We live in networks, although they are almost always very hard to see with our eyes. The most easily visible networks are probably our transportation networks, especially networks of roads. You can see roads branching and intersecting. And you can form a mental map of road networks. Your mental map (if it’s accurate) corresponds to an objectively existing networks of roads and points they connect. Think of the network of interstate highways connecting cities in the United States. These cities are objectively existing physical wholes. Each city is one whole, a single thing, a unit, a network node. The roads that connect them are objectively existing lines between the nodes. The network of American cities and interstates is an enormous connect-the-dots network. The cities and the highways are located in space. And so the network is also located in space. Put a dot, a spatial point, in the center of every city. Link two urban dots with a line if they their two cities are joined by a road. You have a network of points and lines, a purely geometrical network, a purely mathematical network. But this network is located in space.
Parts of our electrical and communications networks are also pretty easy to see. They appear as wires, strung from pole to pole along our roads. These networks carry electrical energy, and signals for telephones and computers. While you can see parts of the electrical network, the entire electrical grid is impossible to visualize without using advanced mapping technologies. Likewise for, say, the main lines of the internet. The cables that carry internet signals span the entire globe, making a network at planetary scale. These networks are physical structures, made of material atoms. But the nodes in these networks also occupy regions of space, and the links occupy regions of space as well. Those regions make a geometrical network of points and lines. This geometrical network of points and lines exists in the same space as the material network of wires and cables. It is a purely mathematical network that is located in space. Given any two points in the electrical network, they probably aren’t connected by a wire that traces out a straight line. But there does exist a straight line that runs between those two points. So there is another electrical network, made of points and straight lines, that is spatially overlaid onto (or that supervenes on) the visible electrical network with its bent wires and cables.
Our bodies contain networks which are visible, but which remain essentially hidden. Your body has a network of blood vessels, but you’ll never see it. Staying alive requires that you don’t see it, that it remains hidden, working beneath the visible surface of your skin. Your lymphatic network, the network of lymph nodes and lymph vessels, is even harder to see. During surgery, a surgeon will see some of the network of your blood vessels, or some of your lymphatic network, but not the entire system. To see these networks, you need to dissect a corpse. Or you can look at pictures in medical books. Of course, a picture of the network isn’t the network. These networks are transvisible, that is, they are not visible to the naked eye, but visible only with technological assistance. Your blood vessels carry oxygen and glucose (along with other molecules), and your cells use oxygen and glucose to generate energy. Energetic stuff flows through your blood network. Here too, in your own body, there are purely geometrical networks, purely mathematical networks, that exist in the space inside your body. They exist whether or not any human thinks about them. Space exists whether or not you think about it. So do points and lines in space.
Likewise, consider the neural network inside your brain, or the network of nerves that spreads through your entire body. Staying alive requires that you don’t see it, that it remains invisible. To observe a neural network requires dissecting a brain, and looking at some neural tissue under a microscope. It requires even more: you have to stain the cells to make them visible. A lot of technology is required to reveal the living networks in human bodies, or other animal bodies. These networks are also transvisible. Like your network of blood vessels, energy flows through your neural networks. Charged ions flow through your nerve cells, generating electrical currents. These electrical currents, flowing through your nerves, generate your whole mental life. And technologists have built artificial neural networks, which power artificial intelligences. Electrical energy, flowing through these networks, also gives rise to something like mind.
Our bodies, since we are social animals, live in social networks. You can’t easily see these networks either. You know who your friends (and enemies) are, and you might map out your social network by drawing it on some paper. But you can’t directly see it. Social networks are actually pretty abstract. All sorts of social signals flow through these social networks. These include linguistic signals, vibrations in the air. But microorganisms flow through social networks. These include bacteria and viruses, both those that cause disease, and those that are beneficial. Our bodies exchange invisible cells, or, at least, cells that become visible only when observed through microscopes. And most of us are not aware that, even when we sit around talking with friends, our bodies are exchanging lots of beneficial microorganisms, like bacteria that colonize our skin and guts. Our microbiomes are social constructions, parts of the social network of microbial life. Obviously, during sex, bodies exchange genetic materials. DNA, energized by chemical reactions, passes from body to body. Social networks are transvisible systems, carrying transvisible energies and signals, generating complex self-organizing societies. Some philosophers say these social networks, like neural networks, give rise to hive minds.
