"Mathematic" has distorted the notion of continuity and for long while made thinkers and calculators confused even afraid about Discreteness.
The simplicity of the fractal paradigm is: everything is discrete until it is continuous! Conversely:everything is continuous until it is discretized,
In practice, our inner sequencing and processing centre acts to discretize. Tha is it acts to conjugate Shunya into 2 discrete regions, snd each region into discrete subregions by a process of comparison/ measurement and distinguishing/naming/discretizing which is typified by the Euclidean factor/commensurabiity algorithm. The parts we store in memory during the sequence of conjugation become adjugates in the regional adjugation/synthesis/aggregation.
Thus sequencing creates and relies upon conjugation and adjugation processes, and of course sequence bracketing of the flow or progression or motion of sequencing. Bracketing directs and delays and stores sequencing processes, and encapsulates different combinatorial actions of conjugation and adjugation which are directly labelled or use the sequence as a label for the action.
Thus we have a sequence acting a a code/reminder for a usually mental sequence of action/associations. This encoding or mnemonic function of the discrete element s and associations within or of a sequence of discrete element is clearly recursive/ logically levelled. The web of recursive and logically levelled associations form the basis for a process called "DEEP" Translation/ Perception by Bandler and Grindler in their book th Structure of Magic, and a process explored developmentally by educator Piaget, and Chomsky in the development of the mother tongue. It is also the basis for the much more general notion of iteration and fractals.
Given that everything is continuous until it is discretized, why did mathematicians spend so much time trying to distinguish continuity?
Newton infact set out the principle par excellence in his method of Fluents. Zeno and parmenides had raised some titillating logical questions about the process of discretizing, which essentially challenged the notion of time, motion and continuity. The Greek response was to establish the principle of exhaustion! simply put: everything can be discretized as fine as one likes, but when one gets tired of doing this what remains is considered whole and continuos.
Using this notion Greeks could logically buiild models of the entire universe as sysntheses of the parts they had analytically discovered, providing duality could be established throughout the synthesis process. When Newton used this to establish his method of fluents he was most careful to observe the principles. Others were not, leading to the ridiculous notionthat something could come from nothing or nowhere! While Newton allowed things to "vanish completely at the end of a process, this was in no way saying that the ratios and comparisons possible up to the point of exhaustion were necessarily equal to zero before he infinite point in time that the process produces vanishing. Thus Newton claimed that th exhaustion principle always produced a continuous quantity that brooked on the limit of assignability. Beyond this limit nothing could be reliably said and indeed ultimately the quantities vanished, removing the issue!
Thus Newton demonstrated that from the comparison of differentials, that is perceptible discrete parts of quantities a whole class of ratios are derivable which may usefully be synthesised to describe behaviours in dynamic situations or to achieve more precise rectifications or more detailed apprehension of dynamic and geometric relationships. These philosophical tools were founded on scientific observations , through microscopy, that showed sequential changes of form in these differential ways. Newton"s genius was to demonstrate these thing in many different waysma thematically and physically and financially and by logarithms etc, to establish the soundness of this quantification approach based on differentials and the flow of time.
Newton"s Fluents are often called calculus of differentials, but they may equally be called the calculus of spacetime, for he usefully brought together space and time to describe the detailed evolution of dynamical systems.
Because of misunderstanding in the intellectual hierarchy, which by the way was synonomous with the established theological powers, and particularly due to th he attack on Kant, who championed Newtons philosophical praxis and approach, Newtons work underwent intense scrutiny, which it passed hands down. However European calculus did not fare so well and this ledto Dedekind proposing an absurd standard called the real number line. The backwash from the European Leibnizian methods of calculus unfairly tarnished Newtons, and Berkelys hectoring misunderstanding of what Newton actually wrote did not help clarify the situation. At the end of the day, theology won out, and discrete items bounded infinitely divisible and ultimately vanishing quantities , thus preserving the distinction between god and man!
The rule is simple, everything is continuous until it is discretized. We currently have a lower bound of discreteness for length called the Planck length. We are justified in considering everything less than a planck length as continuous. Recent experiments in astronomy suggest that space is not smooth even at the planck length scale, but of course we may justifiably assume this for below planck length scales, which currently we cannot resolve.
Quantum Mechanics takes on board this lack of "continuity" as discreteness, and not surprisingly to me discretizes time as well.
However we may usefully resolve any difficulties by attending to Ed Lorenz and his equations for the description of convection in the atmosphere!What he found was for a long time called Chaos theory. I remember it well when it was first outed. I was at university learning about the famous(now not so much) Heine Borel covering Theorem. Somehow a little discussion on discontinuity in continuous curved descriptions was slipped in. Basically we could descrie a system by a continuous function, but its real behaviour appeared to jump from certain maxima/minima peaks, and this was unpredictable. Later i found the differential equations and the famous butterfly 3d curve in which the point spiraled for an unpredictable time and then changed state to spiral in another plane and direction for an unpredictable time. This behaviour actually repeated, and so an aperiodic oscillation was taking place. Later the same equations were used o describe a water clock which again beat aperiodically.
This to me is a clear demonstration of the relation between continuity and discreteness. At all times we can describe the position of a point using our assumption of continuity, but the discrete shift in status is what is transmitted through the actual system.
Think on that point for a while.
Our senses are actually developed to react to change. Thus we tend to habituate to continuous uniform motions. A continuous static universe therefore would not be perceptible. However we have no right to assume that an undulatory and vibratory universe subsists on periodic changes. It is much more reasonable to expect aperiodic , discrete changes to freely occur, with a statistical smoothing effect being applied by the signal receptor! Thus our sensory network tends to smooth out the aperiodic signals resulting in our perception of local order and periodicity. In fact acute observation always reveals this process of smoothing. Thus every four years we adjust our calendar to discretely make up for the drift in the seasons caused by our assumption of continuity
The quantum world is no different. In fact , the notion of Quanta suits us more than it suits "reality". As far as i know we attempt to describe the quantum world in terms of correspondence to the classical dynamics Ed Lorenz equations describe. But we attempt to use the periodic tools, and the set paradigms we are familiar with. This necessarily promotes quantum thinking.
Ed Lorenz equations would allow a so called electron to decay toward a so called proton and then unpredictably change its orbit! This kind of strange aperiodic behaviour might better describe the so called quaantum world, than the assumed quantum frameworks derived in the 1940s