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}=tto 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 $latex\dot \hat u=\hat f(\hat u)$.

A dot-model of the world.

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

Summary of Calculus.

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