The dimensions most commonly attributed to the physical world are the three spatial and the temporal.
If the parts of a universe have a dimensionality equal to the universe that contains them, then clearly the dimensionality of a universe has direct consequences for all of the objects which it contains. For example, if space is defined as a four-dimensional entity, then objects must also be four-dimensional entities (i.e. they must contain their temporal extent). This entails that four-dimensional things are not alterable (or mutable): only objects without a temporal extent can undergo change (or vary as a function of time). If we insist on saying that objects do change, then at least they must change in a dimension other than the four which serve to define them as objects. So: how do we define the universe?
Many people, either consciously or unconsciously, define the universe as “everything that exists” . Note, however, that this definition can be somewhat misleading for at least two reasons:
Everything is often taken to imply only matter, or every material thing, in which case empty space is left out of (or somehow in between) the concept of the universe. This empty space should be included in the concept of everything, so that there is nothing that is not included in our concept of everything (i.e. “empty space” is something, however nebulous, even if it is just the relationship between objects).
“Everything that exists” connotes only the present moment. As such, it cannot contain objects which have a temporal extent (as opposed to objects that exist only in the present). Although it is tempting to say that the physical world is everything that currently exists, the notion of what is current turns out to be a matter of perspective (according to most current physical theories).
An alternative definition of the physical universe is an entity or event which occupies all possible values of all dimensions upon which it is described.^{[}42] This corrects the two shortcomings of the previous definition, both of which are due to their failure to occupy the full extent of some dimension. In the first case, the notion that the universe is material carries the implication that the universe only exists where material exists. In the second case, a universe that exists only at one time (even though it occupies all three spatial dimensions) cannot contain the universe as it is witnessed by multiple observers. A complete universe, by contrast, exists in every part of every dimension. The universe is that thing of which every other thing is a part, where both spatial and temporal parts are considered.
How many dimensions does the universe have? Historically, the world has been described with three spatial dimensions. The temporal dimension was added to these three spatial dimensions relatively recently: time is now commonly known as the fourth dimension (at least to physicists, to whom it is the fourth dimension in a particular type of four-dimensional space called Minkowski space).
Four-dimensional space, or four-space, is required to describe the universe because the simultaneity of events depends on the observer (or the observer's frame of reference). In other words, the order of events (i.e. which events happen at the same time) is not the same for all observers, as would be the case if there were a simultaneous space for all observers. The Euclidean conception of a single extended spatial entity that exists at a single time is physically untenable: there is no single time for all positions, or all observers.
There is a certain amount of circularity, of course, in arguing that the universe has a certain number of dimensions because that is how many are needed in our symbolic formulation of its physical laws. For one thing, this argument ignores that the physical laws could be expressed in different ways. For example, we could express a two-dimensional coordinate using two real numbers and a Euclidean coordinate system, such as the point at [y=1 inch, x=1 inch]. The same point in space, however, could be located in a number of different ways: using polar coordinates, it could be specified as [angle=45 degrees, radius=1.414 inches]. That point could also be described using a single complex number as [1 + i]. All of these formulations have two coordinates, but the coordinate system (or the set of basis vectors) that is used to locate the point is different.
In general, although some number of coordinates may be required, we can choose any dimensions (or basis vectors) that we like. Similarly, although the equations that express Einstein's principles of relativity are easily expressed in Minkowski space, they could be expressed in any number of spaces. To say that the universe has a certain number of dimensions because it gives a convenient formulation for spacetime equations is to make the argument that it will be convenient if we talk about things this way, as opposed to saying that things have a particular dimensionality and can have no other. According to this nominalistic argument, theories of physics which use ten or more dimensions in their equations (such as various kinds of string theory) provide evidence for the universe having a similarly high dimensionality. Similarly, theories of physics with fewer than three spatial dimensions are possible, although it makes the mathematics more difficult. Flatland is an example story about such a world.^{[43]}
In a nutshell, it seems that when events are described in the physical universe, a high number of dimensions (at least higher than three) are necessary to locate them.^{[44]} However, it seems reasonable to assert that the primitive of the physical universe is at least a four-dimensional thing, or an event, as opposed to a substance which undergoes actions (i.e. things in three-space to which we add a temporal dimension). These high-dimensional things, which are non-referential parts of the physical universe, are called objects.
The physical dimensions are most often conceived to be Euclidean.
What is the nature of the physical dimensions? For example, are they circular, or do they extend infinitely in a given direction? Are they orthogonal (at right angles) to one another? Are they infinitely divisible (or continuous)?
Most of us are at least implicitly committed to some idea of the structure of the dimensions of the physical universe. Probably the most common mental model of the dimensions of the physical universe, at least in the western world, are Euclidean dimensions. To say that dimensions are Euclidean is approximately equivalent to saying that given an arbitrarily assigned origin, the dimensions extend to infinity in orthogonal directions. In other words, physical space can be measured by numbered axes which form right angles to one another. However, this understanding of dimensionality is certainly not the only possibility.
One alternative to Euclidean dimensions are circular dimensions (or in the N-dimensional case, hyperspherical dimensions). For example, imagine that you are an ant traveling on the surface of a sphere: if you go far enough in a given direction, even though you are traveling in a straight line with respect to the surface, you will come back around to where you started.^{[}45] This notion of dimensionality does not require infinite extent, and also avoids a number of problems associated with finite dimensions. For example, suppose that the spatial dimensions are finite, and that there is a boundary beyond which one cannot go. This scenario is difficult to comprehend: if there was a boundary, then what happens when an object crosses it? Does the boundary move? In which case, in what sense was it a boundary?
As opposed to this somewhat paradoxical notion of a spatial boundary, holding a corresponding belief in a finite temporal boundary is relatively popular: a large number of people believe in a moment of creation and a moment of destruction (these two beliefs often, but not always, go together). Westerners often understand the temporal dimension to be a linear quantity which extends (possibly “infinitely” ) in both directions. The Hopi, a native American tribe, see time as circular; the Vedic tradition of India also envisions epochs of time as recurring. Physicists talk of the beginning of time (the big bang), and sometimes of its end (the big collapse).
These differing points of view indicate that the dimensions have a nature which is uncertain, precisely because that nature has been viewed with relative certainty in a number of different ways. We can paraphrase this disconnect by saying that the physical dimensions which we use to describe the world have certain properties which are characteristic of our description. At least to some extent, space and time are conceptual contraptions: they are the basis vectors by which we measure objects in space.
^{[42] }This is a bit circular in that it begs the question of which dimensions are used to describe the universe. However, most axiomatic systems have a somewhat awkward beginning, and this definition is well-suited to a nominalistic viewpoint.
^{[43] }Flatland is a two-dimensional world that was originally described in [Abbott]. Three dimensional people do enter that land, but they have unexplainable properties from the point of view of the Flatlanders.
^{[44] }In fact, even if the dimensionality of the physical universe was less controversial, we would still be reluctant to accept it, since physical theories suffer from the unfortunate tendency to change rather often relative to the laws that they attempt to describe. We will not attempt to fix an upper limit to the number of dimensions; in practice, we use as many as are required in order to speak intelligibly about the world.
^{[45] }This model of the spatial dimensions was proposed by Albert Einstein, among others.