Homo Computans of the Read-Write-Execute Society

The recent book Shop Class as Soulcraft by Matthew B. Crawford sings praise to the happy productive engaged craftsman and repair mechanic in direct contact with realities in his shop filled with useful tools and spare parts, as compared to the unhappy unproductive alienated bureaucrat imprisoned in his cubicle filled with useless documents. 

This is Homo faber  = Man the Smith or Man the Maker or the working man using tools.

Homo faber is the craftsman of a pre-industrial society using tools in an inventive fashion according to his own inspiration, as compared the worker at the assembly line of the industrial society repeating a prescribed set of simple routine tasks without inspiration. Homo faber would then connect to Homo ludens, the playing man, while the worker at the assembly line would associate with Homo tristis, the sad man.

If now Homo faber has largely disappeared as production has become mechanized, one may ask if Homo faber may reappear today as the computer wiz using the computer as a tool in an active inventive way, as compared to a Homo consumans passively consuming whatever is presented on the screen. The modern Homo faber could then be 

• Homo Computans =   the working man playing the computer.
  

Other possibilities are  Homo musicus  or Homo coquus = the cooking man.

Homo Computans would be representative of the interactive read-write-execute society now emerging  as the interactive web expands with blogs, youtube and facebook, as compared to the passive read-obey society of the 20th century with state-controled uni-directional radio and television channels. 

Mathematics of Global Warming

Peter Landesman states in the  article The Mathematics of Global Warming in American Thinker: 

• The forecasts of global warming are based on mathematical solutions for equations of weather models. But all of these solutions are inaccurate. Therefore, no valid scientific conclusions can be made concerning global warming. The false claim for the effectiveness of mathematics is an unreported scandal at least as important as the recent climate data fraud. Why is the math important? And why don't the climatologists use it correctly?
  

Good question! But this opens a vast field for efficient correct computational mathematical modeling of weather and climate. This connects to my previous posts on climate simulation including

• How Climate Alarmism Got Started: The Lorenz Equations
• Hurrah! Scientists Back in Business
• Short Time vs Long Time Accuracy
• Climate and Turbulence Modeling.

The World asks for a restart of climate modeling, by computational mathematicians! We are ready! The code words are:

• adaptive finite element methods for turbulent flow with automatic duality based output error control 
  

as presented in Computational Turbulent Incompressible Flow.

Even if it will not be possible to do meaningful longtime predictions, improved climate modeling can give scientific insight into the intricate thermodynamics of weather and climate.

Climate modeling is the only way of reaching understanding, since controled experimenting is impossible. In particular it seems impossible to single out the influence of anthropogenic burning of fossil fuel from unknown other forcings.

The collapse of current climate modeling controled by a few governmental institutions, thus may open the field for a broader participation including computational mathematicians. 

Mathematics of All Souls

The non-physical nature of the human soul or self-consciousness as opposed to the physical nature of the human body, has been studied by many thinkers including Descartes, Spinoza, Kant, Hegel, Freud, Lacan, Dennet and Damasio, as described in particular in The Parallax View by Slavoj Zizek.

Lacan's psychoanalytic theory is mathematical in a sense, and on All Souls Day on Nov 2 it is natural to ask if mathematics can offer some understanding of body-soul duality? Let's seek an answer:

It appears that one aspect of the non-physical or virtual nature of soul can be described as sensitivity of future goal satisfaction with respect to perturbation of the present state. In economy this is represented by derivates relating to changes of underlying assets, like options on stocks, with options representing virtual soul and stocks representing real body.

The important aspect is that the derivate has a non-physical virtual character since it relates to changes of an underlying physical asset. For example, the velocity of a physical body as change of position per unit time step, has a virtual non-physical character, while position is physical. You can directly touch position but not velocity. Being is body and becoming is soul, in the spirit of Heidegger.

The soul thus would appear as a sensitivity, which in mathematical optimization is expressed by duality with the sensitivity being the solution to a linearized dual/adjoint equation obtained by linearizing the state equation. The duality body-soul would thus be reflected in the duality of mathematical optimization.

