The World is Continuous Not Discrete

Calculus was invented to solve a problem of "quadrature" of computation of the total distance $D$ covered when walking with varying step size in space $dx=v(t)\times dt$ with $v(t)$ representing velocity at time $t$ and $dt$ the time required for each step, starting from $t =0$ and ending at $t=T$. The total distance appears as the sum over all steps which takes the form of an integral : 

• $D(T)=\int_0^T v(t)dt$

The "trick" was to find a primitive function $x(t)$ satisfying $\dot x(t) =v(t)$ with $\dot x=\frac{dx}{dt}$ the derivative or $dx=v(t)dt$ to find 

• $D(T)=\int dx = \sum dx = x(T)-x(0)$

allowing $D$ to be computed from knowing a primitive function thus avoiding laborious summation.  For example, if $v(t)=2t$ as increasing velocity with time, then $D(T)=T^2$.

Calculus allowed tedious summation to be replaced be smart analytical mathematics: A tremendous success initiating the scientific revolution in the late 17th century also named the dot-age referring to $\dot x =\frac{dx}{dt}$.

Calculus showed to be more than "quadrature" by allowing a description the world in terms of differential equations depending on continuous space and time variables varying over a continuum of real numbers formalised in the late 19th century. So was continuum physics including electromagnetics formed allowing a description of the world we could fathom with our senses. 

The foundation was a model of space and time as a continuum of real numbers without a smallest scale. It was a world described by fields $\psi (x,t)$ depending on continuous space-time variables $(x,t)$ without smallest scale. 

Such field-models could be discretised  by introducing a smallest scale to allow finitary computation with finite number of digits connecting to "quadrature" performed simply as massive summation. The smallest scale could be refined to resolve increasingly fine details. 

Today this technique in the form of Computational Continuum Physics has been perfected into simulation of increasingly complex phenomena of the macroscopic world. Continuum models allow compact formulation and discretisation makes them computable. This is a world of classical physics made alive by computation. Classical physics as continuum physics.

But it is not the world of modern physics where Quantum Mechanics QM has replaced the continuum of no smallest scale, with a world of quanta of smallest scale $h\nu$ with $h$ Planck's constant and $\nu$ a frequency supposed to be the nature of the microscopics of atoms and molecules. 

This presents a world split into continuous macroscopics and discrete microscopics which comes with many difficulties now manifested in a crisis of modern physics. 

Let us follow the emergence of the split according to this time line:

1. In 1900 Planck introduced quanta of energy $h\nu$ to theoretically explain blackbody radiation. It gave him fame.
2. In 1905 Einstein introduced quanta of light energy $h\nu$ in a heuristic explanation of the photoelectric effect. It gave him the Nobel Prize in Physics in 1921. 
3. In 1915 Bohr introduced quantised discrete energy levels of a Hydrogen atom.
4. In 1925 Schrödinger formulated a model of a Hydrogen atom in the form of classical continuum mechanics.
5. In 1925 Heisenberg introduced a discrete matrix model. 
6. In 1926 Schrödinger's model was extended to atoms with more than one electron as  anew form of multi-d model beyond classical continuum mechanics, which was forcefully sold by Bohr-Heisenberg as Standard Quantum Mechanics StdQM according to the Copenhagen Interpretation. 
7. In 1928 Schrödinger left QM because it did not have the form of classical continuum mechanics.
8. Today the non-classical multi-d model as StdQM dominates completely. 
9. RealQM is a new model in the form of classical continuum mechanics. 

Today physicists speak about "quantisation" as the magic element separating modern physics from classical physics, which has brought so many wonders to the modern world. The idea goes back to the atomists of the Democritus school as smallest building elements of the world today carried in all sorts of particle physics. It appeared in Newton's corpuscular view of light, replaced by Maxwell's wave mechanics in the 19th century to return with Einstein's photons in 1905.  

Is then the split between continuous macro-physics and discrete micro-physics really necessary? Is it impossible to explain blackbody radiation and the photoelectric effect within classical continuum physics? 

No, it is in fact possible as shown in Computational Blackbody Radiation. This was also the message of Willis Lamb Nobel Laureate in Physics in 1955:  

• It should be apparent from the title of this article that the author does not like the use of the word "photon", which dates from 1926. In his view, there is no such thing as a photon. Only a comedy of errors and historical accidents led to its popularity among physicists and optical scientists.

The split has led to many difficulties. If the split can be avoided keeping both macro and micro within a continuum model, it may help out of the present crisis. Why not give continuum physics a new try to cover also microphysics without "quantisation".

The enigma of modern physics is presented as: How to quantise gravitation into a unified quantised theory? No answer in sight. Wrong question. 

A better idea is to de-quantise atom physics into a unified continuum model with gravitation. 

The late Einstein: These days, every Tom, Dick and Harry, thinks he knows what a photon is, but he is wrong. But nobody listened. 

I am pretty sure that Schrödinger would have welcomed RealQM since it follows his basic idea, which was overpowered by Bohr.

Mathematics: Calculus replaced discrete quadrature by understandable analysis, which returned in the form of digital computation giving power to understandable analysis.  

Physics: Calculus allowed classical physics to describe the world as a continuum open to understanding. Modern physics returned to Democritus atomism as a discrete world beyond understanding.