Mathematics and IT

The idea to integrate mathematics into IT education  discussed in a previous post, can really it be seriously considered? Yes, this very natural because

• a laptop is an ideal laboratory for arithmetics, geometry and calculus
• in a couple of  all students from early age will use a laptop/mobile extensively
• computer software is based on mathematics
• a laptop gives feed-back
• programming teaches logic 
• programming can teach computing
• programming can teach mathematical modeling
• programming can teach problem solving 

What more would you like math education to contain? The more I think about this idea, the better it seems. In fact this is the basic idea of Body&Soul.

Reinfeldt to Zuma: Emission Control and Poverty

The recent summit in South Africa co-chaired by President Jakob Zuma and Swedish Prime Minister Fredrik Reinfeldt, who holds the 27-country EU's rotating presidency, dealt with climate change.

Reinfeldt was received with red carpet, military parade and music corps during his first visit to South Africa.

Zuma expressed the view that African countries would veto any climate change deal if rich countries do not meet their demand for money, which some experts said could be up to $200-billion a year.

Reinfeldt expressed the view that developing countries should focus more on the longer-term climate threat than on an economic downturn.

Emission control and poverty in developing countries seems to be message from EU and Reinfelt, which is not acceptable to Zuma and African countries. What does mathematics tell? What will be the deal in Copenhagen, between the rich and the poor? 

If Reinfeldt cannot convince South Africans to stay poor or the EU voters to pay the bill, which seems pretty obvious, will he still insist at the prize of loosing his job, or will he rethink, listen to facts of science and focus on something more constructive? 

Reinfeldt has shown that he is a pragmatist by reforming the old conservative party into the new moderates replacing the socialdemocrats as the party for everybody,  and thus seems to follow reason rather than ideology.

Computer Games: Learning with Feedback

Feedback is most essential in learning. A child learns to speak around the age of two with the help of massive parent feedback. A child cannot learn to speak by listening to a record player, because the communication is one-way without feedback. 

Traditional school education is largely one-way with little feedback and accordingly is inefficient. On the other hand, our modern IT society offers a flood of feed-back on the web, which is attractive to young minds. Computer games offer so exciting feed-back that many young minds get too engaged and even addicted. The risk of getting addicted to reading school books is much smaller. 

It is natural to ask if pedagogics can learn something from computer games? Yes, I believe so. In particular, the Body&Soul reformed mathematics education combining analytical and computational mathematics discussed in previous blogs, can be structured like a computer game including the essential aspects of successively acquiring skills to meet new challenges, with a lot of feed-back. Since computer game technology largely is computational mathematics, it can be fruitful to teach computational mathematics as a form of computer game, or more generally mathematics as a form of IT.

A goal of Icarus Simulation is to develop an interactive web-based version of the Body&Soul program, with features of computer games, to be used in the new Bachelors program in Simulation Technology at KTH. 

If we view interactive simulation as a form of computer game, then we can describe Simulation Technology as an education in the design and construction of computer games based on realistic simulation of physical phenomena, compactly described as 

• an interactive computer game about how to construct interactive computer games
  

which includes mathematics, computation, programming, visualization, physics, mechanics...

More generally, theoretical science can be seen as a game against Nature with the goal of revealing, describing and simulating the secrets of Nature using the language of mathematics...

an addicitive for scientists...

Conservation of Momentum or Newton's 2nd Law?

                                          Incremental or Conservation Party?

Continuing the discussion from the previous post, let us note that the Navier-Stokes equation

expressing conservation of momentum, alternatively can be expressed as Newton's 2nd law 

                                                             F = ma = m dv/dt

with F the force acting on an element of fluid of mass m and acceleration a = dv/dt. We can view these formulations to be equivalent from analytical mathematical point of view, but we may ask if they also are equivalent physically, or computationally? 

Of course they are equivalent, you may say, because mathematics rules the game, but it is not so simple and clear if we recall that the Navier-Stokes equations cannot be solved exactly analytically, only approximately digitally by computers. The equivalence is then not so clear anymore.