The body would thus be the solution of the state equation, while the soul would be the solution of a corresponding dual/adjoint state equation, expressing sensitivity of a goal function with respect to state perturbations. In particular, this captures the aspect of the soul of planning for the future since the dual/adjoint state equation runs backward in time. The physical body thus moves forward in time, while the non-physical soul moves backward. The body moves from the present towards the future, and the soul moves from the future towards the present.

The dual world of the soul would appear as a partly strange Alice in Wonderland reflection of the real world of the body, since in the dual/adjoint equation in linearization picks up certain aspects of the state equation and distorts them in a backward time process. The dual world is a strange fully virtual world with everything moving backwards in a strange distorted fashion. But it can be captured by a computer and thus is open to inspection and a posteriori rationalization and understanding. The rabbit obsessed with time in Alice in the Wonderland maybe lives in the dual world.

Bodily functions may be controled in unconscious direct feed-back, while the soul by duality

would be able to plan and steer towards future goals as a unique capability of homo sapiens.

The sensitivities could be expressed as feelings giving weight to perturbations of the state equation in the optimization process of human life.

The financial crisis shows the danger of souls with too much belief in option derivate fantasy

reflecting unrealistic future gains without connection to realities of underlying body assets.

Another aspect is soul as simulation of body with the soul housing a brain representation of the body, rather than a brain dual of feelings. When the soul back-tracks from some future goal in planning coming activities, simulation of different possible scenarios in forward time plays an important role. In other words, the soul has a double role of both simulating the state equation in forward time and solving the dual equation in backward time. This may be the reason the human brain is so uniquely big compared to the body...

Teoretisk Matematik vs Praktik

Anders Björner, föreståndare för Institut Mittag-Leffler, skriver idag på DN Essä under titeln Matematik i Praktiken:

• Det finns inget Nobelpris i matematik, och intresset för ämnet är svagt -- det uppfattas ofta som teoretiskt och världsfrånvänt.
  
• Men bakom en lång rad av vår tids största tekniska framgångar som Googlesökningar, fiberkablar, kreditkort, satellitsignaler, datortomografi och mobiletelefoner, ligger gamla och nya matematiska rön.
• Det är omomtvistat att matematiken har varit av central betydelse för vår civilisation sedan urminnes tider. 
• Mindre känt är att matematikens ställning och betydelse bland vetenskaperna kraftigt har stärkts under de senaste decennierna.
• Ändå lever matematiken som vetenskapsgren ett liv något i skymundan.
• De ansvariga i vårt land verkar inte inse att kompetens i matematik vid forskningsfronten är en för landets framtid, i äkta mening, strategisk resurs.

Det Björner beskriver är kontradiktioner: 

• Teoretisk och världsfrånvänd matematik har visat sig vara synnerligen praktisk och världstillvänd.
• Att teoretisk matematik är praktisk och världstillvänd, inses inte av de ansvariga i vårt land; den teoretiska matematiken lever i skymundan.
• Om teoretiska matematiker betalas med skattemedel för att utveckla teorier utan tanke på någon praktik, så kommer detta att leda till en rad praktiska tillämpningar av nytta för vårt samhälle, som nya kreditkort och fiberkablar.

Problemet med kontradiktioner är att de är föga övertygande, p g a kancellation. 

Matematik tolererar inte kontradiktioner, och kanske inte heller de ansvariga i vårt land. 

Att säga att tillämpad matematik kan vara av praktisk nytta är inte kontradiktoriskt, och är ganska lätt att argumentera för. Eller hur Anders?

Hurrah! Scientists Back in Spotlight!

100 years ago scientists were leading the debate by forming the new wonderful world of modern physics. Einstein was preparing his entree to the general public to become the most famous person all times. Then came the wars, science was discredited as something useful for evil purposes and the interest decreased,  among students and in the general public.

But with the debate on global warming scientists again are back on track: Today nothing is more important than mathematical climate models and scientific measurements and their accuracy and relevance. Are we approaching a hot hell or not? Only scientists can tell. Politicians listen. The general public waits...

But scientists do not agree, and so there is an intense debate going, in particular as a preparation for the Copenhagen meeting on climate in December, which can be used in a case study of scientific method and practice.  The web is filled with conflicting information, but let us start with something which just appeared:

• The Yamal Implosion about cherry-picking data showing global warming.
• Cooling Down the Cassandras about the plateau in temperature the last decade. 
• Flawed Climate Data about hockey stick mathematics.

