Metaphysics represents a menagerie of dichotomies. Dichotomies are suspicious beasts because we, as humans, think of reality as being more complex, and therefore, reducing to pairs of alternatives would be a case of oversimplifying. The key to unlocking this door of peculiarity lies in the fact that dichotomies are formed by mutual exclusion. An antithesis on its own face. If the symmetry ends up broken you’ll instead have maximal asymmetry. As a result, the question that comes to face at full-speed is where do you go from now that the dichotomy has surfaced? A metaphysical response to that query is to say that we have two extremes–instead of just one–where one extreme is valid and the other one is false. I guess, another way of putting it would be to say that one extreme is fundamental while the other one is emergent.
Understand that this is what’s known as monadism. Monadism is the search of a single essence. In this case, the expectation is that either continuity or discreteness will prove to be the primal view. If you were to ally the response, it would go from monadism to dualism, this, of course, being accepting of both extremes now as fundamental but you’ll also have two essences. So now it’s the dualism of both substance and form or–in the case of the purely metaphysical guilds out there–the body and the mind. That’s a double monadism, kid. A far less frequently-trodden, alternative path would be to accept both extremes–as limited states–and then take a seat and gander at their fruitful interaction. Two extremes may be produced but the interaction that takes place is the story.
We all end up in the land of dichotomies since that is the logical result from trying to divide a reality that is so complex into categories or features that are the simplest forms of that aforementioned reality. Then, you’re left with the choice of having to choose which extreme will be the fundamental extreme. Either that or go triadic and model the two extremes in interaction with one another. However, the other previously mentioned [and alternative] path would be dualism. More or so, an unsatisfactory path to choose, it is a path in which you believe in two fundamentals yet see no possible causal connection between the two extremes.
Symmetry-asymmetry; algebra-geometry; space-time; substance-form; local-global; atom-void; particle-wave; local-nonlocal; signal-noise….these are all dichotomies that we have accepted and have deemed them being as useful. Is dichotomization nothing more than a mirror of epistemology, which is to say a habit that has been imposed on reality by the many inadequacies of human intuition or human curiosity? Or, is it ontic, or the way reality forms itself?
It is true that formal models seem to depend on discrete techniques. Coordinates, measurements, etc. See, physics does things such as measure a start-and-stop point–discrete measurements to plug into an equation–then relies on the continuity of the path in-between that start-and-stop point (one extreme). The models discard the presumed continuity of the world. This is done to condense it as a function that can map one discrete state onto a second discrete state. Formal models of physics are primarily based on a dichotomization in efforts of making a [mixed] reality something that can be described.
Mathematics is much more than symbolic manipulation and a computer that contains only the finite-bit strings of various mathematical papers without containing the semantic meaning has not captured all of the information. Rather, those finite-bit strings refer to an infinite number of possibilities. It would be in folks’ best interest to understand that “either/or” is the conventional bind that people get themselves into–and that is a false dichotomy. What dichotomies tell us is that we would be better interested in understanding how opposing extremes can both be true, both be fundamental and how they both can offer a vantage point when it comes to the production of realities. Think of the category theory–this breaks up the mathematical world into dichotomized fragments of objects and morphisms (or, the discrete object/the continuous morphism). You cannot turn in any direction when it comes to math, science or philosophy without bumping into lurking dualism or dichotomies, which is why it is really important to understand there are other choices of reaction apart from getting stuck in the eternal run-around of “either/or”.
Let me say something from the view of modern epistemology because it’s easy for people to get lost and find themselves bogged down in the confusion between models and simulations. Know that simulation is about recreating what is already “out there”. In other words, you have to represent both the general and the particular. All of the information contained in a system has to be recreated. Models are about the extraction of generals; particulars are discarded in the creation of the models. Think of any equation that stands for a natural “law”. The general relationship is represented by something the likes of
To use the model, you plug-in measurements; you plug-in local particulars like the energy and mass that are relevant to the prediction at hand. Models, on the other hand, involve a reduction of represented information–and the more [information] that can be discarded, the “better” the model. Simulation tends to go the other way. In principle, it would be more preferable [for you] to represent every last bit of information to fully recreate some actual (particular) system. Afterwards, there arises a practical cut-off question because the information that’s representing resources are typically discrete, or digital, and therefore we face an infinite trajectory to arrive at a representation of something that is infinitesimally close to what’s continuous, or analog.
You could argue that object/morphism is not exactly a discrete/continuous distinction but it is close in spirit. Then, discrete/continuous is not itself the most fundamental dichotomy. Local-global would more of a deeper level of generalization. If every field was dependent upon dichotomies–science, math, metaphysics–then why are people, in general, not more familiar with the logical principles involved herein? What’s the reasoning behind the fascination with “either/or” when the disciplines therein rely on the both of them? “Continuous” and “discrete” would both be extremes that would be approached infinitely and infinitesimally closely but never actually achieved. On the global scale, the world would be continuous; on the most local scale, it would be discrete. Or, it would approach both of these extremes on these scales without ever completely being “either/or”. However, instead of “either/or”, it could be both–and by both, I’m implying it in a sense of there being special “limits”. Now, mathematically, you’d be able to model the world in terms of these limit [states]. Just know that the metaphysics can state one thing; apparently, the physics will state another. Some would say that this is now the case for general relativity and quantum mechanics.
Take a look at the number line for a second. There’s an apparent paradox that we can add up to a continuous quantity, or to an infinity, by adding up in discrete steps. Supposedly, the (ancient) Greeks stated–metaphysically–that we would only be able to approach infinity via a discrete counting process, not actually arrive at it–in reality. Yet mathematically, one could simplify in a meaningful and useful way by acting as if the limit state has indeed been achieved. Vice versa, the same paradox applies the same principle though it is mentioned less. On assumption, how about the discreteness of the number 1? Saying 1 would imply, in reality, the meaning of 1.0000000e, not 1.0000000….1. We can approach the idea of a discrete integer infinitesimally closely by constraining the continuous number line. You’ll never metaphysically reach the state of perfect constraint where you’ll be for certain that you’ll have the number 1. Mathematically, you can chip away at the final asymptotic uncertainty and just get by with using the construct.
In the metaphysical sense, it seems obvious that reality tends to lean towards both the continuous extreme and the discrete extreme. See, reality encompasses both the diametrically opposed and mutually contradicting, or, dichotomous extremes. So, which of the two is the more fundamental?
Metaphysically, both are equally fundamental–they just exist in opposite directions–opposite directions of scale. Go large and all seems continuous and connected. If you were to go small, it all breaks up into being discrete and local. Reality, therefore, is a mix in-between. Yet mathematically, it can be found that it’s more effective to separate the mixture, jumping from a nearly dichotomous state to a fully-broken state where the “continuous” becomes the Continuous and the “discrete” becomes the Discrete–and the “mixed” becomes the Mixed as modeling gets even more complicated.
The physics does not have to reflect the metaphysics in exact fashion. However, if the tale of the dichotomous turns out to be true, you should be expecting two complementary viewpoints on the world that’ll emerge in modeling. You see, the two viewpoints will seem irreconcilable since each is defined by being exactly what the other [extreme] is not. If your world is precisely discrete then it cannot be continuous and vice versa.
Irreconcilable mathematics become reconcilable metaphysics when you take a step back and see the one world from which these two modeling extremes emerge, moving in opposite directions of scale.