So which formulation is most suitable to computation?  Newton's 2nd law because it can be solved by time-stepping moving forward in time with small increments of time: The force F gives the acceleration dv/dt = F/m which tells the change of velocity which tells the change in position, from one time level to the next. 

On the other hand, conservation of momentum is not directly ready for time-stepping, since it just expresses that something is conserved, namely momentum.

We are thus led to prefer an interpretation of a law of nature, which is most accessible to computation.  We may prefer such an interpretation also from physical point of view, if we view real physics as some form of analog computation, as discussed in the knol Is the World a Computation?

Light refraction is a result of the wavelike nature of light as propagating electromagnetic waves. Light refraction can alternatively be described as shortest time of travel of light rays. Wave propagation can be time-stepped, while shortest time of travel is a global minimization problem, for which computational solution is less direct. We are led to view light as waves from physical and computational point of view, rather than as rays of particles. 

An equilibrium states may be described as a state of balance of forces without any net force driving change. To find an equilibrium state of a system, we may time-step the system starting from some out-of-balance non-equilibrium state, with the hope that the system by itself approaches equilibrium. A physical law could then express the dynamics of a system computable by time-stepping, rather than a balance of forces at equilibrium, since this balance may not be directly computable. 

A minimization principle in physics, like minimal time of travel of light, would then not qualify as a physical law unless augmented by e.g. time-stepping into computable form.

 

Wigner without Computer is Unreasonable

                               Computational solution of the Navier-Stokes equations.

In his 1963 Nobel Lecture discussed in the previous post: Events, Laws of Nature and Invariance Principles, Eugene Wigner expresses the physicist dream of a Theory of Everything TOE as some fundamental invariance principle or conservation law in the form of a differential equation, to which the World would be the solution.

If we narrow down the World to fluid mechanics, which is a reasonable a approximation as concerns macroscopic phenomena, then we already have a TOE of fluid mechanics

in the form of the Navier-Stokes equations expressing conservation of mass, momentum and energy. From Wigner's point of view this would close the scientific field of fluid mechanics since everything there is to know, is known: The Navier-Stokes equations!

This TOE would seem to represent extremely effective knowledge, since the NS equations can written down in two lines and can be taught to most people in less than an hour. It would be like a very compact two-line genetic code of fluid mechanics.

But thus is too simple, you say, right? Fluid mechanics is more than just jotting down the NS equations! Yes, you are right! The NS equations also have to be solved to tell us anything, and that turns out to be impossible by analytical mathematics: Only very simple analytical solutions are known which tell you very little about fluid mechanics. So the NS equations alone is not a TOE.

But computing solutions of the NS eqautions numerically using computers is today possible, because computers are now powerful enough, which in a sense gives you a TOE for fluid mechanics. However, there is hook: You have to compute solutions one by one with different data and you cannot get all solutions in one shot. 

Today you have a wonderful laboratory in your laptop allowing you to explore the rich field of fluid mechanics successively by computing solutions of the NS equatiopns, studying their properties and hopefully discovering regularities or even laws supporting understanding. Some of what you can learn from this laboratory is presented on my knols on Fluid Mechanics.

Wigner's vision of a TOE represents a pre-computer classical approach to physics, which is beyond reach because even if the basic equation is a simple analytical equation like NS, the World as the solution to the equation is not simple and cannot be described by analytical mathematics, at least not a priori. This does not say that an a posteriori description by analytical mathematics is also impossible. Once solutions have been computed one can start to look for regularities and maybe find some which can be expressed by analytical mathematics, but only a posteriori. Wigner without computer is unreasonable.

The Resonable Ineffectiveness of Mathematics

                                                                The young Wigner.

In the previous post A Critical Analysis of the Ideology of Mathematics we made the observation that the foundation of school mathematics on all levels can be expressed in the words of Physics Nobel Laurate Eugene Wigner as: 

• The unreasonable effectiveness of mathematics in the natural sciences. The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. 
  

In his 1963 Nobel Lecture Events, Laws of Nature and Invariance Principles Wigner further explains:

• Physics does not endeavor to explain nature, it only endeavors to explain the regularities in the behavior of objects, which are called the laws of nature. 

