Mathematics in the Information Society
The theoretical basis of science and technology is mathematical modeling usingcalculus: function, derivative, integral, differential equationlinear algebra: vector, matrix, linear system of algebraic equations.
Calculus was invented by Leibniz and Newton in the late 17th century and reached maturity in the 19th century.Linear algebrawas initiated by Descartesin the form of analytical geometry preparing the development of calculus, andwith the computer developing intonumerical linear algebrain the mid 20th century and more generally into the modern information society as described in Return of Descartes.
Mathematical modeling includes the following key steps: formulate equation using calculus and linear algebra, solve equation by computation, analytical or digital by computers.
The principal tool to solve equations before 1950 was analytical calculus and simple forms of computation using mechanical calculator or slide rule. The principal tool today is computational calculus/linear algebra or computational mathematicsusing computers.
Using analytical calculus only simple equations can be solved. Using computational mathematics virtually any equation can be solved. The usefulness of mathematical modeling is thus drastically expanded using computers because realistic modeling becomes possible.
The result is our modern information society, which is based on computational mathematics as the engine of communication, information, searching… virtual reality: picture, film, computer games, music… science: realistic simulation of physical phenomena: weather, global warming… medicin: tomography, genetics… economy: planning, derivative markets…
Mathematics Education Today and Tomorrow
Mathematics education forms the basis of the modern information society, but its content and form are essentially the sameas 50 years ago before the computer revolution, essentially in the form of classical analytical calculus/algebra.
The result is a crisis of mathematics education: A mathematics teacher can no longer demonstratethe usefulness of analytical calculus, because the information society of the student is missing, and the credibility so important in any form of teaching and learning, is suffering. The knowledge of mathematics is deteriorating in a society where mathematics becomes increasingly important.
A reform of mathematics education is necessary, but strong forces in academics and in the education of mathematics teachers block reform. In a reformed mathematics education the computational power of
the computer in solving equations, is coupled with the power of the mathematical language of formulating equations to solve. This means that calculus/linear algebra is boosted with a turbo into a new powerful tool for realistic simulation by computation, realized in the FEniCS Project .
The power of computational mathematics is demonstrated in resolution of the main open problem of classical mathematics of turbulence by computational solution of the Navier-Stokes/Euler equations, see the Knol articles on d’Alembert’s paradox,second law of thermodynamics,Clay Navier-Stokes millennium problem.
The need of reform is developed in Dreams of Calculus: Perspectives on Mathematics Education.
The Body and Soul Project offers a reformed mathematics education whereBody represents calculus and linear algebra and Soul represents computation, in a synthesis of computational calculus. The program contains books, instructional material and computational software in a presentation of mathematics which is understandable and useful,in contrast to many traditional programs which are not understandable or not useful or both.The reform program is ideally suited for Open Online Learning.
Liberation of Human Body and Soul
The human mind is powerful as concerns matters of principle but it is a poor calculator or computer.
Formulating equations concerns matters of principle: The Navier-Stokes/Euler equations express the principles of conservation of mass, momentum and energyin the form of a system of differential equations. The human mind or intelligence is able to formulate equations based on principles, but is not able to solve equations, because it requires long tedious computation. But computers can do the required computations very fast and don’t get tired.
This means that human intelligence playing with principles combined with a computer doing tedious unintelligent number crunching, is a very powerful tool both for understanding and control of the physical world, and forgenerating new virtual worlds.
Human society with its pyramids, cathedrals and cities has resulted form a combination of human mind expressing principle and human body performing the physical labour required to realize the principle. In the industrial society human body is liberated by machines from doing hard tedious physical work, and in the information society human mind is liberated from doing hard tedious computation. To realize the potentiala reformation of mathematics education is needed.
Soccer Games and Baking Cakes vs Computational Mathematics
Solving a differential equation by computation can be compared to a soccer game consisting ofa set of rules of the game — the principles expressed by the differential equation two teams +one soccer ball +one soccer field — real numbers + derivatives a set of referees — error control checking that the principles/rules are followed the actual game — computing a solution by a computer.
We understand that it is big difference between the setup consisting of rules + teams + ball + field, which islike formulating an equation, and the physical process ofactuallly playing a real game in real time with aspecific choice of players, which is like computing a solution with specific data.
Likewise there is a big difference between a recipee for a cake and an actual cake baked according to therecipee or principle. You can eat a cake but not a recipee!
Computational mathematics can also be compared withbaking a cake according to a recipee. We know thatit is very useful to understand the principles involved, e.g. the effect of beating or heating, and that it is necessary tohave the ingredients available and subject them tobeating and heating in the right order according to the recipee. Without ingredients and without following the recipee, there will be no cake.
Formulating an equation is like writing down a recipee for a cake, and computing a solution is like baking a real cakefollowing the recipee using real ingredients (real numbers!). A computed solution can be inspected and enjoyed, just like a real cake.
Principles of Mathematics and Computation
To understand principles may be possible, while to follow principles may be difficult. In mathematics themost essential is to understand the principles involved, which may not be so difficult. If the more difficult part of actually computing something following the principles, can be left to a computer, mathematics can be made both more understandable and more useful for many.
If you understand the rules of a soccer game, which is not so difficult, then you can enjoy watching a soccer game without being able to actually play that well yourself. If you don´t understand the rules, you get nothing out of it.
Mathematics is based on principles. If you can understand the principles you have come a long way. Part of the difficulty of mathematics is to realize that it is agame based on certain principles, andnothing else.This means that you are not free to invent new principles along the way; you have to stick to the given principles and nothing else.This can be helpful, because the options are restricted, but can also be felt like a straitjacket with no freedom, only impossibility. In any case, understanding the principles and what they allow and forbid is very helpful and not so difficult i many cases. For example, you can understand that the incompressible Navier-Stokes equations just express Newton’s 2nd law F=ma and that the fluid is incompressible (like water).
The Body and Soul program presents mathematics and computation based on principles, which are realized using computers.FEniCSis based on a set of principles for computing solutions of differential equations allowingautomation of computational solution of differential equations:Typing a differential equation on the keyboard into the computer, like the Navier-Stokes equations, together with some input data, you get the solution output by presssing a solve button. This is like getting the value of the exponential function exp(x) for a given x, say x=10, by typing exp(10) on a calculator. In fact, the exponential function is evaluated by solving the simple differential equation du/dt = u for t>0 with u(0)=1.
But it isn’t black box magic: The principles are understandable and can be understood by many, and watching solutions of the Navier-Stokes equations uncovers the secrets of turbulence, just like the secret of exponential growth is revealed by solving the differential equation du/dt = u. Compare withMathematics=Magics?