Ever wondered what the world is made of at the lowest level? Are particles the ultimate building blocks? Or fields? A combination of particles and fields? Or... structures? There are people that are investigating the possibility that structure is all there is. No particles, no fields, structures only. Only relations between - yes, between what, exactly? It is easy to give a mathematical example of a structure. For example the structure consisting of this simple relation: being separated by 1. This generates the integers -3, -2, -1, 0, 1, 2, ... and many other examples. Note that only the relation of being separated by 1 is essential. The numbers -3, -2, ... are not. They can be seen as mere positions, place holders. Not as objects.
I tried to imagine physical reality to be based on structures only. No particles, no fields, nothing of those. This is much more difficult than the previous mathematical example of a structure. After all, we live in a world in which we experience (see, touch, measure, ...) objects like dogs, trees, etc. Objects that may stand in certain relations to each other, like "the dog is 2 meters to the left of the tree". So what we experience in our daily lives are objects that form a structure: objects + structure. Not structure only. To make it plausible that our physical world can be thought of as being made of structures only I will first discuss a mathematical example, the Cantor set. See also: http://en.wikipedia.org/wiki/Cantor_set.
To construct the Cantor set we start with a black line segment extending from 0 (left) to 1 (right). This is the top line segment of this figure:
The second step is to remove the middle third of this segment without the end points. This leaves the two line segments [0, 1/3] and [2/3, 1] of the second line from the top. The third step is to remove the middle thirds from [0, 1/3] and [2/3, 1] in a similar fashion. This gives the four line segments of the third line. Proceeding in this way gives an ever larger number of ever smaller segments. The Cantor set is what remains after an infinite number of iterations. It consists of a large number of points, not segments. For example the points 0, 1, 1/4 and 3/10 belong to the set. In fact the Cantor set contains an uncountable number of points: it has the same cardinality as the length 1 segment we started with. The cardinality is the mathematical measure for the number of points of a set.
There is a significant difference between the results after a finite number of iterations and the Cantor set. After a finite number of iterations we got a structure and objects. The
objects are the line segments of the picture which have a property: their length. However, the Cantor set consist of points only. Points have no properties, they only indicate a position. So to me the Cantor set seems to be a nice example of a structure without objects.
Now for the physical example. I see a parallel with the physical world. At the scale of everyday life we see for example a person to which we can attribute a multitude of properties: length, weight, hair colour, left eye colour, right eye colour, etc. Zooming in it seems that the number of properties decreases. For example, an electron is thought to be a point particle with only three properties: mass, charge and spin. Also at the atomic level we see mostly empty space with atoms and electrons scattered throughout. This suggests that at smaller scales we lose properties and gain empty space. Maybe if we zoom in deeper and deeper the limit is what the Cantor set is: a structure without objects.
We can also go the other way by zooming out. Imagine we look at the Cantor set with a microscope with a certain spatial resolution. All the holes between points with are larger than the resolution will be filled up. This gives one of the iterates above, which consists of segments. So a structure becomes visible as a structure with objects. Because we always probe nature at a finite resolution we always experience objects. This can be generalised from the Cantor set to a general set of points.
The above speculations (which are not entirely water tight, I admit) lead to an amusing thought: that if we zoom in on matter every time we see some object that we like "to get our hands on" it falls apart in separate smaller objects. And at the bottom level there is structure only. No objects. Only a mathematical description of a structure. People advocating that "everything is mathematics" will probably love this view.
Finally, a famous saying is "we are star dust" because a person's mass consists for approximately 93 % of elements produced in stars. Dust is often taken as a model for a set of points. "We are star dust" remains true from the above structural point of view albeit in a different sense.