Daniel S. answered • 12/27/20
UCLA PhD CalArts Faculty - Literature, Writing, and Philosophy
There are different concepts:
1) At the most general level, "dimensionality" simply refers to the number of variables one needs to specify the basic structure of a system of some kind, whether this be a physical, biological, sociological, linguistic system, etc. Mathematically, a structure is an n-tuple, an ordered set minimally composed composed of elements and relations/functions between those elements. These elements correspond to objects studied, and relations/functions that the relations between these elements.
To give such a structural description is understand the problem space of a system at some level of description, and so at different levels of specificity. The same thing can be studied at different levels of description and as having different numbers of dimensions, depending on the problems we want to solve and study.
For example: To specify the problem-space of a 'traffic light system', for example, we need to consider the functional correlations between a set of possible colors (red, yellow, green) and a set of vehicle rules (stops, slows down, moves), so we need a three dimensional structure with three variables: <color, function, rules). The number of dimensions we need to specify the system is therefore relative to the domain and kinds of problems we want to solve. For example, the physical properties concerning the wavelengths and frequencies that determine color differences are irrelevant to the phenomena being studied for traffic light systems.
2) In fundamental physics, the dimensionality of a physical system refers the number of variables that organize the general structure of entities in space and time. The physical world is conceived of having three spatial dimensions which are determined in relation to a fourth, temporal dimension. These four dimensions specify necessary determinations of all physical systems. So physical reality is composed of different values and relations between those four variables: the variable length, width, depth, and temporal determinations of physical beings. Thus we speak of "four dimensional spacetime" as the general structure of the physical world considered as a system.
So, if you think formally of a system with four variables:
S: <x,y,z,t>
you can think of a physical system as
P: <length, width, depth, tense).
Hopefully that helps!
Daniel S.
A "structure" in this context means mathematically a set of elements and relations. The basic definition was given by Marchal in 1957. I quote: "there is a unique and interesting concept [of structure] that underlies the expressed interests of general systems researchers and that it can be given a satisfactory explanation.” S is a system only if S = 〈E,R〉, where: (i) E is an element set (ii) R = 〈 R1, . . ., Rn 〉 is a relation set, i.e., R1, . . . Rn are relations holding among the elements of E." Insofar as structures can serve as models for a system of whatever kind, the elements and relations that compose a structure can correspond to the dimensions of a system. For example, the three variables of a structure S can be used to model a traffic light system, standing for colors, the rules for cars to behave, and a function correlating these two typed variables: S: <color, function, rule> So we can say, for example, that the variable "color" can have three values: red, yellow, green; while the variable "rule" has three values: stops, moves, slows down. The function correlates the values of the two values: if variable "color" has the value "red," the function is correlated to the value in the "rule" set "stops". function(red) = stop One can build all kinds of model-structures for all kinds of systems, of varying complexity and explanatory scope. So, a structure for a system of whatever sort is in short an n-tuple, consisting of elements that correspond to the objects studied by a given domain and at a certain level of description, and to relations/functions between these elements.12/28/20
Michael L.
I noticed you gave a definition of dimensionality and not dimensions. would your definition apply to dimensions as well?01/18/21
Daniel S.
The dimensions of an n-dimensional structure are the variables that stand for the elements and relations (including functions) of the system in question. So if a system S is has 3-dimensions <x, y, z>, the dimensions of the system would be the values of 'x', 'y', 'z' themselves, whatever these might stand for, which depends on the system in question. For example, to simplify, a physical system has four dimensions: length, width, depth, and time. Which is why we speak of four-dimensional spacetime as a singular structure. So you can think of a physical system as determined by the specific values that each of these four dimensions acquire in relation to each other. Depending on the domain in question, different mathematical, logical, and computational structures may be used to model a specific system: from pure set-theory, the predicate calculus, the differential calculus, vector analysis/algebra, all sorts of topologies, category and type theory... Notice that these concepts are part of a specific formal method in contemporary science: the analysis of phenomena understood as structural systems, in which a set of elements and relations, specify the possible behavior of the system in question.01/18/21
Michael L.
What do you mean by "the basic structure of a system of some kind"?12/28/20