Besides socializing with other humans, we also socialize with non-human animals. Our most obvious animal partners are domesticated animals, like cats and dogs. But farm animals are also members of our extended social networks. And so are the commensals, animals that are often parasitic on our bodies, our food, our garbage. These include animals like rats, cockroaches, and other unfriendly parasites. Our larger ecological networks include the synanthropes, animals that have learned to live in human cities and suburbs. The list of synanthropes is very long, including many types of birds, and many mammals. Think of deer, squirrels, foxes, coyotes, raccoons, bears, even cougars, and, in some places, monkeys and other tropical animals. Synanthropes form large-scale ecological networks in human cities. Complex urban ecologies emerge from these networks.
Like most other networks, these synanthropic networks are transvisible. You can’t directly see the predator-prey networks involving hawks, rats, coyotes, squirrels, and so on. You have to map them out scientifically. They are visible to the eye of the mind, that is, to the eye of the brain, an eye which can visualize more abstract relationships. But these transvisible networks also exist in space and time. Say the Great New York Coyote is the whole composed of all the coyotes in New York City. The Great Coyote change its shape as coyotes move around, as old ones die, and new ones are born. The Great Coyote sits beside the Great New York Squirrel, which is the whole composed of all the squirrels in New York. The Great Coyote and the Great Squirrel meet at various places and times, such as when a particular coyote eats a particular squirrel. All these meetings make a link between the Great Coyote and the Great Squirrel, namely, a predator-prey link. This link exists in a high-dimensional ecological space, overlaid on ordinary physical space. Ecological food webs are connect-the-dots networks in ecological space. Coyotes and squirrels exist, and coyotes eat squirrels, whether or not any humans exist. Ecological food webs are not created by human minds. They are purely geometrical networks, that is, purely mathematical networks, that exist in ecological space-time. Ecological space-time is a physical space, which is also a purely geometrical and mathematical space.
Other examples of networks are easy to find. There are networks of roots and fungal fibers (mycelia) in forests, and there are genetic networks in cells. There are networks among stars and planets. The planets in our solar system exchange rocks. And there are very large-scale galactic networks in space. These networks are also transvisible. At the largest cosmic scales, they are ultravisible, visible only as computer simulations. Even more deeply, many of our best current physical theories say that space and time themselves emerge from deeper networks, networks which are not in space or time at all. These theories of emergent space-time are still hypothetical, but they are supported by lots of evidence. If they are right, then space and time are not basic. Loop quantum gravity says space and time emerge from structures called spin networks. Various “it from qubit” theories say space and time (and quantum fields and particles) emerge from networks of entangled quantum bits of information. The networks posited by these theories do not exist in space or time. Rather, they define space and time. Space and time emerge from deeper networks which are both physical and purely mathematical.
Ultimately, these deep physical networks emerge from set-theoretic networks. In these set-theoretic networks, the dots are sets and the connections are instance of the membership relation. At some places, basic quantum mechanical networks emerge within these set-theoretic networks. For instance, the spin networks of loop quantum gravity emerge, or networks of entangled qubits emerge, or networks of nodes in causal dynamical triangles emerge. These networks aren’t made out of any material stuff. They are neither material nor mental. They aren’t made out of any stuff at all. They just exist, and their existence is both physical and mathematical. From those networks, space and time emerge, along with their quantum fields and particles, then atoms and molecules and planets and people. Our own bodies emerge from the regularities of flows of energy in these networks. What are these flows of energy? They are merely ways the networks transform themselves into new networks. But these ways just make a bigger network.
Beneath all the visible networks, and the transvisible networks, there are ultravisible networks. These are deeply hidden networks, networks pulsing with their own energies and signals. The deepest network of all is the purely set-theoretic network, the basement level connect-the-dots network from which all other networks emerge. All the other networks exist inside of the set-theoretic network. Our bodies, our atoms, our quantum mechanical fields, our spin networks, and any other physical networks, exist inside of this purely mathematical network. All the relations in which our bodies participate are links within that network. Nature is an infinite set-theoretic network, an ultravisible network which contains all the transvisible and visible networks, like an opal filled with flames. And this ultimate natural network is that in which we live, move, and have our being.