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 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.
- :: cr.
- :: t.
- starclan
- :: t.
- v4-4_6952_starclan
- ;
- v4.4 scraps/ from Warrior cats to world Bolshevism