This is bad news for climate alarmism, but the debate is not over. Global climate is a complex dynamical system and a real challenge to science. As long as the debate is raging money will flow into climate research and give a boost to classical physics and fluid mechanics...and above all to computational mathematics...and there are new computational tools ready to do their service to humanity...as we will report on later...

An overview is given in the 4-slide presentation The Actual State of Climate Science by Roger Pielke SrRead, also presented on a video featuring 2 alarmists and 2 critics allowing a study of typical arguments of the debate.

The Mathematical Secret of Flight

I have just upon invitation submitted the article The Mathematical Secret of Flight to Normat Nordisk Matematisk Tidskrift.

The article explains in easily understandable form how a wing generates lift at the expense of small drag, based on a stability analysis of potential flow and computations of turbulent solutions of the incompressible Navier-Stokes equations for slightly viscous flow. It is also shown that the state-of-the-art explanation of lift by Kutta-Zhukovsky based on connecting lift to circulation, is physically incorrect.

The new explanation of the mathematics and physics of flight is also presented in the knol

Why It Is Possible to Fly including more references to my work with Johan Hoffman.

Realities of Mathematical Climate Simulation

A new Oxfam report has today warned that at least 4.5 million children could die unless world leaders deliver additional funds to help poor countries fight the growing impact of climate change, rather than diverting it from existing aid promises.

Rich countries must not steal money from poor hospitals and schools in order to pay their climate debt to the developing world, said Jeremy Hobbs, Executive Director of Oxfam International. But is this realistic? 

The idea to strain the already strained economies of poor countries by climate debts, is based on certain alarming predictions obtained by certain computational mathematical models. Suddenly computational mathematics is a question of life and death. If the mathematics alarm is a false alarm, it risks to have meaningless terrible consequences. Who knows if the math is correct? See my posts on climate simulation. All of this is mind-boggling, for a mathematican and human being. Is God Mathematician or is the Devil?

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.
  

Short-Time Accuracy Test of Climate Models?

The climate of the Earth results from turbulent flow of air in the atmosphere and turbulent flow of water in the ocean, which is heated by incoming solar radiation and cooled by outgoing infrared radiation, both of which critically depend on cloud formation, which requires presence of particles in the air or aerosols acting as condensation kernels.

Burning of fossil fuels produces aerosols which can enhance cloud formation. Low clouds  can decrease incoming radiation and thus act as negative feedback to global warming from fossil fuels, while high clouds can decrease outgoing radiation and thus act as positive feed-back. The net effect appears to be unknown and uncertainties in modeling of cloud formation propagate to uncertainties in global climate modeling.

A main goal of climate modeling is to predict climate sensitivity, that is the increase of global temperature from doubling of CO2 in the atmosphere. In the  IPCC 4th Assessment Report climate sensitivity is predicted to likely be between 2 and 4.5 degrees Celcius, which thus could be the global warming in 2100 without CO2 emission control. The UK Met Office explains to us:

• There have been major advances in the development and use of models over the last 20 years and the current models give us a reliable guide to the direction of future climate change.
• Computer models cannot predict the future exactly...
• Current models enable us to attribute the causes of past climate change, and predict the main features of the future climate, with a high degree of confidence.
• As well as producing CO2, burning fossil fuels also produces small particles called aerosols which cool the climate by reflecting sunlight back into space. These have increased steadily in concentration over the 20th century, which has probably offset some of the warming we have seen.

IPCC and the Met Office thus inform us that even if climate model predictions are incorrect over year and decade, they can be correct over centennials. But is this really possible from a mathematical point of view? Are there dynamical systems which allow computational modeling with this surprising property? Long-time accurate while short-time inaccurate? We are familiar with models which are short-time pointwise accurate and long-time pointwise inaccurate, but the opposite?

There are trivial such systems, like modeling an oscillation between -1 and +1 by a constant 0 state, but is the climate such a trivial system? Probably not. 

There are dissipative systems which forget initial data over long-time, so if initial data are incorrect this affects accuracy only for short-time. Climate models are dissipative and thus partly forget initial data, but that is not enough to secure long-time accuracy. 

