fredag 8 november 2013

Mathematics: Backward Magics or Forward Reason

                A primitive function magically pulled out of a hat as an area under a function graph.

There are two approaches to mathematics:
  1. Symbolic mathematics: magics: objects pulled out of hats. 
  2. Constructive mathematics: reason: objects constructed in stepwise computation.  
Let me give two examples:

The Fundamental Theorem of Calculus

The presentation of the Fundamental Theorem of Calculus in standard text books of Calculus, is the following: Consider the integral
  • $u(t) =\int_0^t f(s)\, ds$  for $t > 0$,
defined as the area under the curve determined by the function $s\rightarrow f(s)$ for $s\in [0,t]$.

Compute the derivate $\dot u=\frac{du}{dt}$ of the function $t\rightarrow u(t)$ with respect to $t$, to find that, assuming some suitable continuity property of $s\rightarrow f(s)$:
  • $\dot u (t) = \lim_{\Delta t\rightarrow 0}\frac{u(t+\Delta t) - u(t)}{\Delta t}= \lim_{\Delta t\rightarrow 0}\frac{1}{\Delta t}\int_t^{t+\Delta t} f(s)\, ds = f(t)$ for $t >0$. 
In short, the key argument is to show that the integral $u(t)$, defined as an area, satisfies a differential equation
  • $\dot u(t) = f(t)$ for $t > 0$
or solves an initial value problem
  • $\dot u(t) = f(t)$ for $t > 0$ with $u(0)=0$.        (*)
We thus start with a given function, the integral $u(t)$, which is shown to be the solution of a certain initial value problem. The process leads from solution to equation satisified by the solution. The equation appears as magics without reason, since the reason is put into the specification of the solution or integral $u(t)$, with appeal to a concept of area which has to be defined, and not into the equation.

But this is backwards: The more reasonable forward procedure is to start with the initial value problem (*) expressing that the rate of change $\dot u$ of $u$ is equal to $f$ as balance equation expressing some basic physics, and then proceed to the integral $u(t)$ as the solution to the balance equation constructed by time stepping. This is the approach followed in BodyandSoul. We sum up as follows
  • To proceed from solution to equation is backwards magical. 
  • To proceed from equation to solution by forward time-stepping is reasonable and not magical. 
There are many specific examples of this form including trigonometric and exponential functions and more generally elementary functions all better constructed by time stepping basic differential equations than magically being picked out of hats.

For example, the trigonometric functions $\sin(t)$ and $\cos(t)$ are better defined as solutions to $\ddot u + u =0$, which can be constructed by time stepping, rather than geometrically as in standard calculus as ratios of the lengths of sides of a right-angled triangle, which is not computationally constructive.

Quantum Mechanics

The same situation is met in quantum mechanics:

The backward magical process is to start from a wave function solution and discover an equation satisfied by the solution, a magical Schrödinger equation without physical basis which is a mystery to all physicists.

The more natural procedure is to start from the Schrödinger equation, which can be formulated as a rational balance equation of smoothed particle dynamics, and then construct the solution (the wave function) by forward time stepping.

Concluding Remark: In the discussion of the mathematics program at Chalmers, the standard text book by Adams represents backwards magics, while BodyandSoul represents forward reason. Pick what you think is best. But after all, who cares?

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