• In all things of nature there is something of the marvelous. (Aristotle)
  • It is simplicity that makes the uneducated more effective than the educated when addressing popular audiences. (Aristotle)
  • You cannot teach a man anything; you can only help him find it within himself. (Galileo Galilei (1564-1642))
  • There is no adequate defense, except stupidity, against the impact of a new idea. (Percy Williams Bridgman (1882-1961) Nobel Prize in Physics, 1946)
  • It requires a very unusual mind to undertake the analysis of the obvious. (Alfred North Whitehead (1861-1947))

What is BodyandSoul

BodyandSoul is a new mathematics education program based on combining the power of the human soul with the power of body in the form of the computer as computational workhorse. In short: mathematics boosted by computer.

Basic text: Mathematical Simulation Technology

The computer is now changing society, science and education, and since the computer is based on mathematics it also changes the nature of mathematics, from analytical mathematics based on symbolic computation with pen and paper to computational mathematics controlled by human brains and performed by computers.

Symbolic mathematics represents soul and computation represents body, with computational mathematics a synthesis of symbolic mathematics and number crunching computation, as a synthesis of body and soul.

The objective of mathematics and science is to simulate real or virtual worlds, to reach understanding, prediction and control. Computational mathematics boosted by computers allows simulation of complex phenomena, such as turbulence, which is is impossible by symbolic analytical matematics alone.

With computational mathematics almost anything thinkable is possible, more or less, while with analytical mathematics almost everything is impossible or very difficult and tricky. This is because computational mathematics is like an All Terrain Vehicle ATV allowing you get from A to B regardless of any paved road or track, while analytical mathematics seeks to find an elegant dirtless shortcut from A to B, which may not exist at all or is very difficult to find.

An ATV computational mathematician able to go from any point A to any point B
A classical analytical mathematician able to balance from a point A to point B, on a preset wire.

As you follow the BodyandSoul program you will discover that computational mathematics is based ona quite small set of principles with an amazing range of applicability. It is like having a small set of moralprinciples to guide you through the complexity of life, with body and soul in constructive cooperation.

You will meet these principles over and over again and with each encounter understand more of their potential.

Technology For/With Simulation

You will find that Simulation Technology can be interpreted as

  • Technology For Simulation
  • Technology With Simulation

in the following sense:

  • For: How to make simulations using mathematics and computer.
  • With: How to find out things about the World using simulations.

A Game About Constructing Games

The BodyandSoul program contains texts and software and supporting educational material, all integratedon a webbased platform. The entirely new possibilities in teaching and learning which are now opened by Internet, can broadly be described as new forms of interactive simulation with the student playing different computer games with the teacher/teaching material. It is like an arcade game with the objective of acquiring skills and tools to master (and win) the game.

Mathematical Simulation Technology can be described as the art of constructing computer games, and the BodyandSoul educational program can itself be viewed as a form of interactive computer game, with the objcetive of learning how to construct computer games. BS as a game about games, a \emph{game about constructing games}.

About the Text

Part I-XI gives an introduction to Mathematical Simulation Technology, with the core being Part IV Leibniz’ World of Mathematics and Part V Descartes World of Analytical Geometry. The material is based on the earlier books

and connect to the following new books which can be used as source of inspiration for explorations in different directions:


The text is organized as follows:

  • I: Introduction: Cover story of Icarus and Daedalus
  • II: Newton’s World of Mechanics: Newton’s 2nd Law
  • III: World of Games
  • IV: Leibniz’ World of Mathematics: Calculus and Linear Algebra
  • V: Descartes’ World of Analytical Geometry
  • VI: Tool Bags: Summary of Calculus and Linear Algebra.
  • VII: Sessions: Road Maps to Mastery
  • VIII: World of Differential Equations: Modeling
  • IX: World of Finite Elements: Solving Differential Equations
  • X: Simulators: Vehicles to Drive
  • XI: Technology With Simulation: Understanding, Predicting, Controling
  • XII: 1D Calculus
  • XIII: MultiD Calculus
  • XIV: Complex Calculus.

Parts I-XI can be seen as the main parts with Parts XII-XIV as a form of supplement.

