Philosophical Research:Molecular Democracy/4.4r/6961 starclan
from Warrior cats to world Bolshevism
what a simple, escapist fantasy. cats going from a restrictive, sometimes confusing life in human society to a simple rough-edged life in the wild. in the Warrior cats history we are told that everything was chaos until cats ordered themselves into four clans, and they got this advice from the spectral images of their fallen ancestors, known as StarClan.
that wasn't StarClan. that was mathematics
two-dimensional cats in a one-dimensional forest
imagine that we define a mathematical concept called a Beast. this is not to be confused with any particular instance of a real-world animal or species, and instead is to be understood as a mathematical pattern emerging from particular logical rules that applies to exactly the set of real or fictional animal species and populations it applies to.
say that we define a mathematical Beast as an entity that consumes a particular amount of matter from its environment. we can represent the environment as a mathematical field with varying levels of resources at every point, similar to the fields used for "tracking" the overall effect of many individually-untrackable fundamental particles in quantum mechanics. some points of the field may have high numbers, others may have nothing.
in _Warriors_, it is assumed that there is generally just one major resource: small animals. to stay alive, cats eat a particular mass of small animals over any given period of time. however, a particular small animal cannot be in two places at once. this means each cat must hunt in a particular point in time and space. any particular nonzero point on the prey field can be occupied by just one Beast. perhaps we will stipulate that each Beast contains a scalar measure of its current "equivalent food energy". if so, a Beast has two characteristics, spatial coordinates and energy, which make the Beast structure comparable to a three-dimensional vector; each Beast is like a peak in the overall world map, with an x position, a y position, and a gradually-falling height on the "energy" dimension. the occurrence of nonzero points on the prey field limit the number of Beasts and the energy height of each Beast.
of course, there is something a little odd about two-dimensional space. imagine that an animal is attempting to escape from another animal, but we assume for some reason that space can be divided cleanly into integer units of square area such as meters. if a cat were to pounce toward a mouse in a straight line, the mouse could move 1 meter in the x direction, or 1 meter in the y direction. if two cats were about to get into a fight, once again, a cat could flee in the x direction or the y direction. but whatever direction an animal goes, there is always a certain number of total square meters of space in the forest. it may be that in real life square meters of space are always connected in a certain way, but at the same time, given a particular finite boundary to a spatial area, it would also seem entirely possible to number square meters of space on a single dimension. if a particular terrain is 10 meters by 10 meters, there will always be 100 distinct square meter areas inside it, and square meters 1, 2, and 3 will generally always be there as long as we keep consistently numbering them the same way. this is to say, as long as we can decide on a particular square unit of space which appears small enough to distinguish all the meaningful positions of Beasts and prey animals, it is possible to knock the Beast structure down to just two dimensions of one-dimensional-space-square versus energy.
certainly when we attempt to simplify a physical model to discrete units, _coarse-graining_ the mathematical concept of space, we lose some amount of detail which would exist in the real world. it is fair to say that if we wanted to be the most accurate, our one-dimensional mapping of two-dimensional space-squares should extend itself down to infinitesimal scales such that our forest is divided into approximately-infinite x or y sections, enumerated with some kind of infinite-set, infinite-series, or infinite-calculus-limit trick. as scary as it might be to imagine an infinite ruler zooming in on fractal divisions of measurement in violation of the actual discrete nature of atomic matter, or the kinds of algorithms required to calculate with such a wildly-fractalized grid, if we can so much as get our infinitesimal grid set up the conversion from two dimensions to one always remains just as easy. a 10 by 10 grid has exactly 100 squares, a million by million grid has exactly 1 × 10^12 squares (one trillion), and a grid with countably infinite squares on each side has whatever number of squares that a countable infinity squared is. numbers actually being countable is never as much of a big deal as much as it critically matters that there are exactly as many squares as one side of a square grid squared. and practically speaking, it is very unlikely we will truly need an infinite number of squares. if we are talking about events that happen on the scale of living beings, a physical space containing physical objects made of fermionic matter is very unlikely to contain interesting events at scales smaller than an atom, or at least smaller than an electron. this is to say, it is probably safe in most cases to divide space into discrete units relative to the size of these atomic-scale particles. while it is _conceivable_ a cat could become radioactive and begin influencing its environment on the scale of periodically-breaking neutrons, even that would seem highly unlikely in the context of most real-world ecosystems; although uranium ore does happen to be one of the more common minerals in the earth's crust, radioactive ores with a concentration great enough to harm animals are not a common find. there is most likely a threshold somewhere between the scale of an electron and the scale of a prey animal where it is acceptable to discretely divide space without losing any interesting information relevant to the typical life history of a small carnivore.
