. When I make space upon my countertop to prepare ingredients for cooking, I certainly don’t seem to be generating a “boundless three-dimensional extent”. Alas, my countertop is only so big, to my culinary chagrin.

“Sure,” you might say, “you’re really just subdividing the boundless space into a bounded one on your countertop.” At first, this seems correct, but a closer inspection shows that this is also not quite right either. Where, in this subdivision, are the boundaries of the space I made? Could you measure this space with a ruler? If so, how?

Suppose you make a valiant attempt to measure the space I made upon my countertop. Where would you start? If I were you, I’d start by marking visible boundaries upon the counter so that I can extend my tape measure between them. But this is peculiar, since the boundaries should have already been set when I first made the space that you’re trying to measure. What in the world is going on here?

The deeper meaning of space

For all its boundless grandeur, there is no space in the conventional idea of space for the way human beings “make space”. These cramped quarters are due to the fact that the usual, scientific notion of space is actually a secondary concept rooted in a more fundamental - and more human - understanding of space. It is this latter sense of space that we’re talking about when speak of “making space” for things and others.

While this deeper meaning of space might sound abstract, it is in fact so commonplace that we don’t even notice it at work in all our human dealings. If a friend asks me to make room for him or her on the couch, I understand their request without further ado. We implicitly understand what it means to make space for someone or some thing, and we do so without reference to any idea of a “boundless three-dimensional extent”. Indeed, young children understand “making space” in this sense before words like “dimension” and “boundless” are even part of their conceptual vocabulary.

Rather, we initially experience space as free and open [1]. To “make space” is to make free and open for someone or some thing. Moving over on the couch frees a place for my friend and his or her posterior, making it open for them to sit in. In this case, space is made by opening the possibility of my friend sitting and inhabiting the place beside me on the couch.

To put it another way, all space is free space. Were it not free, we could not place anything “in” it. Note that “free” does not mean “empty”. My mug may be full of coffee, yet it remains free for holding the air that rushes into it as I sip my beloved bitter beverage. So the space in my mug remains free, even when the mug is full.

Similarly for the keyboard I type these words with. The space it “takes up” on my desk is free for my notebook also, which I occasionally place before me after moving my keyboard. Were the space not open for my notebook (or anything else) I wouldn’t be able to place it where my keyboard was. In other words, space is always free and open insofar as it is space at all. In contrast, the “empty” is simply space that is unoccupied. Space need not be empty in order to be free.

To make this a little more concrete, take the leaf drawing at the top of this post. There is an outline of a leaf, with some minor details on the edges. Its center is, however, blank (excepting the incidental palimpsest caused by the sheerness of the paper). That is, the center is free for more drawing, more marks in pencil, in this case. In this, we see that the essence of space is possibility, regardless of dimension. [2]

The “open” container

So, as human beings, we initially understand space as free and open for things, as possibility. This is clearly a very different idea of space from the one taught in schools the world over, though it is always already understood by each and every one of us, even if only implicitly. How, then, do we get to the conventional, scientific notion of space as a boundless three-dimensional container from this basic human understanding of space as free and open?

In a word: abstraction. We all live with the fundamental understanding of space as free for the placement and orientation of things. From this starting point, we can then come to think about space conceptually rather than as a matter of our everyday engagement with the world. We do this by abstracting various conceptual elements from this basic experience of space.

Over the course of intellectual and mathematical history, this has been accomplished through a series of steps. Each step builds upon the last, until we arrive at the edifice that is the conventional scientific notion of space. Let’s go over these steps briefly to plot the course from finite human experience to boundless infinity.

To begin, recall the cooking example from above, where I was to make space for ingredients upon the countertop and you were to measure this space. I recommended first marking out boundary points from the indeterminate open space I cleared. This procedure of marking positions is the first step in the process of spatial abstraction. That is, we begin our path to the full scientific conception of space by distinguishing determinate points in the openness we dwell “in” and measuring the distance between them. In short, we transition from space as open to space as interval.

Once we’ve understood space as interval, we can then go a step further and abstract the familiar three dimensions of height, width, and depth as intervalic axes. With this, we’ve constituted space as extension. Here we can hear the echo of the Wikipedia definition of space as “… boundless three-dimensional extent… " [3]. Each dimension extends toward a point on a horizon that ever recedes, sometimes called the point at infinity. We may then understand discrete intervals as bounded extents within and between the three dimensions.

Though this is all quite literally “abstract”, it should at least be familiar to those who’ve completed a high school math class. Specifically, most of us were acquainted with this notion of space in our in the form of the Cartesian plane. To help understand the mumbo jumbo of the previous paragraph, try drawing a graph with a horizontal (x) axis and a vertical (y) axis and then marking their intersection as zero. Where does each axis end? How far can you extend the line representing each axis? Similarly, if you changed the number at the origin to a number other than zero, could you still use the graph?

Hopefully you noticed that the length or “extent” of each axis is arbitrary, as is the point of intersection. Unlike our immediate experience of space, the conventional concept of space can be shifted around, shrunk, and extended however we like. In contrast, try extending the space upon my countertop this way, or perhaps making space upon the couch for a friend to sit on at an arbitrary location.

If this wasn’t abstract enough for you, we can go yet one step further in our abstraction of space. That is, we can move from the so-called analytic geometrical approach described above to pure mathematical analysis, where space is treated principally in terms of relations that can be understood in an arbitrary number of dimensions. At this point, we’ve left the world of experience completely behind, transcending even our abstracted graph approach for ideas that cannot be directly represented visually at all.

The point of all this is to make it clear that our conventional, scientific idea of space is in fact a product of abstraction rather than a reality we directly experience. That is, we arrive at the conventional notion of space only after a series of conceptual manuevers that take us progressively away from the world we are familiar with.

In light of this, consider the significance of the fact that this highly abstracted idea of space is taught as the fundamental understanding of space without reference to where it came from or how it was derived. Cognitive dissonance, anyone?

Lived space

Now that we’ve seen that the idea of space as a boundless three-dimensional container is secondary to a more fundamental understanding of space, our task is to find our way back down from the top of the tower of abstraction to space as we live and exist in it. Fortunately, free and open space is an inextricable part of our experience, so it is always there even when we don’t think of it.

Next time you find yourself “making space” for something, take note of it. By becoming aware of your own experiences of making space, you can refine your perception of space and develop a habit of dwelling in the space you live everyday. It is only within this lived space that we can meet others and be with them. When was the last time you met someone within a Cartesian plane? [4]

Notes

[1] Martin Heidegger, Zollikon Seminars, trans. Franz Mayr and Richard Askay (Evanston: North Western University Press, 2001), 8-17.

[2] Note in particular that the outline of the leaf drawing exists in two dimensions, not three. Accordingly, the conventional three-dimensional definition of space doesn’t apply here, despite the fact that there is clearly space within this outline that is, moreover, bounded and not boundless.

[3] Emphasis mine.

[3] Perhaps many people feel alienated, lonely, and hopeless because they unwittingly live or think they live in a world of abstractions where it is impossible to meet another human being. Rather, one might take John Frusciante’s advice and “Become your space everyday.”