After browsing the introductory Parts I-II you are invited to the World of Games giving you a direct experience of both mathematics, finite elements and programming, which prepare for a more detailed study of Technology For Simulation in IV-X leading into Technology With Simulation in Part XI. Part XII-XIV gives a more detailed presentation of various aspects of Calculus from 1D to MultiD to Complex Calculus including Fourier Analysis.

Combine Parts III, IV-V and VII

It is a good idea to do Parts III, IV-V and VII in parallel. Part III gives you computational experience ofthe basic concepts of geometry, position, velocity, accelleration, derivative with respect to time/space,physical laws as differential/algebraic equations, which you can connect to physical experience

The mathematics of Part IV-V analyzes these concepts using both a microscope to see details and and a telescope to see gross patterns. Mathematics is partly a play with symbols, and to play the game it is useful to understand the meaning of the symbols. The other aspect of mathematics is number crunching, and to play this game it is useful to understand computational algorithms and to be able to implement them in computer programs.

The Sessions in Part VII help you to learn to master the tools summarized in Tool Bags in VI.

Zapping through BodyandSoul

The IT world and the Internet is based on zapping from one thing to another, by hyperlinks in texts orusing search engines like Google.

To read a massive book from first to last page line after line, is practiced by a decreasing number ofclassically schooled people. Instead an eclectic mode of learning is encouraged, where you quickly can accessinformation about anything from the nearest pizza place to global warming or digital photo, or Calculus and mathematical modeling.

BodyandSoul as an ebook invites the student to zapping, following links in variousdirections, and the text is not really intended to be read page after page. BodyandSoul can be then seen as a set of tools and material stored on shelves of a bike shop, which the student is invited to use to construct new bikes for enjoyment, learning, research or commercial use.

Constructive Mathematics as Turing Machines

You will find that constructive/computational mathematics can be expressed in computer codes. One can argue that the essence of the mathematics then is represented by the code, in the spirit ofTuring and his Turing Machine or universal computer.

In constructive mathematics only constructed (or constructible) mathematical objects exist, in the form ofTuring machines. In constructive mathematics the set of all real numbers does not exist (not even the setof all natural numbers), only specific numbers brought to existence by a Turing machine.

To insist that the set of real numbers does not exist may be shocking to a mathematician brought up inthe ruling paradigm of the logistic/formalist school based on the dogma that it does exist. You cantest the strength of a constructive approach, or weakness of a formalistic/logistic approach by questioning the dogma.

You find material for debate in \hyperref[chapterquarrel]{Do Mathematicians Quarrel?}

Following BodyandSoul you will yourself construct mathematics from scratch by writing the computer codesrepresenting mathematics, and you will thus become the master of your own mathematical bike shop.

BodyandSoul: Games

Through the following sequence of games you will be introduced to Calculus and Linear Algebra and to the construction of simulators and simulation tools:

  • 1d 2d 3d Pong (motion without forces)
  • 1d 2d 3d Ping-Pong (motion with forces: gravity and friction)
  • 1d 2d 3d Elastic Pong (elastic ball)
  • Elastic String/Membrane/Body
  • Elastic String/Membrane (transversal motion)

which prepare to design games of, for example,

  • tennis
  • table-tennis
  • golf, minigolf
  • volleyball…

With this preparation you are then ready to simulate just about anything…In particular, remember that science is a game with the objective of simulating natural phenomena…in words, symbols, images, numbers, by mathematics plus computer…

BodyandSoul: Sessions

The \hyperref[partsessions]{BodyandSoul Sessions} helps you to get in direct interactive contactwith the material by connecting the mathematical theory from start to computational simulation. Going through the sessions you develop understanding of the mathematical principles and you also acquire skill of programming and implement ion of mathematical algorithms. Each Session is self explantory, with references to the text, and treats a specific central topic.

Sessions A gives an introcudtion to programming. Sessions B-D covers the basics of Calculus and Linear Algbra. Sessions E-F concerns mathematics and programjing of FEM.

Herbie Hancock’s chord changes on BodyandSoul in the film Round Midnight.

BodyandSoul: Simulators

With the tools and skills you will acquire, you will be able to construct your simulators for a large variety ofphenomena. BodyandSoul offers prototypes or templates which you can use to get a flying start in your own work. You can build games based on your simulators, or participate in competions between different simulators.