Acoording to Wigner mathematics is thus unreasonably effective as a language of expressing laws of nature interpreted as regularities of nature. 

But is this unreasonable? Is it not completely reasonable that analytical mathematics is effective in expressing regularities, like an elliptic orbit or harmonic oscillation? So if laws of nature express regularities it is fully reasonable that they can be expressed in the language of analytical mathematics.

But what is a law of nature? Is it really a regularity expressible by analytical mathematics as Wigner seems to claim?  

Let us take the same example as Wigner: Consider at planetary system governed by Newton's laws of motion, which no doubt are laws of nature. Is this all there is to say about planetary systems? No, it is not because the planetary motions are not included in Newton's laws. 

The motions result from letting the system evolve forward in time step by step according to Newton's laws from some initial state. In the simplest case of one planet orbiting a heavy sun the orbit is an ellipse, but with more than one planet the motion can be abitrarily complicated and not allow a representation in terms of elementary functions. 

Does this mean that there are no laws for the motion of a planetary system with many planets?

Of course not, but these laws are not explicit to us like Newton's laws, but hidden implicit and there is no golden rule how to find them and express them by elementary fucntions. 

We may compare with turbulent flow which is governed by Newton's laws but exhibits a very complex partly chaotic structure with a variety of interacting vortices on different scales.  But even a turbulent flow can exhibit some regularities in the form of certain meanvalues, which can be computationally predicted even if pointvalues vary chaotically, meanvalues like drag and lift.  However, there is no neat mathematical formula that expresses the drag and lift of a given body. Turbulent flow has to be computed step by step and there is no shortcut to regularity of solutions as in the case of  the elliptic orbit of one planet around a sun.

We are led to the conclusion analytical mathematics is not unresonably effective but rather resonably ineffective, while computational mathematics is resonably effective. 

Analytical vs Computational Mathematics, at KTH...

                                                    Crash simulation of school mathematics.

In the Fall 2010 a new Bachelors Program in Simulation Technology will start at the Royal Institute of Technology KTH based on the Body&Soul Applied Mathematics Reform Program.

In the new B&S program standard analytical mathematics of calculus and linear algebra is combined with computational calculus and linear algebra into an integrated synthesis of analytical and computational mathematics, which opens entirely new possibilities in teaching and learning in science, technology, simulation, visualization and virtual reality. 

This is because calculus and linear algebra boosted by computers gives a new very powerful tool allowing simulation of complex phenomena of real and imagined worlds unreachable by analytical mathematics.  

The shift from standard analytical mathematics to computational mathematics has met strong opposition from the mathematics department at KTH unable or unwilling to reform standard analytical mathematics courses. The fact that KTH anyway has decided to start a program based on computational mathematics, given by people outside the mathematics department, opens to a similar reform in engineering education as a whole... 

Connecting to the previous blogpost Will Mathematics be Replaced by IT in School Education?

one can see the shift at KTH from analytical to computational mathematics as replacing standard analytical mathematics by IT, a shift which can propagate down through the whole school system....

The consequences of the KTH decision thus can be far-reaching, since KTH is a leading university and sets the agenda for school mathematics...in Sweden at least...

Mathematics Integrated with IT in School Education?

As noted in the previous blog, mathematics education of today is in many ways similar to the education in religion of yesterday, which is no longer mandatory in Western schools. 

From this experience we may expect that mathematics will not be mandatory tomorrow. Does this mean that students will no longer learn any mathematics. Not at all! 

A new subject is now entering education on all levels: Information Technology or IT.  Most likely IT will replace mathematics as the core of education together with language. But IT is largely based on logic, programming and computational mathematics, and it is possible to envision an IT education which teaches more mathematics than the present system does. More relevant mathematics for everybody and much more for students with special interest in mathematics and IT.

Integrating mathematics education with IT education is the logical conclusion of the mathematical war in the 1930s, which was won by the constructivists when Gödel hit the logicist/formalist school with his incompleteness theorems. After 80 years of incubation with a rise of the IT age, Gödel's poison now starts to have an effect. After all, IT and constructivist/computational mathematics is the same.

