# The World as IVP

• Dubito ergo cogito; cogito ergo sum. (I doubt, therefore I think; I think therefore I am). (Descartes)
• Man is nothing else but what he makes of himself. Such is the first principle of existentialism. (Jean-Paul Sartre)
• We should all be obliged to appear before a board every five years and justify our existence…on pain of liquidation. (George Bernard Shaw)
• The person lives most beautifully who does not reflect upon existence. (Friedrich Nietzsche)
• There is no place I know that compares to Pure Imagination. (Roal Dahl)

# Summary of Study

We can summarize all our studies as the study of the IVP

• $\dot u(t)=f(u(t))\quad\mbox{for }t>0,\quad u(0)=u^0$,

where $f(u)$ a vector valued function depending on the vector function $u$ and its derivatives, representing the state of a system.

The equation $\dot u(t)=f(u(t))$ connects the rate of change $\dot u(t)$ to the present state $u(t)$ through the function $f(u(t))$, from which the dynamics of the system can be computed by time-stepping.

The model is the function $f(u)$.

The Generalized Fundamental Theorem shows that a World modeled by $\dot u=f(u)$ exists! Convinced? What is the weak point with this argument?

The model $\dot u =f(u)$ thus gives a very compact description of the World.Easy to remember.

# Autonomous and Non-Autonomous IVPs

An IVP with a function $f(u(t),t)$ with an explicit dependence on $t$, referred to as a non-autonomuous IVP,can be rewritten on the autonomuos form (??) by introducing the new dependent variable $u_{d+1}=t$to give $\hat u=(u_1,...,u_d,u_{d+1})$ and adjoining the new equation $\dot u_{d+1}=f_{d+1}\equiv 1$ into an augmented $latex\dot \hat u=\hat f(\hat u)$.

# What Calculus is Most Useful?

The Egg of Calculus is the derivative and the integral is the Hen: First comes the derivative in the formulation of $\dot u (t)=f(t)$ and then comes the integral as the solution $u(t)$, by time-stepping in the same way as the Egg gradually develops into the Hen. Here $f(t)$ acts as the genetic code in interplay with the environment from which the the solution the Hen as the solution $u(t)$.

Classical analytical Calculus of primitive functions concerns techniques for analytical solution of $\dot u(t)=f(t)$ as means to circumvent tedious laborious time-stepping: An analytical primitive function $u(t)$ can be seen as a shortcut replacing a tiresome step-by-step solution. In precomputer times such shortcuts were useful, and thus accordingly highly praised, but with the computer the original motivation has largely dissappeared: Time-stepping solves $\dot u(t)=f(t)$ for any $f(t)$ at little computer cost, and thus in general is much more cost effective than tricky analytical shortcuts.

What is difficult in classical Calculus is analytical integration, since it consists of a bag of tricks with onlylimited power. Replacing analytical integration by time-stepping both simplifies Calculus and makes it more useful.

Human brains are good at formulating problems using principles, but cannot to massive computation and require a lot of training to handle bags of tricks.

The basic philosophy of BodyandSoul is to use the Soul/Brain to formulate equations like $\dot u=f(u)$ and then let the Body/Computer compute the solution by time-stepping. See also