The previous blog Climate and Turbulence Modeling recalled the analysis of turbulent flow in Computational Turbulent Incompressible Flow, showing that long-time meanvalues of turbulent flow may be predictable even if pointvalues are predictable only over short time, because of cancellation effects.  It is likely that climate models can share this property, but it requires short-time accuracy.

Altogether, I see no real reason to expect that short-time inaccurate climate models can be long-time meanvalue accurate, as suggested by IPCC and the Met Office. 

If this observation is correct, one could require climate models to be short-time accurate, which can be tested in short-time,  as a necessary requirement for reliability of long-time meanvalue prediction of climate sensitivity. 

Concerning the predictive capabilities of current climate models, see the US Senate Report: Dissent over Global Warming Claims.

The lecture Considering the Human Influence on Climate by R. A. Pielke Sr, is also informative. 

Mathematics and Religion vs Self-Help

There are similarities between traditional education in religion and contemporary education in mathematics based on an idea that God = Mathematics. To see this, consider the following common statements which express the fundamental beliefs which religious/mathematics education seeks to imprint in the blank minds of the students:

• Religion/Math is a supreme creation of human thought.
• Religion/Math is a high perfect art as daring as the most secret dreams of imagination.
• Religion/Math is a fundament of society.
• Religion/Math helps us to understand the World.
• Religion/Math helps us to cope with the problems we face in our lives.
• God/Math is everywhere present but invisible.
• God/Math is allmighty but with very limited practical utility.
• Religion/Math can be properly understood only by a selected few; the others have to accept it without understanding.
• A life without Religion/Math is a miserable life.
• Religion/Math has an intrinsic beauty and coherence.
• Religion/Math Mathematics plays a pivotal role in the progress of society and its continued growth relies on the encouragement and teaching of the next generation religious/mathematical thinkers, and outreach to the public and schools. 
  

Traditional education in religion was very successful in implanting these beliefs, but nevertheless today confessional religion has vanished from the school system, at least in the Western World. 

Similarly, even if the school system still succeeds in making all students believe in the greatness and importance of mathematics, whether they succeed or fail miserably, mathematics education is today in a state of permanent crisis in the footpaths of religious education. 

Religious education is today being replaced by a flood of self-help psychology and traditional mathematics is being replaced by self-help laptop technology:

• how to handle words, pictures, sound and videos
• how handle social life using facebook and blog 
• how to get information and get to understand the world, using search engines
• how to play computer games and music
• how handle gps, mobile and bank account. 

If not mathematics education is going to vanish from the school system, it has to be reformed into self-help laptop technology, and come down form the higher spheres of religion. This is further motivated on my blogs about mathematics education.

Climate and Turbulence Modeling

Global climate models are based on turbulence models, since the slightly viscous flow of air in the atmosphere and water in the oceans is turbulent. Turbulence modeling, in the form of  analytical mathematical models, is a main unsolved problem of fluid mechanics.

In our book Computational Turbulent Incompressible Flow, with prel. version for download, Johan Hoffman and I present a new approach to turbulence modeling based on ab initio numerical computation with turbulence automatically modeled by the stabilization of the numerics, thus without any explicit analytical turbulence model.  

We show that mean-value quantities of turbulent flow such as drag and mean temperature can be accurately computed without analytical turbulence model, thus circumventing the main unsolved problem of fluid mechanics. We plan to test this approach on climate modeling with hopefully a connection between turbulence and the main unsolved problem of climate modeling: cloud formation. 

We will report as soon as we have something to report on...hopefully before the Copenhagen meeting in December...since the outcome of this meeting critically depends on computational modeling of turbulence and cloud formation...and dark clouds over the meeting are already forming... 

Coin Tossing: Cold or Warm?

Newscientist reports:

• Forecasts of climate change are about to go seriously out of kilter. One of the world's top climate modellers said Thursday we could be about to enter "one or even two decades during which temperatures cool."People will say this is global warming disappearing," he told more than 1500 of the world's top climate scientists gathering in Geneva at the UN's World Climate Conference.
• "I am not one of the sceptics," insisted Mojib Latif of the Leibniz Institute of Marine Sciences at Kiel University, Germany. "However, we have to ask the nasty questions ourselves or other people will do it.
• "Few climate scientists go as far as Latif, an author for the Intergovernmental Panel on Climate Change. But more and more agree that the short-term prognosis for climate change is much less certain than once thought.
•  "In many ways we know more about what will happen in the 2050s than next year," said Vicky Pope from the UK Met Office.