A Critical Analysis of the Ideology of Mathematics

The Ideology of Mathematics as presented in the documents of the previous blogpost can be summarized as:

• Mathematics has a double character: It is both the most original, complex and beautiful free creation of the human spirit with its own internal standards, and a universal practical tool.  The miraculous double character is described by the Physics Nobel Laurate Eugene Wigner as:
• The unreasonable effectiveness of mathematics in the natural sciences. The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. 

Mathematics education is based on this mystification, this true miracle:

It allows mathematics education from early grades and on to focus on the language and logic of mathematics learning the words "and", "or", "not", "there is/are", "for some", "for every", "for all", in the safe conviction that this will be most useful to all students in their practical lives. 

It allows mathematics education to focus on axiomatics in the safe conviction that human knowledge can be axiomatized.

It allows mathematics education to focus just about anything in the safe conviction that mathematics is universally applicable.

But there are no miracles, trivialities are trivialities, axiomatization of knowledge is impossible, and no mathematics is universally applicable. To build education on a mystery which is neither understood nor deserved, is not a good idea, because learning is about understanding and students do not profit from free gifts which they do not understand nor deserve and which they cannot use.

For a discussion of how mathematics education can and should be reformed see my blog posts

on mathematics education. In particular notice Mathematics and Religion vs Self-Help

showing close similarities between mathematics and religion sharing the double character of uplifted divinity and universal practicality. In both cases the idea is to study the language of the Divinity to learn about the World. This is still practiced in Islamic schools, but no longer in Western schools.

The two characters of mathematics clashed in the great war between the logicist/formalist and constructivist schools in 1920-30s. The constructivists won on technical knockout but they were soon cleansed from mathematics departments filled with logicists/formalists still in control and forming the ideology of mathematics today, so well expressed by the Committe on Logical Education. For more war reports, see my knols on mathematics.

The Ideology of Mathematics

What is the ideology of mathematics underlying mathematics education as presented by professional mathematicians? Let us seek an answer in the following typical texts to be analyzed in the next blogpost:

American Association for the Advancement of Science explains in Science for All Americans:

• Mathematics relies on both logic and creativity, and it is pursued both for a variety of practical purposes and for its intrinsic interest. 
• For some people, and not only professional mathematicians, the essence of mathematics lies in its beauty and its intellectual challenge. 
• For others, including many scientists and engineers, the chief value of mathematics is how it applies to their own work. 
• Because mathematics plays such a central role in modern culture, some basic understanding of the nature of mathematics is requisite for scientific literacy. To achieve this, students need to perceive mathematics as part of the scientific endeavor, comprehend the nature of mathematical thinking, and become familiar with key mathematical ideas and skills.
  

The Committee on Logic Eduction offers the following comprehensive summary:

• There are creative tensions in mathematics between beauty and utility, abstraction and application, between a search for unity and a desire to treat phenomena comprehensively.
• Mathematics was originally linked with science and technology; however, it gradually became independent of science and technology, and present-day mathematicians think freely about virtually everything possible.
• Therefore, mathematics is said to be a free creation of the human spirit. Special characteristics of mathematics are the clarity and precision of definitions, including usuage of words in ways that differ from their use in everyday language, and the certainty of mathematical truth based on deductive mathematical reasoning.
• Given what Wigner call the "unreasonable effectiveness of mathematics", all students should learn the basic nature of mathematics and mathematical reasoning and its use in organizing and modeling natural phenomena. 
• In the practice of mathematics, typically some concepts and statements are taken as given. They may be applied or serve as the foundation for the development of further mathematics. Additional concepts can be defined carefully in terms of the given ones. Conjectures can be developed on the basis of experience with examples. Further statements can be proved deductively based on what has been assumed. 
• This process has been repeated extensively, resulting in mathematics having its own intricate structure, with concepts and areas of specialization that require considerable time and study to grasp. Moreover mathematics is interconnected in many interesting ways. 
• It may be useful to think of students learning mathematics along the lines of a generalized structure of reasoning: (1) recognition, (2) analysis, (3) informal deduction, (4) formal deduction, (5)  axiomatics. 
• In early grades, students learn the basic language including the critical logical words "and", "or", "not", "there is/are", "for some", "for every", "for all". They see multidigit numbers being built from single digit numbers. They match the trajectory of a kicked ball with the concept of line. They recognize patterns in sequences of numbers and shapes. In middle grades students develop habits of reasoning "locally", clarifying the assumptions of a particular problems and examining the steps involved in the solution to determine correctness. 
• For example, one of us recently observed a fifth grade teacher asking her students for the definition of polygon. They knew, for example that triangles, squares and hexagons were polygons. It was exciting to see the students wrestling with abstraction, differentiating polygons from circles, and finally focusing on polygons as figures with sides. 
• By the end of high school, students should be aware of the global deductive nature of axiomatic mathematics. They should be familiar with the connections between our number systems and algebra, between algebra and geometry. 
• They should be comfortable reasoning with short sequences of statements, with Venn diagrams and other visual and diagrammatic methods. They should have experience with modeling, recognizing for example that certain natural phenomena obey linear relationships and that linear relationships make prediction so easy that we try to approximate other more complicated phenomena by linear ones. 
• It is important both to understand how algebraic relationships can describe particular problems and to understand the power derived by working abstractly with the mathematics which applies to many different situations.

