# The World from +1

• The present is pregnant with the future. (Leibniz)
• Truth is ever to be found in the simplicity, and not in the multiplicity and confusion of things. (Newton)

The (physical) World is a system of very many atoms connected by forces, which changes in time from one time instant to the next according to Newton’s laws of motion, and thereby creates a wide range of phenomena on a wide range of scales from atoms to galaxies.

The same basic algorithm (defined by Newton’s laws) thus operates on a present state to produce a next state at a next time instant into an evolution over time.

Here mathematics is the language to describe the physical World in the form of mathematical models. Combined with the computer the evolution in time of phenomena can be simulated by time-stepping a model through which the change from one time instant to the next is computed. The prototype of a code for time-stepping as a leading principle of DigiMat, has the form

• x = x + v*dt       (1)                      (same as dx = v*dt)

where = means assignment in the sense that the position variable x (on the left) gets a new value from an old value x on the right + a change v*dt as velocity multiplied by a time step dt.  This means that the basic code has a very simple structure as repetition of (1). See this example showing a black disc moving to the right.

All the codes in Model Workshop basically have this form as an expression of a basic principle of DigiMat: Short simple codes with complex output.

Compare with the genetic code in the 4 letters A, C, G, and T carried inside each cell of a human being with highly complex expression.

In particular, repetion of the code (1) in the form dx = v*dt constructs the x = sum dx = integral of v = sum v*dt where v =dx/dt is the derivative of x. The Fundamental Theorem of Calculus is thus captured in a one-line code releasing the power of Digital Calculus.

See Basic ExampleCosmic Interaction (start by pressing arrow top left, then click screen) and Planetary System. Compare with:

The code (1) is complemented by

• n = n + 1        (2)

which with start n =0 upon repetition of the basic operation of +1  creates/constructs the natural numbers 1, 2 = 1 + 1, 3 = 2 + 1, 4 = 3 + 1 and so on, or more generally the parts of any system one after the other.  See Construction of Natural Numbers (press arrow top left). Compare Creation and Worldometer. For example, it follows by construction that 2 + 3 = 3 + 2 as an instance of the commutative law for natural numbers.

Note that the code n = n + 1 operates with n on the right hand side as input and n on the left hand side as output. A new execution of the code thus uses output from the previous execution as input and so generates the world of natural numbers upon repetition. This is to be compared with repetition of the code n = 1 + 1 (= 2), producing the same output 2 over and over again, which lacks novelty and so interest.

The coding in DigiMat thus can be reduced to different elaborations of (1) +  (2), where (2) constructs the system by adding pieces and (1) puts it in motion.  This means that the programming in DigiMat is not at all the great hurdle it can be in settings different from mathematical models of the World, such as systems for administration.

The basic operation of +1 is brought to life by a child through the magic word “again” as one of the first word mastered after “look”.  Here +1 can be viewed to represent execution of any given algorithm yet another time, which expresses a principle of automation: the same algorithm is repeated over and over with new data.  A winning concept both in the World and in the (Ford) factory.

DigiMat more precisely offers mathematical models of the World formed from the following basic concepts:

• number: natural, integer, rational. Code: n = n + 1;
• space: 1d x-axis, 2d xy-plane, 3d xyz-system, x,y,z numbers,
• time: 1d t-axis, t number,
• motion in space: time stepping with time step dt. Code: x = x + v*dt, x position, v velocity, t time,
• mass-spring mechanics: motion of masses. Code: x = x +v*dt, v = v +a*dt, a = F(x,v)/m, a acceleration, F(x,v) force depending on position x and velocity v, m mass,
• continuum physics: convection: du/dt + du/dx =0, u=u(x,y,t) function of (x,y,t). Code: u(x) = u(x) + dt*(u(x+dx)-u(x))/dx, x =i*dx, i =1, 2, 3,…
• continuum physics: solids. fluids, electromagnetics,… Code: time-stepping: u(x) = u(x) + dt*F(u, du/dx,…),

In short: The World formed from repetition of +1 as adding piece by piece with physics evolving in time by time-stepping as constructive mathematics with properties by construction

In the p5.js code +1 as repetition is expressed in the function draw(){…} which on the screen draws whatever is specified inside {…} as quickly as possible, again and again.