Reading Robert Walker's article on what extraterrestrial mathematics might look like has the wheels in my head a'turning.  We live in a digital civilization, one that specifically evolved toward a binary representation of a decimal-based mathematics.  Our computers count by 1s and 0s, whereas we tend to count by 1s, 2s, 3s, 4s, 5s, 6s, 7s, 8s, 9s, and 0s.  And that is just our conscious countatiousness.  Our bodies count in ways we have yet to enumerate.  I think it's quite likely that any complex biological organism like a jelly fish uses some sort of internal mathematics to regulate itself.  A more supercomplex organism like a mammal -- especially one of the higher-evolved mammals like humans, cetaceans, and the great apes -- must have an incredible amount of math hard-wired into its system.
We might one day unlock these biomathematical processes and use them to devise new theories.  And this is not a new concept.  Researchers have been exploring the potential benefits of chemical computation for many years.  In any computational system you have to be able to combine and separate quantities and to distinguish between THIS and THAT.  An extraterrestrial civilization might use a digital-based math but it doesn't have to use Base 10.  In fact, the earliest documented human math system (that we understand) used Base 60 -- and from that system we derived our clock rules (60 seconds to a minute, 60 minutes to an hour).

Chemistry does without thinking what we train our minds to do consciously.  No human being can consciously match the transformational changes of a chemical reaction as it sweeps through even a small group of molecules.  Chemical math works so quickly and so universally that we have even tapped into it to create marker systems for stars and planets (we call that Spectrography).

In a comment on Robert's article I used ants to illustrate a primitive form of social intelligence.  Like all social insect species ants accomplish many complex tasks when their colonies work together according to biologically-written rules.  Each ant plays an important role within the life of the colony even though the individual ant seems to be relatively unimportant in a human way.  That is, we have not documented individualistic behavior among ants (although maybe one day we will).  An individual has a sense of place-of-self, role-of-self, and possession-of-self.  You collect things that you enjoy because you enjoy them, not because your collection contributes to the survival of your community.

An ant with a personal collection of anything -- grains of sand, space in a colony, favored foods -- would be acting individualistic.  Merely transporting stuff to and from the colony is not sufficient.  The ant needs to have a private space for itself that doesn't really benefit the transformation of the colony on a daily basis.

This model of social intelligence already plays itself out on the Internet.  We lack personal privacy in all things digital because the Internet cannot function with private spaces.  The very connectivity that we crave destroys the privacy we have grown up.  If you type your thoughts into a "private" forum anywhere on the Internet chances are that as many as 100 people can see them (given the right circumstances).

And so without privacy we have no sense of self on the Internet, which makes us social.  And sooner or later someone will invent an Internet of smells and sensations to match our Internet of visual and audio stimuli.  We will communicate more intimately than previously, and more like social insects than ever before.  We will sniff our neighbors for traces of their past experiences and to determine compatibilities between their likes/dislikes and our own.

An alien civilization that hasn't grown up in a digital world might already live this way. And if they invent an Internet of their own then the signals they base their connectivity upon might already resemble sights, sounds, sensations, and smells.  Their computational technology might have been chemical all along, and their collective abilities to solve problems could make our crowd-sourcing look like a child attempting to memorize multiplication tables.

What would it be like to be born with math hard-wired into your body chemistry -- not the math that makes your body work but the math that allows you to manipulate your environment, and to communicate with other people in esoteric ways?  Such an evolved species might not need wait for computational technology to create an Internet.  In their most primitive, least-socially evolved state they might communicate across vast distances without the aid of electromagnetism.

Animals do this already on Earth.  They use sound and scent to call each other over long distances, and to mark boundaries, and to exchange information about health and sexual availability.  They use scent to identify friend and foe, to figure out who is the best prey, who is the leader of a previously unknown group of animals, and many other things.  Communication by sound and scent and even sensation is an integral part of human experience but we have yet to digitize it.

A technology based on a non-digital math might have solved the many complex problems we face in building virtual reality systems.  Today we can sit in an office on one side of the world and have a video conference with an office on another side of the world.  One day we may be able to shake hands with projected avatars in such meetings, perhaps even to share in sensation-based experiences such as smelling food, feeling fabrics, and testing the strength of prototype objects.  We may hear all the subtle rustling of fabric and physical stuff the same way the people in the other office do.

We have experimented with Smell-O-Vision in theaters, Sensurround technologies, and Virtual Reality goggles.  But these things are mere essays in the craft.  One day, maybe, we will create a new technology that is less dependent upon ones and zeroes and more dependent upon basic laws of the universe.  Mixing two chemicals HERE may lead to a transformation THERE because of the meta data we are able to transmit across our connectivity.

If an extraterrestrial civilization were to see our Internet at work today it might not seem like so strange a thing.  To create a civilization the individual members of a community must COMMUNICATE, STORE, and MODIFY information.  An Internet speeds up that process.  An Internet of Smells, Sounds, and Sensations should increase the rate of transformation exponentially compared to what we can do today.

Maybe we just need to free our minds from the boundaries of binary and decimal math and to think more in terms of numberless computation to drive the next era of cultural innovation.