Everybody Counts: A Report to the Nation on the Future of Mathematics Education describes The Nature of Mathematics as follows:

• Mathematics reveals hidden patterns that help us understand the world around us. Now much more than arithmetic and geometry, mathematics today is a diverse discipline that deals with data, measurements, and observations from science; with inference, deduction, and proof; and with mathematical models of natural phenomena, of human behavior, and of social systems.
• As a practical matter, mathematics is a science of pattern and order. Its domain is not molecules or cells, but numbers, chance, form, algorithms, and change. As a science of abstract objects, mathematics relies on logic rather than on observation as its standard of truth, yet employs observation, simulation, and even experimentation as means of discovering truth.
• The special role of mathematics in education is a consequence of its universal applicability. The results of mathematics--theorems and theories--are both significant and useful; the best results are also elegant and deep. Through its theorems, mathematics offers science both a foundation of truth and a standard of certainty.
• In addition to theorems and theories, mathematics offers distinctive modes of thought which are both versatile and powerful, including modeling, abstraction, optimization, logical analysis, inference from data, and use of symbols. Experience with mathematical modes of thought builds mathematical power--a capacity of mind of increasing value in this technological age that enables one to read critically, to identify fallacies, to detect bias, to assess risk, and to suggest alternatives. 
• Mathematics empowers us to understand better the information-laden world in which we live. During the first half of the twentieth century, mathematical growth was stimulated primarily by the power of abstraction and deduction, climaxing more than two centuries of effort to extract full benefit from the mathematical principles of physical science formulated by Isaac Newton. 
• Now, as the century closes, the historic alliances of mathematics with science are expanding rapidly; the highly developed legacy of classical mathematical theory is being put to broad and often stunning use in a vast mathematical landscape.
• Several particular events triggered periods of explosive growth. The Second World War forced development of many new and powerful methods of applied mathematics. Postwar government investment in mathematics, fueled by Sputnik, accelerated growth in both education and research. Then the development of electronic computing moved mathematics toward an algorithmic perspective even as it provided mathematicians with a powerful tool for exploring patterns and testing conjectures.
• At the end of the nineteenth century, the axiomatization of mathematics on a foundation of logic and sets made possible grand theories of algebra, analysis, and topology whose synthesis dominated mathematics research and teaching for the first two thirds of the twentieth century. 
• These traditional areas have now been supplemented by major developments in other mathematical sciences--in number theory, logic, statistics, operations research, probability, computation, geometry, and combinatorics. In each of these subdisciplines, applications parallel theory. 
• Even the most esoteric and abstract parts of mathematics--number theory and logic, for example--are now used routinely in applications (for example, in computer science and cryptography). Fifty years ago, the leading British mathematician G.H. Hardy could boast that number theory was the most pure and least useful part of mathematics. Today, Hardy's mathematics is studied as an essential prerequisite to many applications, including control of automated systems, data transmission from remote satellites, protection of financial records, and efficient algorithms for computation.
• In 1960, at a time when theoretical physics was the central jewel in the crown of applied mathematics, Eugene Wigner wrote about the ``unreasonable effectiveness'' of mathematics in the natural sciences: ``The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.'' 
• Theoretical physics has continued to adopt (and occasionally invent) increasingly abstract mathematical models as the foundation for current theories. For example, Lie groups and gauge theories--exotic expressions of symmetry--are fundamental tools in the physicist's search for a unified theory of force.During this same period, however, striking applications of mathematics have emerged across the entire landscape of natural, behavioral, and social sciences. 
• All advances in design, control, and efficiency of modern airliners depend on sophisticated mathematical models that simulate performance before prototypes are built. From medical technology (CAT scanners) to economic planning (input/output models of economic behavior), from genetics (decoding of DNA) to geology (locating oil reserves), mathematics has made an indelible imprint on every part of modern science, even as science itself has stimulated the growth of many branches of mathematics.
• Applications of one part of mathematics to another--of geometry to analysis, of probability to number theory--provide renewed evidence of the fundamental unity of mathematics. Despite frequent connections among problems in science and mathematics, the constant discovery of new alliances retains a surprising degree of unpredictability and serendipity. 
• Whether planned or unplanned, the cross-fertilization between science and mathematics in problems, theories, and concepts has rarely been greater than it is now, in this last quarter of the twentieth century.

