A Continuous Approximation

## Big Idea

• Let the total number of quarks $N$ get very large, i.e. say of order $10^{50}$. Write this as $\lim_{N\to\infty}$. Example: Eddington number.
• Then the increments $d \overline{r}$, $dt$ and $d \theta$ which are all finite numbers that are inversely proportional to $N$, get very small.
• Then the theoretical difference between quantum vs continuous space-time becomes moot. There is no measurable difference. So we can make a smooth, glossy, continuous approximations for the parameters of space-time; $\overline{r}$, $t$ and $\theta$.
• Then work back through the theorems of calculus (e.g. mean value theorem, Cauchy & Weierstrass theorems) to determine the limits of how differential and integral calculus can be applied to quantum mechanics.

## Displacement

WikiMechanics uses a finite categorical scheme of binary distinctions to describe sensation. So the number of sub-orbital events may be large but not infinite. In principle $N$ is finite and accordingly displacements may be small, negligible or nil, but not infinitesimal. Later we may assume that $N$ is large enough to make an approximation to spatial continuity, then allowing the use of calculus.
ḏ Ḏ ⒟ ⓓ 𝕯 𝓭 𝔡 𝖉 𝞓 𝝙 ⨺ ⫒

## Position

These definitions imply that position, separation and distance are quantized. Their variation is discontinuous because WikiMechanics is based on a finite categorical scheme of binary distinctions. Quantization comes from the logical structure of the descriptive method, even for a continuous sensorium. In principle, motion is always some sort of quantum leaping or jumping from event to event. Phenomena like this have certainly been observed in twentieth-century physics and can, for example, be used to understand and the experiment. For WikiMechanics, smoothly continuous motion is therefore presumed to be a macroscopic approximation. We are cautious about using calculus because the logical foundations of both differential and integral calculus are proven using assumptions about continuity. So WikiMechanics does not require calculus; instead calculations are designed to be implemented on digital computers, in a finite number of discrete steps.

## Phase Angle

The change in $\theta$ during one sub-orbital event is called the phase angle increment and written as \begin{align} d\theta \equiv \delta_{z}\frac{2 \pi}{ N } \end{align}. WikiMechanics uses a finite categorical scheme of binary distinctions to describe sensation. So the number of sub-orbital events $N$ may be large but not infinite. This requirement can be relaxed later to make a continuous approximation, thereby allowing the use of calculus. But in principle $N$ is finite and accordingly changes in $\theta$ may be small but not infinitesimal. For isolated particles the increment in the phase angle does not vary and so there is an equipartition of $\theta$ between sub-orbital events regardless of their quark content.

## Heisenberg Introduction

WikiMechanics is bare naked quantum mechanics. Particle attributes are always quantized because we use a finite categorical scheme of binary distinctions to describe sensation. Quantization comes from the logical structure of the descriptive method, even for a continuous sensorium. Time is quantized too because the time coordinate depends on a chain's event index. Indices are always integers so t changes in steps. Motion is discontinuous in principle, and sometimes this is even observed as quantum leaping and tunnelling. We are cautious about using calculus because the logical foundations of both differential and integral calculus are proven using assumptions about continuity. So WikiMechanics does not require calculus; instead calculations are designed to be implemented on digital computers, in a finite number of discrete steps.

 Werner Heisenberg, 1901—1976.
 Next step: Euclidean space.

Related WikiMechanics articles.

page revision: 24, last edited: 24 Jun 2019 14:19