BodyandSoul: The Value of Proofs

What is the role of mathematical proofs, in mathematics and mathematics education? One answer could be that they are used to prove mathematical theorems expressing mathematical truths, of enough interest to be called theorems. An auxiliary result needed to prove a theorem is called lemma, a theorem of minor importance in itself.

One could the focus on the theorem as the end result of the proof as the most important, or on the proof. The goal or the road to the goal as most important.

What could then the value be of knowing a proof of a theorem, e.g. the proof of Pythagoras Theorem? Isn’t itenough to know the theorem, that a^2+b^2=c^2

The advantage of knowing the proof is that it gives you the ability to answer the question Why?

Why does the length c of the diagonal of a rectangle with sides of length a and b, satisfy c^2=a^2+b^2? And it is useful to know answers to the question Why?, in politics, business, science and life in general.

Why is it useful?

When a child starts to ask the question Why? at the age of three or so, it represents an importantstep in the development to an adult. To answer is a parental duty which can a pleasure, if you know a goodanswer, but also frustrating in the many cases when you do not know an answer. In school the child quickly learnsto not ask the question too often, and not in later professional life either.

In science and mathematics, the question is central Why, because this is what science and mathematics is about.

That is in the ideal case. In practice science and mathematics is too often the opposite, that is to justlearn by heart certain formulas and theorems, without being offered any understandable proofs.

In BodyandSoul we seek to stick to the principles and thus put emphasis on understandable proofs, as understandable answers of the question Why?

Knowing the answer gives you a strong position in arguments and also the ability to understand the meaningof the theorem. This is not simply to read what the theorem states, like a parrot, because the statement has to be properly interpreted, and if you don’t know the proof it is great danger that you misunderstand.

If you understand the proof, then you understand the theorem. If not, then misunderstanding is imminent.

It is well known that Einstein did not do well in mathematics in school, but it is generally believed thatnevertheless he developed a mathematical theory of relativity with stunning theorems about curved-space time with proofs so difficult (obscure) that nobody has ever claimed to understand them, not even Einstein himself. You can read more in Did Einstein Not Understand Mathematics?.

My hope is that you will do better than Einstein in math and science, and that you will spend some timeto understand the proofs you will meet. The number of proofs is kept to a minimum, based on the idea that it isbetter to well understand a couple of central proofs or types of arguments, than to half-understand or misunderstand a larger number. Calculus and Linear Algebra may first seem to be hopeless messof theorems, but you will discover that it carries a long way to master a few central theorems, with proofs.

Proof of Pythagoras Theorem. Can you understand it? Is similarity used to show that \bar b=\frac{b^2}{c}? Compare \hyperref[sectgeominterpret]{below}.}

BodyandSoul: Mathematics vs Music

There are several close relations between mathematics and music, which will be illuminated as we go along.

In short, music is a combination of melodyharmony and rythm formed by sequences of tones and chords from different scales of tones. Compare Carla Bley in Ad Infinitum.

Classical music is usually performed from sheet music, written by a now dead famous composer, as interpretations by classically trained musicians capable of playing the notes according to the sheets.

Jazz music on the other is improvised without sheets to follow, only certain predetermined harmonic and rythmic patterns, like a 12 bar blues pattern $C C C C7 | F7 F7 C7 C7 | G7 F7 C7 G7$ grouped into three 4-bar patterns. A jazz musician creates a direct flow of music drawing from a toolbox of melodic, harmonic and rythmic patterns. The training of a jazz muscian consists of learning how to use certain standard tools and to develop personal tools.

Billie Holiday singing Body and Soul

We shall see that classical analytical mathematics is similar to classical notated music: It is usually very difficult and follows a preset scheme written down by a now dead famous mathematician. The role of the analytical mathematician is to interprete the mathematics of the masters, like a pianist interpreting a Beethoven Sonata by skillfully playing the right notes. An interpreter does not have to know how to compose music, only to play what is already composed.

More interestingly, we will see that a computational mathematician is like a jazz musician creating music while playing:

A computational mathematician plays on the computer using tools from a toolbox, and the training consists in learning how to use certain standard tools and to develop personal tools.

As a good example of webbased jazz piano instruction, take a look at Doug McKenzie’s Youtube jazz2511’s ChannelDoug playing Body and Soul. You see the notes being played by Doug on the keyboard coming up as sheet music automatically, and you can follow how Doug uses standard tools and his owns tools to create a new version of BodyandSoul, everytime he is playing the song. Doug helps you go peek behind the curtains and understand \emph{how} the music is put together, and why certain notes are played rather than others.