The mathematician A N Whitehead, who wrote the bible of the logicist school Principia Mathematica together with Bertrand Russell, explains to us in Mathematics in the History of Thought:

• The science of pure mathematics, in its modern developments, may claim to be the most original creation of the human spirit. 
• The originality of mathematics consists in the fact that in mathematical science connections between things are exhibited which, apart from the agency of human reason, are extremely unobvious. 
• Suppose we project our imagination backwards through many thousands of years, and endeavour to realise the simple-mindedness of even the greatest intellects in those early societies. Abstract ideas which to us are immediately obvious must have been, for them, matters only of the most dim apprehension. For example take the question of number.
• We think of the number 'five' as applying to appropriate groups of any entities whatsoever-to five fishes, five children, five apples, five days. Thus in considering the relations of the number 'five' to the number 'three,' we are thinking of two groups of things, one with five members and the other with three members. But we are entirely abstracting from any consideration of any particular entities, or even of any particular sorts of entities, which go to make up the membership of either of the two groups. We are merely thinking of those relationships between those two groups which are entirely independent of the individual essences of any of the members of either group. This is a very remarkable feat of abstraction; and it must have taken ages for the human race to rise to it. 
• During a long period, groups of fishes will have been compared to each other in respect to their multiplicity, and groups of days to each other. But the first man who noticed the analogy between a group of seven fishes and a group of seven days made a notable advance in the history of thought. He was the first man who entertained a concept belonging to the science of pure mathematics. At that moment it must have been impossible for him to divine the complexity and subtlety of these abstract mathematical ideas which were waiting for discovery. Nor could he have guessed that these notions would exert a widespread fascination in each succeeding generation.
• The tremendous future effect of mathematical knowledge on the lives of men, on their daily avocations, on their habitual thoughts, on the organization of society, must have been even more completely shrouded from the foresight of those early thinkers. Even now there is a very wavering grasp of the true position of mathematics as an element in the history of thought. 
• When we think of mathematics, we have in our mind a science devoted to the exploration of number, quantity, geometry, and in modern times also including investigation into yet more abstract concepts of order, and into analogous types of purely logical relations. The point of mathematics is that in it we have always got rid of the particular instance, and even of any particular sorts of entities. So that for example, no mathematical truths apply merely to fish, or merely to stones, or merely to colours. So long as you are dealing with pure mathematics, you are in the realm of complete and absolute abstraction. 
• All you assert is, that reason insists on the admission that, if any entities whatever have any relations which satisfy such-and-such purely abstract conditions, then they must have other relations which satisfy other purely abstract conditions.
• Mathematics is thought moving in the sphere of complete abstraction from any particular instance of what it is talking about. So far is this view of mathematics from being obvious, that we can easily assure ourselves that it is not, even now, generally understood.