The message is that global climate models cannot predict year or decade meanvalues, but can predict centennial meanvalues. How can this be? What is the mathematics behind such a belief? 

The first idea that come to mind is the law of large numbers of statistics offering prediction of the meanvalue 0.5 of many cointosses between 0 and 1, but no prediction of the meanvalue of a few tosses. But is climate modeling the same as coin tossing between cold and warm? 

Newscientist concludes:

• The world may badly want reliable forecasts of future climate. But such predictions are proving as elusive as the perfect weather forecast.
  

The future of mankind thus seems to lie in the hands of mathematicians running the climate models...but coin tossing statistics does not seem to be enough...what can be done or said? 

Well, let us recall that the 0.5 probability of heads in coin tossing is computed mathematically using the fact that a rotating coin has head up half of the time, that is using a short-time-accurate mathematical model, see the discussion in Chapter 13 Turbulence and Chaos in Computational Turbulent Incompressible Flow. Without a short-time-accurate model, nothing can be be predicted about long-time...Compare with the UK Met Office assurement:

• There have been major advances in the development and use of models over the last 20 years and the current models give us a reliable guide to the direction of future climate change.
• Computer models cannot predict the future exactly...
• Current models enable us to attribute the causes of past climate change, and predict the main features of the future climate, with a high degree of confidence.

What are "the advances in the development and use of models"? What is meant by "direction of future climate change"? Colder or warmer? Does "direction" indicate that the size of the change cannot be predicted? What is the meaning of "computer models cannot predict the future exactly"? That computer models can predict the future almost exactly? Who is the inventor of this form of newspeak? Note the clever construction of the following key statement by Met Office:

• As well as producing CO2, burning fossil fuels also produces small particles called aerosols which cool the climate by reflecting sunlight back into space. These have increased steadily in concentration over the 20th century, which has probably offset some of the warming we have seen.
  

Note the clever use of "probably" and "some of the warming". Very clever doublespeak: Clearly suggesting something, without saying anything! This is not the language of science. 

Can really these semantic tricks help save the World?

US Senators Freeze Despite Global Warming

The Guardian reports that The US Freezes on Climate Change:

• The stalled US climate change debate has killed the hope of reaching a final agreement at the Copenhagen summit
• Without concrete action in the Senate, there will not be an actual deal ready to sign in Copenhagen. With no Senate action, there's no guarantee that the US will commit to binding targets. And with no US targets, there will be no firm agreement from China, India or other emerging powers.
• It won't be a failure if there's no deal in Copenhagen, but it will be hard to gauge success with no new expectations.

What is the true reason that US Senators hesitate? Is it because most US Senators are not convinced about the reliability of present computational climate models predicting catastrophical global warming from CO2 emission? Is this because the Senators knowledge of mathematics and computer simulation is not deep enough? If so, would some education help?

Or is it because the Senators realize that they can only convince voters about the necessity of costly emission controls, if they themselves are convinced about the predictive power of present climate models, and they do not feel convinced? 

It seems that the only possibility is that the Senators sit down and carefully study mathematics and computational simulation, and after this experience check if they are convinced or not. It should be possible to bring them to the research front in a couple of weeks, with good math teachers. I guess this effort cannot be spared? 

I guess we here touch the essence of democracy? You cannot dictate, only convince others by first convincing yourself by first thinking yourself. Not even President Obama can get around this predicament. In his campaing Obama talked like a convinced: "the science is beyond dispute, the facts are clear", but today he has to face a disputable reality of unclear facts. How is Obama going to handle this situation? By joining the Senators math class?

Financial Times reports:

• The cost of reducing China’s total greenhouse gas emissions is likely to reach $438bn a year within 20 years, and developed economies will have to bear much of that cost, according to a group of Beijing’s leading climate economists.The figure, equivalent to about 7.5 per cent of China’s estimated gross domestic product in 2030, is likely to be deployed to support Beijing’s argument at December’s climate change summit in Copenhagen that industrialised nations must share the cost of cutting emissions in developing countries.
  

Another math lesson for both US President Obama and EU Chairman Reinfelt?