The idea of BodyandSoul is the make something similar in mathematics.

Doug McKenzie using tools from his toolbox

Power of Language

We do not learn to walk in school, nor to speak our mothers tongue. Children generally speak grammatically correct at the age of three without being taught any grammar at all, in some form of intuitive self-learning process based on some innate capacity for this complicated task.

Speaking, like musical improvization, means to construct new (more or less) meaningful sentences or phrases from little elements of sound.

To tell something is to describe some real or imagined phenomenon using words, as a simulation in words.You can also in words order someone to do something according to some specification,which may give the desired result if your message is understood and has a proper form.

Mathematics offers you a special language allowing you to construct precise instructions describing real orimagined worlds, which can be understood and executed by computers.

\section{BodyandSoul: Lyrics}

  • My heart is sad and lonely
  • For you I sigh, for you dear only
  • Why haven’t you seen it
  • I’m all for you body and soul
  • I spend my days in longing
  • And wondering why it’s me you’re (ogling)
  • I tell you I mean it 
  • I’m all for you body and soul I can’t believe it
  • It’s hard to conceive it
  • That you turn away romance Are you pretending
  • It looks like the ending
  • And less I could have one more chance to prove, dear
  • My life a wreck you’re making You know
  • I’m yours for just the taking
  • I’d gladly surrender myself to you body and soul
  • My life a wreck you’re making
  • You know I’m yours for just the taking
  • I would gladly surrender myself to you body and soul.

BodyandSoul: Philosophy

Body Soul of Salvator Dali.

BodyandSoul: FEniCS

BodyandSoul is closely related to the FEniCS open source software project aimed at setting a new standard of Automated Computational Mathematical Modeling generality, efficiency and simplicity of mathematical methodology, implementation, and application.

In short, FEniCS offers sofware for automation of (i) formulating mathematical equations (modeling) and (ii) solving equations (computation), with the equations usually taking the form of differential/integralequations. See

FEniCS is based on the same mathematics as you will learn to master in BodyandSoul. You can use FEniCS asa model for your own software development as you develop into an expert user capable of contributingto the further development of FEniCS. In particular, you will be able to see yourself that FEniCS is concrete evidencethat the methodology of BodyandSoul is functional.

FEniCS was born on October 30 2003
Header for FEniCS 10

PS: On Mathematics and Music

  • Musical form is close to mathematics — not perhaps to mathematics itself, but certainly to something like mathematical thinking and relationship. (Igor Stravinsky)
  • The most distinct and beautiful statement of any truth (as of music) must take at last the mathematical form. (Henry David Thoreau)
  • We do not listen with the best regard to the verses of a man who is only a poet, nor to his problems if he is only an algebraist; but if a man is at once acquainted with the geometric foundation of things and with their festal splendor, his poetry is exact and his arithmetic music. (Ralph Waldo Emerson)
  • It is harmony which restores unity to the contrasting parts and which moulds them into a cosmos. Harmony is divine, it consists of numerical ratios. Whosoever acquires full understanding of this number harmony, he becomes himself divine and immortal. (B. L. van der Waerden, mathematician)
  • In the future, we can expect that not much difference will exist between education and entertainment. We just have to put intelligence behind the entertainment. (North Carolina State University’s James Lester, quoted at the 12th International Conference on College Teaching and Learning)
  • Musical training is a more potent instrument than any other, because rhythm and harmony find their way into the inward places of soul, on which they mightily fasten, imparting grace, and making the soul of him who is rightly educated graceful. (Plato)
  • Educators have always known that learning and life are maximal where play and work coincide. (L. W. Gibbs, mathematician)
  • Music is the pleasure the human soul experiences from counting without being aware that it is counting. (Leibniz)
  • Mathematics and music, the most sharply contrasted fields of scientific activity which can be found,and yet related, supporting each other, as if to show forth the secret connection which ties together all the activities of our mind, and which leads us to surmise that the manifestations of the artist’s genius are but the unconscious expressions of a mysteriously acting rationality. (Hermann von Helmholtz, physicist and mathematician)
  • May not music be described as mathematics of the sense, mathematics as music of the reason? (James Joseph Sylvester, mathematician)

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