all right, you may say, I suppose we can _address_ units of square area in one dimension, but what about the way spatial units are spatially connected at their edges? for this, we re-connect all the units of square area using an undirected graph. the connection between space and space then becomes separate from the connection between Beast and space, allowing us to model the fact that cats always stand on a particular object, not on one x coordinate or y coordinate at a time; even standing at the very edge of a perfectly square forest, an animal would still _have_ a zero coordinate on the other axis representing a point in space, not no coordinate. at this point, by essentially mapping realistic physical space to a funny new topology with a different shape and the same rules, we have created a mathematical structure suitable for analyzing the concept of animal territories.
say we have a population of 10 or more cats, represented by 10 Beasts on a line-space of a particular size, and each Beast moves randomly along the graph-paths available to it for movement. we initially assume that Beasts will not attack each other if they merely pass through the same Beast-sized area, and will simply pass each other by, or actively flee. now, what if on a given day there are 10 Beasts and 9 spatial units containing prey animals? one of the Beasts is either going to have to let its energy fall, or chase one of the others off one of the nonzero spaces. how each of the individuals decides whether to fight or flee would realistically require more variables, but one decision function which should work to produce a reasonably-believable simulation is to declare that a Beast challenges another at low energy and runs away at high energy — in a realistic ecosystem, to save effort associated with getting hurt could be considered one of the more rational possible decisions. as we let this particular simulation run, all of the Beasts will decrease in energy, prey animal points will appear periodically at some rate, and some Beasts will snatch prey animals first. if prey animal points are sparse, the less fortunate Beasts will get into more conflicts, or perhaps leave the grid if they happen to have reached the edges and this has been coded in as an option. it is worth noting, however, that depending on how vast our grid is, it may take just as long to get out of the grid as to simply move to another part of it, making the option to migrate to a distant area rather moot.
with a simulation this basic, we already begin to see correspondences to some of the observed real-world behaviors of feral cats. a greater density of cats means that cats move across greater areas, while a lower density of cats requires less movement to find open areas. cats will pass through or inhabit the same broad territory if they have no particular resources to fight over. one thing notably missing from the model is the ability for cats to form social connections. domestic cats, and a select few species of wildcats, are known to form groups starting at adult females. multiple female cats will begin living in the same consistent area so they can all care for and defend their young. males will be admitted if needed, but the overall group keeps out extra males. kittens also live in the group until they are about 6 months old. the characteristics of sexual reproduction begin to emergently create the characteristics of social units.
in order to model the emergence of social units, we need to add some more graph connections to the model. as things currently stand, we have a layer of free-floating Beasts and a layer of spatial units, but only the spatial units have graph connections. we will say that there are three situations that social connections form: adult female to extra adult female, adult female without male to adult male, and juvenile or adult inside social unit to other cat in social unit. cats do not run away or fight each other if they are socially connected, with the exception that the biggest cats eat first. cats bring prey back to their nearest cluster of kittens or adult females.
at this point in the model, we see that social units can easily be reduced to a simple set of rules, and these rules appear to neatly produce the weak emergence of social units. the rule of social connections existing for kittens exists first. the rule of social connections existing for adult females exists second. the rule of social connections existing for adult males is almost extraneous to the creation of social units. it is possible to imagine an alternate species of animals where males do not in fact stay in social units after mating and instead simply keep flowing around outside them, but it would be easy for the rule of females and offspring clustering together to exist regardless of this. however, it is possible to run the simulation to test whether males living in social units or freely flowing around the area leads to more conflicts. one possible outcome is that when there is a sparse prey field all the males increasingly fight each other as well as a few of the females, and some fraction of them die. if this is the case, males within social units would stand a greater chance of surviving because of the existing rule that food is split among cats in social units, albeit usually going to the biggest one that bothers to claim it. this does not by itself lead us to the emergence of neat social units where each individual defends the entire area and social graph contained by the social unit, but it comes very close. the existing set of rules already leads to the logical conclusion that Beasts experience more consistent survival by retreating to not just anywhere but specifically to social units where they can rest and some other individual can go to distant corners of space to track down prey. and from there, it is possible that Beasts will end up defending the rest of their social graph by accident simply through individuals having conflicts over sparse prey at the edge of the graph.
we can begin to see that to some extent, the behavior of an animal population can be neatly attributed to mathematical rules that exist independently of individual living animals, which arise in the interactions between individuals during their lives, and which the genes of animal species simply seem to randomly and stochastically "learn" to fit themselves to. the structure of an animal population can be affected by animals following inborn instincts to carry out particular rules, or conceivably through Animal life making intelligent decisions on its guiding rules, as would the human animal. although real-life cats are not known to have culture or construct proper Social-Philosphical Systems, mathematics would support this social formation as viable had it been the case that cat brains found it _materially_ possible.